Keywords: variance . . . . . . . . . Option pricing, Asymptotic methods, Large deviations, Monte-Carlo, Optimal sampling, Asymptotic expansions, Stochastic volatility, Jump processes, Affine models, Implied volatility

Libor model (Chapter 5) . . . . . . . .

Résumé

Dans cette thèse, nous étudions plusieurs problèmes de mathématiques financières liés à la valorisation des produits dérivés. Par différentes approches asymptotiques, nous développons des méthodes pour calculer des approximations précises du prix de certains types d'options dans des cas où il n'existe pas de formule explicite.

Dans le premier chapitre, nous nous intéressons à la valorisation des options dont le payoff dépend de la trajectoire du sous-jacent par méthodes de Monte-Carlo, lorsque le sous-jacent est modélisé par un processus affine à volatilité stochastique. Nous prouvons un principe de grandes déviations trajectoriel en temps long, que nous utilisons pour calculer, en utilisant le lemme de Varadhan, un changement de mesure asymptotiquement optimal, permettant de réduire significativement la variance de l'estimateur de Monte-Carlo des prix d'options.

Le second chapitre considère la valorisation par méthodes de Monte-Carlo des options dépendant de plusieurs sous-jacents, telles que les options sur panier, dans le modèle à volatilité stochastique de Wishart, qui généralise le modèle Heston. En suivant la même approche que dans le précédent chapitre, nous prouvons que le processus vérifie un principe de grandes déviations en temps long, que nous utilisons pour réduire significativement la variance de l'estimateur de Monte-Carlo des prix d'options, à travers un changement de mesure asymptotiquement optimal. En parallèle, nous utilisons le principe de grandes déviations pour caractériser le comportement en temps long de la volatilité implicite Black-Scholes des options sur panier.

Dans le troisième chapitre, nous étudions la valorisation des options sur variance réalisée, lorsque la volatilité spot est modélisée par un processus de diffusion à volatilité constante. Nous utilisons de récents résultats asymptotiques sur les densités des diffusions hypo-elliptiques pour calculer une expansion de la densité de la variance réalisée, que nous intégrons pour obtenir l'expansion du prix des options, puis de leur volatilité implicite Black-Scholes.

Le dernier chapitre est consacré à la valorisation des dérivés de taux d'intérêt dans le modèle Lévy de marché Libor qui généralise le modèle de marché Libor classique (log-normal) par l'ajout de sauts. En écrivant le premier comme 0 une perturbation du second et en utilisant la représentation de Feynman-Kac, nous calculons explicitement l'expansion asymptotique du prix des dérivés de taux, en particulier, des caplets et des swaptions.

Mots clés : Valorisation d'options, Méthodes asymptotiques, Grandes déviations, Monte-Carlo, Échantillonnage préférentiel, Expansions asymptotiques, Volatilité stochastique, Processus à sauts, Modèles affines, Volatilité implicite

Abstract

In this thesis, we study several mathematical finance problems, related to the pricing of derivatives. Using different asymptotic approaches, we develop methods to calculate accurate approximations of the prices of certain types of options in cases where no explicit formulas are available.

In the first chapter, we are interested in the pricing of path-dependent options, with Monte-Carlo methods, when the underlying is modelled as an affine stochastic volatility model. We prove a long-time trajectorial large deviations principle. We then combine it with Varadhan's Lemma to calculate an asymptotically optimal measure change, that allows to reduce significantly the variance of the Monte-Carlo estimator of option prices.

The second chapter considers the pricing with Monte-Carlo methods of options that depend on several underlying assets, such as basket options, in the Wishart stochastic volatility model, that generalizes the Heston model. Following the approach of the first chapter, we prove that the process verifies a long-time large deviations principle, that we use to reduce significantly the variance of the Monte-Carlo estimator of option prices, through an asymptotically optimal measure change. In parallel, we use the large deviations property to characterize the long-time behaviour of the Black-Scholes implied volatility of basket options.

In the third chapter, we study the pricing of options on realized variance, when the spot volatility is modelled as a diffusion process with constant volatility. We use recent asymptotic results on densities of hypo-elliptic diffusions to calculate an expansion of the density of realized variance, that we integrate to obtain an expansion of option prices and their Black-Scholes implied volatility.

The last chapter is dedicated to the pricing of interest rate derivatives in the Levy Libor market model, that generalizes the classical (log-normal) Libor market model by introducing jumps. Writing the first model as a perturbation of the second and using the Feynman-Kac representation, we calculate explicit expansions of the prices of interest rate derivatives and, in particular, caplets and swaptions. The history of the mathematical pricing of financial derivatives begins with Louis Bachelier [START_REF] Bachelier | Processes of normal inverse Gaussian type[END_REF], who developed a pricing theory based on the assumption that stock prices evolve as a Brownian motion. In the 1970s, this theory made a huge leap forward with the long celebrated Black and Scholes (BS) model [START_REF] Merton | Theory of rational option pricing[END_REF][START_REF] Black | The pricing of options and corporate liabilities[END_REF]) that models the price of a stock as an exponential Brownian motion with SDE dS t = S t (µ dt + σ dW t ) ,

(1.1.1)

where µ ∈ R is the drift of the dynamics of the stock, σ > 0 is its volatility and (W t ) t≥0 is a standard Brownian motion. Under absence of arbitrage opportunities and assuming zero interest rate for simplicity, the price of a derivative with payoff p(S) is expressed as

IE Q [p(S)] ,
where Q is a measure under which (S t ) t≥0 is a martingale. The existence of such a measure is proved in [START_REF] Dalang | Equivalent martingale measures and no-arbitrage in stochastic securities market models[END_REF]. Under this measure, W Q t := W t + µ σ t is a standard Brownian motion and simple calculations show that

S t = S 0 e -σ 2 2 t+σ Wt Q ∼ LN log(S 0 ) - σ 2 2 t , σ 2 t ,
thus allowing to express the price of European derivatives as

IE Q [p(S T )] = 1 √ 2π R p S 0 e -σ 2 2 T +σ √
T w e -w 2 2 dw .

In particular, considering a call option with maturity T and strike K, we obtain the famous Black and Scholes formula

IE Q [(S T -K) + ] = S 0 Φ (d + ) -K Φ (d -) , (1.1.2)
1.1. OPTION PRICING 1 where Φ(•) is the cumulative distribution function of a Gaussian random variable and

d ± = log(S 0 /K) σ √ T ± σ √ T 2
.

(1.1.3)

In the years following its development, the BS model became so popular that, on the markets, option prices are quoted in BS "implied volatility", that is the volatility σ, that yields the price of the option when inserted in eqs. (1.1.2) and (1.1.3).

Beyond the BS model

Even though highly tractable, the BS model suffers from several drawbacks.

In particular, the model fails to replicate some of the behaviours observed on the financial markets, where the implied volatility of options as a function of the strike typically displays the shape of a "smile" (see Figure 1.1.1), whereas in the BS model, the implied volatility is trivially constant and therefore cannot fit the observed data. In addition, the BS setting models the stock log-returns as Gaussian random variables, in which the probability of extreme events is excessively small, much smaller than what is observed on the market, as in Figure 1.1.2, where we can see three peaks between 6.5 and 7 σ, thus leading to an underestimation of the risk. These flaws led to the development of extensions of the BS model which are more flexible, thus allowing to better fit the market data. These models unfortunately induce new difficulties in the pricing and hedging of financial options, often requiring computationally expensive procedures.

Local and stochastic volatility

The first natural extension of the BS model is to allow the volatility to vary with time, that is dS t = S t σ t dW t , (1.1.4)

for some possibly stochastic volatility σ t . Classically, one speaks about local volatility if the volatility σ t = σ(t, S t ) depends only on the price process and about stochastic volatility if σ t depends on a new source of risk such as another Brownian motion. The class of local volatility models allows some flexibility to model the volatility smile, without introducing non-traded sources of risk, thus allowing to hedge financial options with the underlying stocks. Among this class, we find the popular "Constant elasticity of variance" (CEV) model [START_REF] Cox | The constant elasticity of variance option pricing model[END_REF] where αS β-1 t . In general, these models do not have closed-form formulas to price options. The "original" pricing method [START_REF] Dupire | Pricing with a smile[END_REF] consists in solving numerically the PDE ∂ t P (t, s) + 1 2 σ 2 (t, s) s 2 ∂ ss P (t, s) = 0 , P (T, s) = p(s) ,

where P (t, s) = IE Q [ p(S T ) | S t = s ] is the price at time t of a derivative with payoff p(S T ) if S t = s. Later, in [START_REF] Hagan | Equivalent Black volatilities[END_REF], the authors calculate an expansion of the implied volatility of options using singular perturbation techniques.

Even though fairly tractable, market evidence showed that local volatility models have limitations when it comes to replicating the behaviour of market 1.1. OPTION PRICING 1 data. Indeed, when comparing the evolution of the S&P5001 index and the VIX2 index (see Figure 1.1.3), we observe that they are not fully correlated, thus leading to the introduction of a new source of risk. [START_REF] Stein | Stock price distributions with stochastic volatility: an analytic approach[END_REF], where σ t is modelled as a meanreverting Ornstein-Uhlenbeck process [START_REF] Ornstein | On the theory of the brownian motion[END_REF], the Heston model [START_REF] Heston | A closed-form solutions for options with stochastic volatility with applications to bond and currency options[END_REF], where σ 2 t is modelled as a CIR process [START_REF] Cox | An intertemporal general equilibrium model of asset prices[END_REF], the 3/2 stochastic volatility model [START_REF] Carr | A new approach for option pricing under stochastic volatility[END_REF], which uses the interest rate model from [START_REF] Ahn | A parametric nonlinear model of term structure dynamics[END_REF] for σ 2 t and the SABR (Stochastic α, β, ρ) model [START_REF] Hagan | Managing smile risk[END_REF], where σ t = x t S β t and x t is a log-normal process. These models allow for different levels of flexibility and no general method is available for the pricing of options. In some of these models, such as the class of affine models characterized in [START_REF] Duffie | Affine processes and applications in finance[END_REF], the Fourier transform of the stock log-price is known. The pricing problem is therefore generally tackled by inverting the Fourier transform (see also (Carr 1 and Madan, 1999)). Others however require the use of asymptotic formulas or heavy numerical procedures to calculate option prices.

Jumps

Local and stochastic volatility models remedy the BS inability to model the volatility smile.

One of the main properties of the Brownian driven diffusion models is that their trajectories are continuous. However, structurally, market stock prices do not evolve continuously. They instead "jump" from one value to another, sometimes with a large difference. In addition, when time tends to zero, continuous local and stochastic volatility models tend to behave like the BS model, since the volatility does not have the possibility to vary a lot from its initial position. They have therefore limitations when it comes to pricing options with short maturities (see [START_REF] Carr | What type of process underlies options? a simple robust test[END_REF]). Such concerns lead, in many financial models, to the introduction of jumps. We address the reader to [START_REF] Cont | Financial Modelling with Jump Processes[END_REF]) for a general reference on jump processes in finance. Following an idea of [START_REF] Mandelbrot | The variation of certain speculative prices[END_REF], several elements of the class of Lévy processes were proposed in the literature to model stock logprices, such as the Merton model [START_REF] Merton | Option pricing when underlying stock returns are discontinuous[END_REF] with Gaussian jumps, the Kou model [START_REF] Kou | A jump-diffusion model for option pricing[END_REF] with "double exponential" jumps, the Bates model [START_REF] Bates | Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options[END_REF] with jumping stochastic volatility, with finite jump activity, and the Normal-inverse-Gaussian model [START_REF] Bachelier | Processes of normal inverse Gaussian type[END_REF]) and the CGMY model [START_REF] Carr | The fine structure of asset returns: An empirical investigation[END_REF], with infinite jump activity. In Lévy models, when the jump activity is finite, the pricing can be done by conditioning on the jumps, whereas, when the jump activity is infinite, one classically uses the fact that the Fourier transform is known (Lévy-Khintchine formula) to use the fast Fourier transform method [START_REF] Carr | Option valuation using the fast Fourier transform[END_REF].

Non-equity derivatives

We have so far discussed the pricing of stock options. In this thesis, we also discuss the pricing of options that are not based on equity but also on interest rates and realized variance.

Interest rate derivatives and the Libor Market Model

According to the Bank of International Settlements (BIS), derivatives on interest rates amount for more than three quarters of the total volume of derivatives traded in the OTC market. The development of efficient pricing methods for such products is therefore a very important issue for financial institutions, but also for insurance companies and pension funds. Indeed, these companies structurally have to pay deterministic cash flows at future times and interest rate derivatives are a useful tool to hedge interest rate 1.1. OPTION PRICING 1 risk. Besides interest rate swaps, that have a linear payoff, the most common interest rate derivatives are caplets and swaptions. In order to model interest rates, several approaches were proposed in the last 50 years. Starting in the 1970's, the first approach models the "short rate", i.e. the interest rate that one needs to pay to borrow cash on the markets. Among the processes that are used to model the short rate, we cite the Vasicek model [START_REF] Vasicek | An equilibrium characterization of the term structure[END_REF] and the CIR model [START_REF] Cox | An intertemporal general equilibrium model of asset prices[END_REF]. The second approach proposed by [START_REF] Heath | Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation[END_REF] models the instantaneous forward interest rate, i.e. the current interest rate applicable to a future transaction. The third approach called "Libor market model" (LMM) was developed in [START_REF] Brace | The market model of interest rate dynamics[END_REF] as a justification of the Black formula, a version of the BS formula, that was commonly used by practitioners to price caplets, which can be seen as call options on the Libor rates.

Consider a set of times 0 ≤ T 0 < ... < T n for which zero-coupon bonds B t (T k ) are available. Denoting δ j = T j -T j-1 , the forward Libor rate for the period [T j-1 , T j ] is

L j t = 1 δ j B t (T j-1 ) B t (T j ) -1 .
Since Bt(T j-1 )

Bt(T j ) is the price of a traded asset B t (T j-1 ) discounted by B t (T j ), the theory of absence of arbitrage states that L j t should be a martingale under the measure Q T j that uses B t (T j ) as numéraire. The LMM therefore models L j t as a log-normal BS process under Q T j . In the LMM, the price of caplets is calculated using the Black formula, thus justifying the market practice. Swaptions however cannot be priced exactly, but freezing some parameters provides a fairly accurate "Black-type" closed-form pricing approximation. For an overview on the LMM, we refer the reader to [START_REF] Brigo | Interest Rate Modelling -Theory and Practice[END_REF].

As in the case of stocks, following the development of the LMM, several authors have enhanced the log-normal LMM with stochastic volatility [START_REF] Gatarek | Libor market model with stochastic volatility[END_REF][START_REF] Piterbarg | A stochastic volatility forward Libor model with a term structure of volatility smiles[END_REF][START_REF] Hagan | Libor market model with SABR style stochastic volatility[END_REF] and with jumps (Glasserman and [START_REF] Glasserman | The term structure of simple forward rates with jump risk[END_REF][START_REF] Eberlein | The Lévy Libor model[END_REF]. These modifications naturally induce new complications in the option pricing problems and give rise to an important literature. In [START_REF] Eberlein | Calibration of Lévy term structure models[END_REF], the authors tackle the calibration of the Lévy Libor Market model (LLMM) of [START_REF] Eberlein | The Lévy Libor model[END_REF] and several papers [START_REF] Eberlein | The Lévy Libor model[END_REF][START_REF] Kluge | Time-Inhomogeneous Lévy Processes in Interest Rate and Credit Risk Models[END_REF][START_REF] Belomestny | A jump-diffusion Libor model and its robust calibration[END_REF] propose methods to compute numerically caplet prices using Fourier transform inversion.

Worth noting are also the other challenges, such as negative interest rates and multi-curve modelling, that have appeared in the interest rate markets after the sub-prime crisis of 2008. We do not discuss those problems in the scope of this thesis and refer the reader to [START_REF] Grbac | Interest Rate Modeling: Post-Crisis Challenges and Approaches[END_REF].

CHAPTER 1. INTRODUCTION 1

Realized variance and volatility derivatives

The trading of realized variance supposedly first happened at UBS in 1993, but really took off only in 1998, probably due to the elevated volatilities observed this year, (see [START_REF] Carr | Volatility derivatives[END_REF]). Back then, the market offered mostly variance and volatility swaps. It was only in 2005, that the market of derivatives started to offer a wider range of realized-variance-based derivatives, such as options on realized variance. An option on realized variance with strike K is a non-linear derivative with payoff

h(Z) = 1 T T 0 Z 2 t dt -K + (1.1.5)
where Z t is an instantaneous volatility process. The calculation of the expectation of (1.1.5) is generally a complicated task, as (1.1.5) is highly non-linear and the distribution of the squared time-integral of Z is generally unknown.

A number of papers have been written on the subject, in particular [START_REF] Carr | Pricing options on realized variance[END_REF] for pure jump processes, [START_REF] Sepp | Pricing options on realized variance in the Heston model with jumps in returns and volatility[END_REF] for the Heston model with jumps, [START_REF] Kallsen | Pricing options on variance in affine stochastic volatility models[END_REF] in an affine setting, [START_REF] Drimus | Options on realized variance by transform methods: a non-affine stochastic volatility model[END_REF] for the 3/2 model and (Keller-Ressel and Muhle-Karbe, 2013) for exponential Lévy models, almost always using Laplace/Fourier transform methods. In particular, (Keller-Ressel and Muhle-Karbe, 2013) discuss the difference between the discrete and the continuous versions of realized variance. A non-parametric approach to price volatility derivatives is proposed in [START_REF] Carr | Robust replication of volatility derivatives[END_REF].

Based on the replication of the derivative with a portfolio of the stock and vanilla options, the method is exact when asset and volatility are not correlated and is "immune" against non-zero correlation at first order. This is also discussed in (Henry-Labordere, 2017).

Pricing options with asymptotic methods

In general, as soon as we depart from the BS model, we cannot use closedform formulas, thus the need to find alternative approaches. In some models, pricing is done through numerical integration of a known function, in some others, by solving numerically partial (integro-)differential equations. However, these methods are sometimes insufficient, in particular, when pricing path dependent options, when the dimension is large or when we need to compute a large number of prices, as in model calibration procedures. Indeed, in such cases, the amount of required computation can become too large to be executed in a reasonable time. When this is the case, the use of asymptotic methods is often a good solution to obtain accurate approximations of option prices in a decent time. Below, we review some of the important asymptotic methods.

OPTION PRICING

1

Perturbation of differential equations

The perturbation of differential equation is a classical method, widely used in physics and engineering, which consists in writing an unsolvable differential equation as a perturbation of a solvable differential equation in order to approximate the solutions of the unsolvable differential equations by a truncated power series of the perturbation parameter. Perturbations methods are of two kinds, regular and singular. Assume that f λ (x) is the solution of a differential equation

A λ f λ (x) = 0 , f λ (a) = h , (1.1.6)
where A λ is an operator and 0 ≤ λ ≤ λ 1 is a parameter. Classically, the problem of interest corresponds to the case with λ = λ, whereas the case λ = 0 is a simple problem whose solution is explicitly known. Assume that we can write

f λ (x) = ∞ j=0 λ j f j (x) and A λ = ∞ j=0 λ j A j . Then (1.1.6) becomes ∞ k=0 λ k k j=0 A j f k-j (x) = 0 , f λ (a) = h . Hence A 0 f 0 (x) = 0 , f 0 (a) = h (1.1.7)
and

A 0 f k (x) = - k j=1 A j f k-j , f k (a) = 0 , k ≥ 1 . (1.1.8)
Equation (1.1.7) corresponds to (1.1.6) for λ = 0, whose solution is explicitly known and (1.1.8) is of the same "type" as (1.1.7). Provided (1.1.8) can be solved up to order n, it is therefore straightforward to truncate the f λ series to obtain an accurate approximation of fλ.

Many differential equations arise in finance. As an example, the price P of a European financial derivative with payoff h(X T ) is generally expressed as the conditional expectation

P (t, x) = IE Q [h(X T ) | X t = x ]
under a certain probability measure Q. By martingale property of P (t, X t ), the price function verifies the Kolmogorov backward equation 0 = ∂ t P (t, x) + L t P (t, x) ,

(1.1.9)

CHAPTER 1. INTRODUCTION 1
where L t is the infinitesimal generator of X t , under terminal condition P (T, x) = h(x). When pricing options under a model where closed-form formulas do not exist, a possible approach is therefore to write the complicated model as a perturbation of a simple one, such as the BS model. The approach of regular perturbations is used in several papers in the literature. In [START_REF] Sircar | Stochastic volatility, smile & asymptotics[END_REF], the authors use regular perturbations to obtain an asymptotic expansion of call prices when the stochastic volatility varies quickly and [START_REF] Lee | Implied and local volatilities under stochastic volatility[END_REF] extends this methodology to expand the implied volatility of vanilla options for small variations of the volatility and slow variation of the variance.

Regular perturbations techniques are used to obtain asymptotic expansions for option prices in local volatility models [START_REF] Benhamou | Closed forms for european options in a local volatility model[END_REF] and in the time-dependent Heston model [START_REF] Benhamou | Time dependent Heston model[END_REF], as perturbations of the BS model, and in jump-diffusion models [START_REF] Benhamou | Smart expansion and fast calibration for jump diffusions[END_REF], as a perturbation of the Merton model [START_REF] Merton | Option pricing when underlying stock returns are discontinuous[END_REF], combining the approach with Malliavin calculus to calculate the coefficients of the expansions. In [START_REF] Jacquier | From characteristic functions to implied volatility expansions[END_REF], a regular perturbation approach is used to obtain an expansion of implied volatility of vanilla options in models for which the characteristic function is known explicitly. When f λ can be expressed as an infinite series on the whole domain, regular perturbations are an effective way to approximate the solution of differential equations. When this is not possible however, some scaling is required in order to obtain expansions. These methods are called "singular perturbations" and are typically useful when the small parameter multiplies the highest derivative. We present, as an example, the case developed in [START_REF] Widdicks | The Black-Scholes equation revisited: Asymptotic expansions and singular perturbations[END_REF], where the authors calculate an expansion of the BS price when volatility is small, using singular perturbations. Let P (t, s; σ), be the price of a call option with maturity T and strike K, on a BS stock with volatility σ. Then P (t, s) verifies the PDE

∂ t P (t, s) + 1 2 σ 2 s 2 ∂ ss P (t, s) = 0 , P (T, s) = (s -K) + .
Since, when setting σ = 0, the second order derivative vanishes and remains only P (t, s) = (s -K) + , let us define the scaling

P (t, ŝ) = P (t, s) σ , where ŝ = s -K σ .
A simple change of variable in the BS equation then shows that P (t, ŝ) verifies the PDE A σ P (t, ŝ) = 0 , P (T, ŝ) = ŝ+ ,

where

A σ = ∂ t + 1 2 K 2 ∂ ŝŝ + σ K ŝ ∂ ŝŝ + 1 2 σ 2 ŝ2 ∂ ŝŝ .
Since the second order term no longer vanishes when σ = 0, P (t, ŝ) can now simply be expanded using regular perturbation techniques.

OPTION PRICING 1

In the last 20 years, singular perturbations have been widely used to price options. In [START_REF] Fouque | Derivatives in Financial Markets with Stochastic Volatility[END_REF][START_REF] Fouque | Multiscale stochastic volatility asymptotics[END_REF][START_REF] Papanicolaou | Singular perturbations in option pricing[END_REF], the authors use singular perturbation to expand the price of options in the case of fast mean-reverting stochastic volatility model. Singular perturbations are also used in [START_REF] Hagan | Equivalent Black volatilities[END_REF][START_REF] Hagan | Managing smile risk[END_REF] to develop asymptotics for the implied volatility of vanilla options in the famous SABR model. In addition to the presented example, [START_REF] Widdicks | The Black-Scholes equation revisited: Asymptotic expansions and singular perturbations[END_REF] develop asymptotics for the price of American and barrier options. An expansion of the price of options in diffusive stochastic volatility models is obtained in [START_REF] Antonelli | Pricing options under stochastic volatility: a power series approach[END_REF] as a power series of the correlation between the Brownian motions driving the asset and the volatility.

The applications of perturbation theory in finance are naturally not limited to option pricing, but other applications go beyond the scope of this thesis. We refer the reader to the introduction of ( Černỳ et al., 2013) for an overview of other financial applications of perturbation theory, namely hedging and portfolio optimization.

Asymptotic expansions of the transition density

The first result aiming to understand the asymptotic behaviour of the transition density p(s, t, x, y) of a multidimensional diffusion process

X t = x + t s b(X u ) du + t s σ(X u ) dW u
goes back to (Varadhan, 1967a) and (Varadhan, 1967b), who proved that, under uniform ellipticity condition, the solution p(t, x, y) of the heat equation with variable coefficients

∂ t p = 1 2 n i,j=1 σ ij (x)∂ x i x j p with boundary condition p(t, x, y) → δ x (y) as t → 0 satisfies lim t→0 2t log p(t, x, y) = -d 2 (x, y) ,
where d(x, y) is the Riemannian distance induced by σ. This result was generalized to hypoelliptic diffusions under strong Hörmander condition by [START_REF] Léandre | Minoration en temps petit de la densité d'une diffusion dégénérée[END_REF]. Asymptotic expansions of the type

p(s, t, x, y) = (t -s) -n/2 e -d 2 s (x,y) 2(t-s) N j=0 α j (t -s) j + O (t -s) N +1
have been obtained by [START_REF] Molčanov | Diffusion processes and riemannian geometry[END_REF] and [START_REF] Azencott | Densité des diffusions en temps petit: développements asymptotiques[END_REF] for elliptic operators. In [START_REF] Bismut | Large deviations and the Malliavin calculus[END_REF], a first Malliavin calculus approach to this CHAPTER 1. INTRODUCTION 1 problem is discussed to obtain results under the weaker than elliptic "H2" hypothesis. In [START_REF] Watanabe | Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels[END_REF], the author develops a theory of distributions on the Wiener space, thus allowing to study rigorously the density of a functional F (W ) of Brownian motion as f (x) = IE(δ x F (W )), where δ x is a Dirac spike at x. Taking F = F ( , W ), where is a small parameter, this setting allows to study asymptotics of the law f (x) of F ( , W ) as f (x) = IE(δ x F ( , W )) obtaining the transition density of diffusion processes as a particular case. Following [START_REF] Azencott | Densité des diffusions en temps petit: développements asymptotiques[END_REF] and [START_REF] Bismut | Large deviations and the Malliavin calculus[END_REF], Ben Arous obtains in (Ben Arous, 1988a;Ben Arous, 1988b) asymptotic expansions for the density of hypoelliptic diffusion processes using the Laplace method on Wiener spaces and the Mallivin calculus. Later, (Deuschel et al., 2014a) consider the marginal diffusion of the density f of the l-dimensional marginal (X 1t , ..., X lt ) of the n-dimensional diffusion process

dX t = b( , X t ) dt + σ(X t ) dW t ,
where b( , X t ) is typically of the form b(X t ) or 2 b(X t ) and obtain the result

f = e -c 1 / 2 e c 2 / -l (c 0 + O( )) , as → 0 .
We finally cite a recent paper [START_REF] Frikha | A parametrix approach for asymptotic expansion of markov semigroups with applications to multidimensional diffusion processes[END_REF] which combines the parametrix technique with Malliavin calculus to prove an asymptotic expansion of the density of diffusions under weak Hörmander condition, assuming that b and σ are smooth functions with bounded derivatives of all orders and that X satisfies an integrability condition. Around those results, many financial pricing methods were developed. Based on (Ben Arous, 1988a), (Bayer and Laurence, 2013a;[START_REF] Bayer | Asymptotics for at the money local vol basket options[END_REF]) obtain an asymptotic expansion for the implied volatility of basket options in local volatility models out of and at the money. A small time expansion of implied volatility of options in stochastic volatility models is calculated in [START_REF] Henry-Labordère | Analysis, Geometry and Modeling in Finance: Advanced Methods in Option Pricing[END_REF]. In [START_REF] Deuschel | Marginal density expansions for diffusions and stochastic volatility II: Applications[END_REF], the asymptotic implied volatility result for the Stein-Stein model obtained in [START_REF] Gulisashvili | Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models[END_REF] in the uncorrelated case is extended to the correlated case using the results of (Deuschel et al., 2014a).

In [START_REF] Takahashi | An asymptotic expansion approach to pricing financial contingent claims[END_REF], the author expands the density of a diffusion, where the volatility is multiplied by a small parameter, as a "Gaussian" power series of the small parameter. The expansion of the density obtained in [START_REF] Watanabe | Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels[END_REF] is then integrated in order to obtain a small volatility expansion of the price of derivatives in local volatility models. The author calculates explicitly the expansion for vanilla and Asian options. The same methodology is used in [START_REF] Kunitomo | The asymptotic expansion approach to the valuation of interest rate contingent claims[END_REF][START_REF] Kunitomo | The asymptotic expansion approach to the valuation of interest rate contingent claims[END_REF] to price interest rate derivatives in the HJM framework [START_REF] Heath | Bond pricing and the term structure of interest rates: A new methodology for contingent claims valuation[END_REF]. The authors provide an explicit expansion of the price of swaptions and Asian (interest rate) options. The validity of the latter expansions is finally proved in [START_REF] Kunitomo | On validity of the asymptotic expansion approach in contingent claim analysis[END_REF].

OPTION PRICING

1

A later series of papers, among which we can cite [START_REF] Shiraya | Pricing average options on commodities[END_REF][START_REF] Shiraya | Pricing multiasset cross-currency options[END_REF][START_REF] Shiraya | An approximation formula for basket option prices under local stochastic volatility with jumps: An application to commodity markets[END_REF]Shiraya and Takahashi, 2017a;[START_REF] Shiraya | Pricing average and spread options under local-stochastic volatility jump-diffusion models online appendix[END_REF], extends these calculations to certain types of derivatives under stochastic volatility and jump diffusion models.

Methods based on large deviations and geometry

The methods based on large deviations are in the same spirit as the ones based on density expansions. The Freidlin-Wentzell theory [START_REF] Freidlin | Random Perturbations of Dynamical Systems[END_REF] proves, under certain hypotheses, that the diffusion

dX t = b(X t ) dt + σ(X t ) dW t , X 0 = x ,
where (W t ) t≤T is Brownian motion, satisfies log P (X T ∈ A) ∼ I(A) , as → 0 , for the point-set distance

I(A) = inf x 0 =x, x T ∈A 1 2 T 0 • x t -b(x t ) σ(x t ) 2 dt .
Based on the numerous versions and generalizations of this result, a wide range of pricing methods was developed.

In [START_REF] Berestycki | Computing the implied volatility in stochastic volatility models[END_REF], the authors show, using large deviations, that the implied volatility in stochastic volatility models can be expressed as a function of a distance function connected to a Hamilton-Jacobi equation. Asymptotic expansions for the Heston implied volatility are calculed in [START_REF] Forde | Small-time asymptotics for implied volatility under the Heston model[END_REF] for the short-time case and in (Forde and Jacquier, 2011a;[START_REF] Jacquier | Large deviations for the extended Heston model: the large-time case[END_REF] for the long-time case. In [START_REF] Forde | Small-time asymptotics for an uncorrelated local-stochastic volatility model[END_REF], the authors use the Freidlin-Wentzell theory to add some rigour to the implied volatility expansions obtained in [START_REF] Henry-Labordère | Analysis, Geometry and Modeling in Finance: Advanced Methods in Option Pricing[END_REF] and [START_REF] Paulot | Asymptotic implied volatility at the second order with application to the SABR model[END_REF] for stochastic volatility models. In [START_REF] Jacquier | Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models[END_REF], a large deviation principle is shown for affine stochastic volatility models and asymptotics for the implied volatility are obtained.

Monte-Carlo methods, large deviations and optimal sampling

When one wishes to compute the expectation of a random variable, if no simpler method is available, the "last chance" solution is the use of Monte-Carlo methods. These methods were developed in the 1940s and popularised along the years as computer power increased. Let ξ be a square integrable random variable. Our goal is to calculate p := IE(ξ). Assume that we can simulate efficiently from the distribution of ξ and let ξ (1) , ..., ξ (n) be independent and ξ (k) .

The Central Limit Theorem stipulates that √ n(p -p) converges in law to a Gaussian random variable with law N(0, Var(ξ)). Hence, when n is large enough, p behaves like a N(p, n -1 Var(ξ)) random variable and therefore, for a fixed probability α, we can approximate a confidence interval for p at level α by

I α = p + n -1/2 σ q α/2 , p + n -1/2 σ q 1-α/2 , where σ2 = 1 n-1 n k=1 ξ (k) -p
2 and q α is the quantile at the level α of a standard Gaussian random variable. By taking ξ = P (X) as the payoff P of a derivative on a random log-price trajectory X, Monte-Carlo methods then become a powerful option pricing tool. Indeed, they allow to compute numerical estimates of the price of almost any derivative no matter how complicated its payoff is. The biggest drawback of Monte-Carlo methods is that, since the standard deviation of p is proportional to n -1/2 , one needs a large number of simulations to achieve the desired accuracy. To overcome this problem, a possible approach is, instead of simulating the path processes X (1) , ..., X (N ) under the actual risk-neutral probability measure P, to simulate the path processes X (1,Q) , ..., X (N,Q) under an equivalent probability measure Q. The estimator of the price of an option with payoff P (•) on an underlying X then becomes

pQ := 1 N N k=1 P X (k,Q) dP dQ X (k,Q) .
The estimators of the class {p Q : Q ∼ P} are unbiased and have variance

Var Q [p Q ] = 1 N Var Q P (X) dP dQ (X) = 1 N IE P 2 (X) dP dQ (X) -IE 2 [P (X)] ,
thus the interest to find the measure Q ∼ P that minimizes IE P 2 (X) dP dQ (X) . Since this minimization problem is generally complex, some authors, starting with [START_REF] Siegmund | Importance sampling in the Monte Carlo study of sequential tests[END_REF], considered the minimization of an asymptotic version of IE P 2 (X) dP dQ (X) based on the theory of large deviations. Following this idea, [START_REF] Dupuis | Importance sampling, large deviations, and differential games[END_REF] discuss the use of adaptive controltheoretic measure changes. Adaptive schemes are further discussed in the context of Gaussian functionals in [START_REF] Jourdain | Robust adaptive importance sampling for normal random vectors[END_REF]. In [START_REF] Guasoni | Optimal importance sampling with explicit formulas in continuous time[END_REF], the authors combine Laplace method with Schilder's Theorem and Varadhan's Lemma (Dembo and Zeitouni, 1998, Thms. 5.2.3 1.2. SUMMARY OF THE THESIS 1 and 4.3.1) to calculate explicitly the small noise large deviation proxy to the optimal measure to price Asian options when the underlying behaves as a geometric Brownian motion. Then, [START_REF] Robertson | Sample path large deviations and optimal importance sampling for stochastic volatility models[END_REF] generalizes the Freidlin-Wentzell theory (Dembo and Zeitouni, 1998, Section 5.6) proving a trajectorial small-noise large deviations principle for a diffusive stochastic volatility model, thus extending the methodology of [START_REF] Guasoni | Optimal importance sampling with explicit formulas in continuous time[END_REF] to the stochastic volatility framework. Finally, in a more recent work, [START_REF] Genin | Optimal importance sampling for Lévy processes[END_REF]) combine a similar approach with the long-time large deviations results of [START_REF] Léonard | Large deviations for Poisson random measures and processes with independent increments[END_REF] for processes with independent increments to develop an optimal sampling method for path dependent options when the underlying is modelled as an exponential Lévy process.

Summary of the thesis

In this thesis, we tackle four financial pricing problems using asymptotic approaches.

In Chapter 2, we consider the problem of pricing potentially path-dependent derivatives via Monte-Carlo methods when the stock price is given by an affine stochastic volatility model. We start by proving a large-time large deviations principle for the paths of such processes under strong but verifiable hypotheses. We then consider the class of measure changes defined by the time-dependent Esscher transform. By simulating under a measure change, the variance of the Monte-Carlo estimator changes. We write the variance reduction problem, which happens to be unsolvable explicitly. Using the large deviations result, we formulate an asymptotic approximation of the variance reduction problem that we solve to compute an asymptotically optimal measure change and obtain a significant variance reduction when using Monte-Carlo simulations. We test the method on the Heston model with and without jumps to demonstrate its numerical efficiency.

In Chapter 3, we consider the problem of pricing European basket options when stock prices are modelled as a Wishart stochastic volatility process, which is an n-dimensional generalization of the very popular Heston model, using Monte-Carlo methods. Following the approach of Chapter 2, we start by proving that the Wishart stochastic volatility process satisfies a large deviations principle when time tends to infinity. We then write the Monte-Carlo variance minimization problem induced by the Esscher transform class of measure changes and use the large deviation principle to write a solvable approximate minimization problem. We finally test the method numerically and see that the variance reduction obtained allows a significant reduction of the required number of Monte-Carlo simulations. In addition and independently, we use the large deviation result to calculate the asymptotic implied volatility of basket options when time is long and test the convergence numerically using Monte-Carlo simulations.

CHAPTER 1. INTRODUCTION 1

In Chapter 4, we study the pricing of options on realized variance when the instantaneous volatility is modelled as a diffusion process with general drift and constant volatility. We calculate a first order expansion of the density of the integral of the squared volatility process using recent results on asymptotic expansions for marginal densities of hypo-elliptic diffusions. We then integrate the option payoff against the density to expand the option price and compare the result to the expansion of the BS price to calculate an expansion of the associated BS implied volatility.

In Chapter 5, we tackle the issue of pricing interest rates derivatives under the Lévy Libor market model, that generalises the popular log-normal Libor market model by introducing jumps. We expand the generator of the Lévy Libor market model around the generator of the log-normal Libor market model and use the Feynman-Kac formula to calculate an explicit asymptotic expansion of the price of "European-type" options at second order. We compare the numerical results with an accurate Monte-Carlo simulation to demonstrate the efficiency of the method.

The main results of the thesis

Let us now present in more detail the main results of this thesis.

1.3.1 Affine stochastic volatility models, large deviations and optimal sampling (Chapters 2 and 3)

The class of affine models, whose characterization can be found in [START_REF] Duffie | Affine processes and applications in finance[END_REF] is a wide class of models whose Laplace transform is known up to the resolution of an ODE. A particularly interesting subclass of affine models is the class of affine stochastic volatility (ASV) models [START_REF] Keller-Ressel | Moment explosions and long-term behavior of affine stochastic volatility models[END_REF]. It was developed as a generalization of the very popular Heston model [START_REF] Heston | A closed-form solutions for options with stochastic volatility with applications to bond and currency options[END_REF] and combines the tractability of affine models with the flexibility of stochastic volatility models to reflect stylized facts observed on the markets. An ASV process (X t , V t ) t≥0 , where X t is the asset log-price and V t is the instantaneous variance process, is a bivariate affine model whose Laplace transform is of the form IE e uXt+wVt F s = e φ(t-s,u,w)+ψ(t-s,u,w) Vs+u Xs , where φ and ψ satisfy the generalized Riccati equations

∂ t φ(t, u, w) = F (u, ψ(t, u, w)) , φ(0, u, w) = 0 (1.3.1a) ∂ t ψ(t, u, w) = R(u, ψ(t, u, w)) , ψ(0, u, w) = w , (1.3.1b)
and F and R have Lévy-Khintchine forms. This feature granted ASV models an increasing popularity in the industry, owing to the fact that the prices of

THE MAIN RESULTS OF THE THESIS

1 single asset vanilla options can be efficiently computed using Laplace transform inversion methods [START_REF] Carr | Option valuation using the fast Fourier transform[END_REF][START_REF] Duffie | Transform analysis and asset pricing for affine jump-diffusions[END_REF].

Pricing path-dependent derivatives in ASV models

Pricing path-dependent derivatives in ASV models is a more complicated task and often requires Monte-Carlo simulations. We consider the pricing of such derivatives and, in particular, Asian option with payoff

1 T T 0 S t dt -K + .
We propose an optimal sampling method based on an asymptotic large deviations based proxy. A point (single-date) large deviation principle for ASVM was already shown in [START_REF] Jacquier | Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models[END_REF], where the authors use their result to show that the asymptotic long-time implied volatility of a vanilla option with maturity T and strike K on a stock worth S 0 is St) .

σ ∞ = √ 2 h * (x) 1/2 + (h * (x) -x) 1/2 where x = T -1 log(K/S 0 ) and h * (x) := sup θ∈R {θx -h(θ)} is the convex dual of h(θ) := lim t→∞ t -1 log IE e θ log(
Since we want to price path dependent options however, a point large deviation property is insufficient for the scope of our work and we need to prove a trajectorial LDP for ASV models. Let us present the main results shown in Chapter 2. Let (X t , V t ) t≥0 be an ASV process and let I ⊂ R be the set such that for every u ∈ I, (1.3.1b) admits a unique stable equilibrium w(u). Let also J ⊂ I be the domain of h(u) = F (u, w(u)).

Assumption 1. The function h verifies the following condition.

• There exists u < 0, such that h(u) < ∞.

• u → h(u) is essentially smooth.

Assumption 2. One of the following conditions is verified.

1. The interval support of F is J = [u -, u + ] and w(u -) = w(u + ).

2. For every u ∈ R, (1.3.1b) has no unstable equilibrium.

Theorem 1.3.1. Let us define, for ∈ (0, 1], the process X t := X t/ . If Assumptions 1 and 2 are verified, then (X t ) 0≤t≤T satisfies a large deviations 1 property on F([0, T ], R) equipped with the topology of point-convergence, as → 0, with good rate function

Λ * (x) = T 0 h * ( • x t ac ) dt + T 0 H dν t dθ t dθ t , where h * (y) = sup θ∈J {θy -h(θ)} , H(y) = lim →0 h * (y/ ) ,
•

x ac is the derivative of the absolutely continuous part of x, ν t is the singular component of dx t with respect to dt and θ t is any non-negative, finite, regular, R-valued Borel measure, with respect to which ν t is absolutely continuous.

We consider the class of measure changes dP θ dP = e t 0 Xs dθs I E e t 0 Xs dθs , where θ is a finite signed measure on [0, T ] and wish to find the θ that minimizes Var P θ P (X)

dP dP θ = IE P 2 (X) dP dP θ -IE 2 (P (X)) ,
in order to reduce the variance when pricing an option with possibly pathdependent payoff P (X) using Monte-Carlo simulations. This problem is however not explicitly solvable. Denoting H = log P , lim

→0 log IE P 2 (X ) dP dP θ (X ) = sup x∈Vr 2H(x) - T 0 x t dθ t -Λ * (x) + T 0 h(θ([t, T ])) dt .
where V r is the set of trajectories x : [0, t] → R with bounded variation. We therefore call asymptotically optimal a measure θ that minimises the righthand side. A result by [START_REF] Genin | Optimal importance sampling for Lévy processes[END_REF] shows that, under technical hypothesis, for H concave and continuous on its domain with respect to the topology of pointwise convergence, we have

inf θ∈M sup x∈Vr 2H(x) - T 0 x t dθ t -Λ * (x) + T 0 h(θ([t, T ])) dt = 2 inf θ∈M Ĥ(θ) + T 0 h(θ([t, T ])) dt , where Ĥ(θ) = sup x∈Vr H(x) - T 0 x t dθ t .
Furthermore, if θ * minimises the left-hand side of the above equation, it also minimises the right-hand side. We can therefore find the asymptotically optimal measure by minimizing the right-hand side. We consider, in particular,

THE MAIN RESULTS OF THE THESIS

1 the case of a (discretized) Asian option with payoff

P (X) = K - S 0 n n j=1 e Xt j + , t j = T j n ,
in the Heston model

dX t = - 1 2 V t dt + V t dW 1 t , X 0 = 0 dV t = λ t (µ t -V t ) dt + ζ V t dW 2 t , V 0 = V 0 d W 1 , W 2 t = ρ dt , (1.3.2)
where W is 2-dimensional correlated Brownian motion. The asymptotically optimal measure in the discretized Asian option case is the θ supported on {t 1 , ..., t n } that minimizes

log K 1 -Θ 1 - n m=1 (Θ m -Θ m+1 ) log -(Θ m -Θ m+1 ) 1 -Θ 1 nK S 0 + T n n j=1 h (Θ j ) ,
(1.3.3) where Θ j = θ([t j , T ]). Furthermore, under P θ , the dynamics of the P-Heston process (1.3.2) becomes

dX t = Θ τt + ζρ Ψ (τ t -t, Θ τt , ..., Θ n ) - 1 2 V t dt + V t d W 1 t , X 0 = 0 dV t = λt (μ t -V t ) dt + ζ V t d W 2 t , V 0 = V 0 d W 1 , W 2 t = ρ dt ,
where W is 2-dimensional correlated P θ -Brownian motion, where Ψ is defined iteratively as

Ψ (s, Θ j , ..., Θ n ) = ψ (s, Θ j , Ψ (t j+1 -t j , Θ j+1 ..., Θ n )) Ψ (s) = 0
and where, denoting

τ t = inf{s ∈ τ : s ≥ t}, λt = λ -ζΘ τt ρ -ζ 2 Ψ (τ t -t, Θ τt , ..., Θ n ) and μt = λµ λt .
In Section 2.6, we suggest a dichotomy algorithm to minimize (1.3.3).

CHAPTER 1. INTRODUCTION 1

Pricing derivatives on multiple assets in the Wishart ASV model Very early, the problem of pricing derivatives on multiple stocks, such as options on baskets, was formulated. The easiest solution naturally consists in modelling directly the price of the basket. However, the obtained prices are not consistent with the prices of single asset vanilla options and, in order for them to be, the marginals of the process used to model jointly the stock prices should be similar to the single asset model used to model the individual assets. Due to the popularity of the Heston model among practitioners, the Wishart stochastic volatility model was naturally developed.

A Wishart process is a matrix-valued symmetric non-negative definite stochastic process with dynamics

dX t = α a a + b X t + X t b dt + X 1/2 t dW t a + a (dW t ) X 1/2 t ,
where (W t ) t≤T is a matrix Brownian motion. It was invented in [START_REF] Bru | Wishart processes[END_REF] to model perturbations in biological data. Being a matrix version of the CIR process

dX t = λ (ν -X t ) dt + σ X t dW t ,
which is the instantaneous variance process of the Heston model, the Wishart model was quickly used to model the instantaneous covariance matrix in the multi-asset model developed in [START_REF] Gourieroux | Derivative pricing with multivariate stochastic volatility: Application to credit risk[END_REF],

dS t = Diag(S t ) X 1/2 t dZ t ,
where (Z t ) t≤T is an R n -dimensional Brownian motion. The Wishart stochastic volatility model3 generalizes the Heston model to the multivariate setting. Indeed, by taking a, b and X 0 diagonal, S becomes a vector of independent Heston processes. The Wishart stochastic volatility model therefore allows to model each individual price process with a distribution that is close to the one obtained with the Heston model and, at the same time, to have a rich stochastic cross-correlation structure between the price processes. It is therefore a consistent yet flexible instrument to model simultaneously multiple assets. We consider the pricing of options on baskets, i.e. with payoff

n k=1 S k T -K + .
The Wishart stochastic volatility model is affine. Therefore, option pricing is traditionally done using Fourier inversion methods (see (Da Fonseca et al.,1.3. THE MAIN RESULTS OF THE THESIS 1 2007)). However, since these methods require to compute integrals numerically, they suffer directly from the "curse of dimensionality" and become very quickly less tractable as the dimension increases. The use of Monte-Carlo methods then becomes necessary to price derivatives depending on numerous assets. Let us summarize the main results presented in Chapter 3. Let (Y t , X t ) be a Wishart stochastic volatility process, i.e. a (R n , M n )valued process with dynamics

dY t = r1 - 1 2 (a X t a) 11 , ... , (a X t a) nn dt + a X 1/2 t dZ t dX t = (αI n + b X t + X t b) dt + X 1/2 t dW t + (dW t ) X 1/2 t , X 0 = x
where r ∈ R, α > n -1, a is invertible, -b and x are symmetric and positive definite and (Z t ) t≥0 and (W t ) t≥0 are R n and R n×n -dimensional independent standard Brownian motions.

Theorem 1.3.3. For T > 0, the family (Y T ) ∈(0,1] defined by Y t := Y t/ 2 satisfies a large deviation property, when → 0 with good rate function

Λ * (y) = sup θ∈R n θ, y -Λ(θ) ,
where

Λ(θ) := T r θ 1 -α 2 Tr b + φ 1/2 (θ) if θ ∈ U ∞ if θ ∈ U , for φ(θ) := b 2 + a Diag(θ) -θθ a , and
U := θ ∈ R n : φ(θ) ∈ S + n .
The next result uses the large deviations property to characterize the limit behaviour of the implied volatility of basket options when maturity tends to infinity. Recall that the implied volatility of a basket option in a specific model is defined as the value of volatility such that the option price in this model equals the Black-Scholes option price with this volatility level obtained assuming that the entire basket follows the Black-Scholes model. Proposition 1.3.4. Let σ(T, k) be the implied volatility associated with an option on a basket of stocks with payoff

n i=1 ω i S i T -e k + . CHAPTER 1. INTRODUCTION 1 Denote x * = ∂ θ Λ(0) and x * j = [∂ θ Λ(e j )
] j for j = 1, . . . , n and let the constants β * = max j x * j , β * = min j x * j and β * = max j x * j . Then using the scaling y = k/T , if y ∈ ( β * , β * ), the limiting implied volatility is

lim T →∞ σ(T, y T ) = √ 2 ξ L(y) + y + η L(y) ,
where

L(y) =      -y -inf λ∈R n :λ i ≤0,i=1,...,n {Λ(λ) -y λ, 1 } , if y ≤ β * , -max i,j=1,...,n inf λ∈R {-λy + Λ(λe i + e j )} , if y ≥ β * , -y -max i=1,...,n inf λ∈R {-λy + Λ(λe i )} , if β * < y < β * and (ξ, η) =      (-1, 1) , if y ≤ β * , (1, -1) , if y ≥ β * , (1, 1) , if β * < y < β * . In addition, if y ∈ ( β * , β * ), σ(T, yT ) = 2y + N -1 (C ∞ (y)) T -1/2 + O(T -1/2 ) , as T → ∞, where C ∞ (y) = n i=1 ω i 1 x *
i >y and N is the Gaussian distribution function.

We now discuss the variance reduction. We consider the class of measure changes dP θ dP := e θ Y T

I E e θ Y T and wish to find the parameter θ ∈ R n that minimizes

Var P θ P (Y T ) dP dP θ = IE P 2 (Y T ) dP dP θ -IE(P (Y T )) 2
in order to minimize the variance when pricing an European option with payoff P (Y T ) using Monte-Carlo simulations. Since this problem is analytically unsolvable, we use the fact that lim

→0 log IE P 2 (Y T ) dP dP θ (Y T ) = sup y∈R n 2H(y) -θ y -Λ * (y) + Λ(θ) ,
where H = log P , to find an asymptotic proxy of the minimization problem.

We say that θ is asymptotically optimal if it minimizes the right-hand side.

The next result allows to compute the optimal measure without knowing Λ * .

Theorem 1.3.5. Let H be a concave upper semi-continuous function. Then

inf θ∈R n sup y∈R n 2H(y) -θ y -Λ * (y) + Λ(θ) = 2 inf θ∈R n Ĥ(θ) + Λ(θ) ,
where

Ĥ(θ) = sup y∈R n H(y) -θ y .
Furthermore, if θ * minimizes the right-hand side, it also minimizes the lefthand side. , the process (Y t , X t ) has dynamics

dY t = r1 - 1 2 (a X t a) 11 , ... , (a X t a) nn + a X t a θ dt + a X 1/2 t dZ θ t dX t = (αI n + (b + 2 γ θ (T -t))X t + X t (b + 2 γ θ (T -t))) dt + X 1/2 t dW θ t + (dW θ t ) X 1/2 t , X 0 = x ,
where Z θ t t≥0 and W θ t t≥0 are R n and R n×n -dimensional independent standard P θ -Brownian motions and γ θ (t) is the solution of explicitly solvable matrix Riccati equations. Let

P (Y T ) = K - n j=1 ω j e Y j T + ,
where ω j > 0, be the payoff of a put option on a basket of stocks. The asymptotically optimal θ ∈ R n is the one that minimizes Ĥ(θ) + Λ(θ), where

Ĥ(θ) = 1 - n j=1 θ j   log(K) -log 1 - n j=1 θ j +   - n j=1 θ j log(-θ j /ω j ) .

Options on realized variance and density expansion (Chapter 4)

We consider the problem of pricing options on realized variance when volatility is modelled as a diffusion process with general drift and constant diffusion coefficient. Define for every ∈ (0, 1], the joint process (Y t , Z t ) t∈[0,T ] of the integrated variance and the instantaneous volatility

dY t = g(Z t ) dt dZ t = 2 b(Z t ) dt + c dW t , (1.3.4)
where (W t ) t≤T is standard Brownian motion, with initial value Y 0 = 0 and Z 0 = z 0 > 0 and where

g(z) = z 2 e -1 R+1-|z| 1 {|z|<R+1} + (R + 1) 2 e -1 |z|-R 1 {|z|>R} e -1 R+1-|z| 1 {|z|<R+1} + e -1 |z|-R 1 {|z|>R} ,
for R arbitrarily large, is a bounded version of z → z 2 with bounded derivatives of all orders. We calculate the asymptotic expansion of the density of Y T basing our approach on the results presented in (Deuschel et al., 2014a

a ∈ 0, R 2 T 2 1 + z 0 /R arccos(z 0 /R) 1 -z 2 0 /R 2 , the density f Y t admits the expansion f Y T (a) = -1 e -Λ(a)/ 2 (c 0 (a) + o(1)) , as → 0 ,
where

Λ(a) = z 2 0 4 c 2 r 2 r T -sin (2 r T ) 1 + cos (2 r T ) , c 0 (a) = 1 2πA(2rT ) cos 3/2 (r T ) 2 z 0 c T 3/2 e 1 c 2 z T z 0 b(x) dx , A(u) = u 3 + 6 u cos(u) + 3 (u 2 -2) sin(u)
6 u 3 and r is the unique solution of equation

1 + cos (2 r T ) - z 2 0 T a 1 + sin (2 r T ) 2 r T = 0 in the set I := 0, π 2T ∪ iR + ⊂ C .
We finally use the density expansion to obtain asymptotic expansions for the price and implied volatility of options on realized variance. The implied volatility of a realized variance option in a specific model is defined as the value of volatility such that the option price in this model equals the Black-Scholes option price with this volatility value, and initial value equal to the integral of the initial value of the instantaneous variance.

Theorem 1.3.8. Let

P (z 0 , K, T ) = IE (K -T -1 Y T ) +
be the price of a put option on realized variance with maturity T and strike K. Then P admits the asymptotic expansion

P (z 0 , K, T ) = (K -z 2 0 ) + + e -Λ(KT ) 2 3 c 0 (KT ) T (Λ (KT )) 2 + O 3 as → 0 .
The BS implied volatility associated to P (z 0 , K, T ) admits the expansion

σ BS = σ BS,0 + 2 σ BS,1 + O 2 ,
where In this chapter, we study the pricing of interest rate derivatives in the Lévy Libor market model (LLMM) developed in [START_REF] Eberlein | The Lévy Libor model[END_REF] by writing the LLMM as a perturbation of the standard log-normal LMM. Let 0 ≤ T 0 < ... < T n be a tenor structure and denote by L = (L 1 , . . . , L n ) the column vector of the forward Libor rates L j t := L T j t . We assume that the dynamics of L is given by the SDE

σ BS,0 = | log(z 2 0 /K)| √ 2T Λ 1/2 (KT ) and σ BS,1 = | log(z 2 0 /K)| 2 3/2 T 1/2 Λ 3/2 (KT ) log 4 √ π z 0 K 1/2 T 2 c 0 (KT ) (Λ (KT )) 2 Λ 3/2 (KT ) | log(z 2 0 /K)| , as → 0.
dL t = L t-(b(t, L t )dt + Λ(t)dX t ) ,
where X t is a compensated d-dimensional Lévy process with non-zero diffusive part under the terminal measure Q Tn , Λ(t) a deterministic n × d volatility matrix and b(t, L t ) is the drift vector such that L j t is an Q T j -martingale. Under this model, the price of a European derivative with payoff g(L T k ) satisfies

P t = B t (T n ) IE Q Tn n j=k+1 (1 + δ j L j T k )g(L T k ) | F t = B t (T n ) u(t, L t ),
where u is the solution of the partial integro-differential equation (PIDE)

∂ t u + A t u = 0 u(T k , x) = n j=k+1
(1 + δ j x j )g(x) ,

where A t is the generator of L t . In order to approximate the solution u(t, x), we define

dL α t = L α t-(b α (t, L α t )dt + Λ(t)dX α t ) , (1.3.5)
where

X α t := αX t/α 2 and b α (t, L α t ) is the corresponding drift such that L α t is a Q T k -martingale.
The scaling X α t = α X t/α 2 leaves the diffusive part unchanged, while the jump part converges to a Brownian motion as α → 0. Therefore, for α = 1, (1.3.5) is the LLMM, whereas, for α = 0, (1.3.5) corresponds to the standard LMM. The next result approximates the option price u(t, x).

Theorem 1.3.9. Let u α (t, x) be the solution of the PIDE

∂ t u α + A α t u α = 0 , u α (T k , x) = n j=k+1 (1 + δ j x j )g(x) ,
where

A α t = ∞ j=0 α j A j t 1
is the infinitesimal generator of (1.3.5). Then u α (t, x) admits an expansion of the form

u α (t, x) = u 0 (t, x) + α u 1 (t, x) + α 2 u 2 (t, x) + O α 3 ,
where u 0 is the LMM price of the derivative, and the correction terms u 1 and u 2 are the solutions of "LMM-like" PDEs whose explicit solutions are provided in (5.4.13) for u 1 and in (5.4.18) for u 2 .

We apply this result to obtain the expansions for the prices of caplets and swaptions in the LLMM and we test the performance of our approximation at pricing caplets in the model driven by a unidimensional CGMY process.

Chapter 2

Long-time trajectorial large deviations for affine stochastic volatility models and application to variance reduction for option pricing

Introduction

The aim of this paper is to develop efficient importance sampling estimators for prices of path-dependent options in affine stochastic volatility (ASV) models of asset prices. To this end, we establish pathwise large deviation results for these models, which are of independent interest.

An ASV model, studied in (Keller-Ressel, 2011) is a two-dimensional affine process (X, V ) on R × R + with special properties, where X models the logarithm of the stock price and V its instantaneous variance. This class includes many well studied and widely used models such as Heston stochastic volatility model [START_REF] Heston | A closed-form solutions for options with stochastic volatility with applications to bond and currency options[END_REF], the model of Bates [START_REF] Bates | Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options[END_REF], Barndorff-Nielsen stochastic volatility model (Barndorff-Nielsen and Shephard, 2001) and timechanged Lévy models with independent affine time change. European options in affine stochastic volatility models may be priced by Fourier transform, but for path-dependent options explicit formulas are in general not available and Monte Carlo is often the method of choice. At the same time, Monte Carlo simulation of such processes is difficult and time-consuming: the convergence rates of discretization schemes are often low due to the irregular nature of coefficients of the corresponding stochastic differential equations. To accelerate Monte Carlo simulation, it is thus important to develop efficient variancereduction algorithms for these models.

In this paper, we therefore develop an importance sampling algorithm for CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2 ASV models. The importance sampling method is based on the following identity, valid for any probability measure Q, with respect to which P is absolutely continuous. Let P be a deterministic function of a random trajectory S, then

E[P (S)] = E Q dP dQ P (S) .
This allows one to define the importance sampling estimator

P Q N := 1 N N j=1 dP dQ (j) P (S (j) Q ),
where S

Q are i.i.d. sample trajectories of S under the measure Q. For efficient variance reduction, one needs then to find a probability measure Q such that S is easy to simulate under Q and the variance

Var Q P (S)
dP dQ is considerably smaller than the original variance Var P [P (S)].

In this paper, following the work of [START_REF] Genin | Optimal importance sampling for Lévy processes[END_REF] in the context of Lévy processes, we define the probability Q using the path-dependent Esscher transform,

dP θ dP = e [0,T ] Xt•θ(dt) E e [0,T ] Xt•θ(dt)
, where X is the first component of the ASV model (the logarithm of stock price) and θ is a (deterministic) bounded signed measure on [0, T ]. The optimal choice of θ should minimize the variance of the estimator under P θ ,

Var P θ P (S) dP dP θ = E P P 2 (S) dP dP θ -E [P (S)] 2 .
The computation of this variance is in general as difficult as the computation of the option price itself. Following [START_REF] Dupuis | Importance sampling, large deviations, and differential games[END_REF][START_REF] Glasserman | Asymptotically optimal importance sampling and stratification for pricing pathdependent options[END_REF][START_REF] Guasoni | Optimal importance sampling with explicit formulas in continuous time[END_REF][START_REF] Robertson | Sample path large deviations and optimal importance sampling for stochastic volatility models[END_REF] and more recently [START_REF] Genin | Optimal importance sampling for Lévy processes[END_REF], we propose to compute the variance reduction measure θ * by minimizing the proxy for the variance computed using the theory of large deviations.

To this end, we establish a pathwise large deviation principle (LDP) for affine stochastic volatility models. A one dimensional LDP for X t /t as t → ∞ where X is the first component of an ASV model has been proven in [START_REF] Jacquier | Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models[END_REF]. In this paper, we extend this result to the trajectorial setting, in the spirit of the pathwise LDP principles of [START_REF] Léonard | Large deviations for Poisson random measures and processes with independent increments[END_REF] but in a weaker topology.

The rest of this paper is structured as follows. In Section 2.2, we describe the model and recall certain useful properties of ASV processes. In Section 2.3, we recall some general results of large deviations theory. In Section 2.4, we prove a LDP for the trajectories of ASV processes. In Section 2.5, we develop the variance reduction method, using an asymptotically optimal change of measure obtained with the LDP shown in Section 2.4. In Section 2.6, we test the method numerically on several examples of options, some of which are path-dependent, in the Heston model with and without jumps.

Model description

In this paper, we model the price of the underlying (S t ) t≥0 of an option as S t = S 0 e Xt , where we model (X t ) t≥0 as an affine stochastic volatility process. We recall, from (Keller-Ressel, 2011) and [START_REF] Duffie | Affine processes and applications in finance[END_REF], the definition and some properties of ASV models.

Definition 2.2.1. An ASV model (X t , V t ) t≥0 , is a stochastically continuous, time-homogeneous Markov process such that e Xt t≥0 is a martingale and

IE e uXt+wVt X 0 = x, V 0 = v = e φ(t,u,w)+ψ(t,u,w) v+u x , (2.2.1) for all (t, u, w) ∈ R + × C 2 .
Proposition 2.2.2. The functions φ and ψ satisfy generalized Riccati equations

∂ t φ(t, u, w) = F (u, ψ(t, u, w)) , φ(0, u, w) = 0 (2.2.2a) ∂ t ψ(t, u, w) = R(u, ψ(t, u, w)) , ψ(0, u, w) = w , (2.2.2b)
where F and R have the Lévy-Khintchine forms

F (u, w) = u w • a 2 • u w + b • u w + D\{0} e xu+yw -1 -w F (x, y) • u w m(dx, dy) , R(u, w) = u w • α 2 • u w + β • u w + D\{0} e xu+yw -1 -w R (x, y) • u w µ(dx, dy) ,
where

D = R × R + , w F (x, y) = x 1+x 2 0 and w R (x, y) = x 1+x 2 y 1+y 2
and (a, α, b, β, m, µ) satisfy the following conditions CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2

• a, α are positive semi-definite 2×2-matrices where a 12 = a 21 = a 22 = 0.

• b ∈ D and β ∈ R 2 .

• m and µ are Lévy measures on D and D\{0} ((x 2 +y)∧1) m(dx, dy) < ∞.

In the rest of the paper, we assume that there exists u ∈ R such that R(u, 0) = 0, for the law of (X t ) t≥0 to depend on V 0 . Define the function

χ(u) = ∂ w R(u, w)| w=0 = α 12 u + β 2 + D\{0} y e xu - 1 1 + y 2 µ(dx,

dy) .

A sufficient condition for S t = S 0 e Xt to be a martingale (Keller-Ressel, 2011, Corollary 2.7), which we assume to be satisfied in the sequel, is

F (1, 0) = R(1, 0) = 0 and χ(0) + χ(1) < ∞.
In the following theorem, we compile several results of (Keller-Ressel, 2011) that describe the behaviour of the solution to eq. ( 2.2.2) as t → ∞.

Theorem 2.2.3. Assume that χ(0) < 0 and χ(1) < 0.

• There exists an interval I ⊇ [0, 1], such that for each u ∈ I, eq. (2.2.2b) admits a unique stable equilibrium w(u).

• For u ∈ I, eq. (2.2.2b) admits at most one other equilibrium w(u), which is unstable.

• For u ∈ R\I, eq. (2.2.2b) does not have any equilibrium.

We denote B(u) the basin of attraction of the stable solution w(u) of eq.

(2.2.2b) and J = {u ∈ I : F (u, w(u)) < ∞}, the domain of u → F (u, w(u)).

We have that

• J is an interval such that [0, 1] ⊆ J ⊆ I.
• For u ∈ I, w ∈ B(u) and ∆t > 0, we have

ψ ∆t , u, w -→ →0 w(u) .
(2.2.3)

• For u ∈ J, w ∈ B(u) and ∆t > 0, φ ∆t , u, w -→ →0 ∆t h (u) , (2.2.4)
where h(u) = F (u, w(u)) = lim →0 log IE e uX 1/ .

• For every u ∈ I, 0 ∈ B(u).

MODEL DESCRIPTION

2 Definition 2.2.4. A convex function f : R n → R∪{∞} with effective domain D f is essentially smooth if i. D • f is non-empty; ii. f is differentiable in D • f ; iii. f is steep, that is, for any sequence (u n ) n∈N ⊂ D • f that converges to a point in the boundary of D f , lim n→∞ ||∇f (u n )|| = ∞ .
In the rest of the paper, we shall make the following assumptions on the model.

Assumption 3. The function h satisfies the following properties.

1. There exists u < 0, such that h(u) < ∞.

u → h(u) is essentially smooth.

In [START_REF] Jacquier | Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models[END_REF], a set of sufficient conditions is provided for Assumption 3 to be verified: Proposition 2.2.5 (Corollary 8 in [START_REF] Jacquier | Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models[END_REF]). Let (X, V ) be an ASV model such that u → R(u, 0) and w → F (0, w) are not identically 0 and χ(0) and χ(1) are strictly negative. If either of the following conditions holds (i) The Lévy measure µ of R has exponential moments of all orders, F is steep and (0, 0), (1, 0) ∈ D • F .

(ii) (X, V ) is a diffusion, then function h is well defined, for every u ∈ R with effective domain J. Moreover h is essentially smooth and {0, 1} ⊂ J • .

We now discuss the form of the basin of attraction of the unique stable solution of (2.2.2b).

CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2

Lemma 2.2.7. Let f : R → R ∪ {+∞} be a convex function with either two zeros w < w, or a single zero w. In the latter case, we let w = ∞. Assume that there exists y ∈ (w, w) such that f (y) < 0. Then for every x ∈ D f ,

f (x) > 0 , if x < w or w < x , f (x) < 0 , if x ∈ (w, w) .
Proof. By convexity, for every

x ∈ D f such that x < w, y -w y -x f (x) + w -x y -x f (y) ≥ f (w) = 0 and therefore f (x) ≥ -w-x y-w f (y) > 0. Furthermore, for every x ∈ (w, y], f (x) ≤ y -x y -w f (w) + x -w y -w f (y) < 0 . Let s = sup{x ∈ D f : f (x) < 0}. If f is continuous in s, then w = s and for every x > w in D f , f (x) ≥ -w-x y- w f (y) > 0. If f is discontinuous in s however, then by convexity, f (x) = +∞ for x > s. Proposition 2.2.8. Let u ∈ I and consider w(u) the stable equilibrium of (2.2.2b). Then the basin of attraction of w(u) is B(u) = (-∞, w(u))∩D R(u,•)
, where w(u) = ∞ when (2.2.2b) admits only one equilibrium.

Proof. By Lemma 2.2.6, w → R(u, w) is convex. Since w(u) is a stable equilibrium, the hypotheses of Lemma 2.2.7 are verified. Therefore, R(u, w) > 0 for every w < w(u), whereas R(u, w) < 0 for every w ∈ D R(u,•) such that w(u) < w < w(u). This implies that the solution of

∂ t ψ(t, u, w) = R(u, ψ(t, u, w)) , ψ(0, u, w) = w (2.2.5) converges to w(u) for every w ∈ (-∞, w(u)) ∩ D R(u,•) , whereas, if w > w, the solution of (2.2.5) diverges to ∞.

Large deviations theory

In this section, we recall some useful classical results of the large deviations theory. We refer the reader to [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF] for the proofs and for a broader overview of the theory.

Theorem 2.3.1 (Gärtner-Ellis). Let (X ) ∈]0,1] be a family of random vectors in R n with associated measure µ . Assume that for each λ ∈ R n ,

Λ(λ) := lim →0 log IE e λ,X

LARGE DEVIATIONS THEORY

2 as an extended real number. Assume also that 0 belongs to the interior of

D Λ := {θ ∈ R n : Λ(θ) < ∞}. Denoting Λ * (x) = sup θ∈R n θ, x -Λ(θ) ,
the following hold:

(a) For any closed set F ,

lim sup →0 log µ (F ) ≤ -inf x∈F Λ * (x) . (b) For any open set G, lim inf →0 log µ (G) ≥ -inf x∈G∩F Λ * (x) ,
where F is the set of exposed points of Λ * , whose exposing hyperplane belongs to the interior of D Λ .

(c) If Λ is an essentially smooth, lower semi-continuous function, then µ satisfies a LDP with good rate function Λ * .

Definition 2.3.2. A partially ordered set (P, ≤) is called right-filtering if for every i, j ∈ P, there exists k ∈ P such that i ≤ k and j ≤ k.

Definition 2.3.3. A projective system (Y j , p ij ) i≤j∈P on a partially ordered right-filtering set (P, ≤) is a family of Hausdorff topological spaces (Y j ) j∈P and continuous maps

p ij : Y j → Y i such that p ik = p ij • p jk whenever i ≤ j ≤ k.
Definition 2.3.4. Let (Y j , p ij ) i≤j∈P be a projective system on a partially ordered right-filtering set (P, ≤). The projective limit of (Y j , p ij ) i≤j∈P , denoted

X = lim ←-
Y j , is the subset of topological spaces Y = j∈P Y j , consisting of all the elements x = (y j ) j∈P for which y i = p ij (y j ) whenever i ≤ j, equipped with the topology induced by Y. The projective limit of closed subsets F j ⊆ Y j are defined in the same way and denoted

F = lim ←- F j .
Remark 2.3.5. The canonical projections of X , i.e. the restrictions p j : X → Y j of the coordinate maps from X to Y j , are continuous.

Theorem 2.3.6 (Dawson-Gärtner). Let (Y j , p ij ) i≤j∈P be a projective system on a partially ordered right-filtering set (P, ≤) and let (µ ) be a family of probabilities on X = lim ←-

Y j , such that for any j ∈ P, the Borel probability

µ • p -1 j
on Y j satisfies the LDP with the good rate function Λ j . Then µ satisfies the LDP with good rate function

Λ(x) = sup j∈P Λ j (p j (x)) .

CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2

Theorem 2.3.7 (Varadhan's Lemma, version of (Guasoni and Robertson, 2008)). Let (X ) ∈]0,1 ] be a family of X -valued random variables, whose laws µ satisfy a LDP with rate function Λ.

If ϕ : X → R ∪ {-∞} is a continuous function which satisfies lim sup →0 log IE exp γ ϕ(X ) < ∞ for some γ > 1, then lim →0 log IE exp ϕ(X ) = sup x∈X {ϕ(x) -Λ(x)} .

Trajectorial large deviations for affine stochastic volatility model

In this section, we prove a trajectorial LDP for (X t ) when the time horizon is large. Define, for ∈ (0, 1] and 0 ≤ t ≤ T , the scaling X t = X t/ . We proceed by proving first a LDP for X t in finite dimension, that we extend, in a second step to the whole trajectory of (X t ) 0≤t≤T .

Finite-dimensional LDP

Let τ = {0 < t 1 < ... < t n = t}, by convention t 0 = 0, and define

Λ ,τ (θ) = log IE e n k=1 θ k X t k ,
for θ ∈ R n . We start by formulating our main technical assumption.

Assumption 4. One of the following conditions is verified.

1. The interval support of F is J = [u -, u + ] and w(u -) = w(u + ).
2. For every u ∈ R, w(•) = ∞, i.e, the generalized Riccati equations have only one (stable) equilibrium.

The following Lemma gives an intuition on Assumption 4.

Lemma 2.4.1. For every u 1 , u 2 ∈ I, w(u 1 ) ≥ w(u 2 ).

Proof. If Assumption 4(2) holds, then the result is obvious. Assume then that it is Assumption 4(1), that holds. Since u → w(u) is convex and u → w(u) is concave (Keller-Ressel, 2011, Lemma 3.3), then for every u 1 , u 2 ∈ I,

w(u 1 ) ≥ u + -u 1 u + -u - w(u -) + u 1 -u - u + -u - w(u + ) ≥ w(u -) , while w(u 2 ) ≤ u + -u 2 u + -u - w(u -) + u 2 -u - u + -u - w(u + ) = w(u -) .
Therefore w(u 1 ) ≥ w(u 2 ) for every u 1 , u 2 ∈ I.

TRAJECTORIAL LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODEL 2

As a first step to apply Theorem 2.3.1, we prove the following result.

Theorem 2.4.2. Let θ ∈ R n . If Assumption 4 holds, then

Λ τ (θ) := lim →0 Λ ,τ (θ/ ) = n j=1 (t j -t j-1 ) h (Θ j ) if Θ j ∈ J , ∀j ∞ otherwise ,
where Θ j := n k=j θ k . Proof. Since Assumption 4 holds, then, by Lemma 2.4.1, w(Θ j+1 ) ∈ B(Θ j ) for every j. Assume first that Θ j ∈ J for every j. Using the Markov property and eq. ( 2.2.1), we obtain

Λ τ (θ) = lim →0 log IE e n j=1 θ j X t j / = lim →0 log IE e n-1 j=1 θ j X t j / IE e ΘnX tn/ X t n-1 / , V t n-1 / = lim →0 φ t n -t n-1 , Θ n , 0 + log IE e n-2 j=1 θ j X t j / +Θ n-1 X t n-1 / +ψ tn-t n-1 , Θn, 0 V t n-1 / . Since Θ n ∈ J and 0 ∈ B(Θ n ), eqs. (2.2.
3) and (2.2.4) apply and

Λ τ (θ) = lim →0 log IE e n-2 j=1 θ j X t j / +Θ n-1 X t n-1 / +ψ tn-t n-1 , Θn, 0 V t n-1 / + (t n -t n-1 ) h(Θ n ) .
Using the fact that Θ j ∈ J and w(Θ j+1 ) ∈ B(Θ j ) for every j, we can iterating the procedure to obtain

Λ τ (θ) = n j=1 (t j -t j-1 ) h (Θ j ) + lim →0 ψ t 1 -t 0 , Θ 1 , w (Θ 2 ) V 0 + n k=1 θ k X 0 = n j=1 (t j -t j-1 ) h (Θ j ) .
(2.4.1)

Assume now that there exists k such that Θ k ∈ J. Without loss of generality, we take the largest such k. Following the same procedure, we find

Λ τ (θ) = lim →0 log IE e k-2 j=1 θ j X t j / +Θ k-1 X t k-1 / +ψ t k -t k-1 ,Θ k ,w(Θ k+1 ) V t k-1 / + φ t k -t k-1 , Θ k , w(Θ k+1 ) + n j=k+1 (t j -t j-1 ) h(Θ j ) .
Noting that φ(•, u, w) explodes in finite time for u ∈ J then finishes the proof.

CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2

We now proceed to the finite-dimensional large deviations result.

Theorem 2.4.3. Let (X t ) t≥0, ∈(0,1] and τ = {t 1 , ..., t n } as previously. Assuming that Assumption 4 holds, then (X t 1 , ..., X tn ) satisfies a LDP on R n with good rate function

Λ * τ (x) = sup Θ∈J n n j=1 Θ j (x j -x j-1 ) - n j=1 (t j -t j-1 ) h (Θ j ) ,
where

Θ j = n k=j θ k .
Proof. By Assumption 3(1), there exists u ∈ J such that u < 0, which implies that [u, 1] ⊂ J and therefore 0 is in the interior of D Λτ = J n . Theorem 2.4.2 implies that the limit

Λ τ (θ) = lim →0 Λ ,τ (θ/ ) = n j=1 (t j -t j-1 ) h (Θ j ) if Θ j ∈ J , ∀j ∞ otherwise ,
where Θ j := n k=j θ k , exists as an extended real number. Since, by Assumption 3(2), h is essentially smooth and lower semi-continuous, then so is Λ τ . Theorem 2.3.1 then applies and (X t 1 , ..., X tn ) satisfies a LDP, on R n , with good rate function Λ * τ (x) = sup

θ∈R n θ x -Λ τ (θ) .
Furthermore,

Λ * τ (x) = sup θ∈R n θ x -Λ τ (θ) = sup Θ∈J n n j=1 n k=j θ k (x j -x j-1 ) - n j=1 (t j -t j-1 ) h (Θ j ) = sup Θ∈J n n j=1 Θ j (x j -x j-1 ) - n j=1 (t j -t j-1 ) h (Θ j ) ,
which finishes the proof.

Infinite-dimensional LDP Extension of the LDP

We now extend the LDP to the whole trajectory of (X t ) 0≤t≤T on F([0, T ], R) := {x : [0, T ] → R, x 0 = 0}, the set of all functions from [0, T ] to R that vanish at 0, by proving the following general lemma.

TRAJECTORIAL LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODEL 2

Lemma 2.4.4. Let (P, ≤) be the partially ordered right-filtering set

P = ∞ n=1 {(t 1 , ..., t n ) , 0 ≤ t 1 ≤ ... ≤ t n ≤ T }
ordered by inclusion. We consider, on (P, ≤), the projective system (Y j , p ij ) i≤j∈P defined by Y j = R #j and p ij : Y j → Y i the natural projection on shared times. Assume that for any j = {t 1 , ..., t n }, the finite-dimensional process (X t 1 , ..., X tn ) satisfies a large deviation property with good rate function Λ j . Then the family (X t ) 0≤t≤T satisfies a large deviation property on X = F([0, T ], R) equipped with the topology of pointwise convergence, with good rate function

Λ(x) = sup j∈P Λ j (p j (x)) ,
where p τ (x) = (x t 1 , ..., x tn ) is the canonical projection from X to Y τ .

Proof. Let µ be the probability measure generated by (X t ) 0≤t≤T on X . Then, by hypothesis, for any j ∈ P, µ • p -1 j satisfies a LDP with good rate function Λ τ . The result then follows from Theorem 2.3.6.

Theorem 2.4.5. Assume that Assumption 4 holds, then (X t ) 0≤t≤T satisfies a LDP on F([0, T ], R) equipped with the topology of point-convergence, as → 0, with good rate function

Λ * (x) = sup τ Λ * τ (x) ,
where the supremum is taken over the discrete ordered subsets of the form

τ = {t 1 , ..., t n } ⊂ [0, T ].
Proof. The result is a direct application of Lemma 2.4.4.

Calculation of the rate function

We finally calculate the rate function of Theorem 2.4.5.

Theorem 2.4.6. The rate function of Theorem 2.4.5 is

Λ * (x) = T 0 h * ( • x t ac ) dt + T 0 H dν t dθ t dθ t , where h * (y) = sup θ∈J {θy -h(θ)} , H(y) = lim →0 h * (y/ ) ,
•

x ac is the derivative of the absolutely continuous part of x, ν t is the singular component of dx t with respect to dt and θ t is any non-negative, finite, regular, R-valued Borel measure, with respect to which ν t is absolutely continuous.

CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2

Proof. By identifying (Θ 1 , ..., Θ n ) with (θ t 1 , ..., θ tn ), we find for every x ∈ F([0, T ], R),

sup τ Λ * τ (x) = sup τ sup Θ∈J #τ #τ j=1 Θ j (x t j -x t j-1 ) -(t j -t j-1 )h(Θ j ) = sup θ∈F ([0,T ],R) sup τ #τ j=1 θ t j (x t j -x t j-1 ) -(t j -t j-1 )h(θ t j ) = sup θ∈C([0,T ],J) sup τ #τ j=1 θ t j (x t j -x t j-1 ) -(t j -t j-1 )h(θ t j ) .
Note that the supremum can be taken indifferently on F([0, T ], J) or on C([0, T ], J) because the objective function depends on θ only on a finite set.

Since we have assumed that there exists u < 0 in J, then if x has infinite variation, we immediately find that Λ * (x) = ∞. Assume therefore that x has finite variation. We wish to show that

sup θ∈C([0,T ],J) sup τ #τ j=1 θ t j (x t j -x t j-1 ) -(t j -t j-1 )h(θ t j ) = sup θ∈C([0,T ],J) T 0 θ t dx t - T 0 h(θ t )dt . Notice that sup θ∈C([0,T ],J) sup τ #τ j=1 θ t j (x t j -x t j-1 ) -(t j -t j-1 )h(θ t j ) ≥ sup θ∈C([0,T ],J) lim sup τ #τ j=1 θ t j (x t j -x t j-1 ) -(t j -t j-1 )h(θ t j ) = sup θ∈C([0,T ],J) T 0 θ t dx t - T 0 h(θ t )dt .
To prove the other inequality, we use the following construction. Fix τ and let θ ∈ C([0, T ], J). Let also > 0 such that < min(t j -t j-1 ) and define θ ,τ as

θ ,τ t = θ t j-1 + t-t j-1 (θ t j -θ t j-1 ) if t ∈ [t j-1 , t j-1 + ] , θ t j if t ∈ [t j-1 + , t j ] . Then #τ j=1 θ t j (x t j -x t j-1 ) -(t j -t j-1 )h(θ t j ) - T 0 θ ,τ t dx t + T 0 h(θ ,τ t )dt 2.4. TRAJECTORIAL LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODEL 2 = #τ j=1 (θ t j -θ t j-1 ) t j-1 + t j-1 1 - t -t j-1 dx t + t j-1 + t j-1 h(θ ,τ t ) -h(θ t j )dt ≤ #τ j=1 θ t j -θ t j-1 t j-1 + t j-1 1 - t -t j-1 dx t + 2 max |h(θ)| : θ ∈ [θ t j-1 , θ t j ] ≤ #τ j=1 θ t j -θ t j-1 µ x ]0, ] + 2 max |h(θ)| : θ ∈ [θ t j-1 , θ t j ] → →0 0 ,
where µ x is the measure associated with x. Hence

sup θ∈C([0,T ],J) sup τ #τ j=1 θ t j (x t j -x t j-1 ) -(t j -t j-1 )h(θ t j ) ≤ sup θ∈C([0,T ],J) T 0 θ t dx t - T 0 h(θ t )dt and Λ * (x) = sup θ∈C([0,T ],J) T 0 θ t dx t - T 0 h(θ t ) dt .
We will now use (Rockafellar, 1971, Thm. 5.) to obtain the result. Since x has finite variations, the measure dx t is regular. Using the notations of (Rockafellar, 1971), in our case the multifunction D is the constant multifunction t → D(t) = J. Therefore D is fully lower semi-continuous. Furthermore, since [0, 1] ⊂ J, the interior of D(t) is non-empty. The set [0, T ] is compact with no non-empty open sets of measure 0 and for every u in the interior of J, and

V ∈ [0, T ] open, V |h(u)| dt ≤ T |h(u)| < ∞ .
( Rockafellar, 1971, Thm. 5.) then implies that sup

θ∈C([0,T ],J) T 0 θ t dx t - T 0 h(θ t ) dt = T 0 h * ( • x t ac ) dt + T 0 H dν t dθ t dθ t , where h * (y) = lim →0 sup θ∈J {θy -h(θ)} , H(y) = lim →0 h * (y/ ) ,
•

x ac is the derivative of the absolutely continuous part of x, ν t is the singular component of dx t with respect to dt and θ t is any non-negative, finite, regular, R-valued Borel measure, with respect to which ν t is absolutely continuous. Remark 2.4.7. In particular, the proof of theorem 2.4.6 shows that, if x does not belong to V r , the set of trajectories x : [0, t] → R with bounded variation, then Λ * (x) = ∞.

CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2

Variance reduction

Denote P (S) the payoff of an option on (S t ) 0≤t≤T . The price of an option is generally calculated as the expectation IE(P (S)) under a certain risk-neutral measure P. For any equivalent measure Q, the price of the derivative can be written

IE(P (S)) = IE Q P (S) dP dQ .
The variance of P (S) is

Var P (P (S)) = IE P 2 (S) -IE 2 (P (S)) ,
whereas the variance of

Var Q P (S) dP dQ = IE Q P 2 (S) dP dQ 2 -IE Q P (S) dP dQ 2 = IE P 2 (S) dP dQ -IE 2 (P (S)) .
We can therefore choose Q in order to reduce the variance of the random variable, whose expectation gives the price of the derivative.

inf θ∈M IE exp 2H(X) - T 0 X t dθ t + G 1 (θ) , (2.5.1)
where

G (θ) := log IE e 1 T 0 X t dθt .
The optimization problem (2.5.1) cannot be solved explicitly. We therefore choose to solve the problem asymptotically using the two following lemmas. Denote M the set of measures θ ∈ M with support on a finite set of points. We first give a lemma that characterizes the behaviour of G (θ) as → 0, for θ ∈ M as this will be sufficient for the cases that we will consider in Section 2.6 (see Prop. 2.5.5).

Lemma 2.5.1. If Assumption 4 holds, then for any measure θ ∈ M , such that for every t

∈ [0, T ], θ([t, T ]) ∈ J, lim →0 G (θ) = T 0 h(θ([t, T ])) dt .
Proof. Denote τ = {t 1 , ..., t n }, the support of θ. We then obtain lim

→0 log IE e 1 T 0 X t dθt = lim →0 log IE e 1 n j=1 X t j θ (t j-1 ,t j ] = n j=1 (t j -t j-1 ) h θ (t j-1 , t n ] = T 0 h(θ([t, T ])) dt by applying Theorem 2.4.2 to θ = θ (t 0 , t 1 ] , ..., θ (t n-1 , t n ] .
Next, we give a result that characterizes the behaviour of the variance minimization problem 2.5.1 where X has been replaced by X as → 0.

Lemma 2.5.2. Let θ ∈ M such that -θ([t, T ]) ∈ J • for every t ∈ [0, T ].
Assume that the assumptions of Theorem 2.4.3 hold. Assume furthermore that H : F([0, T ], R) → R is bounded from above by a constant C and continuous on D the set of functions x ∈ V r , such that H(x) > -∞, with respect with to the pointwise convergence topology. Then

lim →0 log IE exp 2H(X ) - T 0 X t dθ t + G (θ) = sup x∈D 2H(x) - T 0 x t dθ t -Λ * (x) + T 0 h(θ([t, T ])) dt .
Proof. First note that, by Lemma 2.5.1,

lim →0 log IE exp 2H(X ) - T 0 X t dθ t + G (θ) = lim →0 log IE exp 2H(X ) - T 0 X t dθ t + T 0 h(θ([t, T ])) dt .
We therefore just need to prove that lim

→0 log IE exp 2H(X ) - T 0 X t dθ t = sup x∈D 2H(x) - T 0 x t dθ t -Λ * (x) . Denote ϕ : F([0, T ], R) → R the function ϕ(x) = 2H(x) - T 0 x t dθ t .
Since H is assumed to be continuous and θ has support on τ , ϕ is continuous. Let us show the integrability condition of Theorem 2.3.7. For every γ > 0 lim sup

→0 log IE exp γ ϕ(X ) CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2 = lim sup →0 log IE exp 2γH(X ) -γ T 0 X t dθ t ≤ 2γC + lim sup →0 log IE e 1 T 0 X t d(-γθ)t .
Since -θ([t, T ]) ∈ J • for every t ∈ [0, T ], there exists γ > 1 such that -γθ([t, T ]) remains in J for every t. Therefore Lemma 2.5.1 applies and lim sup

→0 log IE exp γ ϕ(X ) ≤ 2γC + T 0 h(-γ θ([t, T ])) dt < ∞ .
Theorem 2.3.7 then applies and yields the result.

Definition 2.5.3. Let θ ∈ M . We say that θ is asymptotically optimal if it minimises

sup x∈Vr 2H(x) - T 0 x t dθ t -Λ * (x) + T 0 h(θ([t, T ])) dt .
In general, Λ * is not easy to calculate explicitly. To solve this problem, we cite the following theorem of [START_REF] Genin | Optimal importance sampling for Lévy processes[END_REF].

Theorem 2.5.4. Let H be concave and assume that the set {x ∈ V r : H(x) > -∞} is non-empty and contains a constant element. Assume furthermore that H is continuous on this set with respect to the topology of pointwise convergence, that h is lower semi-continuous with open and bounded effective domain and that there exists a λ > 0 such that h is complex-analytic on {z ∈ C : |Im(z)| < λ}. Then

inf θ∈M sup x∈Vr 2H(x) - T 0 x t dθ t -Λ * (x) + T 0 h(θ([t, T ])) dt = 2 inf θ∈M Ĥ(θ) + T 0 h(θ([t, T ])) dt , where Ĥ(θ) = sup x∈Vr H(x) - T 0 x t dθ t .
Furthermore, if θ * minimises the left-hand side of the above equation, it also minimises the right-hand side.

We finally give a result for the case where H depends on x only through x t 1 , ...., x tn .

Proposition 2.5.5. Let τ = {t 1 , ..., t n } and let H : F([0, t], R) → R ∪ {-∞} be a log-payoff depending on x only through x τ . Then for every θ ∈ M such that θ(τ

) = θ([0, T ]), Ĥ(θ) = ∞. Proof. Assume that θ ∈ M is such that θ(τ ) = θ([0, T ]). Then there exists a set A ⊂ [0, T ]\τ , such that θ(A) = 0. Fix x ∈ D. By definition, H(x) > -∞. Then H(x + α 1 A ) - T 0 xt + α 1 A dθ t = H(x) - T 0 xt dθ t -α θ(A) .
By letting α tend to sgn(θ) ∞, one can therefore increase indefinitely H(x) -T 0 x t dθ t . Therefore, Ĥ(θ) = ∞.

Numerical examples

In this section, we apply the variance reduction method to several examples. We first show a result for options on the average value of the underlying over a finite set of points.

Proposition 2.6.1. Let τ = {t 1 , ..., t n } and consider an option with logpayoff

H(x) = log K - S 0 n n j=1 e xt j + .
Then for any θ ∈ M with support on θ = {t 1 , ..., t n },

Ĥ(θ) = log K 1 -n l=1 θ l - n m=1 θ m log -θ m n K/S 0 1 -n l=1 θ l (2.6.1)
where we use the abuse of notation θ j = θ({t j }).

Proof. In this case,

H(x) - T 0 x t dθ t = log K - S 0 n n j=1 e xt j + - n j=1 θ j x t j .
When the option is out or at the money, the log-payoff is -∞. Assume that x is such that H(x) > -∞ and differentiate with respect to x t j . We obtain

0 = ∂ xt j log K - S 0 n n l=1 e xt l - n l=1 x t l θ l = -S 0 n e xt j K -S 0 n n l=1 e xt l -θ j . Therefore the x that maximises H(x) - t 0 x s dθ s satisfies e xt j θ j = -n K S 0 + n l=1 e xt l = -n K S 0 + e xt j θ j n l=1 θ l ,
for every j. Therefore

x t j = log -θ j n K/S 0 1 -n l=1 θ l . Inserting x t j in the value of H(x) - T 0 x t dθ t ,

European and Asian put options in the Heston model

Consider the Heston model [START_REF] Heston | A closed-form solutions for options with stochastic volatility with applications to bond and currency options[END_REF])

dX t = - V t 2 dt + V t dW 1 t , X 0 = 0 dV t = λ(µ -V t ) dt + ζ V t dW 2 t , V 0 > 0 d W 1 , W 2 t = ρ dt , (2.6.2)
where W 1 , W 2 are standard P-Brownian motions. The Laplace transform of (X t , V t ) is IE e uXt+wVt = e φ(t,u,w)+ψ(t,u,w)V 0 +uX 0 , where φ, ψ satisfy the Riccati equations

∂ t φ(t, u, w) = F (u, ψ(t, u, w)) φ(0, u, w) = 0 ∂ t ψ(t, u, w) = R(u, ψ(t, u, w)) ψ(0, u, w) = w (2.6.3) for F (u, w) = λµw and R(u, w) = ζ 2 2 w 2 + ζρ uw -λw + 1 2 (u 2 -u) .
A standard calculation shows that the solution of the Riccati equations (2.6.3) is

ψ(t, u, w) = 1 ζ λ ζ -ρu - γ ζ 2 tanh γ 2 t + η 1 + η tanh γ 2 t φ(t, u, w) = µ λ ζ λ ζ -ρu t -2µ λ ζ 2 log cosh γ 2 t + η sinh γ 2 t ,
(2.6.4)

where γ = γ(u) = ζ λ ζ -ρu 2 + 1 4 -u -1 2 2 and η = η(u, w) = λ-ζρu-ζ 2 w γ(u)
.

Furthermore, for the Heston model, the function h is given by

h(u) = µ λ ζ λ ζ -ρu -µ λ ζ 2 γ(u) .
(2.6.5)

Remark 2.6.2. The log-Laplace transform h of the Heston model converges to the log-Laplace transform of an NIG process (Barndorff-Nielsen, 1997), which is complex-analytic on a strip around the real axis, thus allowing to apply Theorem 2.5.4.

The following proposition describes the effect of the time dependent Esscher transform on the dynamics of the Heston model. Under P θ , the dynamics of the P-Heston process (X t , V t ) becomes

dX t = Θ τt + ζρ Ψ (τ t -t, Θ τt , ..., Θ n ) - 1 2 V t dt + V t d W 1 t , X 0 = 0 dV t = λt (μ t -V t ) dt + ζ V t d W 2 t , V 0 = V 0 d W 1 , W 2 t = ρ dt ,
(2.6.6) where W is 2-dimensional correlated P θ -Brownian motion, Θ j = n m=j θ m , where Ψ is defined iteratively as

Ψ (s, Θ j , ..., Θ n ) = ψ (s, Θ j , Ψ (t j+1 -t j , Θ j+1 ..., Θ n )) Ψ (s) = 0
and where, denoting

τ t = inf{s ∈ τ : s ≥ t}, λt = λ -ζΘ τt ρ -ζ 2 Ψ (τ t -t, Θ τt , ..., Θ n ) and μt = λµ λt . Proof. Denote D(t, X t , V t ) = dP θ dP Ft . Then D(t, X t , V t ) = e τ t -1 j=1 θ j Xt j IE e n j=1 θ j Xt j IE e n j=τ t θ j Xt j F t = e τ t -1 j=1 θ j Xt j +Φ(τt-t,Θτ t ,...,Θn) e Φ(t 1 ,Θ 1 ,...,Θn)+Ψ(t 1 ,Θ 1 ,...,Θn) V 0 +Θ 1 X 0 e Ψ(τt-t,Θτ t ,...,Θn) Vt+Θτ t Xt ,
where Φ is defined iteratively as

Φ (s, Θ j , ..., Θ n ) = φ (s, Θ j , Ψ (t j+1 -t j , Θ j+1 , ..., Θ n )) + Φ (t j+1 -t j , Θ j+1 , ..., Θ n ) Φ (s) = 0 .
The dynamics of D(t, X t , V t ) can then be expressed using Itō's Lemma as

dD(t, X t , V t ) = D(t, X t , V t ) (Θ τt dX t + Ψ (τ t -t, Θ τt , ..., Θ n ) dV t ) + ... dt CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2 = D(t, X t , V t ) V t Θ τt dW 1 t + ζ Ψ (τ t -t, Θ τt , ..., Θ n ) dW 2 t . By Girsanov's theorem, d W 1 t W 2 t = d W 1 t W 2 t -V t Θ τt + ζρ Ψ (τ t -t, Θ τt , ..., Θ n ) Θ τt ρ + ζ Ψ (τ t -t, Θ τt , ..., Θ n ) dt
is a 2-dimensional Brownian motion under the measure P θ . Replacing W in eq. (2.6.2) by W gives the result.

Remark 2.6.4. Prop. 2.6.3 shows that the time-dependent Esscher transform changes a classical Heston process into a Heston process with timeinhomogeneous drift.

Remark 2.6.5. Note that Assumption 4 is verified by the Heston model only when ρ = 0. Indeed, J = [u -, u + ], where

u ± = 1 2 -λ ζ ρ ± 1 2 -λ ζ ρ 2 + λ 2 ζ 2 (1 -ρ 2 ) (1 -ρ 2 ) , while w(u -) = 1 ζ λ ζ -ρu - and w(u + ) = 1 ζ λ ζ -ρu + .
However, since the actual variance reduction problem is itself unsolvable, our goal is to find a good candidate measure that we can test numerically. The fact that we do not have the full theory to justify it is therefore not problematic.

Numerical results for European put options

In this case, by Prop. 2.6.1 with n = 1 and t 1 = T , θ has support on {T }.

Using the abuse of notation θ := θ({T }), we have

Ĥ(θ) + T 0 h(θ([t, T ])) dt = log K 1 -θ -θ log -θ K/S 0 1 -θ + T µ λ ζ λ ζ -ρ θ - γ(θ) ζ .
(2.6.7)

In order to obtain θ, we therefore differentiate (2.6.7) with respect to θ and equate the derivative to 0 by dichotomy . We simulate N = 10000 trajectories of the Heston model with parameters λ = 1.15, µ = 0.04, ζ = 0.2, ρ = -0.4 and initial values V 0 = 0.04 and S 0 = 1, under both P, eq. (2.6.2) and P θ , eq. (2.6.6) with n = 1 and t 1 = T , using a standard Euler scheme with 200 discretization steps. For the P-realisations X (i) , we calculate the European put price as 1 N N j=1 K -S 0 e X (i) T + and for the P θ -realisations X (i,θ) , as

e φ(T,θ,0)+ψ(T,θ,0) V 0 N N j=1 e -θ X (i,θ) T K -S 0 e X (i,θ) T + .
(2.6.8)

Each time, we compute the P θ -standard deviation, the variance ratio and the adjusted variance ratio, i.e. the variance ratio divided by the ratio of simulation time. The latter measures the actual efficiency of the method, given the fact that simulating under the measure change takes in general slightly more time.

In Table 2.1, we fix the strike to the value K = 1 and let the maturity T vary from 0.25 to 3, whereas in Tables 2.2 and 2.3, we fix maturity to T = 1 and to T = 3, while we let the strike K vary between 0.25 and 1.75. We calculate each time the price, the standard error, the variance ratio adjusted and not adjusted by the ratio of simulation time. In all the cases, we can see that the variance ratio becomes very interesting when the option gets deeply out of the money and less significant, yet still very interesting, when the option is at or in the money. This corresponds to the natural behaviour of variance reduction techniques that involve measure changes, as the measure change is going to increase the probability of choosing a trajectory that is eventually going to enter the money. Note that the simulation time is only slightly larger when simulating with the measure change, while the time required for the optimization procedure is negligible compared with the simulation time. In Figure 2.6.1, we fix the maturity to T = 1.5 and plot the empirical variance of the estimator (2.6.8) as a function of θ. Our method provides θ = -0.457 as asymptotically optimal measure change. We can therefore see that the asymptotically optimal θ is very close to the optimal one.

Numerical results for Asian put options

We now consider the case of a (discretized) Asian put option. Here, the log-payoff is

H(X) = log K - S 0 n n j=1 e Xt j +
, where t j = j n T . By Prop. 2.5.5, the support of θ is {t 1 , ..., t n } and we can denote θ j = θ({t j }). Using Prop. 2.6.1 and eq. (2.6.5), the function that we need to minimize

log K 1 -n l=1 θ l - n m=1 θ m log -θ m n K/S 0 1 -n l=1 θ l + T n n j=1 h n l=j θ l or, alternatively, denoting Θ j = n l=j θ l , log K 1 -Θ 1 - n m=1 (Θ m -Θ m+1 ) log -(Θ m -Θ m+1 ) nK/S 0 1 -Θ 1 + T n n j=1 h (Θ j ) .
By differentiating with respect to Θ j , we obtain, for j = 2, ..., n,

0 = ∂ Θ j Ĥ(θ) + T n n m=1 h (Θ m ) = T h (Θ j ) n -log [-(Θ j -Θ j+1 )] + log [-(Θ j-1 -Θ j )] ,
(2.6.9) while, for j = 1, we have

0 = ∂ Θ 1 Ĥ(θ) + T n n m=1 h (Θ m ) = log (1 -Θ 1 ) -log(n K/S 0 ) + T n h (Θ 1 ) -log [-(Θ 1 -Θ 2 )] .
(2.6.10)

Finally, taking the exponential in eqs. (2.6.9) and (2.6.10), we obtain

Θ 2 -Θ 1 = (1 -Θ 1 ) e T n h (Θ 1 ) • S 0 n K Θ 3 -Θ 2 = (Θ 2 -Θ 1 ) e T n h (Θ 2 ) . . . = . . . Θ n -Θ n-1 = (Θ n-1 -Θ n-2 ) e T n h (Θ n-1 ) -Θ n = (Θ n -Θ n-1 ) e T n h (Θn) .
Finally, define T the real function that associates to Θ n

T (Θ n ) = (1 -Θ 1 )e T n h (Θ 1 ) • S 0 n K -Θ 2 -Θ 1 , CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2 where Θ n-1 = Θ n + Θ n e -T n h (Θn) and iteratively, Θ j-2 = Θ j-1 -(Θ j -Θ j-1 ) e -T n h (Θ j-1 ) , j = n, ..., 3 .
Equating T to 0 by dichotomy then gives the asymptotically optimal measure. Again, we simulate N = 10000 trajectories of the Heston model with parameters λ = 1.15, µ = 0.04, ζ = 0.2, ρ = -0.4 and initial values V 0 = 0.04 and S 0 = 1, under both P, eq. (2.6.2) and P θ , eq. (2.6.6) with n = 200 and t j = j n T , using a standard Euler scheme with 200 discretization steps. For the P-realisations X (i) , we calculate the Asian put price as

1 N N j=1 K - S 0 n n j=1 e X (i) t j

+

(2.6.11)

and for the P θ -realisations X (i,θ) , as

e Φ(t 1 ,Θ 1 ,...,Θn)+Ψ(t 1 ,Θ 1 ,...,Θn) V 0 N N j=1 e -n j=1 θ j X (i,θ) t j K - S 0 n n j=1 e X (i) t j + .
(2.6.12) Again, each time, we compute the P θ -standard deviation and the adjusted and non-adjusted variance ratios. In Table 2.4, we fix maturity to T = 1.5 and let the strike K vary between 0.6 and 1.3. The conclusion is the same as for the European put. Indeed, the variance ratio explodes when the option moves away from the money. Due to the timedependence of the measure change, the adjusted variance ratio is consistently around 13% below its non-adjusted version. The adjusted variance ratio remains however very interesting, with values above 3 around the money.

K

European put on the Heston model with negative exponential jumps

We now consider the Heston model with negative exponential jumps

dX t = δ - V t 2 dt + V t dW 1 t + dJ t , X 0 = 0 dV t = λ(µ -V t ) dt + ζ V t dW 2 t , V 0 = V 0 d W 1 , W 2 t = ρ dt ,
(2.6.13)

where W 1 , W 2 are standard P-Brownian motions and (J t ) t≥0 is an independent compound Poisson process with constant jump rate r and jump distribution Neg-Exp(α), i.e. the Lévy measure of (J t ) t≥0 is ν(dx) = r αe αx 1 {x<0} dx.

The martingale condition on S = S 0 e X imposes δ = r α+1 . The Laplace transform of (X t , V t ) is IE e uXt+wVt = e φ(t,u,w)+ψ(t,u,w)V 0 +uX 0 , where φ, ψ satisfy the Riccati equations

∂ t φ(t, u, w) = F (u, ψ(t, u, w)) φ(0, u, w) = 0 ∂ t ψ(t, u, w) = R(u, ψ(t, u, w)) ψ(0, u, w) = w (2.6.14)
for F (u, w) = λµ w + κ(u), where κ(u) = ru(u-1) (α+1)(α+u) and

R(u, w) = ζ 2 2 w 2 + ζρ uw -λw + 1 2 (u 2 -u) .
Again, a standard calculation shows that the solution of the Generalized Riccati equations (2.6.14) is

ψ(t, u, w) = 1 ζ λ ζ -ρu - γ ζ 2 tanh γ 2 t + η 1 + η tanh γ 2 t φ(t, u, w) = µ λ ζ λ ζ -ρu t -2µ λ ζ 2 log cosh γ 2 t + η sinh γ 2 t + tκ(u) ,
(2.6.15)

where γ = γ(u) = ζ λ ζ -ρu 2 + 1 4 -u -1 2 2 and η = η(u, w) = λ-ζρu-ζ 2 w γ(u)
.

Furthermore, for the Heston model, the function h is given by

h(u) = µ λ ζ λ ζ -ρu -µ λ ζ 2 γ(u) + κ(u) .
(2.6.16)

Let us now study the effect of the Esscher transform on the dynamics of the Heston model with jumps.

CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE STOCHASTIC VOLATILITY MODELS 2

Proposition 2.6.6. Let P θ be the measure given by

dP θ dP = e θ X T IE [e θ X T ]
.

Under P θ , the dynamics of the P-Heston process (X t , V t ) becomes

dX t = δdt + θ + ζρ ψ (T -t, θ, 0) - 1 2 V t dt + V t d W 1 t + dJ t , X 0 = 0 dV t = λt (μ t -V t ) dt + ζ V t d W 2 t , V 0 = V 0 d W 1 , W 2 t = ρ dt ,
(2.6.17) where W is 2-dimensional correlated P θ -Brownian motion, φ and ψ are given in (2.6.15), λt = λ -ζθρ -ζ 2 ψ (T -t, θ, 0) and μt = λµ λt and (J t ) t≥0 is a compound Poisson process with jump rate rα α+θ and jump distribution Neg-Exp(α + θ) under P θ .

Proof. Denote D(t, X t , V t ) = dP θ dP Ft = e φ(T -t,θ,0) e φ(T,θ,0)+ψ(T,θ,0) V 0 e ψ(T -t,θ,0) Vt+θ Xt .

The dynamics of D(t, X t , V t ) can then be expressed using Itō's Lemma as

dD(t, X t , V t ) = D(t, X t , V t ) (θdX t + ψ (T -t, θ, 0) dV t ) + ... dt = D(t, X t , V t ) V t θdW 1 t +ζ ψ (T-t, θ, 0) dW 2 t +θ (δdt+dJ t )
and Girsanov's theorem then shows that

d W 1 t W 2 t = d W 1 t W 2 t -V t θ + ζρ ψ (T -t, θ, 0) θρ + ζ ψ (T -t, θ, 0) dt
is a 2-dimensional Brownian motion under the measure P θ . Replacing W in eq. (2.6.2) by W gives eq. (2.6.17). In order to finish the proof, it remains to show that the jump process (J t ) t≥0 has the desired distribution under P θ .

Let us calculate the P θ -Laplace transform of J t .

IE P θ e uJt = IE e uJt IE e θX T F t IE [e θX T ] = e φ(T -t,θ,0) IE [e θX T ]
IE e uJt+ψ(T -t,θ,0) Vt+θXt .

NUMERICAL EXAMPLES

2 By independence of the jumps, IE e uJt+ψ(T -t,θ,0) Vt+θXt = e θδ t IE e (u+θ)Jt IE e ψ(T -t,θ,0) Vt+θ(Xt-δ t-Jt) .

But IE e (u+θ)Jt = e -rt u+θ u+θ+α . Furthermore, (X t -δ t -J t , V t ) t≥0 is a standard Heston process without jump. Therefore comparing (2.6.4) and (2.6.15), we find that IE e ψ(T -t,θ,0) Vt+θ(Xt-δ t-Jt) = e φ(t,θ,ψ(T -t,θ,0))-t rθ(θ-1)

(α+1)(α+θ) +ψ(t,θ,ψ(T -t,θ,0)) V 0 .
Using the fact that ψ(t, θ, ψ (T -t, θ, 0)) = ψ (T, θ, 0) and

φ (T -t, θ, 0) + φ(t, θ, ψ (T -t, θ, 0)) = φ (T, θ, 0)
(see eq. ( 2.1) in (Keller-Ressel, 2011)), we finally obtain

IE P θ e uJt = e θδ t-rt u+θ u+θ+α -t rθ(θ-1) (α+1)(α+θ) = e θ r α+1 t-rt u+θ u+θ+α -t rθ(θ-1) (α+1)(α+θ) = e -rα α+θ t u u+(α+θ) ,
which is indeed the Laplace transform of a compound Poisson process with jump rate rα α+θ and Neg-Exp(α + θ)-distributed jumps.

Numerical results for the European put option

Similarly to the case of the Heston model without jump, denoting θ = θ({T }), we have

Ĥ(θ) + T 0 h(θ([t, T ])) dt = log K 1 -θ -θ log -θ K/S 0 1 -θ + T µ λ ζ λ ζ -ρθ - γ(θ) ζ + T κ(θ)
(2.6.18) and we obtain the asymptotically optimal θ by differentiate (2.6.18) with respect to θ and equating the derivative to 0 by dichotomy .

We simulate N = 10000 trajectories of the Heston model with jumps with parameters λ = 1.1, µ = 0.7, ζ = 0.3, ρ = -0.5, r = 2, α = 3 and initial values V 0 = 1.3 and S 0 = 1, under both P, eq. (2.6.13) and P θ , eq. (2.6.17) using a standard Euler scheme with 200 discretization step. For the P-realisations X (i) , we calculate the standard Monte-Carlo estimator of the European put price and for the P θ -realisations X (i,θ) , we use (2.6.8) where φ and ψ are given in (2.6.15) and compute the same statistics as in the previous examples. In Table 2.5, we fix the strike to the value K = 1 and let the maturity T vary from 0.25 to 3, whereas in Tables 2.6 and 2.7, we fix maturity to T = 1 and to T = 3, while we let the strike K vary between 0.25 and 1.75. When adding negative jumps to the Heston model, one can see that the variance ratio diminishes. When the options are out of the money however it is still sufficiently important to make it interesting to use in applications. In Figure 2.6.2, we fix the maturity to T = 1.5 and plot again the empirical variance of the estimator (2.6.8) as a function of θ for the Heston model with jumps. The method provides θ = -0.312 as asymptotically optimal measure change which is, as in the continuous case very close to the optimal one. Chapter 3

Long-time large deviations for the multi-asset Wishart stochastic volatility model and option pricing

The content of this chapter is based on a paper written in collaboration with Aurélien Alfonsi.1

Introduction

The Heston stochastic volatility model [START_REF] Heston | A closed-form solutions for options with stochastic volatility with applications to bond and currency options[END_REF] is one of the most popular models in quantitative finance. The Wishart stochastic volatility model is its natural extension to a basket of assets, since it coincides with the Heston model in dimension 1 and keeps the affine structure. This model, proposed in [START_REF] Gourieroux | Derivative pricing with multivariate stochastic volatility: Application to credit risk[END_REF], assumes that under the riskneutral probability, the vector of n asset prices is modelled as a diffusion process

dS t = Diag(S t ) r1 dt + X1/2 t d Zt , (3.1.1)
where the n × n volatility matrix ( Xt ) is modelled by a Wishart process with dynamics

d Xt = α a a + b Xt + Xt b dt + X1/2 t d Wt a + a (d Wt ) X1/2 t , (3.1.2)
where Z and W are independent standard Brownian motions of dimensions n and n × n, and Diag(S t ) is the diagonal matrix whose diagonal elements are

INTRODUCTION

3

given by the vector S t ∈ R n . Then, this model has been extended by [START_REF] Da Fonseca | Option pricing when correlations are stochastic: An analytical framework[END_REF] to include a constant correlation between W and Z in a way to preserve the affine structure. The matrix process (3.1.2) has been introduced by [START_REF] Bru | Wishart processes[END_REF] to model the perturbation of experimental biological data. As shown by [START_REF] Bru | Wishart processes[END_REF] and [START_REF] Cuchiero | Affine processes on positive semidefinite matrices[END_REF] in a more general framework, for α ≥ n + 1 (resp. α ≥ n -1), the SDE (3.1.2) has a unique strong (resp. weak) solution. Furthermore, since Xt is positive semi-definite (Bru, 1991, Prop. 4), Wishart processes turn out to be very suitable processes to model covariance matrices. This led several authors to use them in stochastic volatility models, such as [START_REF] Da Fonseca | A multifactor volatility Heston model[END_REF] and [START_REF] Benabid | Wishart stochastic volatility: Asymptotic smile and numerical framework[END_REF] for single asset models and the Wishart stochastic volatility model for multiple assets models. By using the affine property, the Laplace transform of the latter model is given by [START_REF] Da Fonseca | Option pricing when correlations are stochastic: An analytical framework[END_REF].

IE e θ log(St) = exp β θ (t) + Tr γ θ (t) X0 + δ θ (t) log(S t ) , (3.1.3)
where β θ , γ θ and δ θ satisfy the matrix Riccati equations

∂ t β θ (t) = r δ θ (t) 1 + α Tr [γ θ (t)] ∂ t γ θ (t) = b γ θ (t) + γ θ (t) b + 2γ θ (t) a a γ θ (t) - 1 2 Diag(δ θ (t)) -δ θ (t)δ θ (t) ∂ t δ θ (t) = 0 ,
with initial conditions β θ (0) = 0, γ θ (0) = 0 and δ θ (0) = θ. Since the Riccati equations can be solved explicitly, the Laplace transform can be calculated explicitly by the mean of exponential and inversion of matrices.

In the last decade, many works have studied asymptotics of the option prices under the Heston model and extensions through the volatility smile function. In particular, (Forde and Jacquier, 2011a) and [START_REF] Jacquier | Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models[END_REF] have obtained long-time asymptotics by proving a large deviations principle. The goal of the present paper is to extend these results to the Wishart stochastic volatility model (3.1.1) and (3.1.2). Even though the Laplace transform (3.1.3) is given by an explicit formula, it is not easy to calculate longtime asymptotics because of the multi-dimensional setting. Nonetheless, under some assumptions on the coefficients, we can get a simpler formula for the Laplace transform and then prove a large deviations principle. Then, we obtain asymptotics for the smile when the maturity goes to infinity.

Beyond its theoretical interest, this large deviations principle enable us to develop a generic variance reduction method for pricing derivatives. First, let us note that since the Laplace tranform is known explicitly, Fourier inversion methods can be used, as explained in [START_REF] Da Fonseca | Option pricing when correlations are stochastic: An analytical framework[END_REF]. However, Fourier inversion methods are less competitive than in dimension 1 since they require to approximate an integral on R n . When, for complexity reasons, Fourier methods are not an option, the use of a large number of Monte-Carlo simulations is necessary. In [START_REF] Ahdida | Exact and high-order discretization schemes for wishart processes and their affine extensions[END_REF], it is given an CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3 exact simulation method for Wishart processes and a second order scheme for the Gourieroux and Sufana model (3.1.1) and (3.1.2). Thus, it is possible to sample efficiently such processes, and it is relevant to develop variance reduction techniques to reduce computational costs. Following previous works of ( [START_REF] Guasoni | Optimal importance sampling with explicit formulas in continuous time[END_REF], [START_REF] Robertson | Sample path large deviations and optimal importance sampling for stochastic volatility models[END_REF], [START_REF] Genin | Optimal importance sampling for Lévy processes[END_REF] and [START_REF] Grbac | Long-time trajectorial large deviations for affine stochastic volatility models and application to variance reduction for option pricing[END_REF], we develop an importance sampling method based on an asymptotically optimal Esscher transform, using large deviations theory.

In this paper, we denote M n the set of real squared n × n matrices, S n ⊂ M n the set of symmetric matrices and S + n , (resp. S +, * n ), the sets of symmetric and positive semi-definite (resp.) positive definite. The paper is structured as follows.

In Section 3.2, we describe the model, make certain assumptions on the parameters and give some properties of the model. In Section 3.3, we prove that the asset log-price vector satisfies large deviations principle when maturity goes to infinity. In Section 3.4, we calculate the asymptotic put basket implied volatility, following the approach of [START_REF] Jacquier | Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models[END_REF]. In Section 3.5, we develop the variance reduction method using Varadhan's lemma. Finally, in Section 3.6, we test numerically the results of Sections 3.4 and 3.5.

The Wishart stochastic volatility model

Let (S t ) t≥0 be a n-dimensional vector stochastic process with dynamics

dS t = Diag(S t ) r1 dt + a X 1/2 t dZ t , S i 0 > 0, i = 1, . . . , n, (3.2.1)
where 1 = (1, ..., 1) , Diag(S t ) ij = 1 {i=j} S i t , Z t is n-dimensional standard Brownian motion and the stochastic volatility matrix X is a Wishart process with dynamics

dX t = (αI n + bX t + X t b) dt + X 1/2 t dW t + (dW t ) X 1/2 t , X 0 = x . (3.2.2) with α > n -1, a ∈ M n invertible, -b, x ∈ S +, * n
and W is a n × n matrix standard Brownian motion independent of Z. Note again that X t ∈ S + n (Bru, 1991, Prop. 4). Let us also assume that a is such that a a ∈ S [START_REF] Heston | A closed-form solutions for options with stochastic volatility with applications to bond and currency options[END_REF]) and b being negative definite yields the mean reversion property of the stochastic volatility process.

Defining the log-price Y k t := log(S k t ), k = 1, ..., n, a simple application of Itō's lemma gives

dY t = r1 - 1 2 (a X t a) 11 , ... , (a X t a) nn dt + a X 1/2 t dZ t . (3.2.3)
We are interested in the Laplace transform of Y t . In order to calculate it, we first cite the following proposition.

Proposition 3.2.3. (Alfonsi et al., 2016, Prop. 5.1.).

Let α ≥ n-1, x ∈ S + n , b ∈ S n and X with dynamics (3.2.2). Let v, w ∈ S n be such that ∃m ∈ S n , v 2 -mb -bm -2m 2 ∈ S + n and w 2 + m ∈ S + n . If R t := t 0 X s ds, then we have for t ≥ 0 IE exp - 1 2 Tr [wX t ] - 1 2 Tr [vR t ] = exp -α 2 Tr [b] t det [V v,w (t)] α/2 exp - 1 2 Tr V v,w (t)V -1 v,w (t) + b x , with V v,w (t) = ∞ k=0 t 2k+1 ṽk (2k + 1)! w+ ∞ k=0 t 2k ṽk (2k)!
, ṽ = v+b 2 and w = w-b.

If besides, ṽ ∈ S +, * n , then V v,w (t) = ṽ-1/2 sinh ṽ1/2 t w + cosh ṽ1/2 t and V v,w (t) = cosh ṽ1/2 t w + sinh ṽ1/2 t ṽ1/2 . Proposition 3.2.4. Let φ : R n → S n be the function defined by

φ(θ) := b 2 + a Diag(θ) -θθ a ∈ S n , (3.2.4)
Let U ⊂ R n , be the set defined by

U := θ ∈ R n : φ(θ) ∈ S + n . Then, for all θ ∈ U, the Laplace transform of Y t is IE e θ Yt = e θ Y 0 +rθ 1 t-α 2 Tr[b]t-1 2 Tr[(b+φ 1/2 (θ))x-exp(-t φ 1/2 (θ))(b+φ 1/2 (θ))V -1 (t) x] det [V (t)] α/2 , where V (t) = cosh t φ 1/2 (θ) -φ -1/2 (θ) sinh t φ 1/2 (θ) b .

CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3

Proof. By conditioning on the trajectory of X, we have

IE e θ Yt = IE IE e θ Yt (X s ) s≤t ,
where IE e θ Yt (X s ) s≤t = e θ Y 0 +rθ 1 t-1 2 t 0 θ ((a Xs a) 11 , ... , (a Xs a)nn) -θ a Xs a θ ds = e θ Y 0 +rθ 1 t-

1 2 t 0 Tr[Diag(θ) a Xs a]-Tr[θ a Xs a θ] ds = e θ Y 0 +rθ 1 t-1 2 Tr[a (Diag(θ)-θθ )a Rt] . Let m = -b/2. Then m ∈ S + n and a Diag(θ) -θθ a 2 -mb -bm -2m 2 = φ(θ) 2 ∈ S + n . Therefore, by Proposition 3.2.3, IE e θ Yt = e θ Y 0 +rθ 1 t IE e -1 2 Tr[a (Diag(θ)-θθ )a Rt] = e θ Y 0 +rθ 1 t exp -α 2 Tr [b] t det [V (t)] α/2 exp - 1 2 Tr V (t)V -1 (t) + b x (3.2.5)
where

V (t) = cosh t φ 1/2 (θ) -φ -1/2 (θ) sinh t φ 1/2 (θ) b , V (t) = sinh t φ 1/2 (θ) φ 1/2 (θ) -cosh t φ 1/2 (θ) b .
Since φ(θ) ∈ S + n , we can write φ(θ) = P DP , where D is diagonal, P is orthonormal and b = -P b P ∈ S +, * n .

       V (t) = P cosh t D 1/2 + sinh t D 1/2 D -1/2 b P , V (t) = P sinh t D 1/2 D 1/2 + cosh t D 1/2 b P = φ 1/2 (θ)V (t) -exp -t φ 1/2 (θ) b + φ 1/2 (θ) .
Replacing V by the latter expression finishes the proof.

Remark 3.2.5. Note that, when φ(θ) ∈ S + n \S +, * n , φ 1/2 (θ) is not invertible. The notation φ -1/2 (θ) sinh t φ 1/2 (θ) is therefore abusive and is to be interpreted as the finite limit

lim S +, * n φ → φ(θ) φ -1/2 sinh t φ 1/2 = ∞ k=0 φ(θ) k t 2k+1 (2k + 1)! .
Remark 3.2.6. The set U is bounded. Indeed, let θ = λ θ, with λ > 0 and θ = 1. Then, letting u = (a ) -1 θ, we have

u φ(θ)u = b(a ) -1 θ 2 + λ θ Diag( θ) θ -λ 2 ≤ b(a ) -1 2 + λ -λ 2
It follows that U is contained, e.g., in the set θ ≤ λ * with

λ * = max{2, b(a ) -1 θ √ 2}.

LONG-TIME LARGE DEVIATIONS FOR THE WISHART VOLATILITY MODEL 3

Long-time large deviations for the Wishart volatility model

In this section, we prove that the Wishart stochastic volatility model satisfies a large deviation principle when time tends to infinity.

Reminder of large deviations theory

Let us recall some standard definitions and results of large deviations theory.

For a wider overview of large deviations theory, we refer the reader to [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. We consider a family (X ) ∈(0,1] of random variables on a measurable space (X , B), where X is a topological space.

lim inf →0 log P(X ∈ G) ≥ -inf x∈G∩F Λ * (x) ,
where F is the set of exposed points of Λ * , whose exposing hyperplane belongs to the interior of D Λ .

(c) If Λ is an essentially smooth, lower semi-continuous function, then the family (X ) (0,1] satisfies a large deviations principle with good rate function Λ * .

Remark 3.3.5. The function Λ of (3.3.1) is a convex function. Indeed, let λ, µ ∈ R n and u ∈ (0, 1). A direct application of Hölder's inequality yields

IE e uλ+(1-u)µ,X = IE e uλ,X e (1-u)µ,X ≤ IE e λ,X u IE e µ,X 1-u
.

Applying the logarithm then proves that λ → log IE e λ,X and therefore Λ are convex.

Theorem 3.3.6 (Varadhan's Lemma, extension of [START_REF] Guasoni | Optimal importance sampling with explicit formulas in continuous time[END_REF]). Let (X , B) be a metric space with its Borel σ-field. Let (X ) ∈]0,1 ] be a family of X -valued random variables that satisfies a large deviations principle with rate function Λ * . If ϕ : X → R ∪ {-∞} is a continuous function which satisfies

lim sup →0 log IE exp γ ϕ(X ) < ∞
for some γ > 1, then, for any A ∈ B,

sup x∈A • {ϕ(x) -Λ * (x)} ≤ lim inf →0 log A • exp ϕ(z) dµ (z) ≤ lim sup →0 log Ā exp ϕ(z) dµ (z) = sup x∈ Ā{ϕ(x) -Λ * (x)} ,
where µ denotes the law of X

Long-time behaviour of the Laplace transform of the log-price

Let T > 0 and define the transformation Y T := Y T / , which corresponds to the long-time behaviour of Y T . We are interested in the function

θ → lim →0 log IE e -1 θ Y T .
We first give the following lemma.

LONG-TIME LARGE DEVIATIONS FOR THE WISHART VOLATILITY MODEL 3

Lemma 3.3.7. Let A, B ∈ M n such that A + tB is invertible for all t ≥ t 0 . Then, (A + tB) -1 tB is bounded for all sufficiently large t.

Proof. Since A + t 0 B is invertible, for all t ≥ t 0 ,

(A + tB) -1 tB = I + (t -t 0 ) B -1 (t -t 0 ) B t t -t 0 ,
where B = (A + t 0 B) -1 B. Now, the fact that A + tB is invertible for t ≥ t 0 means that the eigenvalues λ i of B satisfy λ i > 0 or λ i = 0 for all i. This implies det[I + (t -t 0 ) B] ∼ t→+∞ ct n for some c = 0, and since the adjugate matrix of I + (t -t 0 ) B has coefficients of order O(t n-1 ), we get that

I+(t-t 0 ) B -1 is bounded for t ≥ t 0 . Therefore, I+(t-t 0 ) B -1 (t-t 0 ) B = I -I + (t -t 0 ) B -1
is bounded, and (A + tB) -1 tB as well, whenever t is sufficiently large.

We now characterise the asymptotic behaviour of the Laplace transform of Y t .

Proposition 3.3.8. Define Λ(θ) := T r θ 1 -α 2 Tr b + φ 1/2 (θ) if θ ∈ U ∞ if θ ∈ U . (3.3.2)
For every θ ∈ U,

lim →0 log IE e -1 θ Y T = Λ(θ) . Proof. Let θ ∈ U. By Proposition 3.2.4, log IE e -1 θ Y T = log IE e θ Y T / = θ Y 0 - 1 2 Tr b + φ 1/2 (θ) x + 1 2 Tr exp -T / φ 1/2 (θ) b + φ 1/2 (θ) V -1 (T / ) x + T rθ 1 - T α 2 Tr [b] - α 2 log det [V (T / )] .
(3.3.3) Write φ(θ) = P DP , where D is diagonal, P is orthonormal and let b = -P b P ∈ S +, * n . Then

V (t) = P cosh t D 1/2 + sinh t D 1/2 D -1/2 b P , CHAPTER 3.

LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3

Let E and Ẽ be n × n square matrices with

E ij = 1 {i=j,D ii =0} and Ẽij = D -1/2 ii 1 {i=j,D ii =0} . We then have cosh t D 1/2 = e tD 1/2 2 I n + e -2tD 1/2 = e tD 1/2 2 I n + E + O t -1 and sinh tD 1/2 D -1/2 = e tD 1/2 2 D -1/2 I n -e -2tD 1/2 = e tD 1/2 2 Ẽ + 2tE + O t -1 .
Therefore,

V (t) = 1 2 P e tD 1/2 (I n + E) + (2tE + Ẽ) b + O t -1 P = - 1 2 P (I n + E) e tD 1/2 b-1 + (t E + Ẽ) + O t -1 P b (3.3.4) and V -1 (t) = -2 b -1 P b-1 + (t E + Ẽ) + O t -1 -1 e -tD 1/2 I n - 1 2 E P
where the invertibility of b-1 + (t E + Ẽ) + O (t -1 ) is guaranteed for every

(t E + Ẽ) + O t -1 = b-1 + (t E + Ẽ) I n + O t -1 and V -1 (t) = -2 b -1 P I n + O t -1 b-1 + (t E + Ẽ) -1 e -tD 1/2 I n - 1 2 E P . But b-1 + t E + Ẽ -1 e -tD 1/2 = b-1 + t E + Ẽ -1 (E + (I n -E)) e -tD 1/2 = t -1 b-1 + t E + Ẽ -1 tE + b-1 + t E + Ẽ -1 (I n -E) e -tD 1/2 , where b-1 + t E + Ẽ -1 tE is bounded by Lemma 3.3.7. Therefore, b-1 + t E + Ẽ -1
e -tD 1/2 → 0

LONG-TIME LARGE DEVIATIONS FOR THE WISHART VOLATILITY MODEL 3

and V -1 (t) → 0 as t → ∞. Using (3.3.4), we find

log det [V (T / )] = T Tr D 1/2 + log det 1 2 (I n + E) I n + ( -1 T E + Ẽ) b + O ( ) = T Tr φ 1/2 (θ) + log det -1 T E b + 1 2 (I n + E) I n + Ẽb + O ( ) = T Tr φ 1/2 (θ) -n log( )+ log det T E b + 2 I n + E + Ẽb + O 2 .
We have

det T E b + 2 I n + E + Ẽb + O 2 ∼ →0 det T E b + 2 I n + E + Ẽb ,
since the latter determinant is a non-zero polynomial of (for = 2T the determinant is clearly positive). Thus, by passing to the limit, lim

→0 log det [V (T / )] = T Tr φ 1/2 (θ) . Furthermore, since φ ∈ S + n , exp -T φ 1/2 (θ) is bounded. Therefore, Tr exp - T φ 1/2 (θ) b + φ 1/2 (θ) V -1 (T / ) x -→ →0 0 .
Finally, passing to the limit in (3.3.3) finishes the proof.

The next proposition proves the essential smoothness of Λ.

Proposition 3.3.9. The function θ → Λ(θ) defined in (3.3.2) is essentially smooth.

Proof. The function Λ defined in (3.3.2) is a lower semi-continuous proper convex function with domain U. Furthermore, since for every θ ∈

• U, φ(θ) ∈ S +, * n , Λ is of class C 1 on • U. Only remains to prove that ||∇ θ Λ(θ)|| → ∞ when θ goes to the boundary of U. Let θ ∈ • U. By Proposition 3.3.8 Λ(θ) = T r θ 1 - α 2 Tr b + φ 1/2 (θ) .
Then for every j ∈ {1, ..., n},

∂ θ j Λ(θ) = T r - α 2 Tr ∂ θ j φ 1/2 (θ) ,
where ∂ θ j φ 1/2 (θ) satisfies

∂ θ j φ(θ) = ∂ θ j φ 1/2 (θ)φ 1/2 (θ) = φ 1/2 (θ) ∂ θ j φ 1/2 (θ) + ∂ θ j φ 1/2 (θ) φ 1/2 (θ).

CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3

Multiplying this equation by φ -1/2 (θ) and using the cyclic property of the trace, we get

Tr ∂ θ j φ 1/2 (θ) = 1 2 Tr φ -1/2 (θ) ∂ θ j φ(θ) .
and therefore

∂ θ j Λ(θ) = T r - α 2 Tr ∂ θ j φ 1/2 (θ) = T r - α 4 Tr φ -1/2 (θ) ∂ θ j φ(θ) ,
(3.3.5) where ∂ θ j φ(θ) = a e j e j -θe j -e j θ a .

We write φ(θ) = P DP with D ∈ S +, * n diagonal and denote w = a P , which is invertible since P is orthonormal and a a ∈ S +, * n . Then

Tr φ -1/2 (θ) ∂ θ j φ(θ) = Tr D -1/2 P ∂ θ j φ(θ) P = Tr D -1/2 w e j e j -θe j -e j θ w = Tr D -1/2 w e j e j -2e j θ w

= n i=1 D -1/2 ii (w 2 ji -2w ji (θ we i )) .
Now, we observe that

D ii = P i φ(θ) P i = ||b P i || 2 + e i w Diag(θ) -θθ we i = ||b P i || 2 + n j=1 θ j w 2 ji -(θ we i ) 2 = ||b P i || 2 + (θ we i ) 2 + n j=1 θ j (w 2 ji -2w ji (θ we i )).
Therefore, we get by the triangular inequality

n j=1 |θ j | Tr φ -1/2 (θ) ∂ θ j φ(θ) ≥ n j=1 θ j n i=1 D -1/2 ii (w 2 ji -2w ji (θ we i )) = n i=1 D 1/2 ii -D -1/2 ii (||b P i || 2 + (θ we i ) 2 ) .
Then, if θ → θ with θ ∈ U\ • U, there exists i such that D ii → 0 and therefore

n i=1 D 1/2 ii -D -1/2 ii (||b P i || 2 +(θ we i ) 2 ) → -∞ since ||b P i || 2 +(θ we i ) 2 ≥ λ(-b) 2 > 0, where λ(-b) is the smallest eigenvalue of -b ∈ S +, *
n . Therefore, Tr φ -1/2 (θ) ∂ θ j φ(θ) → +∞ for some j, which implies then |∂ θ j Λ(θ)| → +∞. Thus, ||∇ θ Λ(θ)|| → ∞ and Λ is therefore essentially smooth.

ASYMPTOTIC IMPLIED VOLATILITY OF BASKET OPTIONS 3

Remark 3.3.10. In fact, Prop. 3.3.8 holds, not only for θ ∈ U, but for every θ ∈ R n . Indeed, by Remark 3.3.5,

θ → lim →0 log IE e -1 θ Y t
is a convex function and by Prop. 3.3.9, Λ admits infinite derivative on U\

• U. Therefore, for every θ ∈ R n \U, lim →0 log IE e -1 θ Y t = Λ(θ) = ∞ .

Long-time large deviation principle for the logprice process

We now state the large deviation principle for the family (Y T ) ∈(0,1] , when → 0. , where Λ is defined in (3.3.2). Finally, Proposition 3.3.9 yields the essential smoothness of Λ. Therefore, by the Gärtner-Ellis Theorem 3.3.4, (Y T ) ∈(0,1] satisfies a large deviation properties, when → 0 with good rate function Λ * .

Asymptotic implied volatility of basket options

In this section, to simplify the formulas and without loss of generality, we assume that Y j 0 = 0 for j = 1, . . . , n and r = 0 so that (e Y j t ) t≥0 is a martingale with initial value 1 (this follows from Proposition 3.2.4). We are interested CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3 in the limiting behaviour far from maturity of basket option prices and the corresponding implied volatilities in the Wishart model. The basket call option price with log strike k and time to maturity T is defined by

C(T, k) = IE n i=1 ω i S i T -e k + ,
and the corresponding put option price is defined by

P (T, k) = IE e k - n i=1 ω i S i T +
where ω ∈ (R * + ) n with n i=1 ω i = 1. The implied volatility of basket options is defined by comparing their price to the corresponding option price in the Black-Scholes model dSt St = σdW t :

C BS (T, k, σ) = N (d 1 ) -e k N (d 2 ), d 12 = -k ± 1 2 σ 2 T σ √ T ,
where N is the standard normal distribution function. The implied volatility for log strike k and time to maturity T is then defined as the unique value σ(T, k) such that C BS (T, k, σ(T, k)) = C(T, k).

It can be equivalently defined using the put option price.

It is well known that in most models, for fixed log strike k, the implied volatility converges to a constant value independent from k as T → ∞ [START_REF] Tehranchi | Asymptotics of implied volatility far from maturity[END_REF]. To obtain a non-trivial limiting smile, we therefore follow [START_REF] Jacquier | Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models[END_REF] and use a renormalized log strike k(T ) = yT . We are interested in computing the limiting implied volatility σ ∞ (y) = lim T →∞ σ(T, yT ).

Asymptotic price for the Wishart model

Introduce the renormalized log-price process in the stochastic volatility Wishart model: Ỹ j T = T -1 Y j T , j = 1, . . . , n. Note that to simplify notation, in this section we avoid using an extra parameter and simply consider the asymptotics when T → ∞. For this reason, the asymptotic Laplace exponent Λ(θ) will be given by equation (3.3.2) with T = 1 and r = 0.

Denote the basket log price by B T := log n j=1 ω j e Y j T , and the corresponding renormalized price by BT := T -1 log n j=1 ω j e Y j T . We first show some LDP-like bounds for this quantity. In the following lemma and below, we will use that Λ(0) = Λ(e j ) = 0, which gives in particular that Λ * (x) ≥ 0 and 3.4. ASYMPTOTIC IMPLIED VOLATILITY OF BASKET OPTIONS 3 Λ * (x) -x j ≥ 0 for all x ∈ R d . Thus, we let x * = Λ (0) and x * j = Λ j (e j ) for j = 1, . . . , n and introduce three constants: β * = max j x * j , β * = min j x * j and β * = max j x * j . It is easy to see from (3.3.5) that x * j = -x * j < 0 since φ(0) = φ(e j ) = b 2 is positive definite and a is invertible. We get β * < 0 < β * ≤ β * .

Lemma 3.4.1. The following estimates hold for BT .

1. If β < β * then lim T →∞ T -1 log P BT ∈ (-∞, β] = -inf x∈(-∞,β] n Λ * (x) = inf λ∈R n ,λ i ≤0,i=1,...,n {Λ(λ) -β λ, 1 } < 0 , (3.4.1) otherwise lim T →∞ T -1 log P BT ∈ (-∞, β] = 0. 2. If β ≥ β * then lim T →∞ T -1 log P BT ∈ (β, ∞) = -inf x / ∈(-∞,β] n Λ * (x) (3.4.2) = max i=1,...,n inf λ∈R {-λβ + Λ(λe i )}, (3.4.3) otherwise lim T →∞ T -1 log P BT ∈ (β, ∞) = 0.
In addition if β ≥ β * and β = x * i for all i, then lim

T →∞ T -1 log P BT ∈ (β, ∞) < -β.
3. Let j ∈ {1, . . . , n}. Then,

lim T →∞ T -1 log IE e Y j T 1 BT ∈(-∞,β] = -inf x∈(-∞,β] n Λ * (x) -x j = β + inf λ j ≤1,λ i ≤0,i =j {Λ(λ) -β λ, 1 }.
(3.4.4)

In addition, if x * j > β then lim T →∞ T -1 log IE e Y j T 1 BT ∈(-∞,β] < 0.

CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3

4. Let j ∈ {1, . . . , n} and assume β > x * j . Then,

lim T →∞ T -1 log IE e Y j T 1 BT ∈(β,∞) = -inf x / ∈(-∞,β] n Λ * (x) -x j = max i=1,...,n inf λ∈R {-λβ + Λ(λe i + e j )} < 0. (3.4.5)
Proof.

1. Since

ω min e max j Y j T ≤ n j=1
ω j e Y j T ≤ nω max e max j Y j T with (ω min , ω max ) := (min j=1,...,n ω j , max j=1,...,n ω j ), we have for every

T > 0 and β ∈ R, ỸT ∈ (-∞, β -T -1 log(n ω max )) n ⊂ ( BT < β) ⊂ ỸT ∈ (-∞, β -T -1 log ω min ) n .
Therefore, we get for every δ > 0 and T sufficiently large,

P ỸT ∈ (-∞, β -δ) n ≤ P( BT < β) ≤ P ỸT ∈ (-∞, β + δ) n .
Passing to the lim sup and lim inf, we get:

lim inf T →∞ T -1 log P ỸT ∈ (-∞, β -δ) n ≤ lim inf T →∞ T -1 log P( BT < β) ≤ lim sup T →∞ T -1 log P( BT < β) ≤ lim sup T →∞ T -1 log P ỸT ∈ (-∞, β + δ) n .
Using the large deviations principle for ỸT (Theorem 3.3.11) further yields:

- inf x∈(-∞,β-δ) n Λ * (x) ≤ lim inf T →∞ T -1 log P( BT < β) ≤ lim sup T →∞ T -1 log P( BT < β) ≤ - inf x∈(-∞,β+δ] n Λ * (x),
and making δ tend to zero, we see that

-inf x∈(-∞,β) n Λ * (x) ≤ lim inf T →∞ T -1 log P( BT < β) ≤ lim sup T →∞ T -1 log P( BT < β) ≤ -inf x∈(-∞,β] n Λ * (x).
The fact that the domain of Λ is bounded (Remark 3.2.6) implies that Λ * is locally bounded from above and therefore continuous. The first 3.4. ASYMPTOTIC IMPLIED VOLATILITY OF BASKET OPTIONS 3 equality of (3.4.1) then follows by continuity of Λ * . The second equality then follows from the definition of Λ * and the minimax theorem (see, e.g., Corollary 37.3.2 in [START_REF] Rockafellar | Convex Analysis[END_REF]) which can be applied because the domain of Λ is bounded (cf. Remark 3.2.6). Finally, the inequality follows from the fact that the function f (λ) = Λ(λ) -β λ, 1 satisfies f (0) = 0 and f (0) = x * -β1. Under the condition β < β * at least one component of the derivative is strictly positive, and hence the minimum of f over the set {λ i ≤ 0, i = 1, . . . , n} is strictly negative.

2. The first equality in (3.4.3) follows similarly to the previous item. If β < β * then x * / ∈ (-∞, β] n and the infimum equals 0. Otherwise by convexity of Λ * the infimum is attained on the boundary of this set. Therefore, we can write:

-inf x / ∈(-∞,β] n Λ * (x) = max i=1,...,n sup x∈R n :x i =β {-Λ * (x)} = max i=1,...,n sup x∈R n :x i =β inf λ∈R n {-λ, x + Λ(λ)} = max i=1,...,n inf λ∈R {-λβ + Λ(λe i )},
since the inf and sup may once again be interchanged in virtue of the minimax theorem and then the supremum on x ∈ R n such that x i = β is clearly +∞ when there is j = i such that λ j = 0. Consider the function

f i : R → R, f i (λ) = -λβ + Λ(λe i ). Since f i (1) = -β and f i (1) = -β + x * i , it follows that β + max i=1,...,n inf λ∈R {-λβ + Λ(λe i )} < 0.
when β = x * i for all i. 3. For the first identity in (3.4.4), remark that, similarly to the first part, for T sufficiently large, all δ > 0 and β ∈ R we have,

IE[e Y j T 1 { ỸT ∈(-∞,β-δ] n } ] ≥ IE[e Y j T 1 { BT ≤β} ] ≥ IE[e Y j T 1 { ỸT ∈(-∞,β+δ] n } ] ,
We can apply Theorem 3.3.6 with the function H : x → x j since Λ(e j ) = 0 and Λ(γe j ) < ∞ for γ > 1 small enough. When δ goes to zero, we get

sup x∈(-∞,β) n {x j -Λ * (x)} ≤ lim inf T →∞ T -1 log IE[e Y j T 1 { BT ≤β} ] ≤ lim sup T →∞ T -1 log IE[e Y j T 1 { BT ≤β} ] ≤ sup x∈(-∞,β] n {x j -Λ * (x)}.
By continuity of Λ * , the lower and the upper bounds are equal. Since Λ * (x) = sup λ∈R n λ + e j , x -Λ(λ + e j ), we get

sup x∈(-∞,β] n {x j -Λ * (x)} = sup x∈(-∞,β] n inf λ∈R n Λ(λ + e j ) -λ, x .

CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3

The second identity in (3.4.4) then follows from the minimax theorem as above. Finally, to show the inequality, remark that inf

λ j ≤1,λ i ≤0,i =j {Λ(λ) -β λ, 1 } ≤ inf λ≤1 f j (λ) and f j (1) = x * j -β > 0.
4. The first identity in (3.4.5) follows as in item (3). We have Λ * (x)x j ≥ 0 and Λ * (Λ (e j )) = Λ j (e j ) = x * j since e j is a critical point of λ → λ, Λ (e j ) -Λ(λ). Since β > x * j and Λ (e j ) ∈ (-∞, β] n , the supremum is attained as in item (2) on the boundary:

sup x∈R n x j -Λ * (x) = max i=1,...,n sup x∈R n :x i =β x j -Λ * (x) = max i=1,...,n sup x∈R n :x i =β inf λ∈R n Λ(λ + e j ) -λ, x .
The second identity in (3.4.5) holds true in virtue of the minimax theorem as above, like in item (2). To prove the negativity, we consider the functions g i (λ) = -λβ + Λ(λe i + e j ). We have that g i (0) = 0 and g i (0) = -β + Λ i (e j ). We have g j (0) = -β + x * j < 0. If g i (0) = 0 for all i, the result is clear. Otherwise, we can find β ∈ (x * j , β) such that β = Λ i (e j ) for all i, and since e Y j T 1 BT ∈(β,∞) ≤ e Y j T 1 BT ∈( β,∞) , we get the claim.

The following theorem characterizes the asymptotic behaviour of basket call prices in the Wishart model. There are different asymptotic regimes to consider, depending on the position of y with respect to these constants.

Theorem 3.4.2. Assume that y = x * i for all i. Then, as T → ∞, the call option price in the Wishart model satisfies holds for every δ > 0 and T > 0. Then by successively taking the expectation, the logarithm and multiplying by T -1 , we find

y + T -1 log(1 -e -δT ) + T -1 log P BT < y -δ ≤ T -1 log IE (e y T -e B T ) + ≤ y + T -1 log P BT < y .
Passing to the limit T → ∞ and using Lemma 3.4.1 part 1, the proof is complete.

Proof of (3.4.8). We use the inequality

e B T (1 -e -δT )1 {y< BT -δ} ≤ e B T -e y T + ≤ e B T 1 {y< BT } .

CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3

Consider for instance the upper bound. Taking the expectation and the logarithm, we obtain log IE[e B T 1 { BT >y} ] = log n j=1 ω j IE e Y j T 1 { BT >y} and thus

T -1 log IE[e B T 1 { BT >y} ] ≤ max j=1,...,n T -1 log IE e Y j T 1 { BT >y} , T -1 log IE[e B T 1 { BT >y+δ} ] ≥ max j=1,...,n T -1 log IE e Y j T 1 { BT >y+δ} + log(ω j )/T.
The result then follows from Lemma 3.4.1, part 4. Proof of (3.4.9). We use the following identity. Consider the function Since, for y ∈ (β * , β * ), f j (0) < 0 and f j (1) > 0, the infimum is attained on the interval (0, 1), and the contribution of this term is less than the one of the first term. The properties of the logarithm allow to conclude the proof.

f i : R → R, f i (λ) = -λy + Λ(λe i ). Since f i (0) = 0 and f i (0) = -y + x * i < 0,

Implied volatility asymptotics

In the Black-Scholes model with volatility σ, we have (see, e.g. (Forde and Jacquier, 2011a), Corollary 2.12) lim where

T →∞ T -1 log(C BS (T, yT, σ) + e yT -1) = - 1 2 σ 2 - y σ 2 , y ≤ - σ 2 2 lim T →∞ T -1 log C BS (T, yT, σ) = - 1 2 σ 2 - y σ 2 , y ≥ σ 2 2 lim T →∞ T -1 log 1 -C BS (T, yT, σ) = - 1 2 σ 2 - y σ 2 , - σ 2 2 < y < σ 2 2 .
L(y) = -y - inf λ∈R n :λ i ≤0,i=1,...,n {Λ(λ) -y λ, 1 }, y ≤ β * L(y) = -max i,j=1,...,n inf λ∈R {-λy + Λ(λe i + e j )}, y ≥ β * L(y) = -y -max i=1,...,n inf λ∈R {-λy + Λ(λe i )}, β * < y < β * .
We deduce (see [START_REF] Jacquier | Large deviations and stochastic volatility with jumps: asymptotic implied volatility for affine models[END_REF] for details) that the limiting implied volatility of a basket option in the Wishart model is given by (3.4.12) where ξ and η are constants with ξ 2 = η 2 = 1, which must be chosen to satisfy the conditions

σ ∞ (y) = √ 2 ξ L(y) + y + η L(y) ,
y ≤ - σ 2 ∞ (y) 2 if y ≤ β * y ≥ σ 2 ∞ (y) 2 if y ≥ β * - σ 2 ∞ (y) 2 < y < σ 2 ∞ (y) 2 if β * < y < β * .
First of all remark that by taking λ = 0 and λ = e i it follows that L(y) ≥ y and L(y) ≥ 0, so that the expressions under the square root sign are positive.

It is easy to see that for y ≤ β * , these conditions imply ξ = -1 and η = 1 since b * < 0 and -y ≤ L(y), and for y ≥ β * one has ξ = 1 and η = -1. For β * < y < β * , we still have |y| ≤ max(L(y), L(y) + y) and to satisfy the conditions in this case and σ ∞ (y) > 0, one must take ξ = η = 1. The case when β * < y < β * requires a specific treatment. It is characterized by the following proposition.

Proposition 3.4.3. Let β * < y < β * . Then, σ ∞ (y) = √ 2y and σ(T, yT ) = 2y + N -1 (C ∞ (y))T -1/2 + O T -1/2
as T → ∞, where C ∞ (y) = n i=1 ω i 1 x * i >y .

CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3

Proof. We follow the arguments of the proof of Theorem 3.3 in [START_REF] Jacquier | Implied volatility in strict local martingale models[END_REF] with some minor changes. The Black-Scholes call option price satisfies

C BS (T, yT, σ) = N -y + σ 2 2 σ √ T -e yT N -y -σ 2 2 σ √ T .
We have by definition of the implied volatility and equation (3.4.6),

C BS (T, yT, σ(t, yT )) = C(T, yT ) → T →+∞ C ∞ (y).
Since y > β * > 0, as T → ∞, we get necessarily

y+ σ(T,yT ) 2 2 σ(T,yT ) √ T → +∞.
Using the classical bound on the Mills ratio N (-x) ≤ x -1 φ(x) for x > 0, where φ is the standard Gaussian density, we have

e yT N -y -σ(T,yT ) 2 2 σ(T, yT ) √ T ≤ φ y -σ(T,yT ) 2 2 σ(T, yT ) √ T σ(T, yT ) y + σ(T,yT ) 2 2 √ T → 0 as T → ∞. Therefore, -y + σ(T,yT ) 2 2 σ(T, yT ) = N -1 (C ∞ (y))T -1/2 + O T -1/2 . (3.4.13)
Consider now the function f (z) = -y z + z 2 . Its inverse which is positive in the neighbourhood of zero is given by

f -1 (x) = x + x 2 + 2y
Applying f -1 to both sides of (3.4.13) and neglecting terms of order O T -1 2 , the proof is complete.

Variance reduction

The general variance reduction problem

Denote P (S T ) the payoff of a European option on (S 1 T , ..., S n T ). The price of an option is generally calculated as the expectation IE(P (S T )) under a certain risk-neutral measure P. When the number of assets n is low, this expectation may be evaluated by Fourier inversion, however, when the dimension is large, as in the case of index options, Monte Carlo is the method of choice. The standard Monte Carlo estimator of IE(P (S T )) with N samples is given by

P N = 1 N N j=1 P (S (j) T ),
where S (j)

T are i.i.d. samples of S T under the measure P. The variance of the standard Monte Carlo estimator is given by

Var[ P N ] = 1 N Var[P (S T )],
and is often too high for real-time applications. To decrease the computational time, various variance reduction methods have been proposed, the most popular being importance sampling.

The importance sampling method is based on the following identity, valid for any probability measure Q, with respect to which P is absolutely continuous.

IE P [P (S T )] = IE Q dP dQ P (S T ) .
This allows one to define the importance sampling estimator

P Q N := 1 N N j=1 dP dQ (j) P (S (j),Q T ),
where S (j),Q T are i.i.d. samples of S T under the measure Q. For efficient variance reduction, one needs then to find a probability measure Q such that S T is easy to simulate under Q and the variance

Var Q P (S T ) dP dQ = IE P P (S T ) 2 dP dQ -IE P [P (S T )] 2
is considerably smaller than the original variance Var P [P (S)].

In this paper we consider the class of measure changes

{P θ : θ ∈ R n }, where dP θ dP = e θ Y T IE e θ Y T .
To find the optimal variance reduction parameter θ * , we therefore need to minimize the variance of the estimator under Q, or, equivalently, the expectation

IE P P (S T ) 2 dP dP θ .
Denoting H(Y T ) := log P e Y T , the optimization problem writes

inf θ∈R n IE exp 2H(Y T ) -θ Y T + G 1 (θ) , ( 3

Asymptotic variance reduction

Since we cannot solve (3.5.1) explicitly, we instead choose to minimize the asymptotic proxy of Proposition 3.5.1, based on Theorem 3.3.6.

Proposition 3.5.1. Let H : R n → R ∪ {-∞} be a continuous function and θ ∈ R n be such that there exists γ > 1 with Furthermore, if θ * minimizes the right-hand side, it also minimizes the lefthand side.

lim sup →0 log IE exp γ 2H(Y T ) -θ Y T < ∞ . (3.5.2) Then lim →0 log IE exp 2H(Y T ) -θ Y T + G (θ) = sup y∈R n 2H(y) -θ y -Λ * (y) + Λ(θ) . Proof. By Theorem 3.3.6, lim →0 log IE exp 2H(Y T ) -θ Y T = sup y∈R n 2H(y) -θ y -Λ * (y) .
Proof. We follow the idea of the proof of (Genin and Tankov, 2016, Theorem 8), with some major simplifications due to the present finite-dimensional setting. By definition of Λ * , inf θ∈R n sup

y∈R n 2H(y) -θ y -Λ * (y) + Λ(θ) = inf θ∈R n sup y∈R n 2H(y) -θ y -sup λ∈R n λ y -Λ(λ) + Λ(θ) = inf θ∈R n sup y∈R n inf λ∈R n 2H(y) -θ y -λ y + Λ(λ) + Λ(θ) .
The function

(y, λ) → 2H(y) -θ y -λ y + Λ(λ) + Λ(θ)
is concave-convex on R n × U where U is bounded by Remark 3.2.6 and both R n and U are convex. Therefore, by the minimax Theorem for concave-convex functions (see, e.g., Corollary 37.3.2 in [START_REF] Rockafellar | Convex Analysis[END_REF]), sup

y∈R n inf λ∈R n 2H(y) -θ y -λ y + Λ(λ) + Λ(θ) = inf λ∈R n sup y∈R n 2H(y) -θ y -λ y + Λ(λ) + Λ(θ) .
This allows us to rewrite

inf θ∈R n sup y∈R n 2H(y) -θ y -Λ * (y) + Λ(θ) = inf θ∈R n inf λ∈R n sup y∈R n 2H(y) -θ y -λ y + Λ(λ) + Λ(θ) = 2 inf θ∈R n inf λ∈R n Ĥ θ + λ 2 + Λ(λ) + Λ(θ) 2 = 2 inf θ∈R n Ĥ(θ) + Λ(θ) ,
where the last equality is justified by the fact that, by convexity,

Λ(λ) + Λ(θ) 2 ≥ Λ λ + θ 2 with equality if λ = θ.
Remark 3.5.5. Similarly to (Genin and Tankov, 2016, Definition 6) and to the discussion in Section 4 of [START_REF] Robertson | Sample path large deviations and optimal importance sampling for stochastic volatility models[END_REF], it can be shown that the asymptotically optimal θ in Theorem 3.5.4 reaches the asymptotic lower bound of the variance on the log-scale over all equivalent measure changes. Let Q ∼ P be an equivalent measure change. Then by Jensen's inequality

lim →0 log IE Q e 2 H(Y T ) dP dQ 2 ≥ 2 lim →0 log IE Q e H(Y T ) dP dQ CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3 = 2 lim →0 log IE e H(Y T )
.

By Theorem 3.3.6, the right-hand side is equal to

2 sup y∈R n {H(y) -Λ * (y)} = 2 sup y∈R n inf θ∈R n H(y) -θ y + Λ(θ) = 2 inf θ∈R n sup y∈R n H(y) -θ y + Λ(θ) ,
where the second equality is obtained by the minimax theorem for concaveconvex functions [START_REF] Rockafellar | Convex Analysis[END_REF], already used in the proof of Theorem 3.5.4. But by the same Theorem 3.5.4, this bound is reached when θ is asymptotically optimal.

3.6 Numerical results

Long-time implied volatility

Let us now fix the parameters of the model to the values b = -1.0 0.7 0.7 0.7 , a = 0.2 0 0 0.3 and α = 1.5, with initial values S 0 = 1 and x = I 2 and consider the problem of pricing a basket put option with log-payoff

H(Y T ) = log K - n j=1
ω j e Y j T + and weights ω i = 1 2 for i = 1, 2. Figure 3.6.1 shows the implied volatility smile for such an option, for T = 1 3 , computed by Monte Carlo over 100'000 trajectories, together with the 95% confidence interval. To sample the paths of the process, we use the exact simulation of the Wishart process described in [START_REF] Ahdida | Exact and high-order discretization schemes for wishart processes and their affine extensions[END_REF], Algorithm 3. Thus, we obtain the values of X t i on the regular time grid t i = i∆t, with i ∈ N and ∆t > 0. Then, for the stock, we use a trapezoidal rule since it gives a second-order weak convergence (see Section 4.3 in [START_REF] Ahdida | Exact and high-order discretization schemes for wishart processes and their affine extensions[END_REF] for details):

Y t i+1 = Y t i - 1 2 diag a X t i + X t i+1 2 a ∆t+Chol a X t i + X t i+1 2 a (Z t i+1 -Z t i ),
where Z is a Brownian motion sampled independently from X and Chol(M ) is the Cholesky decomposition of a definite positive matrix M . We next analyse the convergence of the renormalized implied volatility smile to the long-maturity limit described in Section 3.4.2. Figure 3.6.2, shows the renormalized smiles for different maturities together with the limiting smile. These smiles were computed by Monte Carlo with 100'000 trajectories and a discretization time step ∆t = 0.1. We see that the convergence indeed appears to take place but it is quite slow: even for 50-year maturity using the limit as the approximation for the smile would lead to 10 -15% errors.

Variance reduction

We wish to test numerically the variance reduction method to price basket put options. In order to do so, we first identify the law of the Wishart process under the measure P θ and then calculate the asymptotically optimal measure change to finally test the method through Euler Monte-Carlo simulations.

Change of measure

In order to simulate from the model under P θ , we need the following result.

Proposition 3.6.1. Let θ ∈ R n be such that IE[e θ Y T ] < ∞ and consider the change of measure dP θ dP = e θ Y T I E e θ Y T
. Under P θ , the process (Y t , X t ) has dynamics

dY t = r1 - 1 2 (a X t a) 11 , ... , (a X t a) nn + a X t a θ dt + a X 1/2 t dZ θ t and dX t = (αI n + (b + 2 γ θ (T -t))X t + X t (b + 2 γ θ (T -t))) dt
+ X 1/2 t dW θ t + (dW θ t ) X 1/2 t , X 0 = x , where γ θ (t) = -1 2 (V (t, θ) V -1 (t, θ) + b), V (t, θ) = V (t)
is given in Proposition 3.2.4 and Z θ t t≥0 and W θ t t≥0 are R n and R n×n -dimensional independent standard P θ -Brownian motions.

Proof. By Equation 3.2.5, the Radon-Nikodym density satisfies

ζ t := dP θ dP Ft = IE e θ Y T F t IE e θ Y T = e α 2 Tr[b]t-θ Y 0 -rθ 1t-Tr[γ θ (T ) x] det[V (T, θ)] -α/2 det[V (T -t, θ)] α/2 e θ Yt+Tr[γ θ (T -t)Xt] .
By Itō formula, the martingale property of ζ t , Equations (3.2.2) and (3.2.3), and the properties of the trace, the dynamics of ζ t is

dζ t = ζ t θ a X 1/2 t dZ t + Tr γ θ (T -t) X 1/2 t dW t + Tr γ θ (T -t) (dW t ) X 1/2 t = ζ t θ a X 1/2 t dZ t + 2 Tr X 1/2 t γ θ (T -t) dW t .
Therefore, by Girsanov's theorem,

Z θ t := Z t - t 0 X 1/2 s a θ ds and
W θ t := W t -2 t 0 X 1/2 s γ θ (T -s) ds
are n-dimensional and n×n-dimensional standard P θ -Brownian motions. Replacing dZ t and dW t in (3.2.2) and (3.2.3) by their P θ versions finishes the proof.

We note that X is no longer a Wishart process under the probability P θ , since the dynamics has time-dependent coefficients. To sample paths on the time interval [t i , t i+1 ], we use the exact scheme for the Wishart process with the coefficient b + 2γ θ (T -(t i + t i+1 )/2) instead of b. As explained in (Alfonsi, 2015, Section 3.3.4) on the case of the CIR process with time-dependent coefficient, this leads to a second order scheme for the weak error. Then, we can approximate Y in the same way as under P:

Y t i+1 = Y t i + r1 - 1 2 diag a X t i + X t i+1 2 a + a X t i + X t i+1 2 aθ ∆t + Chol a X t i + X t i+1 2 a (Z t i+1 -Z t i ),
where Z is a Brownian motion sampled independently from X. This gives a second order scheme for (X, Y ).

Optimal variance reduction parameter for the European basket put option

In this section, we compute the asymptotically optimal measure to price basket put options with log-payoff H(Y T ) = log(K -ω e Y T ) + , for some ω ∈ (R * + ) n . It is shown in [START_REF] Genin | Optimal importance sampling for Lévy processes[END_REF], Section 4) that the function H is concave and that its convex conjugate is given by

Ĥ(θ) =      + ∞ θ k ≥ 0 for some k , -1 - k θ k log 1 -k θ k K - k θ k log(-θ k /ω k ) otherwise.
To compute the asymptotically optimal measure change parameter θ * using Theorem 3.5.4 we then minimize Ĥ(θ) + Λ(θ) with a numerical convex optimization algorithm.

Numerical simulations

Let us now fix the parameters of the model to the values b = -0.7 0.3 0.3 0.5 , a = 0.1 0 0 0.12 and α = 4.5, with initial values S 0 = 1 and x = I n and consider the problem of pricing a basket put option with log-payoff

H(Y T ) = log K - n j=1
ω j e Y j T + CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3 and weights ω = 0.5 0.5 . For a wide variety of maturities T and strikes K, listed in Table 3.1, we simulate 100'000 trajectories, with the discretization step described above, with step size ∆ = 1 40 , under both measures P and P θ for the asymptotically optimal θ. The results are presented in Table 3 Table 3.1: The variance ratio as function of the maturity and the strike for the basket put option on the Wishart stochastic volatility model.

The variance ratio is the ratio of the variance under the original measure P to that under the asymptotically optimal measure P θ . As expected, the performance of the importance sampling algorithm is best for options far from the money, when the exercise is a rare event, but even for at the money options the variance reduction factor is significant, of the order of 3-4. The computational overhead for using the variance reduction algorithm is small: it does not exceed 20% for a small number of trajectories decreases with the number of trajectories because some precomputation steps are performed only once.

Chapter 4

An asymptotic approach for the pricing of options on realized variance I would like to thank Archil Gulisashvili1 for the numerous insights he shared with me during his visits to Paris Diderot, and which helped me to write this chapter.

Introduction

The trading of variance and volatility started in the 90s with variance and volatility swaps and increased drastically in the years 2000, probably as a response of the markets to an increasing volatility risk. Gradually, a wider range of more complex financial products based on variance, such as options on realized variance appeared on the markets. An option on realized variance with strike K is a non-linear derivative with payoff

1 T T 0 Z 2 t dt -K + , (4.1.1)
where Z t is the spot volatility process and where 1 T T 0 Z 2 t dt is the continuous version of realized variance. On the markets, only the discrete version of realized variance is traded, but the continuous version is a commonly accepted approximation when Z t is continuous, as the first converges in probability to the second when the number of subdivisions tends to infinity (Jacod and Shiryaev, 2003, Theorem I.4.47) and as the second is the quadratic variation of the log-price process and therefore allows to solve elegantly modelling, pricing and hedging problems (see for example [START_REF] Kallsen | Pricing options on variance in affine stochastic volatility models[END_REF]). We refer CHAPTER 4. AN ASYMPTOTIC APPROACH FOR THE PRICING OF OPTIONS ON REALIZED VARIANCE 4 the reader to (Keller-Ressel and Muhle-Karbe, 2013) for a more detailed discussion on the difference between these. The authors propose an asymptotic correction term to address this issue.

The pricing of an option on realized variance is done calculating the expectation of (4.1.1). Due to the non-linearity and the path-dependence of z → 1 T T 0 z 2 t dt -K + however, the calculation of the option price is a complex task that cannot be solved using straightforward methods. We model the spot volatility as a diffusion process with general drift and constant volatility and thus consider the joint process (Y t , Z t ) t≥0 with dynamics

dY t = Z 2 t dt Y 0 = 0 dZ t = b(Z t ) dt + c dW t Z 0 = z 0 ,
where (W t ) t≥0 is a standard Brownian motion. Even though apparently restrictive, the generality of b allows to model Z as an Ornstein-Uhlenbeck process or as the square root of a CIR process, which are the volatility processes in the popular [START_REF] Stein | Stock price distributions with stochastic volatility: an analytic approach[END_REF] and Heston [START_REF] Heston | A closed-form solutions for options with stochastic volatility with applications to bond and currency options[END_REF] models. The approach that we consider in this chapter relies on the results from (Deuschel et al., 2014a) that provide a short-time/small-noise asymptotic expansion for marginal densities of hypo-elliptic diffusions to obtain an explicit expansion of the density of Y t in short-time.

In Section 4.2, we recall the relevant definitions and results of (Deuschel et al., 2014a). In Section 4.3, we define the process (Y t , Z t ) t≥0 and give some of its properties. In particular, we prove that the joint process of the realized variance and the volatility satisfies the hypothesis of Theorem 4.2.2 below. We then calculate, in Section 4.4, the expansion of the density of the realized variance. We finally calculate, in Section 4.5, the expansion of the price and implied volatility of put options on realized variance.

Expansion of marginal densities

In this section, we recall the main definitions and results of (Deuschel et al., 2014a). Consider a d-dimensional diffusion (X t ) t≥0 solving the equation

dX t = b( , X t ) dt + σ(X t ) dW t , X 0 = x 0 , (4.2.1)
where (W t ) t≥0 is an m-dimensional Brownian motion and the functions b :

[0, 1) × R d → R d , σ = (σ 1 , ..., σ m ) : R d → Lin(R m → R d ) and x • 0 : [0, 1) → R d are
smooth and bounded, with bounded derivatives of all orders. We assume that b( ,

•) → σ 0 := b(0, •) in the sense that for every multi-index α, ∂ α x b( , •) → ∂ α x σ 0 (•) and ∂ b( , •) → ∂ b(0,
•) uniformly on compacts as → 0 and that x 0 = x 0 + x0 + O ( ) as → 0.

EXPANSION OF MARGINAL DENSITIES

4

In order to guarantee that Y T = Π l •X T = X ,1 T , ..., X ,l T admits a smooth density for every T > 0, we assume that the weak Hörmander condition2 is verified at x 0 , i.e. that the linear span of σ 1 , ..., σ m and the Lie brackets of σ 0 , ..., σ m has full rank at x 0 . Let a ∈ R l . In order to define the energy associated to a, a necessary assumption is that the set

K a = {h ∈ H : Π l • φ h T = a} is non-empty 3 .
Here, (H, || • || H ) is the Cameron-Martin space of absolutely continuous trajectories starting at 0 with derivatives in L 2 ([0, T ], R m ) equipped with the norm

||h|| 2 H = m j=1 T 0 | • h j t | 2 dt
and where φ h is the solution of the controlled differential equation

dφ h t = σ 0 φ h t dt + m j=1 σ j φ h t dh j t , φ h 0 = x 0 .
The energy Λ(a) associated to a, that is the minimal energy for φ h T to reach

N a = {(a, •) ∈ R n } in time T , is then Λ(a) = inf 1 2 ||h|| 2 H : h ∈ K a .
We denote K min a ⊂ K a the set of minimizing controls. In order for K a to have a manifold structure around each minimizer h ∈ K min a , we assume the invertibility, at every h ∈ K min a , of the deterministic Malliavin covariance matrix

C(h) = D h φ h T , D h φ h T H
, where D h denotes the Fréchet derivative. A sufficient condition for C(h) to be invertible at every h = 0 is that condition H2 in (Bismut, 1984, p. 28) is verified at x 0 . Condition H2 is verified at x 0 if for every non-zero λ = (λ 0 , ..., λ m ) ∈ R m+1 , denoting V = m j=0 λ j σ j , the linear span of σ 1 , ..., σ m and [σ 0 , V ], ..., [σ m , V ] has full rank.

We define the Hamiltonian

H(x, p) = p, σ 0 (x) + 1 2 m j=1 p, σ j (x) 2 (4.2.2)
and give the following result.

CHAPTER 4. AN ASYMPTOTIC APPROACH FOR THE PRICING OF OPTIONS ON REALIZED VARIANCE 4

Proposition 4.2.1. (Deuschel et al., 2014a, Prop. 2) If h ∈ K min a is a minimizing control and C(h) is invertible, then there exists a unique p 0 such that φ h t = πH t←0 (x 0 , p 0 ) , 0 ≤ t ≤ T , where π denotes the projection from T R d to R d and H t←0 is the flow associated to the vector field (∂ p H, -∂ x H). Moreover , (x(t), p(t)) := H t←0 (x 0 , p 0 ) solve the Hamiltonian ODEs

• x • p = ∂ p H(x(t), p(t)) -∂ x H(x(t), p(t)) ,
with boundary conditions x(0) = x 0 , x(T ) = (a, •) and p(t) = (•, 0). The minimizing control is recovered by

• h j t = σ j (x(t)), p(t)
. We now give the non-degeneracy condition, which is the main hypothesis of the density expansion result (Deuschel et al., 2014a, Theorem 8). This condition generalizes the "not in the cut-locus" condition of (Ben Arous, 1988a).

Condition (ND). We say that {x 0 } × N a satisfies the non-degeneracy condition (ND) if

(i) 1 ≤ #K min a < ∞,
(ii) The deterministic Malliavin covariance matrix C(h) is invertible at every h ∈ K min a . (iii) x 0 is non-focal for N a along h, for every h ∈ K min a , i.e.

∂ (z,q) | (z,q)=(0,0) πH 0←T x T + 0 z , p T + (q , 0)
is non-degenerate, where (x T , p T ) = H T ←0 (x 0 , p 0 (h)).

We now cite the density expansion result.

Theorem 4.2.2. (Deuschel et al., 2014a, Theorem 8) Let X be the solution of (4.2.1), where b( , •) and x 0 converge in the previously given sense. Assume also that X satisfies the weak Hörmander condition at x 0 . Assume finally

that {x 0 }×N a satisfies (ND). Let a ∈ R l . If #K min a = 1, then the energy Λ(a) is smooth, otherwise, if #K min a > 1, we assume that it is. Then there exists c 0 = c 0 (x 0 , a, T ) > 0 such that Y T admits a density f (•, T ) with expansion f (a, T ) = e -Λ(a) 2 e max { Λ (a)• ŶT (h) : h∈K min a } -l (c 0 + O( )) ,
as → 0, where Ŷt (h) = Π l Xt and ( Xt ) t≤T is the solution of the controlled ODE

d Xt = ∂ X b(0, φ h t ) + ∂ X σ(φ h t ) • h t Xt dt + ∂ b(0, φ h t ) dt , X0 = x0 .

DEFINITION AND PROPERTIES OF THE INTEGRATED VARIANCE PROCESS 4

4.3 Definition and properties of the integrated variance process

The integrated variance process

As mentioned, we define X t = (Y t , Z t ) to be the joint process with dynamics

dY t = g(Z t ) dt Y 0 = 0 dZ t = 2 b(Z t ) dt + c dW t Z 0 = z 0 , (4.3.1)
which corresponds to (4.2.1) where

x 0 = (0, z 0 ), b( , (y, z)) = g(z) 2 b(z) -→ →0 σ 0 ((y, z)) = g(z) 0 and σ 1 (x) = 0 c .
In order for b( , x) to be in

C ∞ b ([0, 1) × R 2 , R 2
), the set of bounded functions with bounded derivatives of all orders, we assume that z → b(z) ∈ C ∞ b (R). Furthermore, in order for Y t to correspond to realized variance, while satisfying the hypotheses, we choose

g(z) = z 2 e -1 R+1-|z| 1 {|z|<R+1} + (R + 1) 2 e -1 |z|-R 1 {|z|>R} e -1 R+1-|z| 1 {|z|<R+1} + e -1 |z|-R 1 {|z|>R} (4.3.2)
for R arbitrarily large, following an idea in [START_REF] Kunitomo | The asymptotic expansion approach to the valuation of interest rate contingent claims[END_REF].

Eq. ( 4

.3.2) is a C ∞ b (R) version of the z → z 2 function, i.e. g(z) = z 2 for |z| < R, while g(z) = (R + 1) 2 for |z| > R + 1.
Lemma 4.3.1. The process X t satisfies the Bismut H2 condition at x 0 = (0, z 0 ) and the deterministic Malliavin covariance matrix c(h) is therefore invertible for every non-zero h ∈ H. In particular, the weak Hörmander condition is verified.

Proof. Let λ = (λ 0 , λ 1 ) ∈ R 2 be a non-zero vector and

V (x) = λ 0 σ 0 (x) + λ 1 σ 1 (x) = λ 0 g(z) λ 1 c . Then [σ 0 , V ](x) = λ 0 [σ 0 , σ 0 ](x) + λ 1 [σ 0 , σ 1 ](x) = -λ 1 c g (z) 0 and [σ 1 , V ](x) = λ 0 [σ 1 , σ 0 ](x) + λ 1 [σ 1 , σ 1 ](x) = λ 0 c g (z) 0

DEFINITION AND PROPERTIES OF THE INTEGRATED VARIANCE PROCESS 4

where p t = p is the constant solution of (4.3.5). Following the idea of the proof of (Deuschel et al., 2014b, Lemma 7), we find

| • h t | 2 = c 2 q 2 t = q t • z t = ∂ t (q t z t ) -• q t z t = ∂ t (q t z t ) + 2pz 2 t = ∂ t (q t z t ) + 2p • y t .
The energy of a is therefore

Λ(a) = 1 2 T 0 | • h t | 2 dt = 1 2 T 0 ∂ t (q t z t ) + 2p • y t dt = 1 2 (q T z T -q 0 z 0 + 2py T -2py 0 ) = pa - z 0 2 q 0 ,
where the last equality is obtained by using the boundary conditions.

Below, we solve the Hamiltonian equations. In the rest of this Chapter we denote r = r(p) = √ 2p c ∈ C, where p = p t is the constant solution of (4.3.5).

Lemma 4.3.4. Let s, t ∈ [0, T ]. Then the solution (y t , z t , p t , q t ) t≤T of equations (4.3.4)-(4.3.6) satisfies

z t = z s cos (r (t -s)) + q s c 2 r sin (r (t -s)) (4.3.7)
q t = q s cos (r (t -s)) -z s r c 2 sin (r (t -s)) , (4.3.8)

and

y t = y s + z 2 s 2 1 + sin (2 r (t -s)) 2 r (t -s) (t -s) + z s q s c 2 1 -cos (2 r (t -s)) 4 r 2 + q 2 s c 4 2r 2 1 - sin (2 r (t -s))) 2 r (t -s) (t -s) .
(4.3.9)

Proof. Eq. (4.3.5) implies that p t = p is constant. Eqs. (4.3.4) and (4.3.6) then become a 2-dimensional linear ordinary differential equation, whose solution verifies z s cos (r (u -s)) + q s c 2 r sin (r (u -s))

z t q t = cos (r (t -s)) c √ 2p sin (r (t -s)) - √ 2p c sin (r (t -s)) cos (r (t -s)) z s q s = cos (r (t -s)) c 2 r sin (r (t -s)) -r c 2 sin (
2 du = z 2 s 2 1 + sin (2 r (t -s)) 2 r (t -s) (t -s) + z s q s c 2 1 -cos (2 r (t -s)) 4 r 2 + q 2 s c 4 2 r 2 1 - sin (2 r (t -s))) 2 r (t -s) (t -s) ,
thus proving (4.3.9).

Theorem 4.3.5. The energy associated with a is

Λ(a) = r 2 2 c 2 z 2 0 T a z 2 0 T - tan (r T ) r T ,
where r solves the equation

1 + cos (2 r T ) - z 2 0 T a 1 + sin (2 r T ) 2 r T = 0 . (4.3.10)
Proof. Let (y t , z t , p, q t ) be the solution of (4.3.3)-(4.3.6) with the boundary conditions z 0 = z 0 , y 0 = 0, y T = a and q T = 0. Then by (4.3.8), 0 = q T = q 0 cos (r T ) -z 0 r c 2 sin (r T ) .

Hence q 0 = z 0 r c 2 tan (r T ) and therefore

Λ(a) = pa - z 0 2 q 0 = r 2 2 c 2 z 2 0 T a z 2 0 T - tan (r T ) r T .
Furthermore, (4.3.7) implies that

z 0 = z T cos (r T ) -q T c 2 r sin (r T ) = z T cos (r T ) .
Inserting this in (4.3.9) applied for t = 0 and s = T , we obtain

0 = y 0 = a - z 2 T 2 1 + sin (2 r T ) 2 r T T = a - z 2 0 T 2 cos 2 (r T ) 1 + sin (2 r T ) 2 r T

DEFINITION AND PROPERTIES OF THE INTEGRATED VARIANCE PROCESS

4 = a - z 2 0 T 1 + cos (2 r T ) 1 + sin (2 r T ) 2 r T .
Note that 2 r T = (2k + 1)π for k ∈ Z is not a solution. Indeed, 1 + cos ((2k + 1)π) = 0, while 1 + sin((2k+1)π) (2k+1)π = 1. We can therefore multiply by 1 + cos (2 r T ) to obtain (4.3.10).

Theorem 4.3.5 gives the complete characterization of the energy, up to the calculation of r. Since, p ∈ R, we therefore need to study the solutions of eq. ( 4

f (u) u = 0 , where f (u) = (1 + cosh(u)) u -k (u + sinh(u)) .
Then r > 0 solves (4.3.10) if and only if u > 0 solves f (u) = 0. The two first derivatives of f are

f (u) = sinh(u)u + (1 -k)(1 + cosh(u)) f (u) = cosh(u)u + (2 -k) sinh(u) = f (u) + (k -1) u + 2 sinh(u) .
Assume first that k > 1. Then f (0) = 0 and f (0) = 2 (1 -k) < 0. Therefore, there exists u 1 > 0 such that f (u) < 0 for every u ∈ (0, u 1 ]. Since f (u) → ∞ as u → ∞, there exists u 2 > u 1 such that f (u) < 0 for u ∈ (0, u 2 ) and f (u 2 ) = 0. In particular, such a u 2 verifies f (u 2 ) ≥ 0. Since f (u) = f (u) + (k -1) u + 2 sinh(u), then f (u) > 0 for u ≥ u 2 . Therefore, f (u) > 0 for u > u 2 and u 2 is the unique strictly positive solution of f . Assume now that k ∈ (0, 1]. Then f (0) = 0 and f (u) > 0 for every u > 0. Therefore f (u) > 0 for every u > 0 and f has no strictly positive solution. Then r > 0 solves (4.3.10) if and only if u > 0 solves f (u) = 0. Assume that k ∈ (0, 1). The derivative

f (u) = -sin(u)u + (1 -k) (1 + cos(u)) ,
in (0, π), cancels exactly once at u 1 , where u 1 is the unique solution of u tan(u/2) = 1 -k in (0, π). Therefore, f is strictly monotonous on (0, u 1 ) and (u 1 , π). Since f (0) = 0 and f (u 1 ) > 0, then f is increasing on (0, u 1 ) and does not cancel on this interval. Furthermore, f (π) = -kπ < 0 and therefore f cancels exactly once in (u 1 , π).

Assume now that k ≥ 1. Then the derivative

f (u) = -sin(u)u + (1 -k) (1 + cos(u)) ,
is always strictly negative in (0, π). Since f (0) = 0, then f is always strictly negative in (0, π). Transposing from u to r finishes the proof.

Theorem 4.3.8. The energy associated with a is

Λ(a) = z 2 0 4 c 2 r 2 r T -sin (2 r T ) 1 + cos (2 r T ) ,
where r is the unique solution of equation (4.3.11) in the set

1 + cos (2 r T ) - z 2 0 T a 1 + sin (2 r T ) 2 r T = 0
I := 0, π 2T ∪ iR + ⊂ C .
Proof. The combination of Lemmas 4.3.6 and 4.3.7 proves the uniqueness of the solution of (4.3.11) in I. Due to the form of the Hamiltonian equations, Prop. 4.2.1 guaranties the uniqueness of the optimal control. Note first that if a = z 2 0 T , then h = 0 is the optimal control and r = 0 is therefore the optimal solution. We then consider the case a = z 2 0 T . For h ∈ H, define (y

• ω t = • h t 1 {z h t >0} - • h t 1 {z h t <0}
. We then have that

z ω t = z 0 + c t 0 • h s 1 {z h s >0} - • h s 1 {z h s <0} ds = |z h t |
is simply z h t reflected at 0 and therefore

y ω T = T 0 (z ω t ) 2 dt = a. Furthermore T 0 | • w t | 2 dt = T 0 | • h t | 2
dt and therefore w ∈ K min a . By uniqueness of the optimal control, w = h and z h t ≥ 0. Assume that a < z 2 0 T . Then T 0 (z h t ) 2 dt < z 2 0 T . This implies that there exists a non-trivial interval I ⊂ [0, T ] such that • z h t < 0 for t ∈ I. From eq. (4.3.4), we therefore have q t < 0 for every t ∈ I. Since z ≥ 0, eq. (4.3.6) then implies that the sign of • q is constant. Since, q t < 0 on I and q T = 0, we have • q ≥ 0, which implies that p < 0 and therefore r ∈ i R + .

Assume now that a > z 2 0 T . The same argument allows to conclude that q t is a positive function that decreases strictly to q T = 0. Combining (4.3.7) and (4.3.8), we obtain

q t = z 0 r c 2 cos (r T ) sin (r (T -t)) .
Since the only values of r in R + ∪ i R + such that r sin (r (T -t)) cos (r T ) > 0 , for every t < T are the values of r in 0, π 2T , Theorem 4.3.5 implies that

Λ(a) = r 2 2 c 2 z 2 0 T a z 2 0 T - tan (r T ) r T ,
where r is the unique solution of (4. Proof. As mentioned previously, the linear nature of the Hamiltonian equations implies that #K min a = 1 and the H2 condition implies the invertibility of the Malliavin deterministic covariance matrix. Let us show that (0, z 0 ) is non-focal for N a along the optimal control h. Using Lemma 4.3.4, we find that

πH 0←T a z T + z , (p + q , 0) = a -(z T +z) 2 T 2 1 + sin(2 r T ) 2 r T (z T + z) cos (r T ) where r = r(p + q) = 2(p + q) c. Since r = √ 2c 2 √ p = c 2
r , the Jacobian matrix of the projection of the backward Hamiltonian flow is

∂ (z,q) | (z,q)=(0,0) πH 0←T a z T + z , (p + q , 0) = -z T T 1 + sin(2 r T ) 2 r T - z 2 T T c 2 2 r 2 cos (2 r T ) -sin(2 r T ) 2 r T cos (r T ) -z T T 2 c 2 sin(r T ) r T
.

The determinant of the Jacobian matrix is therefore det

∂ (z,q) | (z,q)=(0,0) πH 0←T a z T + z , (p + q , 0) =z 2 T c 2 T 3 1 + sin (2 r T ) 2 r T sin (r T ) r T + 1 2 r 2 T 2 cos (2 r T ) - sin (2 r T ) 2 r T cos (r T ) =z 2 T c 2 T 3 sin (r T ) r T + 2 cos (r T ) 2 r T -sin (2 r T ) (2 r T ) 3 .
Let us show that the determinant is non-zero. Notice first that, from (4.3.7), we know that z T = z 0 cos(r T ) > 0 for every r ∈ I.

If a = z 2 0 T , then r = 0, z T = z 0 and det ∂ (z,q) | (z,q)=(0,0) πH 0←T a z T + z , (p + q , 0) = 4 z 2 0 c 2 T 3 3 > 0 .
If a > z 2 0 T , then r ∈ 0, π 2 T . In this case, sin(r T ) r T , cos (r T ) and 2 r Tsin (2 r T ) are all strictly positive, hence the determinant is strictly positive. Finally, if a < z 2 0 T , then r

∈ i R * + . Denoting u = -2 i r T ∈ R * + , det ∂ (z,q) | (z,q)=(0,0) πH 0←T a z T + z , (p + q , 0) = z 2 T c 2 T 3 2 sinh (u/2) u + 2 cosh (u/2) sinh (u) -u u 3 ,
where all the terms are individually strictly positive. The determinant of the Jacobian matrix of the projected backward Hamiltonian flow is therefore always strictly positive, thus proving that (0, z 0 ) is non-focal for N a along h. Condition (ND) is therefore verified.

Derivatives of the energy

We know proceed to some calculations concerning the derivatives of the energy. These results will be necessary in the rest of the chapter. As a consequence, Λ (a) ∈ -∞, π 2 8 c 2 T 2 . Furthermore, (4.3.13) when a → z 2 0 T .

Λ (a) = 3 4z 2 0 c 2 T 3 + O |r| 2 ,
Proof. Consider the function a → r(a). It is a 1-1 map from (0, ∞) to I with inverse

r -1 (r) = z 2 0 T 1 + sin(2 r T ) 2 r T 1 + cos(2 r T ) . Since 1 + cos(2 r T ) = 0 on I, r -1 is a differentiable function. Then Λ (a) = r 2 2 c 2 + r c 2 a - z 2 0 2 c 2 tan (r T ) - z 2 0 T c 2 r 1 1 + cos (2 r T ) r = r 2 2 c 2 + a r r c 2 (1 + cos(2 r T )) 1 + cos(2 r T ) - z 2 0 T a 1 + sin(2 r T ) 2 r T ,
where the last factor is 0 by definition of r, thus proving (4.3.12). Furthermore, Λ (a) = r r c 2 where r = 1 (r -1 ) (r) . Then, denoting u = 2 r T , we have

Λ (a) = u 2 2 z 2 0 c 2 T 2 u 1 - (u + sin(u))(1 + cos(u) -u sin(u)) u (1 + cos(u)) 2 -1 = 3 4 z 2 0 c 2 T 3 + O r 2 ,
as r → 0 and therefore as a → z 2 0 T .

Asymptotic expansion of the density

In this section, we provide the asymptotic expansion of the density. The proof relies on (Deuschel et al., 2014a, Theorem 8) with a slight extension of the proof to obtain an explicit value for the c 0 coefficient.

Theorem 4.4.1. Consider the process X = (Y t , Z t ) t∈[0, T ] defined in (4.3.1).

Then, for every a ∈ 0,

R 2 T 2 1 + z 0 /R arccos(z 0 /R) 1 -z 2 0 /R 2
where R is the arbitrarily large constant in (4.3.2), Y t admits a smooth density with expansion f Y T (a) = -1 e -Λ(a)/ 2 (c 0 (a) + o(1)) , as → 0 , where

c 0 (a) = 1 2πA(2rT ) cos 3/2 (r T ) 2 z 0 c T 3/2 e 1 c 2 z T z 0 b(x) dx , and
A(u) = u 3 + 6 u cos(u) + 3 (u 2 -2) sin(u) 6 u 3 .
Proof. Let z t be the solution of (4.3.4). Then for

a ∈ 0, R 2 T 2 1 + z 0 /R arccos(z 0 /R) 1 -z 2 0 /R 2 , |z t | < R for every t ≤ T and therefore g(z t ) = z 2 t .
Then by Lemma 4.3.1, the weak Hörmander condition is verified at (0, z 0 ) and by Theorem 4.3.9, (0, z 0 ) × N a verifies Condition (ND). By Theorem 4.2.2, there exists c 0 = c 0 (x 0 , a, T ) > 0 such that Y T admits a smooth density f Y T (a) with expansion f Y T (a) = -1 e -Λ(a)/ 2 (c 0 + o(1)) , as → 0 , as → 0. In order to calculate the c 0 coefficient we proceed, as in the proof of Theorem 4.2.2 in (Deuschel et al., 2014a), by applying the Laplace method on the Wiener space C([0, T ], R), following the methodology of (Ben Arous, 1988a). By Fourier inversion, for any function F in C ∞ b (R), we have that

f Y T (a)e -F (a)/ 2 = 1 2π R e -iaξ R e iξθ f Y T (θ) e -F (θ)/ 2 dθ dξ = 1 2π R R e iξ(θ-a) e -F (θ)/ 2 f Y T (θ)dθ dξ = 1 2π R IE exp (ξ (Y T -a)) e -F (Y T )/ 2 dξ = 1 2π R IE exp iζ Y T -a e -F (Y T )/ 2 dζ . We choose F ∈ C ∞ b (R)
f Y T (a) = 1 2π R IE exp iζ Y T -a e -F (Y T )/ 2 dζ = 1 2π R IE exp - 1 T 0 • h s dW s - Λ(a) 2 • exp iζ Ȳ T -a e -F ( Ȳ T )/ 2 dζ , (4.4.1)
where

d Ȳ t = g(Z t ) dt Y 0 = 0 d Z t = 2 b(Z t ) dt + c dW t + c dh t Z 0 = z 0 .
In (Deuschel et al., 2014a), the authors expand F ( Ȳ T ) to the first order. This leads to the presence of a O(1) in the definition of c 0 , thus impeding its computation. In order to calculate it, let us expand Y T and F ( Ȳ T ) to the second order.

Ȳ T = a + Ȳ 1 T + 2 2 Ȳ 2 T + o( 2 ) (4.4.2)
and (Ben Arous, 1988b, Lemma 1.43) proves that

F ( Ȳ T ) = F (y T ) + F (y T ) Ȳ 1 T + 2 2 F (y T ) Ȳ 1 T 2 + F (y T ) Ȳ 2 T + o( 2 ) , as → 0, where Ȳ j T = ∂ j j Ȳ t =0 and Zj T = ∂ j j Z t =0 have dynamics d Ȳ 1 t = 2 z t Z1 t dt , Ȳ 1 0 = 0 , d Z1 t = c dW t , Z1 0 = 0 , d Ȳ 2 t = 2 z t Z2 t + Z1 t 2 dt , Ȳ 2 0 = 0 , d Z2 t = 2 b(z t ) dt , Z2 0 = 0 . Since h ∈ K min a ,
F (a) Ȳ 1 T = - T 0 • h s dW s .
Then, since F (y T ) = F (a) = 0, as → 0, where

F ( Ȳ T ) = - T 0 • h s dW s + 2 2 F (a) Ȳ 1 T 2 + F (a) Ȳ 2 T + o( 2 ) .
c 0 (a) = 1 2π R IE e iζ Ȳ 1 T + Λ (a) 2 Ȳ 2 T dζ , for Ȳ 1 T = 2c T 0 z t W t dt , Ȳ 2 T = 2 T 0 2z t t 0 b(z s ) ds + c 2 W 2 t dt .
Combining (4.3.4) and (4.3.6), we obtain

•• z t = -r 2 z t with • z T = 0. Then T 0 z t t 0 b(z s ) ds dt = T 0 b(z s ) T s z t dt ds = - 1 r 2 T 0 b(z s ) T s •• z t dt ds = 1 r 2 T 0 b(z s ) • z s ds = 1 r 2 z T z 0 b(x) dx . Hence Ȳ 2 T = 4 r 2 z T z 0 b(x) dx + 2 c 2 T 0 W 2 t dt
and substituting Λ (a) with (4.3.12),

c 0 (a) = e 1 c 2 z T z 0 b(x) dx 2π R IE e 2c iζ T 0 ztWt dt+ r 2 2 T 0 W 2 t dt dζ .
We then have, using (Gombani and Runggaldier, 2001, Prop. 2.1)4 , that

IE exp 2c iζ T 0 z s W s ds + r 2 2 T 0 W 2 s ds = exp (-A(0)) ,
where A(t) satisfies the following Riccati equations Therefore, since z s = z 0 cos(r (T -s))

C (t) = 2C 2 (t) + r 2 2 , C (
cos(r T ) ,

B(t) = -2c iζ cos (r (T -t)) T t z s cos (r (T -s)) ds = -2c iζ z t (T -t) .
Finally, by (4.4.6)

A(0) = - 1 2 T 0 B 2 (t) dt + T 0 C(t) dt = 2 z 2 0 c 2 cos 2 (r T ) ζ 2 T 0 cos 2 (r (T -t)) (T -t) 2 dt - 1 2 log (cos (r T )) = ζ 2 2 4 z 2 0 c 2 T 3 cos 2 (r T ) A(2 r T ) - 1 2 log (cos (2 r T )) ,
where

A(u) = u 3 + 6 u cos(u) + 3 (u 2 -2) sin(u) 6 u 3 . Then IE exp 2c iζ T 0 z s W s ds + r 2 2 T 0 W 2 s ds = cos (r T ) e -ζ 2 2 4 z 2 0 c 2 T 3 cos 2 (r T ) A(2rT

Application to the pricing of options on realized variance

We now consider the pricing of options on realized variance. In this section, we provide an expansion for the price of a put option on realized variance and for its Black and Scholes implied volatility.

Asymptotics for the price of options on realized variance

In this section, we use Theorem 4.4.1 to calculate the asymptotic expansion of the price of a put option on realized variance. Let us first prove two asymptotic results for the expansion of integrals following the methodology of (Bleistein and Handelsman, 1975, Chapter 5, p. 180-219).

f (x) e -φ(x) 2 dx = e -φ(a) 2 f (a) φ (a) 2 + f (a) (φ (a)) 2 - f (a)φ (a) (φ (a)) 3 4 + O 6 , as → 0.
Proof. With a change of variable, we obtain

G(t) = f (a) φ (a) + f (a) φ (a) (t -a) + O((t -a) 2 ) 1 + φ (a) φ (a) (t -a) + O((t -a) 2 ) = f (a) φ (a) + f (a) φ (a) - f (a)φ (a) (φ (a)) 2 (t -a) + O (t -a) 2 . Since τ := φ(t) -φ(a) = φ (a) (t -a) + O (t -a) 2 , then G(φ -1 (φ(a) + τ )) = f (a) φ (a) + f (a) (φ (a)) 2 - f (a)φ (a) (φ (a)) 3 τ + O τ 2 . Substituting G by its development in φ(b)-φ(a) 0 G(φ -1 (φ(a) + τ )) e --2 τ dτ
and integrating using Wilson's Lemma (Bleistein and Handelsman, 1975, Chapter 4, p. 103) yields the result. Lemma 4.5.2 (Laplace's method). Let a 1 < a < a 2 ∈ R. Let also f ∈ C 3 (R) and φ ∈ C 5 (R), such that f (t) = O e λ t for some λ > 0 as t → ∞ and φ is decreasing on (a 1 , a) and increasing on (a, a 2 ) with a global minimum at a such that φ (a) > 0. Then

a 2 a 1 f (x) e -φ(x) 2 dx = e -φ(a) 2 2π φ (a) f (a) + 3 π 2 G(f, φ, a) + O 5
as → 0, where

G(f, φ, a) = f (a) (φ (a)) 3 - f (a)φ (a) (φ (a)) 5 + 5f (a)(φ (a)) 2 12 (φ (a)) 7 - f (a)φ (4) (a)
4 (φ (a)) 5 .

Proof. First write

a 2 a 1 f (x) e --2 φ(x) dx = a a 1 f (x) e --2 φ(x) dx + a 2 a f (x) e --2 φ(x) dx = I 1 ( ) + I 2 ( ) .
With a change of variable, we obtain

I 2 ( ) = e --2 φ(a)
φ(a 2 )-φ(a) 0 G(φ -1 (φ(a) + τ )) e --2 τ dτ and

I 1 ( ) = -e --2 φ(a) φ(a 1 )-φ(a) 0 G(φ -1 (φ(a) + τ )) e --2 τ dτ
where G = f φ . Since φ (a) = 0 and φ (a) = 0, then

G(t) = f (a) + f (a) (t -a) + 1 2 f (a) (t -a) 2 + O((t -a) 3 ) φ (a)(t -a) + 1 2 φ (a) (t -a) 2 + 1 6 φ (4) (a) (t -a) 3 + O((t -a) 4 ) = f (a) + f (a) (t -a) + 1 2 f (a) (t -a) 2 + O((t -a) 3 ) φ (a)(t -a) 1 + φ (a) 2 φ (a) (t -a) + φ (4) (a) 6 φ (a) (t -a) 2 + O((t -a) 3 ) = f (a) φ (a) (t -a) -1 + f (a) φ (a) - f (a)φ (a) 2 (φ (a)) 2 + O (t -a) 2 + f (a) 2φ (a) - f (a)φ (a) 2 (φ (a)) 2 + f (a)(φ (a)) 2 4 (φ (a)) 3 - f (a)φ (4) (a) 6 (φ (a)) 2 (t -a)
Then note that, since

φ(t) = φ(a)+ φ (a) 2 (t-a) 2 1+ φ (a) 3 φ (a) (t-a)+ φ (4) (a) 12 φ (a) (t-a) 2 +O (t -a) 3 ,
denoting τ = φ(t)-φ(a), we have up to a O((t -a) 2 ) the following equalities,

τ 1/2 = φ (a) √ 2 (t -a) + O (t -a) 2 τ -1/2 = √ 2 φ (a) (t -a) -1 - √ 2φ (a) 6 (φ (a)) 3 + √ 2(φ (a)) 2 24 (φ (a)) 5 - √ 2φ (4) (a) 24 (φ (a)) 3 (t -a) + O (t -a) 2
and hence

(t -a) = √ 2 φ (a) τ 1/2 (t -a) -1 = φ (a) √ 2 τ -1/2 + φ (a) 6 φ (a) - √ 2(φ (a)) 2 24 (φ (a)) 5 - √ 2φ (4) (a) 24 (φ (a)) 3 τ 1/2 .
Therefore, By Watson's Lemma (Bleistein and Handelsman, 1975, Chapter 4, p. 103),

G(φ -1 (φ(a) + τ )) = f (a) 2 φ (a) τ -1/2 + f (a) φ (a) - f (a)φ (a) 3 (φ (a)) 2 + √ 2f (
I 2 ( ) = e --2 φ(a) φ(a 2 )-φ(a) 0 G(φ -1 (φ(a) + τ )) e --2 τ dτ = e --2 φ(a) α -1/2 √ π + α 0 2 + α 1/2 √ π 2 3 + O 4 ,
where α k is the term of order k of G(φ -1 (φ(a) + τ )). The same reasoning allows to obtain

I 1 ( ) = e --2 φ(a) α -1/2 √ π -α 0 2 + α 1/2 √ π 2 3 + O 4 .
Combining the two finishes the proof.

We then proceed to the calculation of the expansion of the option price.

Asymptotics for the Black and Scholes implied volatility of options on realized variance

In order to calculate the asymptotic implied volatility, we start by recalling the asymptotic expansion of the price of a put option in the BS model. See [START_REF] Gatheral | Asymptotics of implied volatility in local volatility models[END_REF] for a complete overview of the methods used in Lemma 4.5.4 and Proposition 4.5.5.

Lemma 4.5.4. Denote Φ the Gaussian distribution function. For x > 0,

Φ(-x) = 1 √ 2π e -x 2 /2 1 x - 1 x 3 + O x -5 as x → ∞ . Proof. By symmetry, √ 2π Φ(-x) = ∞ x e -y 2 /2 dy = ∞ x -1 y (e -y 2 /2 ) dy .
Integrating successively by parts, we then obtain

√ 2π Φ(-x) = 1 x e -x 2 /2 - ∞ x 1 y 2 e -y 2 /2 dy = 1 x e -x 2 /2 - ∞ x -1 y 3 (e -y 2 /2 ) dy = 1 x e -x 2 /2 - 1 x 3 e -x 2 /2 dy + ∞ x 3
y 4 e -y 2 /2 dy , where ∞ x 3 y 4 e -y 2 /2 dy = ∞ x -3 y 5 (e -y 2 /2 ) dy ≤ 3

x 5 e -x 2 /2 , hence the result.

Proposition 4.5.5. Let (S t ) 0≤t≤T with dynamics

dS t = σ BS S t dW t , S 0 = S 0
and let P BS = IE ((K -S T ) + ) be the BS price at time 0 of a put option with strike K and maturity T . Then P BS admits the expansion

P BS (S 0 , K, T ; σ BS ) = (K -S 0 ) + +e - log(S 0 /K) 2 2 2 σ 2 BS T √ S 0 K √ 2π
Proof. The Black-Scholes price is given by

P BS (S 0 , K, T ; σ BS ) = K Φ (-d -) -S 0 Φ (-d + ) ,
where

d ± = log(S 0 /K) σ BS T 1/2 ± 1 2 σ BS T 1/2 . If S 0 > K, d ± → →0 ∞, whereas if S 0 < K, d ± → →0 -∞. Hence P BS (S 0 , K, T ; σ BS ) = (K -S 0 ) + -sgn(K -S 0 ) [K Φ (-|d -|) -S 0 Φ (-|d + |)] . Lemma 4.5.4 indicates that S 0 Φ (-|d + |) = S 0 √ 2π e -d 2 + /2 1 |d + | - 1 |d + | 3 + O |d + | -5 KΦ (-|d -|) = K √ 2π e -d 2 -/2 1 |d -| - 1 |d -| 3 + O |d -| -5
, where

e -d 2 ± /2 = e - log(S 0 /K) 2 2 2 σ 2 BS T ∓ log(S 0 /K) 2 -1 8 2 σ 2 BS T = (S 0 /K) ∓1/2 e - log(S 0 /K) 2 2 2 σ 2 BS T (1 + O 2 ) , 1 |d ± | = σ BS T 1/2 | log(S 0 /K)| ∓ sgn(S 0 -K) 3 σ 3 BS T 3/2 2 | log(S 0 /K)| 2 + O 5 and 1 |d ± | 3 = 3 σ 3 BS T 3/2 | log(S 0 /K)| 3 + O 5 . Therefore, -sgn(K -S 0 ) [KΦ (-|d -|) -S 0 Φ (-|d + |)] = e - log(S 0 /K) 2 2 2 σ 2 BS T √ S 0 K √ 2π 3 σ 3 BS T 3/2 | log(S 0 /K)| 2 + O 5 ,
which proves the result.

With the expansion of the price in the BS model, we now proceed to the calculation of the BS implied volatility of the put option on realized variance. We define the Black-Scholes implied volatility of a realized variance option as the volatility value σ BS such that P BS (z 2 0 , K, T, σ BS ) = P (z 0 , K, T ).

Theorem 4.5.6. The BS implied volatility of the put option on the realized variance admits the expansion σ BS = σ BS,0 + 2 σ BS,1 + O ( 2 ) where

σ BS,0 = | log(z 2 0 /K)| √ 2T Λ 1/2 (KT ) (4.5.6) and σ BS,1 = | log(z 2 0 /K)| 2 3/2 T 1/2 Λ 3/2 (KT ) log 4 √ π z 0 K 1/2 T 2 c 0 (KT ) (Λ (KT )) 2 Λ 3/2 (KT ) | log(z 2 0 /K)| , ( 4
.5.7) as → 0.

Proof. By replacing σ BS by its expansion in (4.5.5), we obtain

P BS (S 0 , K, T ; σ BS ) = (K -S 0 ) + + e - log(S 0 /K) 2 2 2 σ 2 BS,0 T √ S 0 K √ 2π 3 σ 3 BS,0 T 3/2 log(S 0 /K) 2 e log(S 0 /K) 2 σ 3 BS,0 T σ BS,1 + O 5 .
Equating P (z 0 , K, T ) to P BS (z 2 0 , K, T ; σ BS,0 + 2 σ BS,1 ) and identifying the terms then yields (4.5.6) and (4.5.7).

Chapter 5

Approximate option pricing in the Lévy Libor model

Introduction

The goal of this paper is to develop explicit approximations for option prices in the Lévy Libor model introduced by Eberlein and Özkan, 2005. In particular, we shall be interested in price approximations for caplets, whose pay-off is a function of only one underlying Libor rate and swaptions, which can be regarded as options on a "basket" of multiple Libor rates of different maturities.

A full-fledged model of Libor rates such as the Lévy Libor model is typically used for the purposes of pricing and risk management of exotic interest rate products. The prices and hedge ratios must be consistent with the market-quoted prices of liquid options, which means that the model must be calibrated to the available prices / implied volatilities of caplets and swaptions. To perform such a calibration efficiently, one therefore needs explicit formulas or fast numerical algorithms for caplet and swaption prices.

Computation of option prices in the Lévy Libor model to arbitrary precision is only possible via Monte Carlo. Efficient simulation algorithms suitable for pricing exotic options have been proposed in [START_REF] Kohatsu-Higa | Jump-adapted discretization schemes for Lévy-driven SDEs[END_REF][START_REF] Papapantoleon | Efficient and accurate log-Lévy approximations to Lévy driven Libor models[END_REF], however, these Monte Carlo algorithms are probably not an option for the purposes of calibration because the computation is still too slow due to the presence of both discretization and statistical error. [START_REF] Eberlein | The Lévy Libor model[END_REF][START_REF] Kluge | Time-Inhomogeneous Lévy Processes in Interest Rate and Credit Risk Models[END_REF][START_REF] Belomestny | A jump-diffusion Libor model and its robust calibration[END_REF] propose fast methods for computing caplet prices which are based on Fourier transform inversion and use the fact that the characteristic function of many parametric Lévy processes is known explicitly. Since in the Lévy Libor model, the Libor rate L k is not a geometric Lévy process under the corresponding probability measure Q T k , unless k = n (see Remark 5.3.1 below for details), using these methods for k < n requires an additional approximation (some random terms appearing in the compensator of the jump measure of L k are approximated by their values at time t = 0, a method known as freezing).

In this paper we take an alternative route and develop approximate formulas for caplets and swaptions using asymptotic expansion techniques. Inspired by methods used in [START_REF] Černỳ | Hedging in Lévy models and the time step equivalent of jumps[END_REF][START_REF] Ménassé | Asymptotic indifference pricing in exponential Lévy models[END_REF][START_REF] Ménassé | Asymptotic indifference pricing in exponential Lévy models[END_REF] (see also [START_REF] Benhamou | Smart expansion and fast calibration for jump diffusions[END_REF][START_REF] Benhamou | Time dependent Heston model[END_REF] for related expansions "around a Black-Scholes proxy" in other models), we consider a given Lévy Libor model as a perturbation of the log-normal LMM. Starting from the driving Lévy process (X t ) t≥0 of the Lévy Libor model, assumed to have zero expectation, we introduce a family of processes X α t = αX t/α 2 parameterized by α ∈ (0, 1], together with the corresponding family of Lévy Libor models. For α = 1 one recovers the original Lévy Libor model. When α → 0, the family X α converges weakly in Skorokhod topology to a Brownian motion, and the option prices in the Lévy Libor model corresponding to the process X α converge to the prices in the log-normal LMM. The option prices in the original Lévy Libor model can then be approximated by their second-order expansions in the parameter α, around the value α = 0. This leads to an asymptotic approximation formula for a derivative price expressed as a linear combination of the derivative price stemming from the LMM and correction terms depending on the characteristics of the driving Lévy process. The terms of this expansion are often much easier to compute than the option prices in the Lévy Libor model. In particular, we shall see the expansion for caplets is expressed in terms of the derivatives of the standard Black's formula, and the various terms of the expansion for swaptions can be approximated using one of the many swaption approximations for the log-normal LMM available in the literature.

This paper is structured as follows. In Section 5.2 we briefly review the Lévy Libor model. In Section 5.3 we show how the prices of European-style options may be expressed as solutions of partial integro-differential equations (PIDE). These PIDEs form the basis of our asymptotic method, presented in detail in Section 5.4. Finally, numerical illustrations are provided in Section 5.5.

Presentation of the model

In this section we present a slight modification of the Lévy Libor model by [START_REF] Eberlein | The Lévy Libor model[END_REF], which is a generalization, based on Lévy processes, of the Libor market model driven by a Brownian motion, introduced by [START_REF] Miltersen | Closed form term structure derivatives in a Heath-Jarrow-Morton model with lognormal annually compounded interest rates[END_REF][START_REF] Brace | The market model of interest rate dynamics[END_REF][START_REF] Miltersen | Closed form solutions for term structure derivatives with log-normal interest rates[END_REF] Let a discrete tenor structure 0 ≤ T 0 < T 1 < . . . < T n be given, and set δ k := T k -T k-1 , for k = 1, . . . , n. We assume that zero-coupon bonds with maturities T k , k = 0, . . . , n, are traded in the market. The time-t price of a bond with maturity T k is denoted by B

t (T k ) with B T k (T k ) = 1.
For every tenor date T k , k = 1, . . . , n, the forward Libor rate L k t at time t ≤ T k-1 for the accrual period [T k-1 , T k ] is a discretely compounded interest rate defined as

L k t := 1 δ k B t (T k-1 ) B t (T k ) -1 . (5.2.1) For all t > T k-1 , we set L k t := L k T k-1 .
To set up the Libor model, one needs to specify the forward Libor rates L k t , k = 1, . . . , n, such that each Libor rate L k is a martingale with respect to the corresponding forward measure Q T k using the bond with maturity T k as numéraire. We recall that the forward measures are interconnected via the Libor rates themselves and hence each Libor rate depends also on some other Libor rates as we shall see below. More precisely, assuming that the forward measure Q Tn for the most distant maturity T n (i.e. with numéraire B(T n )) is given, the link between the forward measure Q T k and Q Tn is provided by

dQ T k dQ Tn Ft = B t (T k ) B t (T n ) B 0 (T n ) B 0 (T k ) = n j=k+1 1 + δ j L j t 1 + δ j L j 0 , (5.2.2)
for every k = 1, . . . , n -1. The forward measure Q Tn is referred to as the terminal forward measure.

The driving process

Let us denote by (Ω, F, F = (F t ) 0≤t≤T * , Q Tn ) a complete stochastic basis and let X be an R d -valued Lévy process (X t ) 0≤t≤T * on this stochastic basis with Lévy measure F and diffusion matrix c. The filtration F is generated by X and Q Tn is the forward measure associated with the date T n , i.e. with the numéraire B t (T n ). The process X is assumed without loss of generality to be driftless under Q Tn . Moreover, we assume that |z|>1 |z|F (dz) < ∞. This implies in addition that X is a special semimartingale and allows to choose the truncation function h(z) = z, for z ∈ R d . The canonical representation of X is given by 5.2.3) where W Tn = (W Tn t ) 0≤t≤Tn denotes a standard d-dimensional Brownian motion with respect to the measure Q Tn , µ is the random measure of jumps of X and ν Tn (ds, dz) = F (dz)ds is the Q Tn -compensator of µ.

X t = √ cW Tn t + t 0 R d z(µ -ν Tn )(ds, dz), (
= F T k t (dz)dt with F T k t (dz) := n j=k+1 1 + δ j L j t 1 + δ j L j t λ j (t), z F (dz).
(5.2.9) Equalities (5.2.7) and ( 5.2.8) and consequently also the drift condition (5.2.5), are implied by Girsanov's theorem for semimartingales applied first to the measure change from Q Tn to Q T n-1 and then proceeding backwards. We refer to [START_REF] Kallsen | A didactic note on affine stochastic volatility models[END_REF] for a version of Girsanov's theorem that can be directly applied in this case. Note that the random terms δ j L j t 1+δ j L j t appear in the measure change due to the fact that for each j = n, n -1, . . . , 1 we have 5.2.10) We point out that the predictable random terms

d(1 + δ j L j t ) = (1 + δ j L j t-) δ j L j t- 1 + δ j L j t- b j (t, L t )dt + δ j L j t- 1 + δ j L j t- λ j (t)dX t , (
δ j L j t- 1+δ j L j t-
can be replaced with

δ j L j t 1+δ j L j t
in equalities (5.2.5), (5.2.7) and ( 5.2.8) due to absolute continuity of the characteristics of X.

Therefore, the vector process of Libor rates L, given in (5.2.4) with the drift (5.2.5), is a time-inhomogeneous Markov process and its infinitesimal generator under Q Tn is given by

A t f (x) = n i=1 x i b i (t, x) ∂f (x) ∂x i + 1 2 n i,j=1 x i x j (Λ(t)cΛ(t) ) ij ∂f (x) ∂x i ∂x j (5.2.11) + R d f (diag(x)(1 + Λ(t)z)) -f (x) - n j=1 x j (Λ(t)z) j ∂f (x) ∂x j F (dz),
for a function f ∈ C 2 0 (R n , R) and with the function b i (t, x), for i = 1, . . . , n and x = (x 1 , . . . , x n ) ∈ R n , given by b

i (t, x) = - n j=i+1 δ j x j 1 + δ j x j λ i (t), c λ j (t) + R d λ i (t), z 1 - n j=k+1 1 + δ j x j λ j (t), z 1 + δ j x j F (dz).
Remark 5.2.1 (Connection to the Lévy Libor model of [START_REF] Eberlein | The Lévy Libor model[END_REF]. The dynamics of the forward Libor rate L k , for all k = 1, . . . , n, in the Lévy Libor model of [START_REF] Eberlein | The Lévy Libor model[END_REF] (compare also [START_REF] Eberlein | Calibration of Lévy term structure models[END_REF] is given as an ordinary exponential of the following form 5.2.12) for some deterministic volatility vector λk and the drift bk (t, L t ) which has to be chosen such that the Libor rate L k is a martingale under the forward measure Q T k . Here Ỹ is a d-dimensional Lévy process given by

L k t = L k 0 exp t 0 bk (s, L s )ds + t 0 λk (s)d Ỹs , (
Ỹt = √ cW Tn t + t 0 R d z(μ -νTn )(ds, dz),
with the Q Tn -characteristics (0, c, F ), where νTn (ds, dz) = F (dz)ds. The Lévy measure F has to satisfy the usual integrability conditions ensuring the finiteness of the exponential moments. The dynamics of L k is thus given by the following SDE

dL k t = L k t-b k (t, L t )dt + √ c λk (t)dW Tn t + (e λk (t),z -1)(μ -νTn )(dt, dz) = L k t-b k (t, L t )dt + dY k t ,
for all k, where Y k is a time-inhomogeneous Lévy process given by

Y k t = t 0 √ c λk (s)dW Tn s + t 0 R d (e λk (s),z -1)(μ -νTn )(ds, dz) and the drift b k (t, L t ) is given by b k (t, L t ) = bk (t, L t ) + 1 2 λk (t), c λk (t) + R d
(e λk (t),z -1 -λk (t), z ) F (dz).

Option pricing via PIDEs

Below we present the pricing PIDEs related to general option payoffs and then more specifically to caplets and swaptions. We price all options under the given terminal measure Q Tn .

General payoff

Consider a European-type payoff with maturity T k given by ξ = g(L T k ), for some tenor date T k . Its time-t price P t is given by the following risk-neutral pricing formula

P t = B t (T k )IE Q T k [g(L T k ) | F t ] = B t (T n )IE Q Tn B T k (T k ) B T k (T n ) g(L T k ) | F t = B t (T n )IE Q Tn n j=k+1 (1 + δ j L j T k )g(L T k ) | F t CHAPTER 5. APPROXIMATE OPTION PRICING IN THE L ÉVY LIBOR MODEL 5
We now consider a family of Lévy Libor models driven by the processes X α , α ∈ (0, 1), and defined by 5.4.1) where the drift b α is given by (5.2.5) with F replaced by F α . Substituting the explicit form of F α , we obtain

dL α t = L α t-(b α (t, L α t )dt + Λ(t)dX α t ), (
b k α (t, L t ) = - n j=k+1 δ j L j t 1 + δ j L j t λ k (t), c λ j (t) + 1 α R d λ k (t), z 1 - n j=k+1 1 + αδ j L j t λ j (t), z 1 + δ j L j t F (dz) = - n j 0 =k+1 Σ kj 0 (t) δ j 0 L j 0 t 1 + δ j 0 L j 0 t - n-k-1 p=1 α p n j 0 =k+1 n j 1 =j 0 +1
. . . λ p (t), z F (dz) (5.4.3) for all k = 1, . . . , n. We denote the infinitesimal generator of L α by A α t . For a smooth function f : R d → R, the infinitesimal generator A α t f can be expanded in powers of α as follows:

A α t f (x) = n i=1 b i α (t, x)x i ∂f (x) ∂x i + 1 2 n i,j=1 Σ ij (t)x i x j ∂ 2 f (x) ∂x i ∂x j + ∞ k=3 n i 1 ,...,i k =1 α k-2 k! x i 1 . . . x i k ∂ k f (x) ∂x i 1 . . . ∂x i k M k t (λ i 1 , . . . , λ i k ).
Consider now a financial product whose price is given by a generic PIDE of the form (5.3.1) with A t replaced by A α t . Assuming sufficient regularity2 , one

We denote by Y t,y the process starting from y at time t, and by Y t,y,i the i-th component of this process. Let f be a bounded measurable function and define v(t, y) = E[f (Y t,y T )]. Then, for all i 1 , . . . , i m , the process Y t,y,i 1 s . . . Y t,y,im s ∂ m v(s, Y t,y s ) ∂y i 1 . . . ∂y im , s ≥ t, is a martingale.

The proof can be carried out by direct differentiation for smooth f together with a standard approximation argument for a general measurable f . Furthermore, we assume the following simplification for the drift terms:

For all i = 1, . . . , n -1 and p = 1, . . . , n -k -1, the random quantities in the terms b i p (t, L t ) in the expansion of the drift of the Libor rates under the terminal measure are constant and equal to their value at time t, i.e. for all j = 1, . . . , n:

δ j L j s 1 + δ j L j s = δ j L j t 1 + δ j L j t
, for all s ≥ t. (5.4.9) This simplification is known as freezing of the drift and is often used for pricing in the Libor market models.

Coming back now to the first-order term u 1 , we see that it is the solution of ∂ t u 1 + A 0 t u 1 + A 1 t u 0 = 0, u 1 (T k , x) = 0 (5.4.10) with

A 1 t u 0 (t, x) = n j=1
b j 1 (t, x)x j ∂u 0 (t, x) ∂x j (5.4.11)

+ 1 6 n i 1 ,i 2 ,i 3 =1 x i 1 x i 2 x i 3 ∂ 3 u 0 (t, x) ∂x i 1 ∂x i 2 ∂x i 3 M 3 t (λ i 1 , λ i 2 , λ i 3 )
and the drift term M 3 t (λ j , λ j 0 , λ j 1 ) δ j 0 x j 0 1 + δ j 0 x j 0 δ j 1 x j 1 1 + δ j 1 x j 1 .

(5.4.12)

Moreover,

A 0 t u 1 (t, x) = n i=1 b i 0 (t, x)x i ∂u 1 (t, x) ∂x i + 1 2 n i,j=1
Σ ij (t)x i x j ∂ 2 u 1 (t, x) ∂x i ∂x j .

We have Lemma 5.4.2. Consider the model (5.4.1). Under the simplification (5.4.9), the first-order term u 1 (t, x) in the expansion (5.4.4) can be approximated by u 1 (t, x) ≈ 1 6 n i 1 ,i 2 ,i 3 =1

x i 1 x i 2 x i 3 ∂ 3 u 0 (t, x) ∂x i 1 ∂x i 2 ∂x i 3

T k t M 3 s (λ i 1 , λ i 2 , λ i 3 )ds - n j=1 n j 0 =j+1 n j 1 =j 0 +1
δ j 0 x j 0 1 + δ j 0 x j 0 δ j 1 x j 1 1 + δ j 1 x j 1

x j ∂u 0 (t, x) ∂x j T k t M 3 s (λ j , λ j 0 , λ j 1 )ds =: ũ1 (t, x).

(5.4.13)

Proof. Applying the Feynman-Kac formula to (5.4.10), we have,

u 1 (t, x) = 1 6 T k t ds n i 1 ,i 2 ,i 3 =1 M 3 s (λ i 1 , λ i 2 , λ i 3 ) IE Q Tn X t,x,i 1 s X t,x,i 2 s X t,x,i 3 s ∂ 3 u 0 (s, X t,x s ) ∂x i 1 ∂x i 2 ∂x i 3 + T k t ds n j=1 E Q Tn b j
1 (s, X t,x s )X t,x,j s ∂u 0 (s, X t,x s ) ∂x j , (5.4.14) with the process (X t,x s ) defined by (5.4.8). Under the simplification (5.4.9), we can apply Proposition 5.4.1 to obtain (5.4.13).

Similarly, the second-order term u 2 is the solution of ∂ t u 2 + A 0 t u 2 + A 1 t u 1 + A 2 t u 0 = 0, u 2 (T k , x) = 0 (5.4.15) with

A 2 t u 0 (t, x) = n j=1
b j 2 (t, x)x j ∂u 0 (t, x) ∂x j (5.4.16) + 1 24 n i 1 ,i 2 ,i 3 ,i 4 =1

x i 1 x i 2 x i 3 x i 4 ∂ 4 u 0 (t, x) ∂x i 1 ∂x i 2 ∂x i 3 x i 4 M 4 t (λ i 1 , λ i 2 , λ i 3 , λ i 4 ) and the drift

b j 2 (t, x) = - n j 0 =j+1 n j 1 =j 0 +1 n j 2 =j 1 +1
M 4 t (λ j , λ j 0 , λ j 1 , λ j 2 ) δ j 0 x j 0 1 + δ j 0 x j 0 • δ j 1 x j 1 1 + δ j 1 x j 1 δ j 2 x j 2 1 + δ j 2 x j 2 .

(5.4.17)

Lemma 5.4.3. Consider the model (5.4.1). Under the simplification (5.4.9), the second-order term u 2 (t, x) in the expansion (5.4.4) can be approximated by u 2 (t, x) ≈ ũ2 (t, x) := Ẽ1 + Ẽ2 + Ẽ3 + Ẽ4 , (5.4.18) with Ẽ1 := 1 6 n i 1 ,i 2 ,i 3 =1

x i 1 x i 2 x i 3

T k t dsM 3 s (λ i 1 , λ i 2 , λ i 3 ) • 1 6 n i 4 ,i 5 ,i 6 =1
T k s M 3 v (λ i 4 , λ i 5 , λ i 6 )dv δ j 0 x j 0 1 + δ j 0 x j 0 δ j 1 x j 1 1 + δ j 1 x j 1

∂ 3 v i 4 ,i
x j T k t dsM s (λ j , λ j 0 , λ j 1 )

• 1 6 n i 4 ,i 5 ,i 6 =1
T k s M 3 v (λ i 4 , λ i 5 , λ i 6 )dv ∂v i 4 ,i 5 ,i 6 (t, x) ∂x j (5.4.20)

- n j 4 =1 n j 5 =j 4 +1 n j 6 =j 5 +1
T k s M 3 v (λ j 4 , λ j 5 , λ j 6 )dv ∂ 3 vj 4 ,j 5 ,j 6 (t, x) ∂x j Ẽ3 := 1 24 n i 1 ,i 2 ,i 3 ,i 4 =1

x i 1 x i 2 x i 3 x i 4 ∂ 4 u 0 (t, x) ∂x i 1 ∂x i 2 ∂x i 3 ∂x i 4 δ j 0 x j 0 1 + δ j 0 x j 0 δ j 1 x j 1 1 + δ j 1 x j 1 δ j 2 x j 2 1 + δ j 2 x j 2

x j ∂u 0 (t, x) ∂x j

•

T k t M 4 s (λ j , λ j 0 , λ j 1 , λ j 2 )ds (5.4.22) where we define v i,j,l (t, x) := x i x j x l ∂ 3 u 0 (t, x) ∂x i ∂x j ∂x l (5.4.23) for all i, j, l = 1, . . . , n and vi,j,l (t, x) := x i δ j x j 1 + δ j x j δ l x l 1 + δ l x l ∂u 0 (t, x) ∂x i (5.4.24) for all i = 1, . . . , n, j = i + 1, . . . , n and l = j + 1, . . . , n.

Proof. Once again by the Feynman-Kac formula applied to (5.4.15) we have u 2 (t, x) = 1 6

T k t ds n i 1 ,i 2 ,i 3 =1
M 3 s (λ i 1 , λ i 2 , λ i 3 )

• E Q Tn X t,x,i 1 s X t,x,i 2 s X t,x,i 3 s ∂ 3 u 1 (s, X t,x s ) ∂x i 1 ∂x i 2 ∂x i 3

+ T k t ds n j=1 E Q Tn b j
1 (s, X t,x s )X t,x,j s ∂u 1 (s, X t,x s ) ∂x j + 1 24

T k t ds n i 1 ,i 2 ,i 3 ,i 4 =1
M 4 s (λ i 1 , λ i 2 , λ i 3 , λ i 4 ) (5.4.25)

• E Q Tn X t,x,i 1 s X t,x,i 2 s X t,x,i 3 s X t,x,i 4 s ∂ 4 u 0 (s, X t,x s ) ∂x i 1 ∂x i 2 ∂x i 3 ∂x i 4

+ T k t ds n j=1 E Q Tn b j
2 (s, X t,x s )X t,x,j s ∂u 0 (s, X t,x s ) ∂x j

=:

E 1 + E 2 + E 3 + E 4
with the process (X t,x s ) given by (5.4.8), b j 1 (s, x) by ( 5.4.12) and b j 2 (s, x) by (5.4.17).

In order to obtain an explicit expression for u 2 (t, x), we apply Proposition 5.4.1 combined with the simplification (5.4.9) for the drift terms b j 1 and b j 2 above. More precisely, the expressions for the third and the fourth expectation, which are present in the terms E 3 and E 4 , follow by a straightforward application of Proposition 5.4.1 after using the simplification for b j 2 . We get

E 3 ≈ Ẽ3 and E 4 ≈ Ẽ4
with Ẽ3 and Ẽ4 given by (5.4.21) and (5.4.22), respectively.

To obtain explicit expressions for E 1 and E 2 , firstly we insert the expression for u 1 (s, X t,x s ) as given by (5.4.14). After some straightforward calculations, based again on the application of Proposition 5.4.1 and the simplification (5.4.9) for b j 1 , which yields

E 1 ≈ Ẽ1 and E 2 ≈ Ẽ2
with Ẽ1 and Ẽ2 given by (5.4.19) and (5.4.20), respectively. Collecting the terms above concludes the proof.

Summarizing, we get the following expansion for the time-t price P α (t; g) of the payoff g(L T k ) when α → 0.

we see that the zero-order term is given by u 0 (t, x) = P BS (V Cpl t,T k-1 , x k , K) n j=k+1

(1 + δ j x j ), (5.4.27) where V Cpl t,T := T t Σ kk (s)ds. (5.4.28) Now, in complete analogy to the case of a general payoff, the first-order term u 1 (t, x) and the second-order term u 2 (t, x) are given by (5.4.14) and (5.4.25), respectively, with u 0 (t, x) as in (5.4.27). Noting that u 0 (t, x) depends only on x k , x k+1 , . . . , x n , the derivatives of u 0 (t, x) with respect to x 1 , . . . , x k-1 are zero and the sums in (5.4.14) and (5.4.25) in fact start from the index k. An application of Proposition 5.4.1 and simplification (5.4.9) thus yields the following proposition, which provides an approximation of the caplet price P Cpl,α (t; T k-1 , T k , K) when α → 0. Proposition 5.4.5. Consider the model (5.4.1) and a caplet with strike K and maturity T k-1 . Assuming (5.4.9), its time-t price P Cpl,α (t; T k-1 , T k , K) for α → 0 satisfies P Cpl,α (t; T k-1 , T k , K) = P Cpl 0 (t; T k-1 , T k , K) + αP Cpl 1 (t; T k-1 , T k , K) (5.4.29) + α 2 P Cpl 2 (t; T k-1 , T k , K) + O(α 3 ), with

P Cpl 0 (t; T k-1 , T k , K) := B t (T n )δ k u 0 (t, L t ) = B t (T n )δ k P BS (V Cpl t,T k-1 , L k t , K) n j=k+1
(1 + δ j L j t )

P Cpl 1 (t; T k-1 , T k , K)

:= B t (T n )δ k 1 6 n i 1 ,i 2 ,i 3 =k L i 1 t L i 2 t L i 3 t ∂ 3 u 0 (t, x) ∂x i 1 ∂x i 2 ∂x i 3 x=Lt T k-1 t M 3 s (λ i 1 , λ i 2 , λ i 3 )ds - n j=k n j 0 =j+1 n j 1 =j 0 +1 δ j 0 L j 0 t 1 + δ j 0 L j 0 t δ j 1 L j 1 t 1 + δ j 1 L j 1 t L j t ∂u 0 (t, x) ∂x j x=Lt • T k-1 t
M 3 s (λ j , λ j 0 , λ j 1 )ds M 4 s (λ j , λ j 0 , λ j 1 , λ j )ds with V Cpl t,T k-1 given by (5.4.28), u 0 (t, x) by (5.4.27), the terms M 3 s (•) and M 4 s (•) by (5.4.3) and v i 4 ,i 5 ,i 6 (t, x) and vj 4 ,j 5 ,j 6 (t, x) by (5.4.23) and (5.4.24), respectively.

P Cpl 2 (t; T k-1 , T k , K) := B t (T n )δ k 1 6 n i 1 ,i 2 ,i 3 =k L i 1 t L i 2 t L i 3 t T k-1 t dsM 3 s (λ i 1 , λ i 2 , λ i 3 ) • 1 6 n i 4 ,i 5 ,i 6 =k T k-1 s M 3 v (λ i 4 , λ i 5 , λ i 6 )dv ∂ 3 v i 4 ,i
Remark 5.4.6. Recalling that u 0 (t, x) = P BS (V Cpl t,T , x k , K) n j=k+1

(1 + δ j x j )

we see that the functions v and v given by v i,j,l (t, x) := x i x j x l ∂ 3 u 0 (t, x) ∂x i ∂x j ∂x l for all i, j, l = k, . . . , n and vi,j,l (t, x) := x i δ j x j 1 + δ j x j δ l x l 1 + δ l x l ∂u 0 (t, x) ∂x i for all i = k, . . . , n, j = i + 1, . . . , n and l = j + 1, . . . , n, become in fact linear combinations of the terms which are polynomials in x multiplied by derivatives of P BS (•) up to order three.

Approximate pricing of swaptions

Let us consider a swaption defined in Section 5.3.3. For swaption pricing we again use the general result under the terminal measure Q Tn given in Proposition 5.4.4. The price of the swaption P Swn (t; T 0 , T n , K) then satisfies P Swn (t; T 0 , T n , K) = B t (T n )(u 0 (t, L t ) + αu 1 (t, L t ) + α 2 u 2 (t, L t )) + O(α 3 ) =: P Swn 0 (t; T 0 , T n , K) + αP Swn 1 (t; T 0 , T n , K)

+ α 2 P Swn 2 (t; T 0 , T n , K) + O(α 3 ),
where the function u 0 satisfies the equation

∂ t u 0 + A 0 t u 0 = 0, u 0 (T 0 , x) = g(x)
with g(x) = δ n f n (x) -1 n j=1 f j (x)x j -K + . We see that the zero-order term P Swn 0 (t; T 0 , T n , K) corresponds to the price of the swaption in the lognormal LMM model with volatility matrix Σ(t).

The function u 0 related to the swaption price in the log-normal LMM is of course not known in explicit form but one can use various approximations developed in the literature [START_REF] Jäckel | The link between caplet and swaption volatilities in a Brace-Gatarek-Musiela/Jamshidian framework: approximate solutions and empirical evidence[END_REF][START_REF] Schoenmakers | Robust Libor Modelling and Pricing of Derivative Products[END_REF]. To introduce the approximation of [START_REF] Jäckel | The link between caplet and swaption volatilities in a Brace-Gatarek-Musiela/Jamshidian framework: approximate solutions and empirical evidence[END_REF], we compute the quadratic variation of the log swap rate expressed as function of Libor rates: R(t; T 0 , T n ) = R(L 1 t , . . . , L n t ) =

n j=1 δ j L j t j k=1 (1 + δ k L k t ) n j=1 δ j j k=1 (1 + δ k L k t ) . log R(•; T 0 , T n ) T = T 0 d R(•; T 0 , T n ) t R(t; T 0 , T n ) 2 = T 0 n i,j=1 ∂R(L t ) ∂L i ∂R(L t ) ∂L j d L i , L j t R(t; T 0 , T n ) 2 = T 0 n i,j=1 ∂R(L t ) ∂L i ∂R(L t ) ∂L j L i t L j t Σ ij (t)dt R(t; T 0 , T n ) 2 .
The approximation of [START_REF] Jäckel | The link between caplet and swaption volatilities in a Brace-Gatarek-Musiela/Jamshidian framework: approximate solutions and empirical evidence[END_REF] consists in replacing all stochastic processes in the above integral by their values at time 0; in other words, the swap rate becomes a log-normal random variable such that log R(t; T 0 , T n ) has variance

V swap T = n i,j=1 ∂R(L 0 ) ∂L i ∂R(L 0 ) ∂L j L i 0 L j 0 R(0; T 0 , T n ) 2 T 0 Σ ij (t)dt.
The function u 0 (0, x) can then be approximated by applying the Black-Scholes formula (for simplicity, t = 0): u 0 (0, x) ≈ P BS (V swap T , R(0; T 0 , T n ), K) .

Numerical examples

In this section, we test the performance of our approximation at pricing caplets on Libor rates in the model (5.2.4), where X t is a unidimensional CGMY process (Carr et al., 2007). The CGMY process is a pure jump process, so that c = 0, with Lévy measure

F (dz) = C |z| 1+Y e -λ -z 1 {z<0} + e -λ + z 1 {z>0} dz .
The jumps of this process are not bounded from below but the parameters we choose ensure that the probability of having a negative Libor rate value is negligible. We choose the time grid T 0 = 5, T 1 = 6, ... T 5 = 10, the volatility parameters λ i = 1, i = 1, ..., 5, the initial forward Libor rates L i 0 = 0.06, i = 1, ..., 5 and the bond price for the first maturity B 0 (T 0 ) = 1.06 -5 . The CGMY model parameters are chosen according to four different cases described in the following table, which also gives the standard deviation and excess kurtosis of X 1 for each case. Case 1 corresponds to a Lévy process that is close to the Brownian motion (Y close to 2 and λ + and λ -large) and Case 4 is a Lévy process that is very far from Brownian motion. We first calculate the price of the ATM caplet with maturity T 1 written on the Libor rate L 1 with the zero-order, first-order and second-order approximation, using as benchmark the jump-adapted Euler scheme of Kohatsu-Higa and Tankov, 2010. The first Libor rate is chosen to maximize the nonlinear effects related to the drift of the Libor rates, since the first maturity is the farthest from the terminal date. The results are shown in Table 5.1. We see that for all four cases, the price computed by second-order approximation is within or at the boundary of the Monte Carlo confidence interval, which is itself quite narrow (computed with 10 6 trajectories).

Case

Secondly, we evaluate the prices of caplets with strikes ranging from 3% to 9% and explore the performance of our analytic approximation for estimating the caplet implied volatility smile. The results are shown in Figure 5.5.1. We see that in Cases 1, 2 and 3, which correspond to the parameter values most relevant in practice given the value of the excess kurtosis, the second order approximation reproduces the volatility smile quite well (in case 1 there is actually no smile, see the scale on the Y axis of the graph). In case 4, which corresponds to very violent jumps and pronounced smile, the qualitative shape of the smile is correctly reproduced, but the actual values are often outside the Monte Carlo interval. This means that in this extreme case the model is too far from the Gaussian LMM for our approximation to be precise. We also note that the algorithm runs in O(n 6 ), for the second order approximation, due to the number of partial derivatives that one has to calculate. The algorithm may therefore run slowly, should n become too large.

CHAPTER

Figure 1

 1 Figure 1.1.1: The implied volatility smile of call options on UBS stocks on the 31st of May 2018, with maturity 1 year.

1Figure 1

 1 Figure 1.1.2: The daily log-returns of the Intel Corporation stock between June 2013 and May 2018. The standard deviation of the dataset is σ = 6.35 • 10 -3 .

Figure 1

 1 Figure 1.1.3: The daily data of the S&P 500 and the VIX index between 1995 and 2004.

  Under dP θ dP := e θ Y T I E e θ Y T

  Lemma 2.2.6. (Keller-Ressel, 2011, Lemma 2.2.) (a) F and R are proper closed convex functions on R 2 . (b) F and R are analytic in the interior of their effective domain. (c) Let U be a one-dimensional affine subspace of R 2 . Then F |U is either a strictly convex or an affine function. The same holds for R|U . (d) If (u, w) ∈ D F , then also (u, η) ∈ D F for all η ≤ w. The same holds for R.

,

  A flexible class of measure changes introduced in[START_REF] Genin | Optimal importance sampling for Lévy processes[END_REF] is given by path dependent Esscher transform, that is the class P θ such that dP where θ belong to M , the set of signed measures on [0, T ]. Denoting H(X) = log P S 0 e X , the optimization problem writes

  Proposition 2.6.3. Let τ = {t 1 , ..., t n } and P θ the measure given by dP θ dP = e n j=1 θ j Xt j IE e n j=1 θ j Xt j .

Figure 2

 2 Figure 2.6.1: The variance of the Monte-Carlo estimator as a function of θ.

  4: The variance ratio as function of the strike for the Asian put option. λ = 1.15, µ = 0.04, ζ = 0.2, ρ = -0.4, S 0 = 1, V 0 = 0.04, T = 1.5, N = 10000, 200 discretization steps.

Figure 2

 2 Figure 2.6.2: The variance of the Monte-Carlo estimator as a function of θ for the Heston model with jumps.

  x) ≤ lim inf →0 log P(X ∈ A) ≤ lim sup →0 log P(X ∈ A) ≤ -inf x∈ ĀΛ * (x) . Definition 3.3.3. Let f : R n → R∪{+∞} be a convex function with domain D := {x ∈ R n : f (x) < ∞}. f is called essentially smooth if f is differentiable on • D = ∅ and for every x ∈ D\ • D, lim y→x ||∇f (y)|| = +∞.Theorem 3.3.4 (Gärtner-Ellis). Let (X ) ∈(0,1] be a family of random vectors in R n . Assume that for each λ ∈ R n , extended real number. Assume also that 0 belongs to the interior ofD Λ := {λ ∈ R n : Λ(λ) < ∞}. Denoting Λ * (x) = sup λ∈R n λ, x -Λ(λ) ,the Fenchel-Legendre transform of Λ, the following hold.(a) For any closed set F , lim sup→0 log P(X ∈ F ) ≤ -inf x∈F Λ * (x) .CHAPTER 3. LARGE DEVIATIONS FOR THE WISHART STOCHASTIC VOLATILITY MODEL 3 (b) For any open set G,

t ≥ 0

 0 by the existence of the Laplace transform. Since b-1 ∈ S +, * n and (t E + Ẽ) ∈ S + n , b-1 + (t E + Ẽ) ∈ S +, * n and is therefore invertible. Hence b-1 +

  Theorem 3.3.11. The family (Y T ) ∈(0,1] satisfies a large deviation principle, when → 0 with good rate functionΛ * (y) = sup λ∈R n λ, y -Λ(λ) .Proof. First note that φ(0) = b 2 ∈ S +, * n . But since θ → φ(θ) := b 2 + a Diag(θ) -θθ a is a continuous function, there exists a neighbourhood B(0, δ) of 0 such that φ(θ) ∈ S +, * n for every θ ∈ B(0, δ), hence 0 ∈ • U. Furthermore, Proposition 3.3.8 together with the argument in Remark 3.3.10 prove that Λ(θ) = lim →0 log IE e -1 θ Y T

  log IE (e yT -e B T ) + = lim T →∞ T -1 log e yT -1 + IE (e B T -e yT ) + = y -inf z∈(-∞,y] n Λ * (z) < y; (3.4.7) if y > β * , then lim T →∞ T -1 log IE (e B T -e yT ) + = max i,j=1,...,n inf λ∈R {-λy + Λ(λe i + e j )} < 0. (3.4.8) 3.4. ASYMPTOTIC IMPLIED VOLATILITY OF BASKET OPTIONS 3 and if y ∈ (β * , β * ), then lim T →∞ T -1 log 1 -IE[(e B T -e yT ) + ] = y + max i=1,...,n inf λ∈R {-λy + Λ(λe i )}} < min(0, y) . (3.4.9) Proof. Proof of (3.4.6). We remark that IE (e B T -e yT ) + = IE e B T 1 BT >y -e yT P BT > y (3.4.10) and consider the two terms separately. If y < 0, the second term clearly converges to zero. Assume then that y ≥ 0. Since β * ≤ 0, by Lemma 3.4.1 part 2, lim T →∞ T -1 log e yT P BT > y < 0This proves that the second term in (3.4.10) converges to zero. We now focus on the first term, which satisfiesIE e B T 1 BT >y = n i=1 ω i IE e Y i T 1 BT >y .Fix some i ∈ {1, . . . , n}. Then, by Lemma 3.4.1 parts 3 and 4, if y > x * i then lim T →∞ IE e Y i T 1 BT >y = 0, and if y < x * i then lim T →∞ IE e Y i T 1 BT ≤y = 0. Combining these estimates for different i, the proof of (3.4.6) is complete. Proof of (3.4.7) The equality e y T (1 -e -δT )1 { BT <y-δ} ≤ e y T -e B T + ≤ e y T 1 { BT <y}

1

  -IE[(e B T -e yT ) + ] = IE[e B T -(e B T -e yT ) + ] = e yT P[ BT > y] + IE[e B T 1 BT ≤y ]. By Lemma 3.4.1, part 2, lim T →∞ T -1 log e yT P[ BT > y] = y + max i=1,...,n inf λ∈R {-λy + Λ(λe i )} < 0.

  + Λ(λe i )} < y. On the other hand, by Lemma 3.4.1, part 3, lim T →∞ T -1 log IE[e B T 1 BT ≤y ] = y + max j=1,...,n inf λ j ≤1,λ i ≤0,i =j {Λ(λ) -y λ, 1 } ≤ y + max j=1,...,n inf λ≤1 f j (λ).

3. 4 .

 4 ASYMPTOTIC IMPLIED VOLATILITY OF BASKET OPTIONS 3 Under the Wishart model, for the basket option, we can write: lim T →∞ T -1 log IE (e yT -e B T ) + = -L(y), y ≤ β * (3.4.11) lim T →∞ T -1 log IE (e B T -e yT ) + = -L(y), y ≥ β * lim T →∞ T -1 log 1 -IE (e B T -e yT ) + = -L(y), β * < y < β * ,

5 )

 5 and (3.5.4) finishes the proof.Remark 3.5.2. In particular, if H is continuous and bounded from above and θ is such that φ(-θ) ∈ S +, * n , condition (3.5.2) is met.Definition 3.5.3. A parameter θ * ∈ R n is called asymptotically optimal if it achieves the infimum in the minimisation probleminf θ∈R n sup y∈R n 2H(y) -θ y -Λ * (y) + Λ(θ) . (3.5.Theorem 3.5.4. Let H be a concave upper semi-continuous function. Then inf θ∈R n sup y∈R n 2H(y) -θ y -Λ * (y) + Λ(θ)

  Figure 3.6.1: Basket implied volatility smile in the two-dimensional Wishart model. The upper and lower bounds correspond to the 95% confidence interval.

Figure 3

 3 Figure 3.6.2: Convergence of the renormalized implied volatility smile to the theoretical limit in the Wishart model.

  ∞), (4.3.10) has no solution in 0, π 2T . Proof. Denote u = 2rT , k = z 2 0 T a . Then (4.3.10) becomes f (u) u = 0 , CHAPTER 4. AN ASYMPTOTIC APPROACH FOR THE PRICING OF OPTIONS ON REALIZED VARIANCE 4 where f (u) = (1 + cos(u)) u -k (u + sin(u)) .

  Theorem 4.3.9. {(0, z 0 )} × N a satisfies Condition (ND).

  Lemma 4.3.10. Let a ∈ (0, ∞) and let r = r(a) be the unique solution of eq. (4.3.11) in I. Then

  as a smoothly bounded version ofF (y) = λ(y -a) 2 -Λ (a)(y -a) + Λ (a) 2 (y -a) 2 = λ -Λ (a) 2 (y -a) 2 -Λ (a) (y -a) .

F

  (y) = (2λ -Λ (a)) (y -a) -Λ (a) F (y) = 2λ -Λ (a) . The function F (•) + Λ(•) then admits a non-degenerate minimum at a. Let h ∈ K min a be the minimising control. Applying Girsanov's theorem to the martingale -1 t 0 • h s dW s , we find

  and (4.4.3) in (4.4.1) and setting λ = Λ (a)/2 > 0, we obtainf Y T (a) = -1 e -Λ(a)/ 2 (c 0 (a) + o(1))

  (T -s)) ds = -1 2 log (cos (r (T -t))) .

  ) dx , which concludes the calculation of c 0 . CHAPTER 4. AN ASYMPTOTIC APPROACH FOR THE PRICING OF OPTIONS ON REALIZED VARIANCE 4

  Lemma 4.5.1. Let a < b ∈ R. Let also f ∈ C 2 (R) and φ ∈ C 3 (R), such that f (t) = O e λt for some λ as t → ∞ and φ is increasing on (a, b) and φ (a) = 0. Then b a

  e --2 φ(x) dx = e --2 φ(a) φ(b)-φ(a) 0 G(φ -1 (φ(a) + τ )) e --2 τ dτ , where G = f φ . Since φ (a) = 0,

Σ

  ij (t) := (Λ(t)cΛ(t) ) ij + R d λ i (t), z λ j (t), z F (dz), (5.4.2)for all i, j = 1, . . . , n, andM k t (λ 1 , . . . , λ k ) := R d k p=1

Figure 5 .

 5 Figure 5.5.1: Implied volatilities of caplets with different strikes computed using the analytic approximation together with the Monte Carlo bound. Top graphs: Case 1 (left) and Case 2 (right). Bottom graphs: Case 3 (left) and Case 4 (right).

  ).

	CHAPTER 1. INTRODUCTION	1
	Theorem 1.3.7. The process Y t in (1.3.4) admits a smooth density f Y T and,
	for every

  we obtain the result.

	CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE
	STOCHASTIC VOLATILITY MODELS	2

Table 2

 2 Table2.3: The variance ratio as function of the strike for the European put option with maturity T = 3.

	T	Price Std. error Var. ratio Adj. ratio Time, s
	0.25 0.0395 3.72 •10 -4	2.46	2.14	20.2
	0.5 0.0550 4.54 •10 -4	3.12	2.83	19.9
	1	0.0780 5.59 •10 -4	3.92	3.66	19.5
	2	0.111 7.20 •10 -4	4.21	3.89	19.7
	3	0.134 8.48 •10 -4	4.19	3.79	19.8
	Table 2.1: The variance ratio as function of the maturity for at-the-money
	European put options.
	K	Price	Std. error Var. ratio Adj. ratio Time, s
	0.5 0.00014 7.65 •10 -6	26.6	24.5	18.4
	0.75 0.00794 1.34 •10 -4	6.53	5.91	18.7
	1	0.0773 5.60 •10 -4	3.96	3.65	18.5
	1.25 0.261 8.62 •10 -4	4.20	3.78	18.9
	1.5	0.502 7.92 •10 -4	5.84	5.36	18.6
	1.75 0.749 6.84 •10 -4	8.45	7.29	19.7

.2: The variance ratio as function of the strike for the European put option with maturity T = 1.

Table 2 .

 2

		Price	Std. error Var. ratio Adj. ratio Time, s
	0.6 3.466 •10 -5 4.13 •10 -6	16.9	14.6	19.9
	0.7	0.000562	2.60 •10 -5	5.77	4.77	21.1
	0.8	0.00414	9.64 •10 -5	4.36	3.77	20.1
	0.9	0.0185	0.00024	3.48	3.09	20.6
	1	0.0558	0.00043	3.49	3.07	20.1
	1.1	0.120	0.00057	3.69	3.20	20.1
	1.2	0.206	0.00062	4.27	3.80	19.7
	1.3	0.301	0.00059	5.30	4.41	21.0

Table 2 .

 2 Table2.5: The variance ratio as function of the maturity for the European put option on the Heston model with jumps.Table2.6: The variance ratio as function of the strike for the European put option with maturity T = 1 in the Heston model with jumps. 7: The variance ratio as function of the strike for the European put option with maturity T = 3 in the Heston model with jumps.

	CHAPTER 2. PATHWISE LARGE DEVIATIONS FOR AFFINE
	STOCHASTIC VOLATILITY MODELS		2
	T	Price Std. error Var. ratio Adj. ratio Time, s
	0.25 0.0945 9.96 •10 -4	3.28	3.00	23.6
	0.5 0.147 1.28 •10 -3	3.20	2.99	24.5
	1	0.215 1.61 •10 -3	2.95	2.77	24.7
	2	0.309 2.04 •10 -3	2.61	2.43	24.7
	3	0.374 2.30 •10 -3	2.40	2.20	25.0
	K	Price	Std. error Var. ratio Adj. ratio Time, s
	0.25 0.00606 7.83 •10 -5	11.6	10.4	25.8
	0.5 0.0377 4.03 •10 -4	5.42	5.28	24.7
	0.75 0.105 9.44 •10 -4	3.76	3.19	27.3
	1	0.215 1.61 •10 -3	2.93	2.89	26.1
	1.25 0.369 2.26 •10 -3	2.65	2.46	25.4
	1.5	0.550 2.80 •10 -3	2.43	2.24	24.9
	1.75 0.766 3.05 •10 -3	2.57	2.44	24.6
	K	Price Std. error Var. ratio Adj. ratio Time, s
	0.25 0.0280 2.69 •10 -4	5.19	4.99	24.8
	0.5 0.108 8.60 •10 -4	3.32	3.05	25.1
	0.75 0.226 1.58 •10 -3	2.68	2.56	26.3
	1	0.374 2.31 •10 -3	2.39	2.20	27.0
	1.25 0.545 3.01 •10 -3	2.20	2.19	25.2
	1.5 0.730 3.66 •10 -3	2.09	1.94	24.6
	1.75 0.932 4.27 •10 -3	1.97	1.83	24.8

  In dimension one, the model defined by eqs. (3.2.1) and (3.2.2) corresponds to the famous Heston model

	3.2. THE WISHART STOCHASTIC VOLATILITY MODEL	3
	Remark 3.2.2.
						+, * n .
	Remark 3.2.1. The model (S, X) defined in (3.2.1) and (3.2.2) is a (quite
	large) subclass of the one defined in (3.1.1) and (3.1.2). Indeed, defining Xt :=
	a X t a, we have a X	1/2 t	dZ t =	X1/2 t	d Zt , where Zt is another n-dimensional
	standard Brownian motion and
	d Xt = α a a + b Xt + Xt b	dt+	X1/2

t d Wt a+a (d Wt ) X1/2 t , X0 = a x a, where b = a b (a ) -1 and Wt is another n × n-Brownian motion.

  .1.

	Maturity, [Year] Strike Price	Std. dev. Var. ratio Time, [Sec]
	0.50	0.7	2.18 •10 -7 3.37 •10 -8 119	202
	0.50	0.8	3.29 •10 -5 9.5 •10 -7	22.5	167
	0.50	0.9	1.78 •10 -3 1.38 •10 -5 5.28	169
	0.50	1.0	0.02620	6.85 •10 -5 3.15	167
	0.50	1.1	0.10306	9.86 •10 -5 3.96	167
	0.50	1.2	0.20027	8.29 •10 -5 6.68	167
	0.50	1.3	0.30005	6.41 •10 -5 11.3	180
	0.50	1.4	0.39999	5.32 •10 -5 16.5	168
	0.25	1.0	0.01730	5.17 •10 -5 2.42	92
	1.00	1.0	0.04115	9.51 •10 -5 3.76	319
	2.00	1.0	0.06423	1.39 •10 -4 3.86	618
	3.00	1.0	0.08319	1.78 •10 -4 3.63	934
	5.00	1.0	0.11579	2.46 •10 -4 3.22	1522

  .3.10) in r ∈ R + ∪ iR + .

	Lemma 4.3.6. If while if z 2 0 T a ∈ (0, 1), (4.3.10) has no solution in iR + . z 2 0 T a ∈ [1, ∞), (4.3.10) admits a unique solution in iR + ,
	Proof. Let u = -2rT i ∈ R + and k =	z 2 0 T a > 0. Notice that 0 solves (4.3.10)
	if and only if k = 1. Let us write (4.3.10) as

  5 ,i 6 (t, x) ∂x i 1 ∂x i 2 ∂x i 3(5.4.19) (λ j 4 , λ j 5 , λ j 6 )dv∂ 3 vj 4 ,j 5 ,j 6 (t, x) ∂x i 1 ∂x i 2 ∂x i 3

	n	n	n	T k
	-v Ẽ2 := -j 4 =1 j 5 =j 4 +1 j 6 =j 5 +1 s M 3 n n n
	j=1	j 0 =j+1	j 1 =j 0 +1

  5 ,i 6 (t, x) ∂x i 1 ∂x i 2 ∂x i 3 x=Lt (λ j 4 , λ j 5 , λ j 6 )dv∂ 3 vj 4 ,j 5 ,j 6 (t, x) ∂x i 1 ∂x i 2 ∂x i 3 x=Lt (λ j , λ j 0 , λ j 1 ) (λ i 4 , λ i 5 , λ i 6 )dv ∂v i 4 ,i 5 ,i 6 (t, x)∂x (λ j 4 , λ j 5 , λ j 6 )dv∂ 3 vj 4 ,j 5 ,j 6 (t, x) ∂x j ,i 2 ,i 3 ,i 4 =k L i 1 t L i 2 t L i 3 t L i 4 t ∂ 4 u 0 (t, x) ∂x i 1 ∂x i 2 ∂x i 3 ∂x i 4 x=Lt T k-1 t M 4 s (λ i 1 , λ i 2 , λ i 3 , λ i 4 )ds

			n		n			n		T k-1
	-v -j 4 =k j 5 =j 4 +1 j 6 =j 5 +1 s M 3 n j=k n j 0 =j+1 n j 1 =j 0 +1 δ j 0 L j 0 t (1 + δ j 0 L j 0 t δ j 1 L j 1 t (1 + δ j 1 L j 1 t	t L j	T k-1
	•	1 6	n i 4 ,i 5 ,i 6 =k		s	T k-1	M 3 v j	x=Lt
			n		n			n		T k-1
		-	j 4 =k	j 5 =j 4 +1	j 6 =j 5 +1	s	M 3 v x=Lt
	n n j 0 =j+1 i 1 -1 + 24 n j=k	n j 1 =j 0 +1	n j 2 =j 1 +1	δ j 0 L j 0 t 1 + δ j 0 L j 0 t	δ j 1 L j 1 t 1 + δ j 1 L j 1 t	δ j 2 L j 2 t 1 + δ j 2 L j 2 t	L j t	∂u 0 (t, x) ∂x j	x=Lt
			T k-1
		•
			t

t dsM s

Table 5 .

 5 1: Price of ATM caplet computed using the analytic approximation together with the 95% confidence bounds computed by Monte Carlo over 10 6 trajectories.

		Case 1	Case 2	Case 3	Case 4
	Order 0	0.008684	0.006392	0.003281	0.007112
	Order 1	0.008677	0.006361	0.003241	0.006799
	Order 2	0.008677 0.006351 0.003172 0.006556
	MC lower bound 0.008626	0.006306	0.003178	0.006493
	MC upper bound 0.008712	0.006361	0.003204	0.006578

  5. APPROXIMATE OPTION PRICING IN THE L ÉVY LIBOR MODEL5

	0.218 0.220 0.222					MC lower MC upper order 2 order 1 order 0		0.155 0.160
	0.03 0.210 0.212 0.214 0.216	0.04	0.05	0.06	0.07	0.08	0.09	0.03 0.145 0.150	0.04	0.05	0.06	0.07	0.08 order 0 order 1 order 2 MC upper MC lower	0.09
	0.095							0.20
	0.090							0.19
								0.18
	0.085							0.17
	0.080							0.16
	0.03 0.065 0.070 0.075	0.04	0.05	0.06	0.07	0.08 order 0 order 1 order 2 MC upper MC lower	0.09	0.15 0.14 0.03 0.12 0.13	0.04	0.05	0.06	0.07	order 0 order 1 order 2 0.08 MC upper MC lower	0.09

Index based on 500 large American companies.

Volatility index based on the S&P 500 (SPX), that estimates expected volatility by aggregating, weighted prices of SPX puts and calls over a wide range of strikes. The detailed calculation methodology can be found in(Chicago Board Options Exchange, CBOE, 2018).

Note that the appellation "Wishart stochastic volatility model" also refers to the single asset modelsdS t = S t Tr X 1/2 t dZ t ,where (Z t ) t≤T is a matrix Brownian motion (see[START_REF] Benabid | Wishart stochastic volatility: Asymptotic smile and numerical framework[END_REF]), which is a more flexible version of the Heston model.

Université Paris-Est, Cermics (ENPC), INRIA, F-77455 Marne-la-Vallée, France.

Ohio University, Athens, United States of America

The weak Hörmander condition is a classical hypothesis for X T to admit a smooth density. (See[START_REF] Nualart | The Malliavin Calculus and Related Topics[END_REF] for example.)

A sufficient condition for K a = ∅ is the strong Hörmander condition, i.e. that Lie(σ 1 , ..., σ n )| x has full rank for every x ∈ R d . (See(Jurdjevic, 1997, p. 106)

The result in[START_REF] Gombani | A filtering approach to pricing in multifactor term structure models[END_REF] concerns the price of zero-coupon bonds in exponentially quadratic term structure models. The exact same proof however holds by replacing the "HJM drift condition" by the martingale property of

σ 3 BS T 3/2 |log(S 0 /K)| 2 + O 5 ,(4.5.5) as → 0.

See[START_REF] Ménassé | Asymptotic indifference pricing in exponential Lévy models[END_REF] for rigorous arguments in a simplified but similar setting.

Acknowledgements. This research benefited from the support of the "Chaire Risques Financiers", Fondation du Risque.

CHAPTER 4. AN ASYMPTOTIC APPROACH FOR THE PRICING OF OPTIONS ON REALIZED VARIANCE 4

Since R is arbitrarily large, g (z 0 ) = 2z 0 = 0. Therefore, since λ = 0, then σ 1 (x 0 ), [σ 0 , V ](x 0 ) and [σ 1 , V ](x 0 ) span R 2 . Condition H2 is then verified at x 0 and the invertibility of C(h) for every non-zero h ∈ H then follows from (Bismut, 1984, Theorem 1.10). The weak Hörmander condition is obtained by taking λ = (0, 1).

Remark 4.3.2. In general, the H2 condition is much stronger than the weak Hörmander condition. When m = 1 however, the two are equivalent.

Hamiltonian equations and optimal control

In this section, we formulate and solve the Hamiltonian equations. The Hamiltonian is H y z , (p , q) : = p q , σ 0 (x) + 1 2 p q , (σ 1 σ 1 )(x) p q

Given a certain a, provided R is taken large enough for the (z t ) t≤T trajectory to stay in the ball B(0, R), Prop. 4.2.1 yields the Hamiltonian equations

• z t = c 2 q t (4.3.4)

• p t = 0 (4.3.5)

• q t = -p t g (z t ) = -2p t z t , (4. 3.6) with the boundary conditions z 0 = z 0 , y 0 = 0, y T = a and q T = 0.

Lemma 4.3.3. Let (y t , z t , p, q t ) be the solution of (4.3.3)-(4.3.6) with the associated boundary conditions. Then the minimizing control is

The energy associated to a is then

Proof. By Prop. 4.2.1, the minimizing control associated to the solution (y t , z t , p, q t ) of (4.3.3)-(4.3.6) with the associated boundary conditions is given by

Theorem 4.5.3. Let

be the price of a put option on realized variance with maturity T and strike K. Then P admits the asymptotic expansion

as → 0.

Proof. First notice that, Λ (a) > 0 ⇔ a > z 2 0 T and Λ (a) < 0 ⇔ a < z 2 0 T . If K < z 2 0 , then using Lemma 4.5.1 and Theorem 4.4.1, we have

By Lemma 4.5.1 again, we have

The model

Denote by L = (L 1 , . . . , L n ) the column vector of forward Libor rates. We assume that under the terminal measure Q Tn , the dynamics of L is given by the following SDE 5.2.4) where b(t, L t ) is the drift term and Λ(t) a deterministic n×d volatility matrix.

We write Λ(t) = (λ 1 (t), . . . , λ n (t)) , where λ k (t) denotes the d-dimensional volatility vector of the Libor rate L k and assume that λ k (t) = 0, for t > T k-1 .

One typically assumes that the jumps of X are bounded from below, i.e. ∆X t > C, for all t ∈ [0, T * ] and for some strictly negative constant C, which is chosen such that it ensures the positivity of the Libor rates given by (5.2.4).

The drift b(t

) is determined by the no-arbitrage requirement that L k has to be a martingale with respect to Q T k , for every k = 1, . . . , n. This yields

The above drift condition follows from (5.2.2) and Girsanov's theorem for semimartingales noticing that

where

where u is the solution of the following PIDE 1

and g denotes the transformed payoff function given by g

In what follows we shall in particular focus on two most liquid interest rate options: caps (caplets) and swaptions.

Caplet

Consider a caplet with strike K and payoff ξ = δ k (L k T k-1 -K) + at time T k . Note that here the payoff is in fact a F T k-1 -measurable random variable and it is paid at time T k . This is known as payment in arrears. There exist also other conventions for caplet payoffs, but this one is the one typically used.

The time-t price of the caplet, denoted by P Cpl t is thus given by

where u is the solution to

For the second equality in (5.3.2) we have used the measure change from Q T k to Q Tn given in (5.2.2).

1 A detailed proof of this statement is out of scope of this note. Here we simply assume that Equation (5.3.1) admits a unique solution which is sufficiently regular and is of polynomial growth. The existence of such a solution may be established first by Fourier methods for the case when there is no drift and then by a fixed-point theorem in Sobolev spaces using the regularizing properties of the Lévy kernel for the general case (see (De Franco, 2012, Chapter 7) for similar arguments). Once the existence of a regular solution has been established, the expression for the option price follows by the standard Feynman-Kac formula.

CHAPTER 5. APPROXIMATE OPTION PRICING IN THE L ÉVY LIBOR MODEL 5

Remark 5.3.1. Noting that the payoff of the caplet depends on one single underlying forward Libor rate L k , it is often more convenient to price it directly under the corresponding forward measure Q T k , using the first equality in (5.3.2). Thus, one has

where u is the solution to

with g(x) := (x k -K) + and where A T k is the generator of L under the forward measure Q T k . In the log-normal LMM this leads directly to the Black's formula for caplet prices. However, in the Lévy Libor model the driving process X under the forward measure Q T k is not a Lévy process anymore since its compensator of the random measure of jumps becomes stochastic (see (5.2.9)). Therefore, passing to the forward measure in this case does not lead to a closed-form pricing formula and does not bring any particular advantage. This is why in the forthcoming section we shall work directly under the terminal measure Q Tn .

Swaptions

Let us consider a swaption, written on a fixed-for-floating (payer) interest rate swap with inception date T 0 , payment dates T 1 , . . . , T n and nominal N = 1. We denote by K the swaption strike rate and assume for simplicity that the maturity T of the swaption coincides with the inception date of the underlying swap, i.e. we assume T = T 0 . Therefore, the payoff of the swaption at maturity is given by P Sw (T 0 ; T 0 , T n , K)

+ , where P Sw (T 0 ; T 0 , T n , K) denotes the value of the swap with fixed rate K at time T 0 given by

where is the swap rate i.e. the fixed rate such that the time-t price of the swap is equal to zero. Here we denote

Note that n j=1 w j (t) = 1. Dividing the numerator and the denominator in (5.3.5) by B t (T n ) and using the telescopic products together with (5.2.1) we see that w j (t) = f j (L t ) for a function f j given by

Therefore, the swaption price at time t ≤ T 0 is given by

where u is the solution to

Approximate pricing

Approximate pricing for general payoffs under the terminal measure

Following an approach introduced by Černỳ et al., 2013, we introduce a small parameter into the model by defining the rescaled Lévy process X α t := αX t/α 2 with α ∈ (0, 1). The process X α is a martingale Lévy process under the terminal measure Q Tn with characteristic triplet (0, c, F α ) with respect to the truncation function h(z) = z, where

APPROXIMATE PRICING

5 may expand the solution u α in powers of α:

α p u p (t, x). (5.4.4) Substituting the expansions for A α t and b α into this equation, and gathering terms with the same power of α, we obtain an 'open-ended' system of PIDE for the terms in the expansion of u α .

The zero-order term u 0 satisfies

with .4.6) Hence, by the Feynman-Kac formula

where the process X t,x = (X i,t,x ) n i=1 satisfies the stochastic differential equation (5.4.8) with W a d-dimensional standard Brownian motion with respect to Q Tn and σ an n × d-dimensional matrix such that σσ = (Σ i,j ) n i,j=1 . To obtain an explicit approximation for the higher order terms u 1 (t, x) and u 2 (t, x) given above, we consider the following proposition.

Proposition 5.4.1. Let Y be an n-dimensional log-normal process whose components follow the dynamics

where µ and σ are measurable functions such that ) and a European-type payoff with maturity T k given by ξ = g(L T k ). Assuming (5.4.9), its time-t price P α (t; g) for α → 0 satisfies P α (t; g) = P 0 (t; g) + αP 1 (t; g) + α 2 P 2 (t; g) + O(α 3 ), (5.4.26) with

where P LM M (t; g) denotes the time-t price of the payoff g(L T k ) in the lognormal LMM with covariance matrix Σ and the drift given by (5.4.6), M 3 s (λ i 1 , λ i 2 , λ i 3 ) and M 3 s (λ j , λ j 0 , λ j 1 ) are given by (5.4.3), u 0 (t, x) by (5.4.7) and ũ1 (t, x) and ũ2 (t, x) by (5.4.13) and (5.4.18), respectively.

Approximate pricing of caplets

Recalling that the caplet price is given by (5.3.2), where u is the solution of the PIDE (5.3.3), we can approximate this price using the development u α (t, x) = u 0 (t, x) + αu 1 (t, x) + α 2 u 2 (t, x) + O(α 3 ) where the zero-order term u 0 satisfies ∂ t u 0 + A 0 t u 0 = 0, u 0 (T k-1 , x) = (x k -K) + n j=k+1

(1 + δ j x j )

with

Σ ij (t)x i x j ∂ 2 u 0 (t, x) ∂x i ∂x j and b i 0 (t, x) = - Σ ij (t) δ j x j 1 + δ j x j .

The solution to the above PDE can be found via the Feynman-Kac formula, where the conditional expectation is computed in the log-normal LMM model with covariation matrix (Σ ij ) n i,j=1 as in Section 5.4.1. Performing a measure change from Q Tn to Q T k and denoting by P BS (V, S, K) the Black-Scholes price of a call option with variance V ,