Mohamed Sidi
email: sidi@math.lth.se

Aly

MOMENT EXPLOSIONS, IMPLIED VOLATILITY AND LOCAL VOLATILITY AT EXTREME STRIKES

Keywords: 2000 Mathematics Subject Classification : 60E10, 62E20, 40E05 Tauberian theorems, local volatility, implied volatility, extreme strikes, Moments, SVI, Heston model

We consider a stochastic volatility model where the moment generating function of the logarithmic price is finite only on part of the real line. Using a new Tauberian result obtained in [1] and [2], we show that the knowledge of the moment generating function near its critical moment gives a sharp asymptotic expansion (with an error of order o(1)) of the local volatility and implied volatility for small and large strikes. We apply our theoretical estimates to Gatheral's SVI parametrization of the implied volatility and Heston's model.

introduction

In [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF] and [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF] we derived a Tauberian result that gives a sharp asymptotic formula for the complementary cumulative distribution function of a random variable whose moment generation function (MGF) is finite only on part of the real line. The corresponding formula depends only on the behavior of the MGF near its critical moment. This result is very useful in the case where the moment generating function is known (as it is the case for the CIR process, Heston's model and many time changed Lévy processes, cf. [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF], [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF]). In [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF] we have shown that this Tauberian result may also be used in cases where MGF is unknown in order to derive an asymptotic expansion of this MGF near its critical moment and then (using the theorem once more) to derive a sharp asymptotic formula for the cumulative distribution; indeed we proved in [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF] that the MGF of V

2(1-p) t

, where V is given by the stochastic differential equation dV t = (a -bV t )dt + σV p t dW t , explodes at the critical moment µ * t = b σ 2 (1-p)(1-e -2b(1-p)t ) t and derived and sharp asymptotic formula for the MGF of V 2(1-p) t near µ * t as well as the complementary cumulative distribution function of V 2(1-p) t . In the present work we give another application to the Tauberian result of [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF] to the well known Dupire's local volatility surface: Σ 2 (t, k) = 2∂ T C(T, K)/K 2 ∂ KK C(T, K)| K=e k where C(T, K) denotes the price of a European Call option with strike K and maturity T . Indeed, if we denote X t = ln(S t /S 0 ) with S t referring to the stock price, then the Call price is given as (assuming without loss of generality that the interest and dividend rates are 0)

C(T, K) = E(e X T -K) + .
Differentiating with respect to K we have ∂ K C(T, K) = -P(X T > ln(K)).

On the other hand, our Tauberian result states that if the logarithm of the MGF: Λ : p → ln Ee pX is a regularly varying function of 1 µ * -p with a positive index near its critical moment µ * (plus other non restrictive assumptions) the Fenchel-Legendre transform Λ * of Λ is a good asymptotic approximation of the logarithm of cumulative distribution of X with lim sup-lim inf arguments:

(1.1) P(X ≥ x) = e -Λ * (x) p * ′ (x)

p * (x) √ 2π - 2 + α (α+1) 2 24µ * √ 2π 1 x 2 p * ′ (x) + o( 1
x 2 p * ′ (x) ) .
Here p * is such that Λ * (x) = p * (x)x + Λ(p * (x)) and α is the index of the regularly varying function x → Λ(µ * -1 x ). In particular if the logarithmic price X t = ln(S t /S 0 ) satisfies the assumptions of the theorem then we immediately have a sharp asymptotic formula for the price of Call option as well as its derivative with respect to the strike; thus a sharp asymptotic formula can be derived for Dupire's local volatility function Σ(T, K) by integrating and differentiating (1.1) with respect to t and x. It will turn out in fact that a sharp asymptotic formula for local volatility function can be obtained in terms of Λ and its derivates with respect to t and µ via another method. We find indeed that under the same assumptions as for (1.1), the local volatility for small/large strikes is given as

(1.2) Σ 2 (t, ±y) = σ ± 0 (t)y + ∂ t ∆ ± (t, ν ± (y)) ∓ q ∓ σ ± 0 (t)(1 ∓ 2µ * ± ± 1 ν ± (y) )y/ν ± (y) 1 2 ((µ * ± -1 ν ± (y) ) 2 ∓ (µ * ± -1 ν ± (y) )) 1 + (γ 2 ± -γ ± ) ν ± (y) 2 2y 2 ν ′ ± (y)
+ o [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF],

where σ ± 0 = -2∂tµ * ± (t) µ * ± (t)∓µ * ± (t)
, with µ * ± (t) referring to the critical moment of (±X t ) and ν ± is given in terms of the moment generating function of (±X t ). In particular, the second term in the right hand side is O(y α ± α ± +1 ), where α + (resp. α -) is the index of the regularly varying function ln

Ee (µ * + -1 x )Xt (resp. ln Ee -(µ * --1 x )Xt ).
The expansion (1.2) of the local volatility (and (1.1)) applies to a large class of time changed Lévy models (see e.g. [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF] and [START_REF] Benaim | Regular Variation and Smile Asymptotic[END_REF]). Several affine stochastic volatility models satisfy also the assumption of moment explosion (cf. [START_REF] Keller-Ressel | Moment explosions and long-term behavior of affine stochastic volatility models[END_REF]). The most famous example of the later family is the Heston model, whose MGF satisfies the assumption of the main Tauberian theorem of [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF]. In particular (1.1) holds for the logarithmic price, as highlighted in [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF]; this is a slight improvement on a result obtained in [START_REF] Friz | On Refined Volatility Smile Expansion in the Heston Model[END_REF]. As regards the local volatility asymptotics, Friz and Gerhold show in [START_REF] Friz | Extrapolation analytics for Dupire's local volatility[END_REF] that Σ(t, y) ∼ σ + 0 (t)y. We clearly see that (1.2) applied to Heston's model gives a substantial improvement on [START_REF] Friz | Extrapolation analytics for Dupire's local volatility[END_REF]. We draw attention that the authors of [START_REF] Friz | Extrapolation analytics for Dupire's local volatility[END_REF] use Saddle point and Hankel contour integration methods to obtain an equivalence of the local volatility function for the specific Heston and NIG models, whereas our result is model independent.

Our Tauberian result also allows to derive a sharp asymptotic formula for the Black-Scholes implied volatility, similar to the one obtained for the local volatility function. Indeed, we show that the implied volatility for small/large stokes is given as function of k := ln(K/S 0 ) by:

(1.3) tσ 2 (t, ±k) = 4Λ * ± (t, k)+2c ± t (k)∓2k -4 (Λ * ± (t, k) + c± t (k) 2 )(Λ * ± (t, k) + c± t (k) 2 ∓ k)+O(k -α ± ∧1 α ± +1 ).
where Λ * ± (t, .) is Fenchel-Legendre transform of Λ ± (t, .) and c± t ∼ ln(k) is given in terms of Λ. We draw attention to the fact that under the theorems assumption, Λ *

± (t, x) = µ * ± (t)x + O(x α ± α ± +1
). In particular, dividing both sides of (1.3) by k and letting k → ±∞ we find Lee's moment formula [START_REF] Lee | The Moment Formula for Implied Volatility at Extreme Strikes[END_REF].

While our results apply to any model whose moment generating function of the logarithmic price is given explicitly, we choose to study two examples. In the first one, we consider the SVI parametrization of the implied volatility; we show that this will lead to an explosion of the MGF and that an asymptotic formula for the distribution tail as well as the local volatility may be easily obtained. The second example that we will study in details is the Heston model; we use the explicit formula for ES p t , for p < 0 and p ≥ 1 obtained in [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF] (without any restriction on the parameters). We then apply our Tauberian results to derive sharp asymptotic expansions for the left/right wings of local volatility as well as implied volatility. We also show that a similar formula holds for the Stein-Stein model.

The literature around the implied/volatility is very vast and the number of papers dealing with issues related to the implied/local volatility asymptotic in stochastic volatility model is countless. For the local volatility case, the most relevant work to this paper is [START_REF] Friz | Extrapolation analytics for Dupire's local volatility[END_REF] mentioned above (see also [START_REF] De Marco | Rational shapes of local volatility[END_REF] for the numerical and practical aspects of [START_REF] Friz | Extrapolation analytics for Dupire's local volatility[END_REF]). As regards implied volatility and moment explosion, this work extends the famous Lee's moment formula obtained in [START_REF] Lee | The Moment Formula for Implied Volatility at Extreme Strikes[END_REF] by looking more closely to the moments of the stock prices. Benaim and Friz [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF] have shown the equivalence of the regular variation property between the logarithm of density and the implied volatility. Other results deal with the implied volatility of some stochastic volatility models, such as [START_REF] Friz | On Refined Volatility Smile Expansion in the Heston Model[END_REF], [START_REF] Gulisashvili | Asymptotic Analysis of Stock Price Densities and Implied Volatilities in Mixed Stochastic Models[END_REF], [START_REF] Gulisashvili | Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models[END_REF], [START_REF] Gathera1 | asymptotics of implied volatility in local volatility models[END_REF]. This paper is organized as follows: In section 2 we recall the main Tauberian result of [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF] that we will use to formulate and proof a new Tauberian result which will be the key to link the local volatility asymptotics to the MGF near its critical moment; this link will be given as a theorem that is formulated and proved in Section 3. In section 4 we give a result relating the (sharp) asymptotics of the implied volatility for large strike and the MGF near its critical moment. Section 5 gives the application of our results to the SVI parametrization of the implied volatility, while Section 6 studies extensively the application the Heston model case. We then give the implication to Stein-Stein model.

Tauberian relation between the moment explosion and the distribution tails

Throughout this paper (ω, F, (F t ) t≥0 , P) is a complete filtered probability space satisfying the usual conditions and E refers to the expectation under P.

In this section, we first recall the Tauberian theorem that has been first formulated in [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF] and then extended in [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF]. We then present a new Tauberian result that will be crucial when deriving the local volatility from the moment generating function. We present the version that has been given in [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF]; it concerns a random variable X satisfying the following assumption: Assumption 1. There exist µ * > 0 and α > 0 such that

(i) ∀µ ∈ [0, µ * [, Λ(µ) := ln E e µX < +∞, (ii) The function f (x) : x -→ Λ(µ * -1 x ) is R α and C 2 ([M, ∞[), for some M sufficiently large. (iii) The function x → 1 µ * -p * (x) , with p * (x) = Λ * ′ (x), is smoothly varying with index 1 α+1
, where Λ * is the Fenchel-Legendre transform of Λ.

We use the same notation as [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF]; R α refers to the set of regularly varying function with index α. It is also worth noticing that (see Remark 2.1 in [START_REF] Aly | Moment Explosion and tail asymptotics for the CKLS process[END_REF]) Λ is convex. In particular, Λ * is well defined and is given as Λ * (x) = p * (x)x -Λ(p * (x)), with p * is the unique solution to Λ ′ (p * (x)) = x. Theorem 2.1. Let Assumption 1 hold for some random variable X. Consider the function χ x (.) defined, for x sufficiently large, by

(2.1) χ x (z) = (p * (x) -p * (xz))xz + Λ(p * (xz)) -Λ(p * (x)),
and let g be smoothly varying with index γ ∈ R. Then for any β ∈]0, 1[, we have the following expansion as x → ∞

(2.2)

∞ x β-1 g(xz) e ψx(z) dz = g(x) √ π 2x 2 p * ′ (x) 2 + γ 2 + γ α+1 + c α x 2 p * ′ (x) + o( 1 x 2 p * ′ (x)
)

,
where

c α = - 1 4 (1 + 1 α + 1 )(2 + 1 α + 1 ) + 5 12 (1 + 1 α + 1 ) 2 .
Furthermore,

(2.3) lim sup x→∞ x 2 p * ′ (x) ln P(X ≥ x) + Λ * (x) - p * ′ (x) p * (x) √ 2π ≥ - 2 + α (α+1) 2 24µ * √ 2π
and

(2.4) lim inf x→∞ x 2 p * ′ (x) ln P(X ≥ x) + Λ * (x) - p * ′ (x) p * (x) √ 2π ≤ - 2 + α (α+1) 2 24µ * √ 2π .
In particular, if "lim sup" equals "lim inf" then we have

(2.5) P(X ≥ x) = e -Λ * (x) p * ′ (x) p * (x) √ 2π - 2 + α (α+1) 2 24µ * √ 2π 1 x 2 p * ′ (x) + o( 1
x 2 p * ′ (x)
) .

The only use we will make of this theorem in this paper is to prove the following result which will be the key to link the local volatility asymptotics to the moment generating function near its critical moment. The proposition is followed by its proof. Proposition 2.1. Let X be a random variable satisfying Assumption 1. Assume that "lim sup" and "lim inf" in (2.4) and (2.3) are equal. Let g be smoothly varying with index γ ∈ R such that E|g(X)| < ∞. We have, for x sufficiently large,

(2.6) E(g(X)e (µ * -1 x )X ) E(e (µ * -1 x )X ) = g(Λ ′ (µ * - 1 x )) 1 + (γ 2 -γ) Λ ′′ (µ * -1 x ) 2Λ ′ 2 (µ * -1 x ) + o( Λ ′′ (µ * -1 x ) 2Λ ′ 2 (µ * -1 x )
) .

Remark 2.1. Note that under the theorem's assumption,

Λ ′′ (µ * -1 x ) 2Λ ′2 (µ * -1 x ) = O(x -α ).
Proof. We shall not assume that X admits a smooth density; if that would be the case we could express E(U e px ) by integrating with respect to this density function. Instead, we proceed as in [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF] by using the following representation of the exponential function:

g(U )e pU = g(U )1 U ≤c e pU + 1 U >c g(U )e pU = g(U )1 U ≤c e pU + 1 U >c g(c)e pc + U c e pz (g ′ (z) + pg(z))dz = g(U )1 U ≤c e pU + 1 U >c g(c)e pc + ∞ c e pz (g ′ (z) + pg(z))1 z≤U dz = g(U ∧ c)e p U ∧c + ∞ c e pz (g ′ (z) + pg(z))1 z≤U dz.
which holds for any U, p, c ∈ R. Applying this to U = X we have

E[g(X)e pX ] = E[g(X ∧ c)e p X∧c ] + ∞ c e pz (g ′ (z) + pg(z))P(z ≤ X)dz. (2.7)
Let's denote, for a function h with polynomial growth,

I h (p) := ∞ c h(z)e pz P(z ≤ X)dz.
With this notation we can write E[g(X)e pX ] as

(2.8) E[g(X)e pX ] = E g (c) + I g ′ (p) + pI g (p)
,

where

E g (c) = E[g(X ∧ c)e p X∧c ].
The idea here is to choose c to be large but so that E g (c) is negligible with respect to the two terms I g ′ and I g . It is worth noticing that under the assumption E|g(X)| = m ≤ ∞, we have

(2.9) |E g (c)| ≤ E|g(X)|e pc + g(c)e pc = (m + g(c))e pc .
Now applying Theorem 2.1 to X we have

P(z ≤ X) = e Λ * (z) f (z),
where f is given as

f (z) = p * ′ (z) p * (z) √ 2π - 2 + α/(α + 1) 2 24p * (z) √ 2π + o( 1 z 2 p * ′ (z)). For x sufficiently large, define Z ≡ Z(x) by (2.10) Z = Λ ′ (t, µ * - 1 x )
In particular, we have

µ * - 1 x = p * (Z).
It follows that, for any function h (for which I h is well defined), I h (µ * -1 x ) ≡ I h (p * (Z)) may be written as

I h (p * (Z)) = ∞ c h(z)e p * (Z) e Λ * (z) f (z)dz = Ze Λ(p * (Z)) ∞ c/Z h(zZ)f (zZ)e χx(z) dz.
We then choose c = Z β , where β = α 2(α+1) (∈]0, 1[). In particular we have Z β ∼ Λ(p * (Z)) (and hence e Z β is negligible with respect to e Λ(p * (Z)) ). Now we can easily see that

f ∈ R -(α+2) 2(α+1) . Hence gf ∈ R γ-α+2 2(α+1)
. The first statement of Theorem 2.1 ensures that

I g (p * t (Z)) = e Λ(p * (Z)) g(Z)f (Z) √ 2π p * ′ (Z) 1 + (γ -α+2 2(α-1) ) 2 + (γ -α+2 2(α+1) )/(α + 1) + c α Z 2 p * ′ (Z) + o( 1 Z 2 p * ′ (Z)
) .

A similar statement holds for I g ′ (p * (Z)). It follows that for x sufficiently large we have (2.11)

I g I z →1 (p * (Z)) = g(Z) 1 + γ 2 -γ 2Z 2 p * ′ (Z) + o( 1 Z 2 p * ′ ) ,

and

(2.12)

I g ′ I z →1 (p * (Z)) = g ′ (Z) 1 + (γ -1) 2 -(γ -1) 2Z 2 p * ′ (Z) + o( 1 Z 2 p * ′ ) ,
where we used the fact that g ′ is smoothly varying with index γ -1. The notation z → 1 refers to the "constant" function mapping R into {1}. Now using the fact that β is such that Z β is negligible with respect to Λ(p * (Z)) we have

E[g(X)e p * (Z)X ] E[e p * (Z)X ] = E g (Z β ) + I g ′ + p * (Z)I g E z →1 (Z β ) + p * (Z)I z →1 = 1 p * (Z) I g ′ I z →1 + I g I z →1 + o(e Z β -Λ(p * (Z)) ),
where we used (2.9) which ensures that E g (Z β )| ≤ (m+g(Z β ))e pZ β . In particular

Rg (Z β ) I z →1 = o(e pZ β -Λ(p * (Z)) ) decays exponentially to 0 as Z goes to ∞ (recall that Z β ∼ Λ(p * (Z))). Hence E[g(X)e p * (Z)X ] E[e p * (Z)X ] = g(Z) 1 + γ 2 -γ 2Z 2 p * ′ (Z) + o( 1 Z 2 p * ′ ) . Now from the definition of p * we have Λ ′ (p * (Z)) = Z. It follows that p * ′ (Z) = 1 Λ ′′ (p * (Z)) = 1 Λ ′′ (µ * -1 x )
.

On the other hand,

p * ′ ∈ R -α+2 α+1 . It follows that Z(x) 2 p * ′ (Z(x)) ∼ Z(x) α α+1 ∼ x -α .
The proof is completed by replacing Z by Λ ′ (µ * -1 x ) and p * ′ (Z) by 1/Λ ′′ (µ * -1 x ).

From moment explosion to the local volatility asymptotics

In this section we study the link between the local volatility asymptotics and and the moment explosion. We consider a stock price process S t and denote X t := ln(S t /S 0 ). Let's also denote, for t, p > 0, (3.1) Λ + (t, p) = ln E(e pXt ) and Λ -(t, p) = ln E(e -pXt ).

Suppose that for every t > 0, Λ + (t, .) and Λ -(t, .) are finite only on part of the real line; i.e. there exist µ * + (t) and µ * -(t) such that 0 < p < µ * ± (t), Λ ± (t, p) < 0, and lim

p→µ * ± (t) Λ ± (t, p) = ∞.
Consider the local volatility function Σ(t, x) such that the process Y defined by the stochastic differential equation

dY t = (q - 1 2 Σ 2 (t, Y t ))dt + Σ(t, Y t )dW t ,
generates the same marginal distributions as X. The following result links Σ(t, ±y) for large y to the behavior of Λ + (t, .) and Λ -(t, .) near the critical moments µ * + and µ * -.

Theorem 3.1. If Λ + and Λ -satisfy (ii)-(iii) in Assumption 1, then for y sufficiently large we have

(3.2) Σ 2 (t, ±y) = σ ± 0 (t)y + ∂ t ∆ ± (t, ν ± (y)) ∓ q ∓ σ ± 0 (t)(1 ∓ 2µ * ± ± 1 ν ± (y) )y/ν ± (y) 1 2 ((µ * ± -1 ν ± (y) ) 2 ∓ (µ * ± -1 ν ± (y) )) 1 + (γ 2 ± -γ ± ) ν ± (y) 2 2y 2 ν ′ ± (y) + o( ν ± (y) 2 2y 2 ν ′ ± (y) )
,

where γ ± = α ± α ± +1 , ∆ ± (t, x) = Λ ± (t.µ * ± (t) -1 x ), with σ ± 0 (t) = -2∂ t µ * ± (t) µ * ± 2 (t) ∓ µ * ± (t) and ν ± (t, y) := (∂ µ Λ ± (t, µ * ± (t) - 1 (.) )) -1 (y).
Remark 3.1. Note that the second term in the right hand side of (3.2) is O(y

-α ± α ± +1 ). It is also worth noticing that ν ± (y) 2 2y 2 ν ′ ± (y) ∼ y -α ± α ± +1 .
Proof. Applying Itô's formula to the process e pYt we find that, for t > 0 and p < µ * + (t),

(3.3) ∂ t ψ 1 + (t, p) = qψ 1 + (t, p) + 1 2 (p 2 -p)ψ Σ 2 (t,.) + (t, p),
where ψ 1 + (t, p) := E (e pYt ), and for a function f the notation ψ f (.) refers to

ψ f + (t, p) := E (f (Y t )e pYt ). Define ∆ + (t, x) := ln ψ 1 + (t, µ * + (t) -1 x
)); we have

∂ t ∆ + (t, x) = ∂ t ψ 1 + (t, µ * + (t) -1 x ) ψ 1 + (t, µ * + (t) -1 x ) + ∂ t µ * + (t)∂ µ ψ 1 + (t, µ * + (t) -1 x ) ψ 1 + (t, µ * + (t) -1 x )
.

Noticing that ∂ x ∆ + (t, x) = ∂µψ 1 + (t,µ * + (t)-1 x )) x 2 ψ 1 + (t,µ * + (t)- 1
x )) and dividing both sides of (3.3) by

ψ 1 + (t, µ * + (t) -1 x ) we have (3.4) ∂ t ∆ + (t, x) = q + 1 2 ((µ * + (t) - 1 x ) 2 -(µ * + (t) - 1 x )) ψ Σ(t,.) + ψ 1 + (t, µ * + (t) - 1 x ) + ∂ t µ * + x 2 ∂ x ∆ + (t, x).
As highlighted in the introduction, we can actually derive an asymptotic formula for the local volatility by integrating and differentiating (2.5) with respect to t and x. Without going through the formula, we can prove that Σ 2 (t, .) is smoothly varying which allows us to apply Proposition 2.1. This gives

ψ Σ 2 (t,.) ψ 1 (t, µ * + - 1 x ) = Σ(t, x 2 ∂ x ∆(t, x))(1 + O(x -α + )). It follows that Σ 2 (t, y) = -2∂ t µ * + y + 2∂ t ∆ + (t, ν(y)) µ * + 2 (t) -µ * + (t) + 1 ν(y) (1 -2µ * + + 1 χ(y) ) (1 + O(y -α + α + +1 )), where ν(y) = [(.) 2 ∂ x ∆ + (t, .)] -1 (y) ∼ y 1 α + +1 . Hence Σ 2 (t, y) = -2∂ t µ * + (t) µ * + 2 (t) -µ * + (t) y + O(y α α+1 ).
This means that

Σ 2 (t, y) = -2∂ t µ * + (t) µ * + 2 (t) -µ * + (t) y + Σ 1 (t, y),
where Σ 1 (t, y) = O(y α α+1 ). Rewriting the partial differential equation (3.4) we have

∂ t ∆ + (t, x) = q + 1 2 ((µ * + - 1 x ) 2 -(µ * + - 1 x ))(σ 0 (t) ψ Id + ψ 1 + + ψ Σ 1 (t,.) + ψ 1 + (t, µ * + - 1 x )) + ∂ t µ * + x 2 ∂ x ∆ + (t, x),
where Id : z → z and

σ 0 (t) = -2∂ t µ * + (t) µ * + 2 (t) -µ * + (t)
.

Observe that, for any µ > 0,

ψ Id + (t, µ) := E(Y t e µYt ) = ∂ µ ψ 1 + (t, µ). In particular ψ Id + (t, µ * + - 1 x )/ψ 1 + (t, µ * + - 1 x ) = x 2 ∂ x ∆ + (t, x).
It follows that

∂ t ∆ + (t, x) = q + 1 2 (1 -2µ * + + 1 x )σ 0 (t)x∂ x ∆ + (t, x) + 1 2 ((µ * + - 1 x ) 2 -(µ * + - 1 x )) ψ Σ 1 (t,.) + ψ 1 + (t, µ * + - 1 x ).
Using Proposition 2.1 once more we have

ψ Σ 1 (t,.) + ψ 1 + (t, µ * + - 1 x ) = Σ 1 (t, x 2 ∂ x ∆(t, x))(1 + (γ 2 -γ) Λ ′′ (µ * -1 x ) 2Λ ′ 2 (µ * -1 x ) + o( Λ ′′ (µ * -1 x ) 2Λ ′ 2 (µ * -1 x ) )),
where γ is the index of Σ 1 (t, .). It follows that

∂ t ∆ + (t, ν + (y)) = σ 0 (t)(1 -2µ * + + 1/ν + (y))y/ν + (y) + 1 2 ((µ * + - 1 ν + (y) ) 2 -(µ * + - 1 ν + (y) ))Σ 1 (t, y)(1 + (γ 2 -γ) ν + (y) 2 2y 2 ν ′ + (y) + o( ν + (y) 2 2y 2 ν ′ + (y)
)).

Hence γ = α + α + +1 and we finally have

Σ 1 (t, y) = ∂ t ∆ + (t, ν + (y)) -q -σ + 0 (t)(1 -2µ * + -1 ν + (y) )y/ν + (y) 1 2 ((µ * + -1 ν + (y) ) 2 -(µ * + -1 ν + (y) )) 1 + (γ 2 -γ) ν + (y) 2 2y 2 ν ′ + (y) + o( ν + (y) 2 2y 2 ν ′ + (y) )
.

To prove the result for Σ(t, -k), we consider the process Ỹt = -Y t . This process is given by the stochastic differential equation

d Ỹt = (-q + 1 2 Σ2 (t, Ỹt ))dt + Σ(t, Ỹt )d Wt ,
where Wt = -W t and Σ(t, y) = Σ(t, -y). Applying Itô's formula to the process e p Ỹt we find that

∂ t ψ 1 -(t, p) = -qψ 1 -(t, p) + 1 2 (p 2 + p)ψ Σ2 (t,.) - (t, p),
where

ψ f -(t, p) := E(f ( Ỹt e p Ỹt ). Define ∆ -(t, x) := ln ψ 1 -(t, µ * -(t) -1 x )
). We have

∂ t ∆ -(t, x) = -q + 1 2 ((µ * -(t) - 1 x ) 2 + (µ * -(t) - 1 x )) ψ Σ2 (t,.) - ψ 1 + (t, µ * -(t) - 1 x ) + ∂ t µ * -x 2 ∂ x ∆ -(t, x).
We find in the same way as for Σ(t, y) that

Σ 2 (t, -y) = σ - 0 (t)y + ∂ t ∆ -(t, ν -(y)) + q + σ - 0 (t)(1 + 2µ * --1 ν -(y) )y/ν -(y) 1 2 ((µ * --1 ν -(y) ) 2 -(µ * --1 ν -(y) )) 1 + (γ 2 --γ -) ν -(y) 2 2y 2 ν ′ -(y) + o( ν -(y) 2 2y 2 ν ′ -(y)
) with σ - 0 and ν -are given by the theorem.

From moment explosion to Implied volatility asymptotics

Under Black Scholes model, the prices of standard European options are given explicitly via the famous Black-Scholes formula. In particular the price of an European Call option with strike K and maturity t is given by:

E(S 0 e -σ 2 2 t+σWt -K) + = N ( 1 σ √ t (-k + 1 2 σ 2 t)) -e k N ( 1 σ √ t (-k - 1 2 σ 2 t)),
where N denotes the cumulative distribution function of the standard Gaussian law, k = ln(K/S 0 ) and σ is the volatility parameter. For any price quote there exists a unique volatility parameter (that depends on t and k) such that the Black-Scholes formula gives the same price: this is called the Black-Scholes implied volatility.

Suppose that the market is described by a model X (for (log(S t /S 0 ))) under the measure P. Today's price of a Call option with maturity t and and strike e k is then given as function of the Black-Scholes implied volatility as follows:

(4.1) E(e Xt -e k ) + = N ( 1 √ tσ(t, k) (-k + 1 2 tσ 2 (t, k))) -e k N ( 1 √ tσ(t, k) (-k - 1 2 tσ 2 (t, k))).
Differentiating both sides with respect to k we get

-e k P(X t ≥ k) = ∂ k C BS (k, √ tσ(t, k)) + √ t∂ k σ(t, k)C BS (k, √ tσ(t, k)),
where the notation C BS (k, v) refers to

C BS (k, v) := N ( 1 v (-k + 1 2 v 2 )) -e k N ( 1 v (-k - 1 2 v 2 )). Hence P(X t ≥ k) = Ñ ( 1 √ tσ(t, k) (k + 1 2 tσ 2 (t, k))) - √ t∂ k σ(t, k)N ′ ( 1 √ tσ(t, k) (k + 1 2 tσ 2 (t, k))),
where Ñ (x) = 1 -N (x). We clearly see that when |k| is large, the quantity d(k) :=

1 √ tσ(t,k) (k + 1 2 tσ 2 (t, k)) → ∞.
The following bounds hold for any positive y and obtained via an integration by part:

N ′ (y) y (1 - 1 y 2 ) ≤ Ñ (y) ≤ N ′ (y) y .
In particular, we have

Ñ (y) = N ′ (y) y (1 + O(y -2 )).
Applying this to the black Scholes formula we get

(4.2) P(X t ≥ k) = 1 √ 2π e -1 2 d 2 (k) 1 d(k) - √ t∂ k σ(t, k) + O(d(k) -3 ) , Note that d 2 (k) = k 2 tσ 2 (t, k) + 1 4 tσ 2 (t, k) + k.
The left wing is obtained in a very similar way; let k be large enough and consider a European Put option with strike e -k and maturity t. Its price is given by

(4.3) E(e -k -e Xt ) + = e -k N (-d(k)) -N (-d(k) - √ tσ(t, k)),
where σ(t, k) = σ(t, -k) and

d(k) = 1 √ tσ(t, k) (k - 1 2 tσ 2 (t, k))
Differentiating both sides with respect to k we get

P(-X t ≥ k) = Ñ ( d(t, k)) - √ t∂ k σ(t, k)N ′ ( d(t, k)).
The bounds derived above hold also for d(t, k), which gives

(4.4) P(-X t ≥ k) = 1 √ 2π e -1 2 d2 (k) 1 d(x) - √ t∂ k σ(t, k) + O( d(k) -3 ) .
Using the bounds (4.2) and (4.4), the next theorem links the left/right wings of the Black-Scholes implied volatility to the moment generating function of -X t /X t . Theorem 4.1. Suppose that for any t > 0, X t and -X t satisfy Assumption 1. For k sufficiently large we have

(4.5) tσ 2 (t, ±k) = 4Λ * ± (t, k)+2c ± t (k)∓2k -4 (Λ * ± (t, k) + c± t (k) 2 )(Λ * ± (t, k) + c± t (k) 2 ∓ k)+O(k -α ± ∧1 α ± +1 ).
with the notations Λ ± (t, µ) := ln Ee µ(±X) , Λ * ± (t, .) is the Fenchel-Legendre transform of Λ ± (t, .), α ± is the index of the regularly varying function x → Λ ± (t, µ * ± (t) -1 x ) and c± is defined by

(4.6) c± t (k) = -ln(k∂ kk Λ * ± (t, k)) + 2 ln µ * ± (t) 2 - µ * ± (t) 2 ∓2 + 4µ * ± (t) -4 µ * ± (µ * ± ∓ 1) .
Proof. We have from Theorem 2.1

P(X t ≥ k) = e -Λ * + (t,k) ∂ x p * + (t, k) p * + (t, k) √ 2π - 2 + α + (α + +1) 2 24µ * + √ 2π 1 k 2 ∂ x p * + (t, x) + o( 1 k 2 ∂ x p * + (t, k)
) .

Comparing this with (4.2) we see that

(4.7) k 2 tσ 2 (t, k) + 1 4 tσ 2 (t, k) + k -2 ln( 1 d(k) - √ t∂ k σ) + O(k -1 ) = 2Λ * + (t, k) -2 ln( ∂ x p * + p * + (t, k) ) + O(k -α + α + +1 ). Hence 1 4tσ 2 (t, k) × tσ 2 (t, k) -2(2Λ * + (t, k) -k -2 Λ * + (t, k)(Λ * + (t, k) -k)) × tσ 2 (t, k) -2(2Λ * + (t, k) -k + 2 Λ * + (t, k)(Λ * + (t, k) -k)) = c + t (k) + O(k -α + α + +1 ),
where

(4.8) c + t (k) := -2 ln( p * + ′ (t, k) p * + (t, k) ) + 2 ln( 1 d(k) - √ t∂ k σ(t, k)) + O(k -1 ) = O(ln(k)).
We emphasize that Λ

* + (t, k) = µ * + (t)k + O(k α + α + +1
). On the other hand, Lee's moment formula states that

tσ 2 (t, k) ∼ k(-2 + 4µ * + -4 µ * + (µ * + -1)). Noticing that 2 2Λ * + (t, k) -k -2 Λ * + (t, k)(Λ * + (t, k) -k) ∼ k -2 + 4µ * + -4 µ * + (µ * + -1) ,
we conclude that

tσ 2 (t, k) = 2 2Λ * + (t, k) -k -2 Λ * + (t, k)(Λ * + (t, k) -k) + O(ln(k)).
Plugging this value into the definition of c + t (k), (4.8), we find that

c + t (k) = c+ t (k) + O(k -1 α+1 ), with c+ t (k) = -ln(k∂ kk Λ * + (t, k)) + 2 ln µ * + 2 - µ * + 2 -2 + 4µ * + -4 µ * + (µ * + -1) .
Now if we go back to (4.7) and plug the value of c + t (k) we have

k 2 tσ 2 (t, k) + 1 4 tσ 2 (t, k) + k = 2Λ * + (t, k) + c+ t (k) + O(k -α + ∧1 α + +1 ).
We finally find that

tσ 2 (t, k) = 4Λ * + (t, k) + 2c + t (k) -2k -4 (Λ * + (t, k) + c+ (k) 2 )(Λ * + (t, k) + c+ (k) 2 -k) + O(k -α∧1 α+1 ).
The left wing is obtained in the same way: we write, using Theorem 2.1,

P(-X t ≥ k) = e -Λ * -(t,k)   ∂ x p * -(t, k) p * -(t, k) √ 2π - 2 + α - (α -+1) 2 24µ * - √ 2π 1 k 2 ∂ x p * -(t, x) + o( 1
k 2 p * - ′ (t, k) )   .
We obtain, by comparing this with (4.4),

k 2 tσ 2 (t, k) + 1 4 tσ 2 (t, k) -k -2 ln( 1 d(k) + √ t∂ k σ) + O(k -1 ) = 2Λ * -(t, k) -2 ln( ∂ x p * - p * -(t, k) ) + O(k -α - α -+1 ).
We find in the same way as for the right wing that

tσ 2 (t, k) = 4Λ * -(t, k) + 2c - t (k) + 2k -4 (Λ * -(t, k) + c- t (k) 2 )(Λ * -(t, k) + c- t (k) 2 + k) + O(k -α -∧1 α -+1 ), with c-(t, k) = -ln(k∂ kk Λ * -(t, k)) + 2 ln µ * - 2 - µ * - 2 2 + 4µ * --4 µ * -(µ * -+ 1) .

From SVI parametrization of implied volatility to moment explosion and local volatility

We remain in the same setup as the previous section, where S denotes the stock price and X t = ln(S t /S 0 ). We consider Gatheral's SVI parametrization (cf. [START_REF] Gatheral | A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives[END_REF]):

(5.1)

σ 2 (t, k) = a + b ρ(k -m) + (k -m) 2 + η 2 .
It is worth noticing that all the parameters depend on t.

Proposition 5.1. Suppose that the Black-Scholes implied volatility function generated from the option prices is given as (5.1). The moment generating functions of X t and (-X t ) explode at critical moment µ * + and µ * -respectively, where

(5.2) µ * ± (t) = 1 2 ( 1 bt(1 ± ρ) + 1 4 bt(1 ± ρ) ± 1).
Furthermore we have

(5.3) E e (µ * ± -1 x )(±Xt) = µ * ± ξ ± e d ± 0 +µ * ± m (2x) 1 2 + O(1)
, where

d ± 0 = 1 2 (-m + a b 2 t(1 ± ρ) 2 ∓ 2m bt(1 ± ρ) - at 4
),

and

ξ ± = ((2µ * ± ) -1 2 - 1 2 bt(1 ± ρ)).
Proof. From (5.1) we have, for k sufficiently large,

tσ 2 (t, k) = at + bt(1 + ρ)(k -m) + b 2 η 2 k -m + O(k -m) -3 ,
It follows that d 2 (y + m) may be written as

(5.4) -1 2 d 2 (y + m) = - 1 2 ( 1 bt(1 + ρ) + 1 4 bt(1 + ρ) + 1)y + d + 0 + O(y -1 ),
where

d + 0 = 1 2 (-m + a b 2 t(1 + ρ) 2 - 2m bt(1 + ρ) - at 4
),

Note also that

1 d(y + m) - √ t∂ k σ(y + m) = ((2µ * ) -1 2 - 1 2 bt(1 + ρ))y -1 2 + ( d + 0 2 (2µ * ) -3 2 + a 4 bt(1 + ρ) )y -3 2 + O(y -5/2 ),
where

µ * + = 1 2 ( 1 bt(1 + ρ) + 1 4 bt(1 + ρ) + 1).
We may differentiate with respect to k and consider the density function and then express the moment generating function in terms of the density. But to stay in general setting (i.e. without assuming the existence of a density function for X t ) we first use following representation of the exponential function as in (2.7)

∀c, p, k, X ∈ R, e pX = (1 ∧ c) e p(X∧c) + p ∞ c
e pz 1 X≥z dz.

It follows that for any p > 0 such that Ee pXt < ∞ and for any c > 0 we have

Ee pXt = (1 ∧ c)E e p(X∧c) + p ∞ c e pz P(X t ≥ z)dz.
It is easy to see that Ee µ * Xt = ∞. For x sufficiently large, we have

E e (µ * + -1 x )Xt = (1 ∧ c)Ee (µ * + -1 x )c∧Xt + 1 √ 2π (µ * + - 1 x )e (µ * + -1 x )m ∞ c-m e -1 x z+d + 0 +O(z -1 ) (ξ + z -1 2 + O(z - 3
2 ))dz,

where

ξ + = ((2µ * + ) -1 2 -1 2 b(1 + ρ)). Choosing c = 1 + m, we find that E e (µ * + -1 x )Xt = (1 ∧ c)Ee (µ * + -1 x )c∧Xt + 1 √ 2π x(µ * + - 1 x )e d + 0 +(µ * + -1 x )m ∞ 1 x e -z (ξ + (zx) -1 2 + O((zx) -1 ))dz.
Hence

E e (µ * + -1 x )Xt = (1 ∧ c)Ee (µ * + -1 x )c∧Xt + x(µ * + - 1 x )e d + 0 +(µ * + -1 x )m (ξ + ( 1 2 x) -1 2 + O(x -1 )).
Observing that Ee (µ * + -1

x )c∧Xt ≤ e µ * + |c| , we finally have

E e (µ * + -1 x )Xt = µ * + ν 1 e d + 0 +µ * + m (2x) 1 2 + O(1).
For (-X t ) we proceed in a very similar way; we write, for k sufficiently large, (

) P(-X t ≥ k) = 1 √ 2π e -1 2 d2 (k) 1 d(x) - √ t∂ k σ(t, k) + O( d(k) -3 ) , where d(k) = 1 √ tσ(t, k) (k - 1 2 tσ 2 (t, k)), and σ2 (t, k) := σ 2 (t, -k) = a + b -ρ(k + m) + (k + m) 2 + η 2 5.5
We find in the same way as before that

(5.6) -1 2 d2 (y -m) = - 1 2 ( 1 bt(1 -ρ) + 1 4 bt(1 -ρ) -1)y + d - 0 + O(y -1 ),
where

d - 0 = 1 2 (-m + a b 2 t(1 -ρ) 2 + 2m bt(1 -ρ) - at 4 ), Note also that 1 d(y -m) √ t∂ k σ(t, y -m) = ((2µ * -) -1 2 - 1 2 bt(1 -ρ))y -1 2 + O(y -3/2 ),
where

µ * -= 1 2 ( 1 bt(1 -ρ) + 1 4 bt(1 -ρ) -1).
The rest is just repeating what we have done for X t .

We draw attention to the fact that a sharp asymptotic expansion of the price of standard European options (namely Call and Put options) is easily derived under the SVI parametrization of the implied volatility surface. It is given by the following result: Proposition 5.2. The following expansions holds for European Call and Put options with strike e k and e -k respectively:

(5.7) E(e Xt -e k ) + = 1 √ 2π bt(1 + ρ) 2µ * + -2µ * + bt(1 + ρ) k -1 2 e -(µ * + -1)k+d + 0 +mµ * + (1 + O(k -1 )), and
(5.8) E(e -k -e Xt ) + = 1 √ 2π bt(1 -ρ) 2µ * --µ * -bt(1 -ρ) k -1 2 e -(µ * -+1)k+d - 0 +mµ * -(1 + O(k -1 )),
with the same notations as in Proposition 5.1.

Proof. We have

C(t, k) = E(e Xt -e k ) + = e k Ñ (d(k)) -Ñ (d(k) - √ tσ(t, k)) = N ′ (d(k) - √ tσ(t, k) d(k) - √ tσ(t, k) (1 + O((d(k) - √ tσ(t, k)) 2 )) -e k N ′ (d(k)) d(k) (1 + O(d(k) 2 )) = 1 √ 2π e k-µ * + (k-m)+d + 0 +O(k -1 ) 1 d(k) - √ tσ(t, k) - 1 d(k) + O(k -3 2 ) = 1 √ 2π bt(1 + ρ) 2µ * + -2µ * + bt(1 + ρ) e k-µ * + (k-m)+d + 0 k -1 2 (1 + O(k -1 )).
The same analysis applies to Put option with small strike; we write

P (t, k) = E(e -k -e Xt ) + = e -k Ñ ( d(k)) -Ñ ( d(k) + √ tσ(t, -k)) = 1 √ 2π e -k-µ * -(k-m)+d - 0 +O(k -1 ) 1 d(k) - 1 d(k) + √ tσ(t, -k) -+O(k -3 2 ) = 1 √ 2π bt(1 -ρ) 2µ * --µ * + bt(1 -ρ) e -k-µ * -(k-m)+d - 0 k -1 2 (1 + O(k -1 )).
From this expansion we can derive the large asymptotics of the local volatility function. We emphasize that Dupire's local volatility is defined as

(5.9) Σ 2 (t, k) = 2∂ t E(e Xt -e k ) + (∂ kk -∂ k )E(e Xt -e k ) + = 2∂ t E(e k -e Xt ) + (∂ kk -∂ k )E(e k -e Xt ) + . and (5.10) Σ 2 (t, -k) = 2∂ t E(e Xt -e -k ) + (∂ kk + ∂ k )E(e Xt -e -k ) + = 2∂ t E(e -k -e Xt ) + (∂ kk + ∂ k )E(e -k -e Xt ) + .
Corollary 5.1. Suppose that all prices of European Call options can be recovered from the SVI parametrization (5.1). Then the local volatility function is given by

Σ 2 (t, ±y) = -2∂ t µ * ± (t) µ * ± 2 (t) ∓ µ * ± (t) (y - 2µ * ± ∓ 1 2(µ * ± 2 ∓ µ * ± ) )) + ∂ t c ± (t) + O(y - 1
2 ), where c ± (t) = ln( 1

√ 2π √ bt(1±ρ) 2µ * ± - √ 2µ * ± bt(1±ρ)
) + d ± 0 + mµ * ± , with the same notations as in Proposition 5.1.

Proof. It follows from (5.7), (5.7), (5.9) and (5.10)

Remark 5.1. This result could have been proved differently using Theorem 3.1 and Proposition 5.1.

Heston's model and Stein-Stein model

In

d(ln(S t )) = - V t dt + V t dB t , dV t = (a -bV t )dt + σ V t dW t , d W, B t = ρdt. (6.1)
This model is known to have the moment explosion property. In the case of negative correlation, Friz and Gerhold [START_REF] Friz | Extrapolation analytics for Dupire's local volatility[END_REF] derive an equivalence for the local volatility corresponding to the Heston model as

Σ 2 (t, y) ∼ 2 µ * + (t)(µ * 1 -1)R 1 (t)R 2 (t) y, as k → ∞.
with an explicit expression for R 1 and R 2 . It is clear that Theorem 3.1 gives much more accurate expansion for the local volatility. It also applies to all cases (:k → -∞ and ρ > 0). In order to apply Theorem 3.1 to the Heston model, we need first to calculate the moment generating function of X t = ln(S t /S 0 ). For µ > 0 we have

E e µXt = E e -µ 2 t 0 Vsds+µρ t 0 √ VsdWs+µ √ 1-ρ 2 t 0 √ VsdW 2 s = E e µ 2 (1-ρ 2 )-µ 2 t 0 Vsds+µρ t 0 √ VsdWs E e µ √ 1-ρ 2 t 0 √ VsdW 2 s - µ 2 (1-ρ 2 ) 2 t 0 Vsds (W s ) s≤t = E e µρ t 0 √ VsdWs-µ 2 ρ 2 2 t 0 Vsds e µ 2 -µ 2 t 0 Vsds = E Q e µ 2 -µ 2 t 0 Vsds ,
where we used the law of iterated conditional expectation and the fact that V s is measurable with respect to W . The last inequality is a consequence of Girsanov theorem, where under Q, the process V satisfies the stochastic differential equation (6.2)

dV t = (a -(b -ρσµ)V t ) dt + σ V t dW Q t with Q-Brownian motion W 1,Q .
Calculating E e µXt is then reduced to the calculation of the moment generating function of the time average of the CIR process V under Q; this is given explicitly in [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF] by

E Q e µ 2 -µ 2
t 0 Vsds = e aϕ(t;µ)+v 0 ψ(t;µ) , where ϕ(t; µ) = t 0 ψ(s; µ)ds and (6.3)

ψ(t) =          ψ 1 (t; µ) := b-ρσµ σ 2 - √ c 1 (µ) σ 2 c 2 (µ)e √ c 1 (µ)t +1 c 2 (µ)e √ c 1 (µ)t -1 , if 0 ≤ µ 2 -µ 2 ≤ (b-ρσµ) 2 2σ 2 , ψ 2 (t; µ) := b-ρσµ σ 2 + √ -c 1 (µ) σ 2 tan -c 1 (µ) t 2 + arctan( -b+ρσµ √ -c 1 (µ) ) , if µ 2 -µ 2 > (b-ρσµ) 2 2σ 2
, where c 1 and c 2 are defined by (6.4)

c 1 (µ) = (b -ρσµ) 2 -σ 2 (µ 2 -µ), and c 2 (µ) = b -ρσµ + c 1 (µ) b -ρσµ -c 1 (µ) .
The function ϕ(.) is given by (6.5)

ϕ(t) =                ϕ 1 (t; µ) := b-ρσµ+ √ c 1 (µ) σ 2 t -2 σ 2 ln c 2 (µ)e √ c 1 (µ)t -1 c 2 (µ)-1 , if 0 ≤ µ 2 -µ 2 ≤ (b-ρσµ) 2 2σ 2 , ϕ 2 (t; µ) := b σ 2 t -2 σ 2 ln   cos √ -c 1 (µ) t 2 +arctan( -b+ρσµ √ -c 1 (µ) ) cos arctan( -b+ρσµ √ -c 1 (µ) )   , if µ 2 -µ 2 > (b-ρσµ) 2 2σ 2
. Remark 6.1. We deliberately did not consider the case µ 2 -µ 2 < 0. The reason is that when µ 2 -µ 2 < 0, the ψ(t, µ) might take a third form depending on other parameters (see Appendix A in [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF]). This, however, will not affect our analysis as µ 2 -µ 2 < 0 happens only when 0 < µ ≤ 1. As it is well known that S is a true martingale, the "positive" critical moment of X is always larger than 1.

From (6.3) and (6.5) we clearly see that there exists µ * + (t) > 1 (resp µ * -(t) > 0) such that the moment generating function of X t (resp. -X t ) is finite between 0 and µ * + (t) (resp. µ * -(t)) and explodes at µ * + (t) > 0 (resp. µ * -(t)). The next result is nothing but rewriting the moment generating function of X t and -X t in terms of (6.3) and (6.5). Proposition 6.1. Define μ+ and μ-by

(6.6) μ+ = min {µ > 0 : c 1 (µ) = 0} = σ 2 -2bρσ + (σ 2 -2bρσ) 2 + 4b 2 σ 2 (1 -ρ 2 ) 2σ 2 (1 -ρ 2 ) , and (6.7)
μ-= min {µ > 0 : c 1 (-µ) = 0} = -(σ 2 -2bρσ) + (σ 2 -2bρσ) 2 + 4b 2 σ 2 (1 -ρ 2 ) 2σ 2 (1 -ρ 2 ) .
where c 1 is defined by (6.4). For any µ > 0 we have (6.8) E e µXt 1 µ≥1 = e aϕ 1 (t;µ)+v 0 ψ 1 (t;µ) 1 1≤µ≤μ + ∧µ * + (t) + e aϕ 2 (t;µ)+v 0 ψ 2 (t;µ) 1 μ+ <µ<µ * + (t) , and (6.9) E e -µXt 1 µ>0 = e aϕ 1 (t;-µ)+v 0 ψ 1 (t;-µ) 1 0<µ<μ -∧µ * -(t) + e aϕ 2 (t;-µ)+v 0 ψ 2 (t;-µ) 1 μ-<µ<µ * -(t) , where ϕ 1,2 and ψ 1,2 are given in (6.3) and (6.5) Proof. This follows immediately from (6.3) and (6.5) and the fact that for µ > 1 or µ < 0, we have 0

≤ µ 2 -µ 2 and µ 2 -µ 2 < (b -ρσµ) 2 2σ 2 ⇐⇒ c 1 (µ) > 0.
The moment generating function of X and -X near the critical moments µ * + and µ * -are given in the following result. Proposition 6.2. Denote Λ ± (t, µ) ln Ee ±µXt . For µ sufficiently close to µ

* + (t) (resp. µ * -(t)) Λ ± (t, µ) is given as (6.10) Λ ± (t, µ) = ω ± (t) µ * ± (t) -µ + 2a σ 2 ln( 1 µ * -µ ) + m ± (t) + i≥1 d ± i (t)(µ * ± (t) -µ) i
Proof. See Theorem 3.2 in [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF]. A similar result is found in [START_REF] Friz | On Refined Volatility Smile Expansion in the Heston Model[END_REF].

Remark 6.2. It is difficult, or even impossible to obtain an explicit expression for µ * + and µ * -. However, all other coefficients of (6.10) may be obtained explicitly (or numerically) in terms of the model parameters and µ * ± ; to calculate numerically, one needs just to see them as

ω ± (t) = lim µ→µ * ± (t) (µ * ± (t) -µ) Λ ± (t, µ) - 2a σ 2 ln( 1 µ * -µ ) , m ± (t) = lim µ→µ * ± (t) Λ ± (t, µ) - ω ± (t) µ * ± (t) -µ - 2a σ 2 ln( 1 µ * -µ
) , and for i ≥ 1,

d ± i (t) = lim µ→µ * ± (t) 1 (µ * -µ) i Λ ± (t, µ) - ω ± (t) µ * ± (t) -µ - 2a σ 2 ln( 1 µ * -µ ) -m ± (t)
We clearly see that the Heston model satisfies the assumption of Theorem 3.1. The next result gives a sharp asymptotic expansion of the local volatility under Heston's model. Proposition 6.3. The local volatility corresponding to the Heston model satisfies

(6.11) Σ 2 (t, ±y) = σ ± 0 (t)y + ω ′ ± (t)ν ± (y) + m ′ ± (t) ∓ q ∓ σ ± 0 (t)(1 ∓ 2µ * ± ± 1 ν± (y) )y/ν ± (y) 1 2 ((µ * ± -1 ν± (y) ) 2 ∓ (µ * ± -1 ν± (y) )) 1 -1 4 ν± (y) 2 2y 2 ν′ ± (y) + o(1), where σ ± 0 (t) = -2∂ t µ * ± (t) µ * ± 2 (t) ∓ µ * ± (t)
, and (6.12) ν± (t, y) = 1 ω ± (t) (y + a 2 σ 4 ω ± (t) )

1 2 -a σ 2 ω ± (t) .

Proof. The first statement of the proposition follows from a direct application of Theorem 3.1. For the approximation (6.12) we observe that ν ± (t, y) is the unique solution of

ω ± (t)x 2 + 2a σ x -d ± 1 - i≥1 (i + 1)d ± i+1 (t)x -i = y.
This can be approximated by taking the unique positive solution to ω ± (t)x 2 + 2a σ x = y : x = ν± (y) = 1

ω ± (t) (y + a 2 σ 4 ω ± (t) ) 1 2 - a σ 2 ω ± (t)
.

It is easy then to see that the error of this approximation of order y -1

2 ; that is

ν ± (t, y) = ν± (y) + O(y - 1
2 ). Choosing ν ± or ν± is then equivalent in the formula since the error is o(1). Remark 6.3. The error in the expansion (6.11) is actually O(y -1 4 ) which is much more precise than o(1) (the difference between this approximation and the actual local volatility tends to 0 as y → ∞ anyway). Indeed, the term o(1) comes from the product o( ν± (y) 2 2y 2 ν′ ± (y) ), which is equivalent to o(y -1

2 ) and the numerator of the second term in the right hand side of (6.11), which is ≈ y 1 2 . We can see by looking at Theorem 2.1 and its proof in [START_REF] Aly | From Moment Explosion to the Asymptotic Behavior of the Cumulative Distribution for a Random Variable[END_REF] and Remark 8 in [START_REF] Friz | On Refined Volatility Smile Expansion in the Heston Model[END_REF] that o( ν± (y) 2 2y 2 ν′ ± (y) ) has to be O(y -3 4 ). This will lead to an approximation of order O(y -1 4 ) instead of o(1) in (6.11). Remark 6.4. The expansion (6.11) does not use the coefficients d ± i as they appear only on higher order terms which are not covered by Theorem 3.1. There will be therefore no need to calculate them if the only goal is to apply Theorem 3.1.

A sharp asymptotic formula for the implied variance is easily obtained by applying Theorem 4.1. The result is similar to the result obtained in [START_REF] Gulisashvili | Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models[END_REF]. Proposition 6.4. The following expansion holds for the implied volatility under Heston's model This model was introduced by Stein and Stein in [START_REF] Stein | Stock price distributions with stochastic volatility: An analytic approach[END_REF]. In [START_REF] Deuschel | Marginal density expansions for diffusions and stochastic volatility, part II: Applications[END_REF], Deuschel et al derive an asymptotic formulae for the density of X t := ln(S t ) as f (x) = e -B 1 (t)x+B 2 (t) √ x-1 2 ln(x)+B 3 (t) (1 + O(x -1

(
2 )). From this we immediately have

P(X t ≥ x) = e -B 1 (t)x+B 2 √ x-1 2 ln(x)+B 3 (t)-ln(B 1 (t)) (1 + O(x - 1
2 )). We can then easily see that the (positive) moment generating function of X t : Ee We derive then a very similar expansion of the local volatility as well as the implied volatility (the positive wings) to the case of Heston's model by applying the formulas in Proposition 6.3 and Proposition 6.4.

6 . 1 .

 61 this section we consider the application of the results obtained in the previous sections to two examples of stochastic volatility model: Heston's model and Stein-Stein model Heston's model. The Heston model is defined by the stochastic differential equation:

µXt 1

 1 Xt>0 explodes at µ * t = B 1 and that Λ(µ) := ln Ee µXt =

  6.13) tσ 2 (t, ±k) = 4 Λ * Proof. It follows from a direct application of Theorem 4.1 by replacing Λ with Λ. It's worth noticing that Λ * ± (t, .) is the Fenchel-Legendre transform of Λ. 6.2. Stein-Stein model. The dynamics of the stock price in the Stein-Stein model is given by: dS t = µS t dt + |Y t |S t dW t , dY t = q(m -Y t )dt + σdZ t , d W, Z t = ρdt.

			c± t (k) 2	)( Λ * ± (t, k) +	c± t (k) 2	∓ k) + O(k	-1 2 ).
	where Λ *
	(6.14)	Λ± (t, µ) = µ ∂ k ν(t, k) ω ± (t) µ * ± (t) -µ + 2a σ 2 ln( ν2 (t, k) ) + 2 ln µ * ± (t) 2 -µ * ± (t) 2	1 ∓2 + 4µ * ± (t) -4 µ * ± (µ * ± ∓ 1)

± (t, k) + 2c ± t (k) ∓ 2k -4 ( Λ * ± (t, k) + ± (t, k) = (µ * ± (t) -1 ν± (t,k) )k -Λ(t, µ * ± (t) -1 ν± (t,k)

), with ν defined by (6.12), Λ is defined by

* -µ ) + m ± (t),

where ω ± , m ± are given in Proposition 6.2 and c± t (k) is given by c± t (k) = -ln(k


Mohamed Sidi

Aly

FROM MOMENT EXPLOSION TO THE ASYMPTOTIC BEHAVIOR OF THE CUMULATIVE DISTRIBUTION FOR A RANDOM VARIABLE

Keywords: 2000 Mathematics Subject Classification : 60E10, 62E20, 40E05 Regular variation, Tauberian theorems, Moment generating function, Tail asymptotic, CIR process

We study the Tauberian relations between the moment generating function (MGF) and the complementary cumulative distribution function of a random variable whose MGF is finite only on part of the real line. We relate the right tail behavior of the cumulative distribution function of such a random variable to the behavior of its MGF near the critical moment. We apply our results to an arbitrary superposition of a CIR process and the time-integral of this process.

Introduction

Tauberian theory is a powerful tool to deal with the problem of obtaining asymptotic information about a function from a priori knowledge of the asymptotic behavior of some integral transform of the function. The study of Tauberian type theorems has been historically stimulated by their potential applications in diverse fields of mathematics. In mathematical finance, they allow to derive closed form approximate solutions for large values asymptotic analysis.

The present paper investigates the Tauberian relations between the moment generating function of a random variable and its complementary cumulative distribution. Intuitively, if we assume that we have a positive random variable Z whose density is given by P Z (x) = 1 ν e -νx 1 x≥0 , then it is easy to see that MGF of Z is only finite on [0, ν[. It is however much more difficult to prove the other way: i.e. if the MGF of a random variable is finite only on part of the real line THEN the density (or the complementary cumulative distribution function) is exponential. This paper derives sufficient condition under which this last statement is true. Furthermore we show that we only need the asymptotic behavior of MGF near the critical moment to obtain the right tail asymptotics of the random variable.

We consider a real-valued random variable Z whose moment generating function (MGF : µ -→ E e µZ ) is finite only on part of the real line and explodes at some critical moment µ * < ∞. We investigate the link between the behavior of MGF near µ * and the (right) tail distribution of Z. In [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF], Benaim and Friz develop criteria, checkable by looking at the moment generating function of Z, which guarantees that ln P(Z > x) ∼µ * x, or equivalently (1.1) lim

x→∞ ln P(Z > x) x = -µ * .
A similar statement, but under stronger conditions, appears in [START_REF] Nakagawa | Application of Tauberian theorem to the exponential decay of the tail probability of a random variable[END_REF]. The purpose of this paper is to show that a sharp asymptotic formula of the form (1.2) lim sup

x→∞ ln P(Z > x) -h(x) log(x) = c
can be derived by only looking at the moment generating function of Z near µ * . The link between the asymptotics of a function f (which here represents the density or the complementary cumulative distribution of the random variable) and its integral transforms (Laplace-Stieltjes transform for example : f (x) := ∞ 0 e -su df (u)) is the main purpose of Abelian (: results in which we pass from a function to its integral) and Tauberian (from integrals to functions) theorems. For example, Karamata's Taberian theorem gives the equivalence between f being regularity varying and f (1/.) being so, under some suitable conditions known as "Tauberian conditions" (see eg. [START_REF] Bingham | Regular Variation[END_REF] Theorem 4.12.7). For the Tauberian theorems regarding the moment generating functions, Kasahara's Tauberian theorem (cf. [START_REF] Bingham | Regular Variation[END_REF] Theorem 4.12.1) links the regular variation property of log Ee xZ with that of log P(Z > x), as x → ∞, but it is only applicable when the moment generating function is finite for every real number. In fact, the oldest results we could find in the vast literature of Tauberian theory that relates the tail behavior of Z to the behavior of MGF near µ * are the relatively recent works [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF] and [START_REF] Nakagawa | Application of Tauberian theorem to the exponential decay of the tail probability of a random variable[END_REF] mentioned above.

Our study is motivated by applications in finance, in particular by the study of models for financial securities whose returns have a known moment generating functions, but whose distribution functions are unknown. It is now well known that MGF explode for a large class of distributions used for modeling returns in finance (see eg. [START_REF] Andersen | Moment explosions in stochastic volatility models[END_REF]). On the other hand, Lee's celebrated moment formulae [START_REF] Lee | The moment formula for implied volatility at extreme strikes[END_REF] establishes an explicit link between µ * and the large-strike asymptotics of the Black-Scholes implied volatility. And more recently, Benaim and Friz related the asymptotic behavior of the tails of the distribution functions of returns to the small-strike and large-strike asymptotics of the implied volatility.

The main part of this work is devoted to show a Tauberian Theorem relating the right tail behavior of the cumulative distribution function of Z to the behavior of the MGF near µ * . As an application of our result, we give sharp asymptotic formulas for any superposition of a CIR process and its time-integral. Our results can also be applied to some time-changed Lévy models, where several examples have been given in [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF].

This paper is organized as follows: In section 2 we give the Tauberian relation between the Laplace transform of a random variable and the complementary cumulative distribution function of this variable. In section 3 we give an application to the CIR process. We give some proofs in the appendix.

Moment explosion and distribution tails

Throughout this paper we consider a random variable Z for which there exists µ * > 0 such that E e µZ < ∞, for any µ ∈ [0, µ * [ and lim µ→µ * E e µZ = +∞. Our purpose is to derive a sharp asymptotic formula for the cumulative distribution function of Z. An upper bound for this distribution can be easily derived using Markov argument:

(2.1) P(Z > x) = P(e µZ > e µx ) ≤ e -µx+Λ(µ) , whenever µ > 0. It follows that (2.2)

P(Z > x) ≤ e -sup 0<µ<µ * (µx-Λ(µ)) = e -Λ * (x) ,
where

(2.3) Λ(p) := ln E e pZ
and Λ * is the Fenchel-Legendre transform of Λ. We suspect this upper bound to be a good approximation of the P(Z > x). The main purpose of this paper is to investigate how this upper bound is far from the distribution function under suitable conditions. Let's define the function

(2.4) ϕ(x) : x -→ ln E e (µ * -1 x )Z = Λ(µ * - 1 x )
In [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF], it's shown that if ϕ is regularly varying, then ln P(Z > x) ∼ x→∞ -µ * x : lim

x→∞ ln P(Z > x) + µ * x x = 0.
The main result of this paper shows that if we assume that ϕ is regularly varying and that ϕ is sufficiently differentiable-monotone at infinity (Assumption 1) then the Fenchel-Legendre transform of Λ gives a very sharp expansion of the complementary cumulative distribution of Z.

2.1. Regular variation theory. Let us first recall briefly some facts from the theory of regular variation. Let f : [0, ∞[-→ [0, ∞[ be a measurable function. We say f has regular variation of order α at infinity, U ∈ R α , if there exists a real number α such that for every x > 0,

(2.5) lim t→∞ f (tx) f (t) = x α .
The definition may be extended to allow α = ±∞ where

x ±∞ = lim t→±∞ x t .
To see which function f can satisfy (2.5), we recall the representation-characterization theorem, which can be found eg. in [START_REF] Bingham | Regular Variation[END_REF] (cf. Threorem 1.3.1 and Threorem 1.4.1)

Theorem 2.1 (Representation-Characterization Theorem). The function f > 0 is measurable such that (2.5) holds for all x > 0 if and only if it may be written in the form

(2.6) f (x) = x α exp c(x) + x a ǫ(u) u du (x > a)
for some a > 0, where c(.) is measurable and c(x

) → c ∈ R, ǫ(x) → 0 as x → ∞.
In particular every regularly varying function with index α can be written as

f (x) = x α l(x)
, where l is slowly varying (l ∈ R 0 ).

For a function f defined, locally bounded on [X, ∞[, for some X > 0, and such that lim x→∞ f (x) = +∞, the generalized inverse of f ), ∞[ and is monotone increasing to ∞. In particular, if f ∈ R α with α > 0, then Theorem 1.5.12 in [START_REF] Bingham | Regular Variation[END_REF] asserts that f ← ∈ R 1 α .

f ← (y) := inf {x ∈ [X, ∞[ : f (x) > y} is defined on [f (X

Function with smooth variation.

A positive function g defined on [X, ∞[, for some X sufficiently large, varies smoothly with index α > 0 (we denote g

∈ SR α ) iff h(x) := log(g(e x )) is C ∞ and h ′ (x) -→ α, h (k) (x) -→ 0 for n = 2, 3, . . . as x → ∞.
In particular SR α ⊂ R α . (for more of this, see [START_REF] Bingham | Regular Variation[END_REF] Section 1.8). The next result will be useful later in this article Proposition 2.1. Let α > 0 and g ∈ SR α . Then we have

(2.7) g ′ (x) 1 x g(x) -→ α, g ′′ (x) 1 x 2 g(x) -→ -α + α 2 , as x → ∞.
Proof. We have, by definition of SR α ,

∂ ∂ x log(g(e x )) = e x g ′ (e x ) g(e x ) = h ′ (x).
It follows that g ′ (x)

1 x g(x) -→ α, as x → ∞. Differentiating g ′ (x) = g(x)
x h ′ (log(x)) we have

g ′′ (x) = g(x) x 2 -h ′ (log(x)) + g ′ (x)x g(x) h ′ (log(x)) + h ′′ (log(x)) = 1 x 2 g(x) -h ′ (log(x)) + h ′2 (log(x)) + h ′′ (log(x)) . Thus g ′′ (x) 1 x 2 g(x) -→ -α + α 2 , as x → ∞.
2.3. New Tauberian result. We next present our main result concerning the asymptotic behavior of the cumulative distribution of a random variable Z satisfying Assumption 1.

Assumption 1. There exist µ * > 0 and α > 0 such that

• ∀µ ∈ [0, µ * [, E e µZ < +∞, • the function ϕ(x) : x -→ ln E e (µ * -1 x )Z is R α and C 1 ([X, ∞[), for some X,
sufficiently large, such that the function

x -→ x 2 ϕ ′ (x) is R α+1 and monotone increasing to ∞. • The function ǫ : x -→ [(.) 2 ϕ ′ (.)] ← (x) is smoothly varying with index 1
α+1 . We emphasize that the main idea in this paper is to investigate how the upper bound (2.2) is far from the distribution function. Let us here consider the function p * defined such that:

Λ * (x) = p * (x)x -Λ(p * (x)).
We can easily see that under Assumption 1, p * can be written as

(2.8) p * : x ∈ [X, ∞[ -→ Λ ′ ← (x) = µ * - 1 [(.) 2 ϕ ′ (.)] ← (x)
.

Assumption 1 ensures that the function p * is twice-differentiable and its derivatives are monotonic for large x. The hypothesis that ϕ has regular variation appears also in [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF] and is sufficient to derive a statement like (1.1) for the distribution tail of Z. Here we add the hypothesis of differentiability only in order to ensure the differentiability of p * and to obtain the asymptotic expansion of its derivatives. This assumption is satisfied by a large family of function with regular variation. For example we can easily check that the case ϕ(x) = x α satisfies all the criteria of Assumption 1. It can also be proved that the case ϕ(x) = x α + f (x) where f varies smoothly with index β < α satisfies the assumption as well. In particular, in all examples treated in this paper, the function ϕ takes the latter form.

We next present our main Tauberian result which relates the behavior of the the right tail behavior of the complementary cumulative distribution function of Z to the behavior of the MGF near µ * . But first we present a kay lemma that will be crucial to prove our main theorem: Lemma 2.1. Let Assumption 1 hold for some random variable X. Denote Λ(p) := ln Ee pX and let x be large enough. Then for any γ ∈ R and β ∈]0, 1[, we have

(2.9) ∞ x β-1 z γ e ψx(z) dz = x -α 2(α+1) f (x),
where f is slowly varying (f ∈ R 0 ) and

(2.10) ψ x (z) := (p * (x) -p * (xz)) xz + Λ(p * (xz)) -Λ(p * (x)).
Proof. We split the integral into two parts:

I(γ) := 1 R β-1
z γ e ψx(z) dz and J(γ

) := ∞ 1
z γ e ψx(z) dz.

We will first show that (2.11)

I(γ) ∼ f 1 (x)x -α 2(α+1) , f 1 ∈ R 0 . The function z ∈]0, 1] -→ ψ x (z) is decreasing with ψ x (0) = -Λ(p * (x)
) and ψ x (1) = 0. In particular, for any h ∈ [0, 1], we have

I(γ) - 1 1-h z γ e ψx(z) = I(γ) - 1 1-h z γ e ψx(z) = 1-h x β-1 z γ e ψx(z) ≤ e ψx(1-h) 1-h x β-1 z γ dz .
On the other hand, ψ x is three times differentiable and its derivatives are given by (2.12)

ψ x ′ (z) = (p * (x) -p * (xz)) x, ψ (n)
x (z) = -x n p * (n-1) (xz), n = 2, 3. It follows that for h sufficiently small, we have

ψ x (1 -h) = ψ x (1) -ψ x ′ (1)h + ψ x ′′ (1) h 2 2 -ψ x (3) (1) h 3 6 + . . . = -x 2 p * ′ (x) h 2 2 + x 3 p * ′′ (x) h 3 6 + . . .
In particular, we have

sup z∈[0,h] ψ x (1 -z) + x 2 p * ′ (x) z 2 2 ≤ h 3 6 x 3 sup z∈[1-h,1] |p * ′′ (zx)| = h 3 6 x 3 |p * ′′ (x(1 -h))|. (2.13)
Under assumption 1, we have p

* (x) = µ * -1 ǫ(x) , where ǫ(x) = [(.) 2 ϕ ′ (.)] ← (x) ∈ SR 1 α+1
. Using Proposition 2.1 we can easily show that there exist f 0 , g 0 ∈ R 0 (slowly varying) such that

x 2 p * ′ (x) ∼ x α α+1 f 0 (x) and x 3 |p * ′′ (x)| ∼ x α α+1 g 0 (x). Indeed as ǫ ∈ SR 1 α+1 ⊂ R 1 α+1
, there exists a slowly varying function f such that ǫ(x) = x 1 α+1 f (x). On the other hand, using Proposition 2.1 , which gives ǫ ′ (x)

1 x ǫ(x) ∼ 1 α+1 and ǫ ′′ (x) 1 x 2 ǫ(x) ∼ -1 α+1 + ( 1 α+1 ) 2 , we have p * ′ (x) = ǫ ′ (x) ǫ 2 (x) = ǫ ′ (x) 1 x ǫ(x) 1 xǫ(x) ∼ 1 α + 1 1 xǫ(x) = 1 x 1+ 1 α+1 1 (α + 1)f (x)
,

where the function f 0 (x) := 1 (α+1)f (x) is slowly varying. Similarly, we have

p * ′′ (x) = ǫ ′′ ǫ 2 -2 ǫ ′ 2 ǫ 3 = 1 x 2 ǫ ( ǫ ′′ 1 x 2 ǫ -2( ǫ ′ 1 x ǫ ) 2 ) ∼ c 1 x 2 ǫ , where c = -1 α+1 + ( 1 α+1 ) 2 -2( 1 α+1 ) 2 = -1 α+1 -( 1 α+1 ) 2 . Hence sup z∈[0,h] ψ x (1 -z) + x 2 p * ′ (x) z 2 2 ≤ c 1 g 0 (x) h 3 3 x α α+1 , (2.14) with a constant c 1 > 0. Let's set (2.15) h ≡ h(x) = x -5 12 α α+1 (2f 0 (x)) 1 2
.

For this particular h, we have

sup z∈[0,h] ψ x (1 -z) + x 2 p * ′ (x) z 2 2 ≤ c 1 x -1 4 α α+1 g 0 (x) (f 0 (x)) 3 2
.

It follows that

ψ x (1 -h) = -x 2 p * ′ (x) h 2 2 + O 0 (x -1 4 α α+1 g 0 (x) (f 0 (x)) 3 2
) ∼ -x

I(γ) - 1 1-h z γ e ψx(z) = 1-h x β-1
z γ e ψx(z) ≤ e ψx(1-h)

1-h x β-1 z γ dz ≤ e -x 1 6 α α+1 1-h x β-1 z γ dz ≤ Q(x)e -x 1 6 α α+1 ,
for some polynomial function Q. On the other hand, we have, using (2.14),

1 1-h z γ e ψx(z) - 1 1-h z γ e -x 2 p * ′ (x) (1-z) 2 2 dz ≤ 1 1-h z γ e -x 2 p * ′ (x) (1-z) 2 2 dz e h 3 6 x 3 |p * ′′ (x(1-h))| -1 .
We emphasize that

h 3 6 x 3 |p * ′′ (x(1 -h))| = x -1 4 α α+1 g 1 (x) with g 1 ∈ R 0 . It follows that I(γ) ∼ 1 1-h z γ e ψx(z) ∼ 1 1-h z γ e -x 2 p * ′ (x) (1-z) 2 2 dz = h(x) 0 (1 -z) γ e -x 2 p * ′ (x) z 2 2 dz ∼ h(x) 0 e -x 2 p * ′ (x) z 2 2 dz ∼ 1 x 2 p * ′ (x) h(x) √ x 2 p * ′ (x) 0 e -z 2 /2 dz.
Here f (x) ∼ g(x) means that lim x→∞ f (x) g(x) = 1. Now by definition of h we have for x sufficiently large

h(x) x 2 p * ′ (x) = x -5 12 α α+1 (2f 0 (x)) 1 2 × x α 2(α+1) f 0 (x) = x 1 12 α α+1 . Hence h(x) x 2 p * ′ (x) -→ +∞ as x → ∞. Thus (2.16) I(γ) ∼ 1
x 2 p * ′ (x) = x -α 2(α+1) 1 f 0 (x)
.

We can show by the same way that (2.17)

J(γ) := ∞ 1 z γ e ψx(z) dz = x -α 2(α+1) g(x), whee g ∈ R 0 .
The main Tauberian theorem of this paper is the following: Proof. The upper bound is obtained using (2.2), as a consequence of Markov's inequality:

(2.20) lim sup

x→∞ ln P(Z > x) + Λ * (x) ln x ≤ 0.
Lower Bound for 'limsup': In order to derive a lower bound for the lim sup in (2.18) we proceed as follows. We suppose that lim sup

x→∞ ln P(Z > x) + Λ * (x) ln(x) = ν * < - α + 2 2(α + 1)
.

We will show that this will lead to a contradiction. First, we can easily see that e pZ can be written in terms of any c > 0 as It follows that for any p < µ * , β ∈]0, 1[ and x sufficiently large,

E e pZ = E(e p(x β ∧Z) ) + pE ∞ x β e pk 1 Z>k dk = E(e p(x β ∧Z) ) + p ∞ x β e pk P(Z > k)dk ≤ e px β + p ∞ x β
e pk e -Λ * (k)+ν ′ ln(k) dk,

whenever ν ′ ∈]ν * , -α+2 2(α+1) [. Choosing p = p * (x) we have E e p * (x)Z ≤ e p * (x)x β + p * (x)e Λ(p * (x)) x ν ′ +1 ∞ x β-1 z ν ′ e ψx(z) dz, where (2.22) ψ x (z) := (p * (x) -p * (xz)) xz + Λ(p * (xz)) -Λ(p * (x)).
Now for β = α 2(1+α) and using Lemma 2.1 we have, for x sufficiently large,

∞ x β-1 z γ e ψx(z) dz = x -α 2(α+1) f (x),
where f is slowly varying. Hence (α+1) . On the other hand we have Indeed, using (2.8) we have

e Λ(p * (x)) ≤ e p * (x)x β + p * (x)f (x) e Λ(p * (x)) x ν ′ +1 x -α 2
p * (x) = Λ ′ ← (x) = µ * - 1 [(.) 2 ϕ ′ (.)] ← (x) , with ϕ ∈ R α . We emphasize that ϕ(x) := Λ(µ * -1 x ). It follows that Λ(p * (x)) = ϕ [(.) 2 ϕ ′ (.)] ← (x) . Assumption 1 implies that {x -→ x 2 ϕ ′ (x)} ∈ R α+1 and [(.) 2 ϕ ′ (.)] ← (.) ∈ R 1 α+1 . It follows that Λ(p * (.)) ∈ R α α+1 and hence lim x→∞ p * (x)x β
Λ(p * (x)) = 0. Now using (2.23) we have, for x sufficiently large,

e Λ(p * (x)) ≤ 2f (x) e Λ(p * (x)) x ν ′ +1 x -α 2(α+1) . Which is impossible as ν ′ < -α+2 2(α+1) = -1 + α 2(α+1) . Thus (2.24) lim sup x→∞ ln P(Z > x) + Λ * (x) ln(x) ≥ - α + 2 2(α + 1)
.

Upper Bound for 'liminf': Suppose that lim inf

x→∞ ln P(Z > x) + Λ * (x) ln(x) = ν * > - α + 2 2(α + 1)
.

We obtain, by the same way as before,

e Λ(p * (x)) = E e p * (x)Z ≥ p * (x)e Λ(p * (x)) x ν ′ +1 ∞ x β-1 z ν ′ e ψx(z) dz ∼ C f (x)e Λ(p * (x)) x ν ′ +1 x -α 2(α+1) , whenever ν ′ ∈] -α+2 2(α+1) , ν * [, which is yet impossible. Thus (2.25) lim inf x→∞ ln P(Z > x) + Λ * (x) ln(x) ≤ - α + 2 2(α + 1)
. x c . Hence Assumption 1 holds here. Our result applies also to the NIG model as well as for double exponential model (see section 5 in [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF] for more details).

Application: CIR process

In this section, we study the asymptotic behavior of the cumulative distribution of a superposition of the CIR process and its time integral. The CIR process has been proposed by Cox, Ingersoll and Ross [START_REF] Cox | A Theory of the Term structure of Interest Rates[END_REF] to model the evolution of interest rates. It is defined as the unique solution of the following stochastic differential equation:

(3.1) dV t = (a -bV t )dt + σ V t dW t , V 0 = v,
where a, σ, v ≥ 0 and b ∈ R (see [START_REF] Ikeda | Stochastic Differential Equations and Diffusion Processes[END_REF] for the existence and uniqueness of the SDE). The density transition V t is known and is given by (cf. [START_REF] Lamberton | Introduction to Stochastic Calculus Applied to Finance[END_REF])

(3.2) P t (v, z) = e bt 2c t ze bt v ν-1 2 exp - v + ze bt 2c t B ν 1 c t √ vze bt ,
where c t = (e bt -1)σ 2 4b

, ν = 2a/σ 2 and B ν is the modified Bessel function of order ν. We set

(3.3) I t = t 0 V u du.
Gulisashvili and Stein [START_REF] Gulisashvili | Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models[END_REF] (cf Lemma 6.3 and Remark 8.1) show that I t admits a distribution density (see also Dufresne [6] for a similar result). A sharp asymptotic formulas for the distribution density of I t is obtained in [START_REF] Gulisashvili | Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models[END_REF] (cf Theorem 2.4) as

(3.4) p I t (y) = A t e -Cty+Bt √ y y ν 2 -3 4 1 + O(y -1 2 ) ,
where A, B and C are positive constants. A similar result exits also for the cumulative distribution of the stock price in Heston model (see Friz et al [START_REF] Friz | On Refined Volatility Smile Expansion in the Heston Model[END_REF]). We will show that a similar formula can be obtained for any superposition of the CIR process and its time integral.

Moment generating function of a superposition of V and I:

To compute the moment generating function of a superposition of V and I, we use the additivity property of V with respect to (a, v). Indeed, if we fix parameters b, σ and denote by V v,a t the solution of (3.1) corresponding to parameters (a, b, v, σ) in which W is replaced by an independent Brownian motion, then for any (v 1 , v 2 ), (a 1 , a 2 ) ∈ R 2 + , the process V v 1 +v 2 ,a 1 +a 2 has the same law as

(V v 1 ,a 1 t + V v 2 ,a 2 t
). This additivity property of CIR processes is well known; it follows from the fact that a CIR process is a squared Bessel process subjected to a deterministic change of time, and from the additivity property of squared Bessel processes, which is a celebrated result of Shiga and Watanabe (see [START_REF] Shiga | Bessel diffusions as a one-parameter family of diffusion processes[END_REF]). As consequence of this property, if we define F a λ 1 ,λ 2 (t, v), for λ 1 and λ 2 ∈ R, by

F a λ 1 ,λ 2 (t, v) = E e λ 1 V v,a t +λ 2 It , we have F a 1 +a 2 λ 1 ,λ 2 (t, v 1 + v 2 ) = F a 1 λ 1 ,λ 2 (t, v 1 )F a 2 λ 1 ,λ 2 (t, v 2 ). It follows that the function F a λ 1 ,λ 2 is given by F a λ 1 ,λ 2 (t, v) = e aϕ λ 1 ,λ 2 (t)+vψ λ 1 ,λ 2 (t)
, where the functions ϕ λ 1 ,λ 2 (t) and ψ λ 1 ,λ 2 (t) do not depend on a and v. Under the assumption (λ 2 ≤ b 2 2σ 2 and 2a ≥ σ 2 ), an explicit formulas for the Laplace transform of the joint distribution of a CIR process and its time-integral is obtained in [START_REF] Hurd | Explicit formulas for Laplace transforms of stochastic integrals[END_REF]. The next result gives the explicit formulas for ϕ and ψ without any restrictions on the parameters of the CIR process.

Theorem 3.1. For λ 1 , λ 2 ∈ R, let ψ λ 1 ,λ 2 be the maximal solution of

(3.5)    ψ ′ λ 1 ,λ 2 (t) = σ 2 2 ψ 2 λ 1 ,λ 2 (t) -2 b σ 2 ψ λ 1 ,λ 2 (t) + 2 λ 2 σ 2 , ψ λ 1 ,λ 2 (0) = λ 1 , defined over [0, t * λ 1 ,λ 2 [, with t * λ 1 ,λ 2 ∈]0, ∞] such that if t * λ 1 ,λ 2 < ∞ then lim t→t * λ 1 ,λ 2 ψ λ 1 ,λ 2 (t) = +∞. Also let ϕ λ 1 ,λ 2 be the time integral of ψ λ 1 ,λ 2 (.): ϕ λ 1 ,λ 2 (t) := t 0 ψ λ 1 ,λ 2 (u)du.
Then, for any T < t * λ 1 ,λ 2 , we have

(3.6) E e λ 1 V v,a T +λ 2 I T = e aϕ λ 1 ,λ 2 (T )+vψ λ 1 ,λ 2 (T ) Proof. Let's define F a λ 1 ,λ 2 (t, v) by F a λ 1 ,λ 2 (t, v) = e aϕ λ 1 ,λ 2 (t)+vψ λ 1 ,λ 2 (t) . The function F a λ 1 ,λ 2 is, by definition, the maximal solution of        ∂F a λ 1 ,λ 2 ∂t (t, v) = σ 2 2 v ∂ 2 F a λ 1 ,λ 2 ∂v 2 (t, v) + (a -bv) ∂F a λ 1 ,λ 2 ∂v (t, v) + λ 2 vF a λ 1 ,λ 2 (t, v), 0 < t ≤ T, F a λ 1 ,λ 2 (0, v) = e λ 1 v . This is indeed equivalent to ϕ λ 1 ,λ 2 (0) = 0, ψ λ 1 ,λ 2 (0) = λ 1 and aϕ ′ λ 1 ,λ 2 (t) + vψ ′ λ 1 ,λ 2 (t) = σ 2 2 vψ 2 λ 1 ,λ 2 (t) + (a -bv)ψ λ 1 ,λ 2 (t) + λ 2 v = aψ λ 1 ,λ 2 (t) + v σ 2 2 ψ 2 λ 1 ,λ 2 (t) -bψ(t) + λ 2 .
This system is equivalent to

ϕ ′ λ 1 ,λ 2 (t) = ψ λ 1 ,λ 2 (t) and ψ ′ λ 1 ,λ 2 (t) = σ 2 2 ψ 2 λ 1 ,λ 2 (t) -2 b σ 2 ψ λ 1 ,λ 2 (t) + 2 λ 2 σ 2 .
Hence for any

T < t * λ 1 ,λ 2 , the process e λ 2 s 0 V v,a u du F a λ 1 ,λ 2 (T -s, V v,a s )
s≤T is a positive local martingale; it is therefore a super-martingale. In particular, for any stopping time τ ∈ T 0,T , we have

E e λ 2 τ 0 V v,a u du F a λ 1 ,λ 2 (T -τ, V v,a τ ) ≤ F a λ 1 ,λ 2 (T, V v,a 0 ). Hence (3.7) sup τ ∈T 0,T E e λ 2 τ 0 V v,a u du+aϕ λ 1 ,λ 2 (T -τ )+V v,a τ ψ λ 1 ,λ 2 (T -τ ) ≤ e aϕ λ 1 ,λ 2 (T )+vψ λ 1 ,λ 2 (T ) .
We will show that there exists p > 1 such that sup

τ ∈T 0,T E e p(λ 2 τ 0 V v,a u du+aϕ λ 1 ,λ 2 (T -τ )+V v,a τ ψ λ 1 ,λ 2 (T -τ )) < +∞.
We will first show that ψ λ 1 ,λ 2 is increasing with respect to λ 1 and λ 2 . This means that if

λ 1 < λ1 (resp λ 2 < λ2 ), we have ψ λ 1 ,λ 2 (t) ≤ ψλ 1 ,λ 2 (t), for every t < t * ( λ1 , λ 2 ) (resp ψ λ 1 ,λ 2 (t) ≤ ψ λ 1 , λ2 (t)
, whenever t < t * (λ 1 , λ2 )). For this, we define the functions f and g by

f = ψλ 1 ,λ 2 -ψ λ 1 ,λ 2 and g = ψ λ 1 , λ2 -ψ λ 1 ,λ 2 .
We have

f (0) = λ1 -λ 2 > 0, g(0) = 0, f ′ (t) = σ 2 2 ψ 2 λ1 ,λ 2 (t) -ψ 2 λ 1 ,λ 2 - 2b σ 2 f (t) = σ 2 2 f (t) ψλ 1 ,λ 2 (t) + ψ λ 1 ,λ 2 - 2b σ 2 and g ′ (t) = σ 2 2 ψ 2 λ1 ,λ 2 (t) -ψ 2 λ 1 ,λ 2 - 2b σ 2 g(t) + 2 λ2 -λ 2 σ 2 ≥ σ 2 2 g(t) ψλ 1 ,λ 2 (t) + ψ λ 1 ,λ 2 - 2b σ 2 .
The functions f and g satisfy (f e A ) ′ (t) = 0, with f (0) > 0 and (ge A ) ′ (t) ≥ 0, with g(0) = 0, where

A ′ (t) = σ 2 2 ψλ 1 ,λ 2 (t) + ψ λ 1 ,λ 2 - 2b σ 2 . It follows that f ≥ 0 and g ≥ 0. Hence ψλ 1 ,λ 2 (t) -ψ λ 1 ,λ 2 (t) = ( λ1 -λ 1 ) e bt-t 0 (ψλ 1 ,λ 2 +ψ λ 1 ,λ 2 )(s)ds ,
whenever t < t * ( λ1 , λ 2 ). It follows that for every T < t * λ 1 ,λ 2 and for ǫ > 0 sufficiently small, there exists

λ ǫ 1 > λ 1 such that (1 + ǫ)ψ λ 1 ,λ 2 (t) ≤ ψ λ ǫ 1 ,λ 2 (t), ∀t ≤ T. So for any stopping time τ ∈ T 0,T , E e (1+ǫ)(λ 2 τ 0 V v,a u du+aϕ λ 1 ,λ 2 (T -τ )+V v,a τ ψ λ 1 ,λ 2 (T -τ )) ≤ e aǫϕ λ ǫ 1 ,λ ǫ 2 (T )+vψ λ ǫ 1 ,λ ǫ 2 (T ) ,
where a ǫ = (1 + ǫ)a and λ ǫ 2 = (1 + ǫ)λ 2 . Thus (3.8) sup

τ ∈T 0,T E e (1+ǫ)(λ 2 τ 0 V v,a u du+aϕ λ 1 ,λ 2 (T -τ )+V v,a τ ψ λ 1 ,λ 2 (T -τ )) < +∞.
In conclusion, the local martingale

M s := e λ 2 s 0 V v,a u du F a λ 1 ,λ 2 (T -s, V v,a s )
s≤T is uniformly integrable; it is therefore a true martingale. In particular, we have

E e λ 2 t 0 V v,a u du+aϕ λ 1 ,λ 2 (T -t)+V v,a t ψ λ 1 ,λ 2 (T -t) = e aϕ λ 1 ,λ 2 (T )+V v,a t ψ λ 1 ,λ 2 (T ) , whenever t ≤ T . Thus E e λ 2 T 0 V v,a u du+λ 1 V v,a
T = e aϕ λ 1 ,λ 2 (T )+V v,a t ψ λ 1 ,λ 2 (T ) .

Moment explosion. The function

F a λ 1 ,λ 2 is defined over [0, t * λ 1 ,λ 2 [ so that lim t→t * λ 1 ,λ 2 F a λ 1 ,λ 2 (t, v) = +∞ if t * λ 1 ,λ 2 < ∞. For λ 1 , λ 2 ∈ R, let us set t * λ 1 ,λ 2 (µ) = t * µλ 1 ,µλ 2 . Let us also define µ * -(t) and µ * + (t) by (3.9) µ * -(t) = -sup µ < 0 : t * λ 1 ,λ 2 (µ) = t and µ * + (t) = inf µ > 0 : t * λ 1 ,λ 2 (µ) = t , with sup ∅ = -∞ and inf ∅ = +∞. The moment generating function of λ 1 V t + λ 2 I t is defined over ] -µ * -(t), µ * + (t)[ and if µ * -(t) < ∞ (resp µ * + (t) < +∞) then lim µ→µ * -(t) E e -µ(λ 1 Vt+λ 2 It) = +∞ (resp. lim µ→µ * + (t) E e µ(λ 1 Vt+λ 2 It) = +∞).
When either µ * -(t) or µ * + (t) is finite, the behavior of the moment generating function near the critical moment is given in the next result, whose proof can be found in the appendix.

Theorem 3.2. Let t > 0 and λ 1 , λ 2 ∈ R such that max(λ 1 , λ 2 ) > 0 (resp. min(λ 1 , λ 2 ) < 0). Then µ * + (t) < +∞ (resp. µ * -(t) < +∞). Furthermore, ∃ω + t > 0 (resp. ∃ω - t > 0) such that µ -→ ln E e µ(λ 1 Vt+λ 2 t 0 Vudu) -H(µ) resp. µ -→ ln E e -µ(λ 1 Vt+λ 2 t 0 Vudu) -H(µ) is bounded near µ * + (t) (resp. µ * -(t) ), where H(µ) = ω + t µ * + (t)-µ -2a σ 2 ln 1 µ * + (t)-µ (resp. H(µ) = ω - t µ * -(t)-µ -2a σ 2 ln 1 µ * -(t)-µ ). Corollary 3.1. Let t > 0 and Z := λ 1 V t + λ 2 I t , for λ 1 , λ 2 ∈ R such that max(λ 1 , λ 2 ) > 0.
Then we have (3.10) lim sup

R→∞ ln P(Z > R) + µ * + (t)R -2 ω + t R ln(R) ∈ a σ 2 - 3 4 , a σ 2 and (3.11) lim inf R→∞ ln P(Z > R) + µ * + (t)R -2 ω + t R ln(R) ≤ a σ 2 - 3 4 .
Remark 3.1. A similar result can be obtained for the complementary cumulative distribution of the stock price in Heston model. The analysis relies on the explicit calculation of the moment generating function of X. Let's define the quantity (3.12) F p (t) = E e pXt , p ∈ R.

We have for all p ∈ R

F p (t) = E e -p 2 t 0 Vsds+pρ t 0 √ VsdW 1 s +p √ 1-ρ 2 t 0 √ VsdW 2 s = E e p 2 (1-ρ 2 )-p 2 t 0 Vsds+pρ t 0 √ VsdW 1 s E e p √ 1-ρ 2 t 0 √ VsdW 2 s - p 2 (1-ρ 2 ) 2 t 0 Vsds (W 1 s ) s≤t = E e pρ t 0 √ VsdW 1 s -p 2 ρ 2 2 t 0 Vsds e p 2 -p 2 t 0 Vsds = E Q e p 2 -p 2 t 0 Vsds ,
where we used the law of iterated conditional expectation and the fact that V s is measurable with respect to W 1 . The last inequality is a consequence of Girsanov theorem, where under Q, the process V satisfies the stochastic differential equation

(3.13) dV t = (a -(b -ρσp)V t ) dt + σ V t dW 1,Q t with Q-Brownian motion W 1,Q .
We are then reduced to the calculation of the moment generating function of the time average of the CIR process V under Q. It follows that F p (t) = e aϕp(t)+vψp (t) , where ϕ p (t) = t 0 ψ p (s)ds and ψ p is given by theorem 3.1.

Conclusions

We have presented a Tauberian result that relates the right tail behavior of the cumulative distribution function to the behavior of the moment generating function, for a random variable, whose moment generating function is finite only on part of the real line. We have shown that if the function ϕ : x -→ Λ(µ * -1

x ) := ln E e (µ * -1

x )Z is regularly varying, then the Fenchel-Legendre transform of Λ gives a log-equivalent (we say f and g are logequivalent if lim sup x→∞ f (x)-g(x) log(x)

= 0) of the logarithm of the complementary cumulative distribution, while the most recent result of this kind gives only an equivalence of type (lim x→∞ f (x)-g(x)

x = 0). Our results apply to financial models that generate distributions satisfying Assumption 1 (several examples can be found in [START_REF] Benaim | Smile Asymptotics II: Models with known moment generating function[END_REF]). We applied our results to an arbitrary superposition of the CIR process and its time integral. It would be interesting to have a similar result to Theorem 2.2 for R d random variables. Such a result allows more understanding of the joint distribution of CIR process and its time integral as well as the conditional distribution of the CIR with respect to its time average. This will be subject to further investigation.

Appendix A. Solution of (3.5) To solve (3.5), we write

ψ 2 λ 1 ,λ 2 (t) -2 b σ 2 ψ λ 1 ,λ 2 (t) + 2 λ 2 σ 2 = ψ λ 1 ,λ 2 (t) - b σ 2 2 + 2λ 2 σ 2 -b 2 σ 4 .
There are three situations : Finally, the solution of (3.5) is given by the following table , where β = √ 2λ 2 σ 2b 2 /σ 2 . and t * λ 1 ,λ 2 is given explicitly in terms of λ 1 and λ 2 by

Case 2λ 2 σ 2 < b 2 : Setting ψλ 1 ,λ 2 (t) = ψ λ 1 ,λ 2 (t) - b σ 2 and α = b 2 -2λ 2 σ 2 /σ
λ 2 < b 2 2σ 2    ψ λ 1 ,λ 2 (t) = b σ 2 -α Ce ασ 2 t +1 Ce ασ 2 t -1 , if |λ 1 -b σ 2 | > α, ψ λ 1 ,λ 2 (t) = b σ 2 -α Ce ασ 2 t
λ 2 < b 2 2σ 2 and λ 1 > b σ 2 + √ b 2 -2λ 2 σ 2 /σ 2 t * λ 1 ,λ 2 = 1 √ b 2 -2λ 2 σ 2 ln λ 1 -b/σ 2 + √ b 2 -2λ 2 σ 2 /σ 2 λ 1 -b/σ 2 - √ b 2 -2λ 2 σ 2 /σ 2 . λ 2 < b 2 2σ 2 and λ 1 ≤ b σ 2 + √ b 2 -2λ 2 σ 2 /σ 2 t * λ 1 ,λ 2 = +∞. λ 2 = b 2 2σ 2 t * λ 1 ,λ 2 = 2 σ 2 (λ 1 -b σ 2 ) . λ 2 > b 2 2σ 2 t * λ 1 ,λ 2 = 2 √ 2λ 2 σ 2 -b 2 π 2 -arctan λ 1 σ 2 -b √ 2λ 2 σ 2 -b 2 .
We show by the same as before that for ǫ sufficiently small, ln E e where e(ǫ) is bounded.

  |O 0 (x)| ≪ |x| for x sufficiently small. i.e lim x→0 O 0 (x) x = 0. It follows that

Theorem 2 . 2 .

 22 Let Assumption 1 hold for some random variable Z. Denote Λ(p) = ln Ee pZ and Λ * the Fenchel-Legendre transform of Λ. Then we have lim sup x→∞ ln P(Z > x) + Λ * (x) ln(x)

( 2 .

 2 21) e pZ = e p(c∧Z) + p1 Z>c Z c e pk dk.

Theorem 2 .

 2 2 applies to several examples where the risk neutral log-price is modeled by Lévy process such as the variance gamma model whose moment generating function is given as MGF (p) = gm (m-p)(p+g) c . This function explodes at µ * = m and for x small enough, MGF (µ *x) = gm m-x+g c 1

- 1 2 - 2 ψ λ 1 ,λ 2 ( 2

 122122 Ce ασ 2 t +1 , if |λ 1 -b σ 2 | ≤ α, where α = √ b 2λ 2 σ 2 /σ 2 and C = λ 1 -b σ 2 -α λ 1 -b σ 2 +α . λ 2 = b 2 2σ > b 2 2σ 2 ψ λ 1 ,λ 2 (t) = b σ 2 + β tan β σ 2 t 2 + arctan λ 1 σ 2 -b βσ 2

2 t 0 1 ǫ - 2a σ 2

 2012 (µ * + -ǫ)λ 1 Vt+(µ * + -ǫ)λ Vudu = x 2µ * + σ 2b 2 σ 2 ∂ µ g(t, µ * + ) ln(ǫ) + e(ǫ),

  ,λ 2 (t)α ψλ 1 ,λ 2 (t) + α = ασ 2 t + ln λ 1 -b σ 2α λ 1 -b σ 2 + α . ,λ 2 (t) = -α (Ce ασ 2 t +1) 2 C 2 e 2ασ 2 t -1 , if |λ 1 -b σ 2 | > α, ψλ 1 ,λ 2 (t) = -α (Ce ασ 2 t -1) 2 C 2 e 2ασ 2 t -1 , if |λ 1 -b σ 2 | ≤ α, ,λ 2 (t) = b σ 2α Ce ασ 2 t +1 Ce ασ 2 t -1 , if |λ 1 -b σ 2 | > α, ψ λ 1 ,λ 2 (t) = b σ 2α Ce ασ 2 t -1 Ce ασ 2 t +1 , if |λ 1 -b σ 2 | ≤ α. Case 2λ 2 σ 2 = b 2 :The equation (3.5) can be written asψ′ λ 1 ,λ 2 (t) = Case 2λ 2 σ 2 > b 2 : We set β = √ 2λ 2 σ 2b 2 /σ 2 .The equation (3.5) can be written as ψ′ λ 1 ,λ 2 (t) =

	Thus
		ψ λ 1 ,λ 2 (t) =	λ 1 -b σ 2 1 -σ 2 2 (λ 1 -b σ 2 )t	+	b σ 2 .
				σ 2 2	ψ2 λ 1 ,λ 2 (t) + β 2 .
	the equation (3.5) becomes ψ′ λ 1 ,λ 2 (t) = It follows that ψ′ λ 1 ,λ 2 (t) ψ2 λ 1 ,λ 2 (t) + β 2 = σ 2 2 ψ2 Integrating both sides of this equation we obtain σ 2 2 . λ 1 ,λ 2 (t) -α 2 , which gives 1 2α 1 ψλ 1 ,λ 2 (t) -α -1 ψλ 1 ,λ 2 (t) + α ψ′ λ 1 ,λ 2 (t) = 1 β arctan( ψλ 1 ,λ 2 (t)/β) = σ 2 t + Cst. 2 Thus Hence ln ψλ 1 It follows that ψ λ 1 ,λ 2 (t) = b σ 2 + β tan β σ 2 t 2 + arctan λ 1 σ 2 -b βσ 2	σ 2 2	2 , . .
	
	      ψλ 1 where C = λ 1 -b σ 2 -α λ 1 -b σ 2 + α	. Thus
	    	ψ λ 1 σ 2 2	ψ2 λ 1 ,λ 2 (t),
	which gives
		1 ψλ 1 ,λ 2 (0)	-	1 ψλ 1 ,λ 2 (t)	=	σ 2 t 2	.

Acknowledgments

This research has been supported by a grant from Riksbankens Jubileumsfond (P10-0113:1). The author would also like to thank Professor Damien Lamberton for many useful discussions and anonymous referee who read the first version and helped me improve the presentation.

Appendix B. Proof of Theorem 3.2

The critical moment µ * + (t) is defined as solution of t * λ 1 ,λ 2 (µ * + (t)) = t. There are only two situations for t * λ 1 ,λ 2 :

In this case we have, for ǫ positive and sufficiently small,

,

We obtain, by writing the Taylor expansion of Ce ασ 2 t with respect to ǫ, that

can be written as

where e 1 (ǫ) is bounded. It follows that, for ǫ sufficiently small, ln E e (µ *

On the other hand, we have

We obtain that ϕ λ 1 (µ *

, where e 2 (ǫ) is bounded.

).

Sidi Mohamed Aly, Department of Mathematics, Uppsala University, S-751 06 Uppsala, Sweden E-mail address: souldaly@math.uu.se, souldaly@gmail.com