Huu Thai

Nguyen
email: huu.nguyen@etu.univ-rouen.fr.

Serguei Pergamenshchikov
email: serge.pergamenchtchikov@univ-rouen.fr.

Approximate hedging problem with transaction costs in stochastic volatility markets *

Keywords: Leland strategy, transaction costs, stochastic volatility, quantile hedging, pricing option, approximate hedging, high frequency markets Mathematics Subject Classification (2010): 91G20, 60G44, 60H07 JEL Classification G11, G13

. We analyze possible relationships between the present setting and high frequency markets with transaction costs. Possibilities to improve the convergence rate and reduce the option price inclusive transaction costs are also discussed.

Introduction

In the theory of hedging options, Leland's strategy provides a simple way to eliminate efficiently risks caused by transaction costs. This prescription is based on the idea that transaction costs can be compensated by enlarging the volatility parameter in the delta Black-Scholes strategy. The pioneering work in this field was first given in [START_REF] Leland | Option pricing and replication with transactions costs[END_REF], where a discrete approximation was used to study the asymptotic behavior of the hedging error (difference of the terminal portfolio value and the payoff) as the number of transactions goes to infinity. It was then shown in [START_REF] Lott | Ein verfahren zur eplikation von optionen unter transaktionkosten in stetiger Zeit[END_REF] that the hedging error vanishes if the transaction cost percentage converges to zero at a power rate. Unfortunately, this property does not 1 hold for the interesting case when the proportional cost is a constant. In [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF], the authors found the explicit limit of the hedging error but unexpectedly, it is a negative quantity. It means that the option is actually under-hedged in limit if the investor follows the Leland strategy. The convergence problem was investigated in the paper [START_REF] Granditz | Leland's approach to option pricing: The evolution of discontinuity[END_REF] and then, a complete answer was provided in [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF] with the corresponding limit theorem allowing to identify the asymptotic distribution of the hedging error. Recently, a modified strategy with non-uniform revisions has been suggested in [START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF][START_REF] Darses | Limit theorem for a modified Leland hedging strategy under constant transaction costs rate[END_REF] and it turns out that the rate of convergence is improved. For related results, see further in [START_REF] Lépinette | Mean square error for the Leland-Lott hedging strategy: convex payoffs[END_REF][START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF][START_REF] Gamys | Mean square error for the Leland-Lott hedging strategy[END_REF][START_REF] Granditz | Leland's approach to option pricing: The evolution of discontinuity[END_REF].

Many empirical studies show that the constant volatility assumption in the classical Black-Scholes is not realistic and the Black-Scholes formula constructed under this assumption has an inaccuracy in anticipating option prices. The discrepancy between Black-Scholes option prices and market-traded ones, known as smile curve, can be explained by using stochastic volatility (SV) models which have been used to describe complex markets e.g. when fat-tailed returns are taken into account. It is well-known that modeling SV markets contains some intrinsic difficulties [START_REF] Fouque | Derivatives in Financial Markets with Stochastic Volatility[END_REF]. In fact, the incompleteness property makes the pricing problem more challenging to deal with. Hence, derivatives may not be perfectly hedged with only trading the underlying assets and asymptotic analysis is in general an efficient tool for studying such models. See [START_REF] Fouque | Derivatives in Financial Markets with Stochastic Volatility[END_REF] and the references therein for detailed discussions.

In this work, we study the problem of hedging European style options in SV markets in the presence of transaction costs using a simpler form of adjusted volatility. We will show that the payoff can be approximately replicated by establishing limit theorems for both Leland's strategy and Lépinette's one in a general SV setting. In particular, these asymptotic results recover the existing works in [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF][START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF][START_REF] Darses | Limit theorem for a modified Leland hedging strategy under constant transaction costs rate[END_REF][START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF] and also provide the possibility to improve the rate of convergence. It turns out that superhedging is attainable and both of mentioned strategies are close to the well-known buy-and-hold strategy. We finally point out that the option price can be reduced following the spirit of quantile hedging.

Let us emphasize that the classical form for enlarged volatility σ proposed in [START_REF] Leland | Option pricing and replication with transactions costs[END_REF] and then applied in [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF][START_REF] Kabanov | Markets with transaction costs: Mathematical Theory[END_REF][START_REF] Lépinette | Marché avec côuts de transaction: approximation de Leland et arbitrage[END_REF][START_REF] Lépinette | Leland's approximations for concave pay-off functions[END_REF][START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF] is no longer applicable in SV models from a practical point of view. The reason is that the quantity λ t = 1 t σ 2 u du appearing in the Black-Scholes formula is substantially dependent on future realizations of the random process driving the volatility. Therefore, the strategy is not available for investors in this case. To surpass this issue, we suggest to use an adjusted volatility which is independent of the initial volatility and much more simple than the one used in the previous works. In particular, the same asymptotic results are obtained for SV contexts and the rate of convergence can be improved by controlling the model parameter. Furthermore, note that in the existing works, asymptotic analyses are mainly based on moment estimates. This technique does no longer work in general SV models unless some intrinsic conditions are imposed on the model parameters, see [START_REF] Andersen | Moment explosions in stochastic volatility models[END_REF][START_REF] Lions | Correlations and bounds for stochastic volatility models[END_REF]. This undesirable property can be avoided by establishing convergences in probability to keep the model setting as general as possible. This can be considered as the main contribution of this note in the literature of discrete hedging with proportional transaction costs.

As discussed in [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF], the option price of Leland's strategy is too high because it includes transaction costs. Another practical advantage of our method is that a simple method can be proposed to lower the option price as long as the option seller is willing to take a risk in option replication. This approach is inspired from the theory of quantile hedging [START_REF] Föllmer | Quantile hedging[END_REF].

The remainder of the paper is organized as follows. In Section 2, we briefly give a general view of Leland's approach then formulate the problem and present our principal results in Section 3. The new choice of adjusted volatility allows us to propose a reasonable way in Section 4 to fix the underhedging situation (shown in [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF]) and reduce the option price in the presence of transaction costs. Section 5 discusses some common SV models for which our condition on volatility function is fulfilled. A numerical result for Hull-White model is also provided for illustration. Section 6 discusses a connection of the present context to high frequency markets with proportional transaction costs. The proofs of Main Results are reported in Section 7 and auxiliary lemmas can be found in the Appendix.

2 Hedging with transaction costs: a review on the Leland approach In a complete no-arbitrage model (i.e. there exists a unique equivalent martingale measure under which the stock price is a martingale), options can be completely replicated by a self-financing trading strategy. Option price, defined as the replication cost, is the initial capital that the investor must introduce into his portfolio to obtain a complete hedge. It can be computed as the expectation of the discounted claim under the unique equivalent martingale measure. Let us consider a continuous time model of two-asset financial market on the time interval [0, 1], where the bond price is a constant over the time and equals to one. The stock price dynamics follows the stochastic differential equation

dS t = σ 0 S t dW t , (2.1)
where σ 0 > 0 is a positive constant and (W t ) 0≤t≤1 is a standard Wiener process. As usual we denote F t = σ{W u , 0 ≤ u ≤ t}. We recall that a financial strategy (β t , γ t ) 0≤t≤1 (the fractions of wealth invested in bond and stock respectively) is called an admissible self-financing strategy if it is (F t ) -adapted, integrable with t 0 (|β t | + γ 2 t ) dt < ∞ a.s. and the portfolio value satisfies the equality

V t = β t + γ t S t = V 0 + t 0 γ u dS u , t ∈ [0, 1].
The classical hedging problem is to find an admissible self-financing strategy (β t , γ t ) whose terminal portfolio value exceeds the payoff h(S 1 ) = (S 1 -K) + ; that is

V 1 = V 0 + 1 0 γ u dS u ≥ h(S 1 ) a.s.,
where K is the option strike. For this problem, Black and Scholes [START_REF] Black | The pricing of options and corporate liabilities[END_REF] proposed a dynamically replicating self-financing strategy with γ t = C x (t, S t ) (partial derivative with respect to the space variable), where the option price C(t, S t ) reads the famous formula

C(t, x) = C(t, x, σ 0 ) = xΦ( v(t, x)) -KΦ( v(t, x) -σ 0 √ 1 -t) , (2.2)
where

v(t, x) = v(σ 2 0 (1 -t), x) and v(λ, x) = ln(x/K) √ λ + √ λ 2 . (2.3)
Here Φ is the standard normal distribution function. In the sequel, we denote by ϕ the N (0, 1) density, i.e. ϕ(z) = Φ ′ (z). One can check directly that

C x (t, x) = Φ( v(t, x)) and C xx (t, x) = ϕ( v(t, x)) xσ 0 √ 1 -t . (2.4)
Clearly, hedging via discrete strategies is especially attractive since dynamically adjusted portfolios are impossible in practice. However, discrete time hedging, in turn, will face to intrinsic problems because of the presence of transaction costs. In particular, transaction costs are random and path-dependent, so they significantly effect the hedging error. Additionally, despite of the fact argued by Black and Scholes that the hedging error may be relatively small if trading activities take place reasonably frequently, transaction costs may increase without limit as portfolio revisions are frequent, so it may lead to an explosion.

Constant volatility case

The above considerations lead us to the Leland approach [START_REF] Leland | Option pricing and replication with transactions costs[END_REF], which provides an efficient technique to compensate transaction costs. This method is simply based on the intuition that the option price should include transaction costs as a reasonable extra fee necessary for the option seller to cover the option return. In some situations (discussed in the next two sections), this strategy successfully replicates the payoff including transaction costs by simply adjusting the volatility parameter in Black-Scholes's model.

Let us shortly describe the Leland approach in [START_REF] Leland | Option pricing and replication with transactions costs[END_REF][START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF]. Suppose that for each trading activity, the investor has to pay a fee directly proportional to the trading volume measured in dollar value. Naturally, we suppose that the proportional transaction cost is given by the law κ * n -α , where n is the number of revisions, 0 ≤ α ≤ 1/2 and κ * > 0 are two fixed parameters. To compensate transaction costs the investor is suggested to enlarge the volatility as

σ 2 = σ 2 0 + ̺ n 1/2-α and ̺ = κ * σ 0 8/π . (2.5)
We assume further that the portfolio is revised discretely at t i = i n , i ∈ {1, 2, .., n}, by following the strategy (which is a piecewise process so-called Leland's strategy)

γ n t = n i=1 C x (t i-1 , S t i-1 )1 (t i-1 ,t i ] (t), C(t, x) = C(t, x, σ). (2.6)
It means that the number of shares held in the interval (t i-1 , t i ] is the delta strategy calculated at the left bound of this interval. Then, the portfolio value takes the following form

V n 1 = V n 0 + 1 0 γ n u dS u -κ * n -α J n , (2.7)
where the total trading volume J n is given by

J n = n i=1 S t i |γ n t i -γ n t i-1
|, measured in dollar value. The option price is now given by the initial time-value of the solution C(t, x) of the Black-Scholes PDE with the adjusted volatility σ

C t (t, x) + 1 2 σ 2 x 2 C xx (t, x) = 0 , 0 ≤ t < 1; C(1, x) = h(x) . (2.8)
Using Itô's formula we can represent the hedging error V n 1h(S 1 ) as

1 0 γ n t -C x (t, S t ) dS t + 1 2 ( σ 2 -σ 2 0 ) 1 0 S 2 t C xx (t, S t )dt -κ * n -α J n .
(2.9)

Remark 1 (Leland). The specific form (2.5) results from the following intuition: the Lebesgue's integral in (2.9) is clearly well-approximated by the Riemann sum of the terms σ 0 S 2

t i-1 C xx (t i-1 , S t i-1 )∆t, while S t i |γ n t i -γ n t i-1
| is approximated by

≈ σ 0 S 2 t i-1 C xx (t i-1 , S t i-1 )|∆W t i | ≈ σ 0 2/(nπ) S 2 t i-1 C xx (t i-1 , S t i-1 ), since E|∆W t i | = 2/π √ ∆t = 2/(πn).
Hence, it is reasonable to expect that choosing the modified volatility as in (2.5) may give an appropriate approximation to compensate transaction costs.

Leland [START_REF] Leland | Option pricing and replication with transactions costs[END_REF] conjectured that if the proportional transaction cost is a constant, i.e. α = 0 then, the portfolio value of strategy (2.6) converges in probability to the payoff h(S 1 ) as n → ∞. He also gave a remark without proof that this result is still true for the case α = 1/2. The latter remark is correct and was completely proved by Lott in [START_REF] Lott | Ein verfahren zur eplikation von optionen unter transaktionkosten in stetiger Zeit[END_REF], where one can find a rigorous explanation why the Leland strategy is important in practice.

Theorem 2.1 (Leland-Lott). For α = 1/2, strategy (2.6) defines an approximately replicating strategy as the number of revision intervals n tends to infinity, i.e.

Plim

n→∞ V n 1 = h(S 1 ) .
This result was then extended by Ahn et al in [START_REF] Ahn | Option replication with transaction costs: General diffusion limits[END_REF] to general diffusion models. Kabanov and Safarian [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF] also observed that the Leland approach is still valid as long as the cost proportion converges to zero as n → ∞.

Theorem 2.2 (Kabanov-Safarian). For any 0 < α ≤ 1/2, Plim n→∞ V n 1 = h(S 1 ) . It is, of course, possible to study the Leland-Lott approximation in sense of L 2convergence. Such a result1 was established in [START_REF] Lépinette | Mean square error for the Leland-Lott hedging strategy: convex payoffs[END_REF][START_REF] Kabanov | Markets with transaction costs: Mathematical Theory[END_REF] for the case α = 1/2. Theorem 2.3 (Kabanov-Lépinette). Let α = 1/2. The mean-square approximation error for Leland's strategy with ̺ defined in (2.5) satisfies the following asymptotic equality

E V n 1 -h(S 1 ) 2 = B(S 1 )n -1 + o(n -1 ) as n → ∞,
where B is some positive function.

The above result suggests that the normalized replication error n 1/2 (V n 1h(S 1 )) converges in law as n → ∞.

Theorem 2.4 (Lépinette-Kabanov [START_REF] Kabanov | Markets with transaction costs: Mathematical Theory[END_REF]). For α = 1/2, the processes Y n = n 1/2 (V nh(S 1 )) converge weakly in the Skorokhod space D[0, 1] to the distribution of the process Y • = • 0 B(S t )dZ t , where Z is an independent Wiener process. Remark 2. An interesting connection of this case with the problem of hedging under proportional transaction costs in high frequency markets is discussed in Section 6.

It is crucial to note that the Leland approximation in Remark 1 is not mathematically accurate and so, his first conjecture is not correct. In fact, as n → ∞, the trading volume J n may be approximated by the following sum (which converges in probability to

J(S 1 , ̺) given in (2.11)) -n i=1 λ -1/2 i-1 S t i-1 ϕ(λ i-1 , S t i-1 )|σ 0 ̺ -1 Z i + q(λ i-1 , S t i-1 )| ∆λ i , where λ i = λ t i = σ 2 (1 -t i ), Z i = ∆W t i / ∆t i and ϕ(λ, x) = ϕ(v(λ, x)), q(λ, x) = ln(x/K) 2λ - 1 4 .
(2.10)

In approximation procedures, one should also pay attention to the fact that C(•, •) and its derivatives substantially depend upon n. This property leads to the following important result: there is a non trivial discrepancy between the limit of the terminal portfolio value and the payoff in the practically interesting case α = 0.

Theorem 2.5 (Kabanov-Safarian). If α = 0 then, V n 1 converges to h(S 1 ) + min(S 1 , K)κ * J(S 1 , ̺) , in probability, where

J(x, ̺) = x +∞ 0 λ -1/2 ϕ(λ, x) E | ̺Z + q(λ, x)| dλ , (2.11)
with ̺ = σ 0 ̺ -1 and Z ∼ N (0, 1) independent of S 1 .

Under-hedging: It is important to observe that the problem of option replicating is not solved in this case. Indeed, taking into account that E | ̺Z| = 1/(2κ * ) and the identity

x ∞ 0 λ -1/2 ϕ(λ, x)dλ = 2 min (x, K) , (2.12)
we obtain (for the parameter ̺ given in (2.5)) that min(x, K)κ * J(x, ̺) = xκ * equals to

+∞ 0 λ -1/2 ϕ(λ, x) (E | ̺Z| -E | ̺Z + q(λ, x)|)
dλ. Now, Andreson's inequality (see, for example [START_REF] Ibragimov | Statistical estimation: asymptotic theory[END_REF], page 155) implies directly that for any q ∈ R,

E | ̺Z + q| ≥ E | ̺Z| . Therefore, P -lim n→∞ (V n 1 -h(S 1
)) ≤ 0, i.e. the option is asymptotically underhedged in this case. Another important point should be noted here is that the coefficient ̺ appearing in (2.5) can be chosen in an arbitrary way. We now state the main result in [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF], which also provides the convergence rate for the hedging error.

Theorem 2.6 (Pergamenshchikov). Consider the Leland strategy (2.6) with α = 0 and let ̺ in (2.5) be some fixed positive constant. Then, the sequence of random variables

n 1/4 (V n 1 -h(S 1 ) -min(S 1 , K) + κ * J(S 1 , ̺)) (2.13)
weakly converges to a centered mixed Gaussian variable as n → ∞.

This result is important because it not only gives the asymptotic information of the hedging error but also provides a reasonable way to fix the underhedging issue. More precisely, as discussed in [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF], by choosing a suitable value of ̺ the investor can get a portfolio whose terminal value exceeds the option return as desired.

Darses and Lépinette [START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF] noted that one can modify the Leland strategy to improve the convergence rate in Theorem 2.6. In particular, one can apply a non-uniform revision times (t i ) 1≤i≤n defined by

t i = g (i/n) , g(t) = 1 -(1 -t) µ for some µ ≥ 1 (2.14)
and then adjust the volatility as

σ 2 t = σ 2 0 + κ * σ 0 8/π nf ′ (t)
, where f is the inverse function of g. It was also suggested in [START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF] to use the following modified discrete strategy to release the discrepancy appearing in Theorems 2.5 and Theorem 2.6:

γ n t = n i=1 C x (t i-1 , S t i-1 ) - t i-1 0 C xt (u, S u )du 1 (t i-1 ,t i ] (t) . (2.15)
Theorem 2.7. Let V n 1 be the terminal portfolio value of the strategy (2.15) with α = 0. Then, for any 1 ≤ µ < µ max the sequence n β (V n 1h(S 1 )) weakly converges to a centered mixed Gaussian variable as n → ∞, where

β = µ 2(µ + 1)
and

µ max = 3 + √ 57 8 . (2.16)

Time-depending volatility case

We assume in this subsection that the stock price is driven by dS t = σ(t)S t dW t , where σ is some positive deterministic function. Under the non-uniform rebalancing plan (2.14) the investor should modify the volatility as

σ 2 t = σ 2 (t) + κ * σ(t)n
P -lim n→∞ V n 1 = H(S 1 ) + H 1 (S 1 ) -κ * H 2 (S 1 ),
where H 1 (•) and H 2 (•) are positive functions depending on the payoff H.

Remark 3. Theorem 2.7 still holds in the setting of Theorem 2.8 [START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF].

It is clear that the Leland algorithm is important for option pricing and hedging thanks to its easy practical implementation. The most interesting case α = 0 still needs to be investigated in more general situations, for instance, where volatility depends on other external random factors or jumps in stock prices are taken into account. It is worth noticing that the methodology used in the existing works needs a delicate treatment and seemingly, it is difficult to apply for such models.

Forms of adjusted volatility

Recall from Remark 1 that choosing the modified volatility as in (2.5) would give an appropriate approximation to compensate transaction costs. However, it is not always the case since the option price inclusive transaction costs C(t, S t ) now depends intrinsically on the rebalancing number n. In more general models, this specific choice can cause to technical issues. For example, in local stochastic models [START_REF] Lépinette | Marché avec côuts de transaction: approximation de Leland et arbitrage[END_REF], proving the existence of solution to (2.8) requires an effort since now σ is computed in terms of the stock price and time. This feature makes the Cauchy problem more challenging to deal with. Nevertheless, it is interesting to point out that the true volatility σ 2 (t) plays no role in the approximation procedure. In fact, all results reviewed above for the case α = 0 can be recovered by using the form

σ 2 t = κ * σ(t)n 1/2 f ′ (t)8/π
, where the first term σ 2 (t) has been removed. More general, we can completely remove σ(t) out of the formula of enlarged volatility by taking the new form

σ 2 t = ̺ nf ′ (t), (2.18)
for some positive constant ̺. Of course, the limit of transaction costs will slightly change since ̺ is no longer related to the terminal value of volatility, see Theorem 2.6. This important observation follows from the fact which can be proved similarly as Lemma 1.2.8 in [START_REF] Kabanov | Markets with transaction costs: Mathematical Theory[END_REF] (page 16)

1 0 σ 2 (t)S k t ∂ k C ∂x k (t, S t )dt = O( σ -1 ) = O(n -1/4 ) as n → ∞, (2.19)
for all k ≥ 2. The asymptotic representation (2.19) still holds if σ = σ(y t ) for some extra random process y t . Let us emphasize that using the new form (2.18) has two folds of importance. From a technical point of view, it allows us to carry out a much more simple approximation than what have been done in the existing literature. More importantly, when volatility depends on some external factor, says y t , the Leland strategy is no longer available for practitioners. The reason is that the quantity λ t = 1 t σ 2 u du, which is substantially dependent on future realizations of y t (from now, at time t, to the terminal date t = 1), is impossible to obtain from practical point of view. In contrast, the simpler form is still helpful in this context since it is a deterministic function of t.

Model and Main Results

Let (Ω, F 1 , (F t ) 0≤t≤1 , P) be the standard filtered probability space with two standard independent (F t ) 0≤t≤1 adapted Wiener processes (W

(1) t ) and (W (2)
t ) taking their values in R. Our financial market consists of one risky asset governed by the following equations on the time interval [0 , 1]:

dS t = σ(y t )S t dW (1) t ; dy t = F 1 (t, y t )dt + F 2 (t, y t )(rdW (1) t + 1 -r 2 dW (2) t ), (3.1)
where -1 ≤ r ≤ 1. It is well-known in the literature of SDEs, for example [START_REF] Friedman | Stochastic differential equations and applications[END_REF][START_REF] Liptser | Statistics of random processes I: General Theory[END_REF], that if F 1 (t, y) and F 2 (t, y) are measurable in (t, y) ∈ [0, T ] × R, linearly bounded and locally Lipschitz then, there exists a unique solution y to the last equation of system (3.1), see Theorem 5.1. We assume in this model that the bond interest rate equals to 0, i.e. the non-risky asset is chosen as the numéraire.

As discussed in the previous section, we use the adjusted volatility given by

σ 2 t = ̺ nf ′ (t) = 1 √ µ ̺ √ n(1 -t) 1-µ 2µ , 1 ≤ µ < 2. (3.2)
The parameter ̺ > 0 plays an important role in controlling the rate of convergence and it will be specified later. As discussed in details below, the limit of the total trading volume J n is essentially related to the dependence of ̺ on the number of revisions n. For convenience, recall that C(t, x) is the solution of the Cauchy problem (2.8) with two first derivatives given as in (2.4):

C x (t, x) = Φ(v(λ t , x)) and C xx (t, x) = x -1 λ t -1/2 ϕ (λ t , x)
, where

λ t = 1 t σ 2 s ds = µ ̺ √ n(1 -t) 1 4β
and µ = 2 √ µ/(µ + 1) .

Remark 4. We will also see in Section 4.1 that the underhedging situation pointed out in [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF] can be fixed by controlling the parameter ̺.

We will make use of the following condition on the volatility function.

(C 1 ) Assume that σ(y) is a C 2 -function and there exists a positive constant σ min such that

0 < σ min ≤ σ(y) for all y ∈ R and sup 0≤t≤1 E[σ 2 (y t ) + |σ ′ (y t )|] < ∞.
Assumption (C 1 ) is not too restrictive and it is indeed fulfilled in almost all popular SV models of the existing literature, see Section 5 and [START_REF] Pham | Smooth solutions to optimal investment models with stochastic volatilities and portfolio constraints[END_REF].

Asymptotic results for Leland's strategy

Let us consider the option hedging problem for the model (3.1) in the case of constant proportional cost via Leland's strategy γ n t defined in (2.6). This strategy yields a portfolio whose terminal value V n 1 is defined as in (2.7), where rebalancing times (t i ) are given by (2.14). Now, by Itô's formula we obtain

h(S 1 ) = C(1, S 1 ) = C(0, S 0 ) + 1 0 C x (t, S t )dS t - 1 2 I 1,n , (3.4)
where

I 1,n = 1 0 σ 2 t -σ 2 (y t ) S 2 t C xx (t, S t )dt. Setting V 0 = C(0, S 0 )
we can represent the hedging error as

V n 1 -h(S 1 ) = 1 2 I 1,n + I 2,n -κ * J n , (3.5)
where

I 2,n = 1 0 γ n t -C x (t, S t ) dS t and J n is defined in (2.7
). The goal is to find the limit of the hedging error and point out the convergence rate as n → ∞. To this end, we investigate the limit of the terms that contribute in V n 1h(S 1 ) using the essential property σ → ∞ as n → ∞. In our setting, I 2,n convergences to zero faster than n β with β defined in (2.16), whereas the gamma error I 1,n approaches to 2 min(S 1 , K) at the same rate. On the other hand, the total trading volume J n converges in probability to the random variable J(S 1 , y 1 , ̺) defined by

J(x, y, ̺) = x +∞ 0 λ -1/2 ϕ(λ, x) E σ(y)̺ -1 Z + q(λ, x) dλ , (3.6)
where Z ∼ N (0, 1) independent of S 1 and y 1 .

In order to determine the asymptotic distribution we need to find the martingale remaining part of the above terms. The most challenging issue in our analysis is that the rest term of total transaction costs naturally takes a discrete form whereas the one obtained by studying I 1,n has a continuous form. To combine these two quantities into a unified form that permits one to apply the theory of limit theorem for martingales, we use a special discretization procedure set up in Section 7.

We now state our first asymptotic result for Leland's strategy.

Theorem 3.1. If condition (C 1 ) is fulfilled then for any ̺ > 0 the sequence n β (V n 1 -h(S 1 ) -min(S 1 , K) + κ * J(S 1 , y 1 , ̺))
weakly converges to a centered mixed Gaussian variable as n → ∞.

Remark 5. This theorem is a generalization including an improved convergence rate of the results in [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF][START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF] where the uniform revision is taken and the volatility is assumed to be a constant.

Remark 6. For classical European call option with payoff h(x) = (x -K) + , one easily observes that h(x) + min(x, K) = x. Then, one deduces from Theorem 3.1 that the wealth process V n 1 approaches to S 1κ * J(S 1 , y 1 , ̺) as n → ∞. In fact, this is not a big surprise because the option is now sold at high price. The reason is that C(0, S 0 , σ) → S 0 as σ → ∞. In other words, Leland's strategy now converges to the well-known buy-and-hold one [START_REF] Karatzas | Methods of mathematical finance[END_REF], i.e. to cover the option the seller just takes the trivial strategy: buy a stock share at time t = 0 for price S 0 and keep it until the expiry.

By letting ̺ → ∞ we observe that lim ̺→∞ J(x, y, ̺) = x +∞ 0 λ -1/2 ϕ(λ, x)|q(λ, x)|dλ := J * (x), (3.7)
which is independent of y. This suggests that the rate of convergence in Theorem 3.1 can be improved if ̺ is taken as a function of n. Our next result is established under the following condition on ̺.

(C 2 ) The parameter ̺ = ̺(n) is a function of n such that lim n→∞ ̺(n) = ∞ and lim n→∞ ̺ n -µ 2(µ+2) = 0 .
The specific choice for ̺ in condition (C 2 ) provides the possibility to drop the dependence on volatility in the asymptotic result of the hedging error.

Theorem 3.2. Under conditions (C 1 ), (C 2 ), the sequence

θ n (V n 1 -h(S 1 ) -min(S 1 , K) + κ * J * (S 1
)) with θ n = n β ̺ 2β weakly converges to a centered mixed Gaussian variable as n → ∞.

Remark 7. The asymptotic distributions in both Theorem 3.1 and Theorem 3.2 are explicitly determined in their proofs in Section 7. Furthermore, these results still hold if σ 2 t = σ 2 (y t ) + ̺σ(y t ) nf ′ (t) and the limit of transaction costs is now given by

J ′ (x, ̺) = x +∞ 0 λ -1/2 ϕ(λ, x) E Z̺ -1 + q(λ, x) dλ. (3.8)
However, such a use for enlarged volatility is far away from practical significance as discussed in Subsection 2.3.

Asymptotic result for Lépinette's strategy

Let us consider the modified strategy γ n t defined in (2.15), which produces a portfolio whose terminal values

V n 1 defined by V n 1 = V n 0 + 1 0 γ n t dS t -κ * J n ,
where

J n = n i=1 S t i |γ n t i -γ n t i-1 | . (3.9)
Now by Itô's formula, one presents the hedging error as

V n 1 -h(S 1 ) = 1 2 I 1,n + I 2,n -κ * J n , (3.10)
where

I 2,n = I 2,n + i≥1 (S t i -S t i-1 ) t i-1 0
C xt (u, S u )du. We obtain the following result using the form (3.2) for enlarged volatility. Theorem 3.3. Suppose that (C 1 ) is fulfilled. Then, for any ̺ > 0, the sequence

n β (V n 1 -h(S 1 ) -η min(S 1 , K)) with η = 1 -κ * σ(y 1 )̺ -1 8/π
weakly converges to a centered mixed Gaussian variable as n → ∞. Remark 8. If volatility is a constant then it is interesting to see that Theorem 2.7 can be recovered from Theorem 3.3 with ̺ = κ * σ 8/π. Also note that in our model, the parameter µ takes its values in the interval [1 , 2), that is slightly more general than the condition imposed in Theorem 2.7. Moreover, if the classical form of adjusted volatility is applied for Lepinette's strategy γ n t then the option can be completely replicated by taking ̺ = κ * 8/π, even in SV models and we recover again the result established in [START_REF] Darses | Limit theorem for a modified Leland hedging strategy under constant transaction costs rate[END_REF].

In the context of condition (C 2 ), the cumulated cost κ * J n converges to 0 whereas the hedging error approaches to the terminal value S 1 of the buy-and-hold strategy. Hence, the option is over replicated in this case, see Remark 6. This section presents some applications of the results in Section 3 for the problem of option pricing with transaction costs. We first emphasize that it is impossible to obtain a non-trivial perfect hedge with the presence of transaction costs even in constant volatility models. In other words, to cover completely the option return, the seller can take the buy-and-hold strategy, but this makes the option price too expensive. However, once the investor accepts to take a risk in his hedging problem, the option price can be lowered in a way so that the payoff will be covered with a given probability.

Superhedging with transaction costs

To stand on the safe side, the investor will search for strategies providing the terminal value that exceeds the payoff. Such strategies usually concern solutions to dynamic optimization problems. More precisely, let H be a general contingent claim and denote by A(x) the set of all admissible strategies π with the initial capital x and V π,x T the terminal value of strategy π. Then, the super-replication cost of H is determined as

U 0 = inf x ∈ R : ∃π ∈ A(x), V π,x T ≥ H a.s. , (4.1)
see [START_REF] Karatzas | Methods of mathematical finance[END_REF] and the references therein.

In the presence of transaction costs, Cvitanić and Karatzas [START_REF] Cvitanić | Hedging and portfolio optimization under transaction costs: a martingale approach[END_REF] show that the buy-andhold strategy is the unique choice if one wishes to successfully replicate the option and then S 0 is the super-replication price. In this section, we will show that this property still holds in the sense of approximate superhedging via Leland's spirit. The following observation is just a direct consequence of Theorem 3.2 when ̺ is used as a function of n. Proof. Note first that J * (x) ≤ min(x, K), for all x > 0. Hence, by Theorem 3.2

P -lim n→∞ (V n 1 -h(S 1 )) ≥ (1 -κ * ) min(S 1 , K).
The term in the left hand side is obviously non negative since κ * < 1 hence the conclusion follows. The conclusion for Lépinette strategy directly follows from Theorem 3.3.

Asymptotic quantile pricing

As seen above, the superhedging cost is too high from the buyer's point of view though it indeed gives the seller a successful hedge with probability one. More practically, one can ask that how much initial capital can be reduced by accepting a shortfall probability in replication objective. More precisely, the seller may take a risk and look for hedges with the minimal initial cost defined by

inf x ∈ R, ∃π ∈ A(x) : P V π,x T ≥ H ≥ 1 -ε ,
with a given significance level 0 ≤ ε ≤ 1. See [START_REF] Föllmer | Quantile hedging[END_REF][START_REF] Novikov | Hedging of options with a given probability[END_REF][START_REF] Baran | Quantile hedging on markets with proportional transaction costs[END_REF][START_REF] Bratyk | The generalization of the quantile hedging problem for price process model involving finite number of Brownian and fractional Brownian motions[END_REF][START_REF] Barski | Quantile hedging for multiple assets derivatives[END_REF] for discussions in details.

Let us adapt this idea to the hedging problem in the presence of transaction costs. As seen above, the super-hedging price is S 0 if Leland's algorithm is used to replicate the option. On the seller's side we propose to sell the option at the price δS 0 < S 0 , (where 0 < δ < 1 will be properly chosen) and follow Leland's strategy as before for replication.

To be safe at the terminal moment, we need to choose the parameter ̺ such that the probability that the terminal portfolio exceeds the sum of the real objective (i.e. the payoff) and the additional amount (1δ)S 0 is greater than 1ε, where ε is a significance level predetermined by the seller. We easily observe that this purpose can be achieved by Proposition 4.1. To determine the option price it now remains to choose value δ. We suggest to define it by

δ ε = inf {a > 0 : Υ(a) ≥ 1 -ε} , (4.2)
where Υ(a) = P ((1κ * ) min(S 1 , K) > (1a)S 0 ). The quantity δ ε is called quantile price of the option at level ε and the difference (1δ ε )S 0 is the reduction amount of option price (initial cost for quantile hedging). Clearly, the smaller value of δ ε is, the cheaper the option is.

We show that the option price is significantly reduced, compared with powers of parameter ε. Proposition 4.2. Assume that σ max = sup y∈R σ(y) < ∞ . Then, for any r > 0 and δ ε defined by

(4.2), lim ε→0 (1 -δ ε )ε -r = +∞ . (4.3)
Proof. Observe that 0 < δ ε ≤ 1 and δ ε tends to 1 as ε → 0. Set b = 1κ * . Then for sufficiently small ε such that δ ε > a > 1 -bK/S 0 one has

1 -ε > P(b min(S 1 , K) > (1 -a)S 0 ) = 1 -P(S 1 /S 0 ≤ (1 -a)/b).
Therefore,

ε < P (S 1 /S 0 ≤ (1 -a)/b) ≤ P (X 1 ≤ -z a ) , (4.4)
where X t = t 0 σ(y t )dW

(1) t and z a = ln(b/(1-a))-σ 2 max /2. To estimate this probability we note that for any integer m ≥ 1, E (X 1 ) 2m ≤ σ 2m max (2m -1)!! (see, for example, [29, Lemma 4.11, page 130]). Setting now R(υ) = 2υσ 2 max , we obtain that for any 0 < υ < 1/2σ 2 max ,

E e υX 2 1 = ∞ m=0 υ m m! E (X 1 ) 2m ≤ ∞ m=0 υ m m! σ 2m max (2m -1)!! ≤ 1 1 -R(υ)
.

Therefore, for ε > 0 sufficiently small one has

ε ≤ P(X 1 ≤ -z a ) = P(-X 1 ≥ z a ) ≤ e -υz 2 a E e υX 2 1 ≤ e -υz 2 a 1 -R(υ) .
One then deduces that 1a ≥ b e -ι ε (υ) , where

ι ε (υ) = |ln ε(1 -R(υ))| /υ + σ 2 max /2. Letting now a → δ ε one obtains 1 -δ ε ≥ be -ι ε (υ) , which implies (4.3).
The boundedness of volatility function is essential for the above comparison proposition. If one wishes to relax this assumption, the price reduction is now less free than in Proposition 4.2.

Proposition 4.3. Suppose that E exp{α 1 0 σ 2 (y s )ds} < ∞ for some constant α > 1/2. Then, for r α = (2 √ 2α + 1)/2α, lim inf ε→0 ε -rα (1 -δ ε ) > 0 . (4.5)
Proof. For any positive constant L we set

τ = τ L = inf t > 0 : t 0 σ 2 (y s )ds ≥ L ∧ 1, (4.6)
which is understood as the first time that the log-price's variance passes the level L. Then, one deduces from (4.4) that

ε ≤ P E -1 1 (σ) ≥ u a , 1 0 σ 2 (y s )ds ≤ L + P 1 0 σ 2 (y s )ds ≥ L , (4.7)
where

E t (σ) = e R t 0 σ(ys)dW (1) s -1 2 R t 0 σ 2 (ys)ds , u a = (1 -κ * )/(1 -a) and δ ε > a > 1 -bK/S 0 .
Note that for any p > 0, the process χ t = E τ ∧t (-pσ) is a martingale, i.e. Eχ t = 1. Therefore, the first probability in the right side of (4.7) can be estimated as

(u a ) -p E E -p τ (σ) = (u a ) -p E χ 1 e p R τ 0 σ 2 (ys)ds ≤ (u a ) -p e pL ,
where p = (p 2 + p)/2. By hypothesis and Chebysev's inequality one obtains

P 1 0 σ 2 (y s )ds ≥ L ≤ C α e -αL with C α = E exp α 1 0 σ 2 (y s )ds .
Hence, ε ≤ (u a ) -p e pL + C α e -αL . Choosing L = α -1 ln(2C/ε) and letting a → δ ε , one deduces that for any p > 0 and for some positive constant Cα ,

1 -δ ε ≥ Cα ε γ * (p)
, where γ * (p) = (p + 1)/(2α) + p -1 .

Note that r α = min p>0 γ * (p) = γ * ( √ 2α). Therefore, taking in the last inequality p = √ 2α we obtain the property (4.5).

Remark 9. It is clear that r α < 1 if α > 3/2 + √ 2.
The condition used in Proposition 4.3 holds for such α when σ is linear bounded and y t follows an Orstein-Uhlenbeck process, see the Appendix C. The same quantile pricing results can be established for Lépinette strategy.

Examples

In this section, we list some well-known SV models for which condition (C 1 ) is fulfilled. For this aim, we will need some moment estimates for solutions to general non-linear SDEs

dy t = F 1 (t, y t )dt + F 2 (t, y t )dZ t , y(0) = y 0 , (5.1)
with Z is a standard Wiener process and F 1 , F 2 are two smooth functions. We first recall the well-known result in theory of SDEs, see for example [START_REF] Friedman | Stochastic differential equations and applications[END_REF], Th.2.3, p.107.

Theorem 5.1. Suppose that F 1 (t, y) and F 2 (t, y) are measurable in (t, y) ∈ [0, T ] × R, linearly bounded and locally Lipschitz. If E |y 0 | 2m < ∞ for some integer m ≥ 1 then, there exists a unique solution y t to (5.1) and

E |y t | 2m < (1 + E |y 0 | 2m )e αt , E sup 0≤s≤t |y s | 2m < M (1 + E |y 0 | 2m ),
where α, M are positive constants depending on t, m.

We will see that in the context of the previous theorem, condition (

In this case σ(y) = y 2 + σ min and condition (C 1 ) is obviously verified throughout Theorem 5.1.

Stein-Stein models:

dS t = y 2 t + σ min S t dW t and dy t = (aby t )dt + dZ t .

(5.4)

We have σ(y) = y 2 + σ min and condition (C 1 ) is also verified by Theorem 5.1.

Heston models: Heston [START_REF] Heston | A closed-form solution for options with stochastic volatility with applications to bond and currency options[END_REF] proposed a SV model where volatility is driven by a CIR process, which is also called squared root process. This kind of model can be used in our context. Indeed, assume now that the price dynamics is given by the following dS t = y t + σ min S t dW t and dy t = (aby t )dt + √ y t dZ t , y 0 ≥ 0.

(5.5)

For any a and b > 0, there exists a unique strong solution y t > 0. Note that the Lipschitz condition of diffusion coefficient in Theorem 5.1 is violated but using stopping times method, we can directly show that E y * < ∞ hence, condition (C 1 ) is satisfied.

Similarly, one can verify that (C 1 ) also holds for Ball-Roma's models [START_REF] Ball | Stochastic volatility option pricing[END_REF] or, more generally, for a class of processes of bounded diffusion holding the following condition.

(A) There exist positive constants a, b, M such that yF 1 (t, y) ≤ aby 2 and |F 2 (t, y)| ≤ M, for all t > 0, y ∈ R. Proposition 5.1. Under condition (A), there exists a constant α > 0 such that E e α|y| 2 1 < ∞, where |.| 2 1 stands for sup 0≤t≤1 y 2 t . Proof. A proof can be made using the method in Proposition 1.1.2 in [START_REF] Kabanov | Two-scale stochastic systems, Asymptotic Analysis and Control[END_REF].

Scott models: Let us consider the situation where volatility follows an Orstein-Uhlenbeck as in Stein-Stein's models. Assume now that the function σ takes the exponential form dS t = (e δy t + σ min )S t dW (1) t and dy t = (aby t )dt + dZ t , (

where a, b and σ min > 0 are constants and δ > 0 is chosen such that 2δ ≤ α defined as in Proposition 5.1. Here σ(y) = e δy + σ min and then condition (C 1 ) is fulfilled since

E sup 0≤t≤1 |σ(y)| 2 ≤ 2σ 2 min + 2E (e 2δ 1 {|y t |≤1} + e 2δ|y| 2 1 1 {|y t |>1} ) < ∞.
Numerical result for Hull-White's model: We provide a numerical example for Lépinette's strategy γn t used for the Hull-White model (5.2) discussed above. Here the correlation coefficient of two Brownians is assumed to be 0.05 and other initial values are given by S 0 = K = 1, y 0 = 2, σ min = 2, a = -2, b = 1. We first recall from Theorem 3.3 that the sequence n β (V n 1h(S 1 )η min(S 1 , K)) converges weakly to a centered mixed Gaussian, where η = 1-κ * σ(y 1 )̺ -1 8/π and the payoff h(x) = max(x-K, 0). Therefore, the quantity V n 1 -max(S 1 -K, 0)-η min(S 1 , K) can be considered as the theoretical error. The final gain/loss is measured by the difference of the portfolio value V n 1 and the payoff max(S 1 -K, 0). In order to see the performance of strategy γn t , we compute both gain/loss and theoretical error. In fact, theoretical error values will be estimated for each value of revision number n including the corresponding 95% intervals defined by lower bounds and upper bounds by simulating N = 500 trajectories in the crude Monte-Carlo method. Moreover, the initial amounts of shares to hold are also given in the last column of Table 1 and Table 2.

It turns out that strategy γn t converges quite rapidly to the buy-and-hold and the option prices approach to the superhedging price S 0 . In contrast, convergence of the replication error to 0 is somehow slow. In fact, increasing values of ̺ can provide a better convergence but this unexpectedly leads to the superhedge more rapidly. This evidence again emphasizes the importance of price reduction discussed in Subsection 4.2.

High frequency markets

We now assume that purchases of the risky asset are carried out at a higher ask price S t + ε t whereas sales only earn a lower bid price S tε t , where the mid price S t is given as in model (3.1) and ε t is the halfwidth of the bid-ask spread. Then, for any trading strategy ψ t of finite variation the wealth process can be determined by

V t = V 0 + t 0 ψ s dS s - t 0 ε s d|ψ| s , (6.1)
where |ψ| is the total variation of strategy ψ t . Here the first two terms are the classical components in frictionless frameworks, which respectively describe the initial capital and gains from trading. The last integral accounts for transaction costs incurred by trading activities by weighting the total variation2 of the strategy with the halfwidth of the spread. For problems of optimal investment and consumption with small transaction costs [START_REF] Kallsen | The general structure of optimal investment and consumption with small transaction costs[END_REF], the additional terms should be added in the formulation of V t . In such cases, approximate solutions are usually determined throughout an asymptotic expansion around 0 of the halfwidth spread ε, where the leading corrections are obtained by collecting the inputs from the frictionless problem.

In this section, we are only interested in the replication purpose using discrete strategies in the Leland spirit. Assume that for his replication aim, the option seller will apply a discrete hedging strategy ψ n,ε t that will be executed at n dates defined by t i = g(i/n) as in Section 3. The corresponding wealth process is now given by

V n,ε t = V n,ε 0 + t 0 ψ n,ε s dS s - n i=1 ε t i |ψ n,ε t i -ψ n,ε t i-1 |. (6.2)
In order to partially eliminate the influence of transaction costs in the replication error, we intend to apply again the increasing volatility principle for the present context. Note that in high frequency markets, the bid-ask spread is in general of the same order of magnitude as price jumps and hence ε t is assumed to be of the form κ * n -1/2 S t , for some positive constant κ * . Then, it is interesting to see that this case corresponds to the Leland-Lott framework with α = 1/2. 3In our context, it is interesting to see that the enlarged volatility σ 2 t = ̺ nf ′ (t) is still helpful if the option seller uses the Leland strategy or the Lépinette one in the place of ψ n,ε t .

Proposition 6.1. Assume that ε t = κ * n -1/2 S t . If the option seller uses the enlarged volatility of the form σ 2 = ̺ nf ′ (t) and follows the Leland or the Lépinette strategy then, the sequence of portfolio values V n,ε 1 converges in probability to h(S 1 ) + min(S 1 , K) = S 1 . In particular, n β (V n,ε 1 -S 1 ) converges to a mixed Gaussian variable as n → ∞.

Proof. The proof is just a direct consequence of Theorem 3.1 in Section 3 because the total transaction cost now converges to zero.

It is worth noting that the case α = 0 studied in Section 3 would correspond to the assumption ε t = κ * S t . This specific form means that the market is more illiquid and the bid-ask spread is now proportional to the current asset price at every trades. Clearly, the results in Section 3 are recalled for such a case.

We conclude the section by mentioning the case where the stock spreads remain constant all the time regardless of the current stock price, i.e. ε t = κ * for some positive constant κ * . This can be explained that transaction costs are now based on the volume of traded shares, instead of on the traded amount of money as treated in the literature and in Section 3. It is interesting to see that our methodology still works for such cases. The following result is just an analogue of Theorem 3.1 with a small modification in the limit of transaction costs, defined by

J 0 (x, y, ̺) = +∞ 0 λ -1/2 ϕ(λ, x) E σ(y)̺ -1 Z + ln(x/K) 2λ - 1 4 dλ , (6.3)
where Z ∼ N (0, 1) independent of S 1 , y 1 Proposition 6.2. Suppose that ε t = κ * > 0 and σ 2 = ̺ nf ′ (t). For Leland's strategy under condition (C 1 ), the sequence

n β (V n,ε 1 -h(S 1 ) -min(S 1 , K) + κ * J 0 (S 1 , y 1 ̺))
weakly converges to a centered mixed Gaussian variable as n → ∞. Furthermore, if Lépinette's strategy is used then n β V n,ε 1h(S 1 ) -(1η 0 ) min(S 1 , K) weakly converges to a centered mixed Gaussian variable, where η 0 = σ(y 1 )̺ -1 S -1

1 8/π.
Proof. The proof is similar to that of Theorem 3.1, see Section 7.

Remark 10. When ̺ → ∞ under condition (C 2 ) one obtains an improved-rate version of the above results as in Theorem 3.2 and the initial volatility is completely removed out of the limit of transaction costs.

Proofs

The limit theorems in Section 3 are proved in the following generic procedure.

Step 1: Determine the principal term of the hedging error. In particular, we will point out that the gamma term I 1,n converges to 2 min(S 1 , K) while the cumulated transaction cost approaches to its limit J defined in (3.6). Both convergences are at order of θ n = n β ̺ 2β .

Step 2: Represent the residual terms, which are in the form of stochastic integral, at order of θ n as martingales. Since the residual terms resulting from the analysis of transaction costs are naturally discrete, we need to discretize all the stochastic integrals using a special procedure set up below in Subsection 7.2.

Step 3: Determine the limit distribution of the residual using limit theorem results for martingales established in [START_REF] Hall | Martingale limit theory and its applications[END_REF]. This result is the key tool but we need in fact some special versions compatible with our context. These will be explicitly constructed in Subsection 7.3.

Preliminary

Note that C(t, x) and its derivatives can be represented as functions of λ t and x, where

λ t = λ 0 (1 -t) 1 4β := λ 0 ν(t) and λ 0 = µ̺ √ n. (7.1)
Moreover, the function ϕ(λ, x) (appearing in all k-th (k ≥ 2) degree derivatives of C with respect to the space variable and also for derivatives in time via the relation (2.8)) is exponentially decreasing to 0 as λ approaches to 0 or ∞. This property motives our analysis in terms of variable λ. In particular, let us fix two functions l * , l * and let 1 ≤ m 1 < m 2 ≤ n be two integers such that l * = λ 0 ν(g(m 2 /n)) and l * = λ 0 ν(g(m 1 /n)). Then, all terms corresponding with index j / ∈ [m 1 , m 2 ] can be ignored in the approximation analysis at a certain order depending on the choice of l * and l * . For our purpose, the desired order is θ n ∼ λ 2β 0 . Therefore we will take for example l * = 1/ ln 3 n, l * = ln 3 n and define

m 1 = n -n (l * /λ 0 ) 2/(µ+1) and m 2 = n -n (l * /λ 0 ) 2/(µ+1) , (7.2)
where the notation [x] stands for the integer part of a number x. Below we focus on the subsequence (t j ) of trading times and the corresponding sequence λ j defined as

t j = 1 -(1 -j/n) µ and λ j = λ 0 (1 -t j ) 1 4β , m 1 ≤ j ≤ m 2 . (7.3)
Note that t j is an increasing sequence with values in [t * , t * ], where t * = 1 -(l * /λ 0 ) 4β and t * = 1 -(l * /λ 0 ) 4β , whereas λ j is decreasing in [l * , l * ]. Therefore, in the sequel we make use the notations ∆t j = t jt j-1 whereas ∆λ j = λ j-1λ j , for m 1 ≤ j ≤ m 2 to avoid the negative sign in discrete sums. Below, Itô integrals will be discretized throughout the following sequences of independent normal random variables

Z 1,j = W (1) t j -W (1) t j-1 t j -t j-1 and Z 2,j = W (2) t j -W (2) t j-1 t j -t j-1 . (7.4)
We set

p(λ, x, y) = ̺ σ(y) ln(x/K) 2λ - 1 4 (7.5)
and write for short p j-1 = p(λ j-1 , S t j-1 , y t j-1 ). This reduced notation is also frequently applied for functions appearing in the approximation procedure. With the sequence of revision times (t j ) in hand, we consider the centered sequences

   Z 3,j = |Z 1,j + p j-1 | -E |Z 1,j + p j-1 | | F j-1 , Z 4,j = |Z 1,j | -E |Z 1,j | | F j-1 = |Z 1,j | -2/π. (7.6)
The sequences (Z 3,j ) and (Z 4,j ) will serve in finding the Dood decomposition of considered terms. To represent the limit of transaction costs, we introduce the functions

   G(a) = E (|Z + a|) = 2ϕ(a) + a (2Φ(a) -1) , Λ(a) = E (|Z + a| -E |Z + a|) 2 = 1 + a 2 -G 2 (a), (7.7)
for a ∈ R and Z ∼ N (0, 1). We also write o(a -r n ) for generic sequences of random variables (X n ) satisfying Plim n→∞ a r n X n = 0.

Approximation for stochastic integrals

For any L > 0, we consider the stopping time

τ * = τ * L = inf t ≥ 0 : σ(y t ) + |σ ′ (y t )| > L , (7.8)
and denote by S * t = S τ * ∧t and y * t = y τ * ∧t the stopped processes. We present here the approximation procedure for Itô's stochastic integrals throughout the sequences (Z 1,j ) and (Z 2,j ). In particular, the discrete approximation concerns the class of functions holding the below technical condition. (H) A : R + ×R + ×R → R is a continuously differentiable function satisfying the following: there exist γ > 0 and a positive function U such that

sup λ>0 min(λ γ , 1)|A(λ, x, y)| ≤ U (x, y) and sup 0≤t≤1 E (S * t ) m U 2r (S * t , y * t ) < ∞,
for any -∞ < m < +∞, r ≥ 0 and L > 0.

Remark 11. We can check directly that ∂ k x C(λ, x) = x k-1 λ -k/2 ϕ(λ, x)P (ln(x/K)), where P is some polynomial. Therefore, all functions A appearing in our approximation are of the form λ -k/2 x m σ(y)P (ln(x/K)), where σ can be a power of σ or its two first derivatives σ ′ , σ ′′ . In constant or bounded volatility settings, it can be shown with some computational effort e.g. [START_REF] Darses | Limit theorem for a modified Leland hedging strategy under constant transaction costs rate[END_REF][START_REF] Lépinette | Marché avec côuts de transaction: approximation de Leland et arbitrage[END_REF][START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF] that

sup 0≤t≤1 ES m t ln 2r S t < ∞, for any m ∈ R, r ≥ 0. (7.9)
However, it is not always fulfilled for SV models with unbounded volatility and natural conditions on the correlation and the coefficients of the equation of y t are of course necessarily required, see for instant [START_REF] Andersen | Moment explosions in stochastic volatility models[END_REF][START_REF] Lions | Correlations and bounds for stochastic volatility models[END_REF] for discussion on this interesting direction. This undesirable property prevent to carry out an asymptotic analysis using L 2 estimates as in the existing works. It is important to note that (7.9) is true for processes stopped at τ * . Therefore, only convergences in probability can be provided in the below approximation.

For simplicity, in the sequel we use the notation Š = (S, y). The following technique is used frequently in our asymptotic analysis.

Proposition 7.1. Let A(λ, x, y) = A 0 (λ, x, y) ϕ(λ, x), where A 0 = A 0 (λ, x, y) is a function satisfying (H). Then, for i = 1, 2,

1 0 σ 2 t 1 t A(λ t , Šu )dW (i) u dt = ̺ -1 m 2 j=m 1 A j-1 Z i,j ∆λ j + o(θ -1 n ), (7.10)
which converges to zero since

1 t * ( f * u ) 2 du ≤ Cλ -4β 0 l * . Hence R 3,n = o(θ -1 n ).
It remains to discretize the integral term R 2,n using the sequence (Z i,j ). The key steps for this aim are the followings. First, we represent R

2,n = t * t * Ǎu dW (i) u = m 2 j=m 1 t j t j-1 Ǎu dW (i)
u and replace the Itô integral in the last sum with A j-1 Z i,j ∆t j . Next, Lemma A.1 allows to substitute ∆t j = ̺ -1 ∆λ j into the last sum to obtain the martingale M m 2 defined by

M k = ̺ -1 k j=m 1 A j-1 Z i,j ∆λ j . We need to show that |R 2,n -M m 2 | = o(θ -1 n ) or equivalently, m 2 j=m 1 B j,n = o(θ -1 n ), where B j,n = t j t j-1 A u,j dW (i) u and A u,j = Ā(λ u , Šu ) - Ā(λ j-1 , Št j-1 ). For this aim, set Γ 2,b = sup t * ≤u≤1 sup z∈R |A(z, Šu )| + ∂ x Ā(z, Šu ) + ∂ y Ā(z, Šu ) ≤ b .
Then, for any ε > 0,

P θ n | m 2 j=m 1 B j,n | > ε is bounded by P(Γ c 2,b ) + P(τ * < 1) + P θ n | m 2 j=m 1 B j,n | > ε, Γ 2,b , τ * = 1 . Put B j,n = t j t j-1 A u,j dW (i)
u , where

A u,j = A u,j 1 {| e A u,j |≤b(|λu-λ j-1 |+|S * u -S * t j-1 |+|y * u -y * t j-1 |)} .
Then, the latter probability is equal to

P θ n | m 2 j=m 1 B j,n | > ε , which is smaller than ε -2 θ 2 n m 2 j=m 1 E B 2 j,n by the Chebychev inequality. Clearly, E B 2 j,n is bounded by 2b 2 t j t j-1 ((λ u -λ j-1 ) 2 + E(S * u -S * t j-1 ) 2 + E(y * u -y * t j-1 ) 2 )du ≤ (∆λ j ) 3 + (∆t j ) 2 .
up to a multiple constant. Consequently,

θ 2 n m 2 j=m 1 E B 2 j,n ≤ Cθ 2 n m 2
j=m 1 (∆λ j ) 3 + (∆t j ) 2 , which converges to 0 by Lemma A.1 and condition (C 2 ). On the other hand, by Lemma A.4 one has lim b→∞ lim n→∞ P(Γ c 2,b ) = 0 and hence, the proof is completed.

Limit theorems for approximations

We first recall the following result in [START_REF] Hall | Martingale limit theory and its applications[END_REF], which is extremely useful for studying asymptotic distribution of discrete martingales.

Theorem 7.1. [Theorem 3.2 and Corollary 3.1, p.58 in [START_REF] Hall | Martingale limit theory and its applications[END_REF]] Let M n = n i=1 X i be a zero-mean, square integrable martingale and ς be an a.s. finite random variable. Assume that the following convergences are satisfied in probability:

n i=1 E X 2 i 1 {|X i |>δ} |F i-1 -→ 0 for any δ > 0 and n i=1 E X 2 i |F i-1 -→ ς 2 .
Then, (M n ) converges in law to X whose characteristic function is E exp(-1 2 ς 2 t 2 ), i.e. X has a Gaussian mixture distribution.

Below we will establish some special versions of Theorem 7.1. In particular, our aim is to study the asymptotic distribution of discrete martingales resulting from approximation (7.10) in Proposition 7.1. More precisely, consider (M k ) defined as

M k = k j=m 1 υ j , m 1 ≤ k ≤ m 2 , (7.14)
where υ j = 3 i=1 A i,j-1 Z i,j ∆λ j , A i,j = A i (λ j , Št j ) and Z i,j are defined as in (7.4) and (7.6). To describe the asymptotic variance of M, let us introduce the following function

L(λ, x, y) = A 2 1 (λ, x, y) + 2A 1 (λ, x, y)A 3 (λ, x, y)(2Φ(p) -1) + A 2 3 (λ, x, y) Λ(p) + A 2 2 (λ, x, y) , (7.15)
where p is defined in (7.5). Set

μ = 1 2 (µ + 1) µ 2 µ+1
and µ = (µ -1)/(µ + 1).

(7.16) Proposition 7.2. Let A 0 i = A 0 i (λ, x, y), i = 1, 2, 3 be functions having property (H) and A i (λ, x, y) = A 0 i (λ, x, y) ϕ(λ, x). Then, for any fixed ̺ > 0 the sequence (n β M m 2 ) n≥1 weakly converges to a mixed Gaussian variable with mean zero and variance ς 2 defined as

ς 2 = ς 2 ( Š1 ) = μ̺ 2 µ+1 +∞ 0 λ b µ L(λ, Š1 )dλ.
The same property still holds if some (or all) of the functions A i are of the form

∞ λ A 0 i (z, x, y) ϕ(z, x)dz.
Proof. Note that the square integrability property is not guaranteed for the random variables (υ j ). To overcome this issue let us take their "stopped version" υ * j obtained by substituting Št j-1 by Š * t j-1 in the functions A i , i.e. υ * j = 3 i=1 A i (λ j , Š * t j

) Z i,j ∆λ j . In this sense, we denote by

M * k = k j=m 1
υ * j the corresponding stopped martingale. First, we show throughout Theorem 7.1 that for any L > 0 this martingale weakly converges to a mixed Gaussian variable with mean zero and variance ς * 2 (L) = ς 2 ( Š * 1 ) defined in the proposition. To this end, setting Γ 1,η = {inf t * ≤u≤1 | ln(S * u /K)| > η} and a * j = E (υ * 2 j 1 n˛υ * j ˛>δ o |F j-1 ), we obtain

P   n 2β | m 2 j=m 1 a * j | > ε   ≤ P   n 2β | m 2 j=m 1 a * j | > ε, Γ 1,η   + P(Γ c 1,η ). (7.17)
It suffices to show the convergence to 0 of the first probability in the right side of (7.17).

Along with the proof of Proposition 7.1, one has on the set Γ 1,η and for t * ≤ u ≤ t * that max i=1,2,3

A i (λ u , Š * u ) ≤ U ( Š * u )(1 + λ -γ u ) (7.18)
for some γ > 0 and

U ( Š) = S -1/2 U ( Š). Set υ * j = υ * j 1 Γ 3,j and a * j = E ( υ * 2 j 1 n˛b υ * j ˛>δ o |F j-1 )
, where

Γ 3,j = max 1≤i≤3 A i (λ u , Š * u ) ≤ U ( Š * u )(1 + λ -γ u ) .
We then observe that

P   n 2β | m 2 j=m 1 a * j | > ε, Γ 1,η,L   = P   n 2β | m 2 j=m 1 a * j | > ε   ≤ ε -1 n 2β m 2
j=m 1 E a * j by Markov's inequality. Using the Chebychev inequality and then again the Markov inequality, one gets that E a * j is smaller than

E υ * 4 j P(| υ * j | > δ) ≤ δ -2 E υ * 4 j ≤ 9δ -2 (1 + λ -γ u ) 4 (∆λ j ) 4 E U 4 ( Š * u ) 3 i=1 Z 4 i,j .
Taking into account that all of Z i,j have bounded moments and using (7.18), we ob-

tain that ε -1 n 2β m 2 j=m 1 E a * j is bounded by 9ε -1 δ -2 n 2β m 2 j=m 1
(1 + λ -γ u ) 4 (∆λ j ) 4 , which converges to 0 by Lemma A.1. Let us verify the limit of the sum of conditional variances E(υ * 2 j |F j-1 ). Set υ * i,j = A * i,j-1 Z i,j ∆λ j . Since Z 1,j and Z 2,j are independent,

E υ * 1,j υ * 3,j |F j-1 = E υ * 2,j υ * 3,j |F j-1 = 0. It follows that E(υ * 2 j |F j-1 ) = E(υ * 2 1,j |F j-1 ) + E(υ * 2 2,j |F j-1 ) + E(υ * 2 3,j |F j-1 ) + 2E(υ * 1,j υ * 2,j |F j-1 ).
Observe that for Z ∼ N (0, 1) and some constant a, E(Z |Z + a|) = 2Φ(a) -1 and Therefore, taking into account that ς * 2 (L) converges a.s. to ς 2 as L → ∞, we conclude that n β M m 2 converges in law to N (0, ς 2 ), which completes the proof.

E (Z + a) 2 -(E|Z + a|) 2 = Λ(
Let us consider martingales of the following form resulting from the approximation for Lépinette's strategy,

M k = k j=m 1 A 1,j-1 Z 1,j + A 2,j-1 Z 2,j + A 4,j-1 Z 4,j ∆λ j . (7.19)
Their limiting variance is defined throughout the function

L(λ, x, y) = A 2 1 (λ, x, y) + A 2 2 (λ, x, y) + (1 -2/π)A 2 4 (λ, x, y). (7.20)
Then, the following result is similar to Proposition 7.2.

Proposition 7.3. Let A 0 i = A 0 i (λ, x, y), i = 1, 2, 4 be functions having property (H) and A i (λ, x, y) = A 0 i (λ, x, y) ϕ(λ, x). Then, for any fixed ̺ > 0 the sequence (n β M m 2 ) n≥1 weakly converges to a mixed Gaussian variable with mean zero and variance ς 2 given by ς 2 = μ ̺ 2 µ+1 +∞ 0 λ b µ L(λ, Š1 )dλ. The same property still holds if some (or all) of the functions A i are of the form ∞ λ A 0 i (z, x, y) ϕ(z, x)dz. Proof. The conclusion follows directly from the proof of Proposition 7.2 and the observation that E Z 2 4,j = E(|Z 1,j | -2/π) 2 = 1 -2/π, and E (Z i,j Z 4,j ) = 0, for i = 1, 2 and

m 1 ≤ j ≤ m 2 .
The following result is established for discrete martingales

Mk = k j=m 1 A 1,j-1 Z 1,j + A 3,j-1 Z 3,j ∆λ j := k j=m 1 υj ,
which result from approximation in the case ̺ diverges to infinity, where A i (λ, x, y) = A 0 i (λ, x, y) ϕ(λ, x) with A 0 i , i = 1, 3 are functions having property (H). where Ľ(λ, x, y) = A 2 1 (λ, x, y) + 2A 1 (λ, x, y)A 3 (λ, x, y) + A 2 3 (λ, x, y). The same property still holds if some (or all) of the functions A i are of the form

∞ λ A 0 i (z, x, y) ϕ(z, x)dz.
Proof. Let us determine the limit of conditional variances of n β ̺ -1

µ+1 Mm 2 . We first observe that

n 2β ̺ -2 µ+1 E(υ 2 j |F j-1 ) = μ(1 + o(1)) λ b µ j-1 Q(λ j-1 , Št j-1 )∆λ j , (7.21)
where Q(λ, x, y) = A 2 1 (λ, x, y)+A 2 3 (λ, x, y) Λ(p)+2A 1 (λ, x, y)A 3 (λ, x, y) (2Φ(|p|) -1) . One can check directly that the function G(•) defined in (7.7) satisfies the following inequalities: |a| ≤ G(a) ≤ |a| + 2ϕ(a) , for any a ∈ R. This implies that |Λ(a) -1| ≤ 4|a|ϕ(a) + ϕ 2 (a) and hence, sup a∈R |Λ(a)| < ∞. Note also that Q → Ľ a.s. as n → ∞ since p(λ, x, y) → ∞ as ̺ = ̺(n) → ∞ for any x > 0 and λ = 2 ln(x/K). Using now Lemma A.5, we claim that the sum in the right hand side of (7.21) converges in probability to ς2 and the proof is completed by running again the argument in the proof of Proposition 7.2.

Proof of Theorem 3.1

The term I 1,n approximates to 2 min(S 1 , K) at order θ n . In particular, setting Ī1,n =

1 0 λ t -1/2 σ 2 t (S t ϕ(λ t , S t ) -S 1 ϕ(λ t , S 1 )) dt and changing variables v = 1 t σ 2 s ds we can represent I 1,n as I 1,n = S 1 λ 0 0 v -1/2 ϕ(v, S 1 ) dv + Ī1,n + o(θ -1 n ) .
The first integral in the right side converges a.s. to 2 min(S 1 , K) by (2.12) while Ī1,n is approximated by (1) u dt, where H = (2 -1 λ -1/2λ -3/2 ln(x/K)) ϕ(λ, x). Discretization technique of Proposition 7.1 is applied to replace the latter double integral by U 1,m 2 defined as

1 0 σ 2 t 1 t σ(y u )S u H(λ t , S u )dW
U 1,k = ̺ -1 k j=m 1 σ(y t j-1 )S t j-1 Ȟj-1 Z 1,j ∆λ j , m 1 ≤ k ≤ m 2 , (7.22)
where Ȟ(λ, x) = ∞ λ (z -1/2 /2z -3/2 ln(x/K)) ϕ(z, x)dz. We summarize the asymptotic form of I 1,n in the following. Proposition 7.5. If ̺ either is constant or satisfies condition (C 2 ) then,

P -lim n-→∞ θ n I 1,n -2 min(S 1 , K) -U 1,m 2 = 0.
Next, we claim that the term I 2,n is θ n -negligible. Let us study the trading volume J n . It is easy to check that for v ≥ 0, 1 -Φ(v) ≤ Cv -1 ϕ(v) and t * 0 ϕ(λ u , S u )du + 1 t * ϕ(λ u , S u )du almost surely converges to 0 more rapidly than any power of n. Therefore, one can truncate the sum and keep only the part corresponding to index m 1 ≤ j ≤ m 2 . In other words, J n is approximated by

J 1,n = m 2 j=m 1 S t j-1 ∆Φ j . Putting b j = ∆Φ j -ϕ j-1 ∆v j , we can represent J 1,n as J 1,n = J ′ 1,n + ε 1,n + ε 2,n , where J ′ 1,n = m 2 j=m 1 S t j-1 ϕ j-1 ∆v j , ε 1,n = m 2 j=m 1 ∆S t j-1 ∆ j Φ and ε 2,n = m 2 j=m 1 S t j-1 b j .
In view of (A.1) and condition (C 2 ), we can easily show that ε 1,n = o(θ -1 n ) as n → ∞. Furthermore, using the Taylor expansion we obtain ε 2,n ≤ CS sup m 2 j=m 1 ∆v j 2 for some constant C > 0 and S sup = sup 0≤t≤1 S t . Taking into account

E v j-1 -v j 2 ≤ 1 nλ j-1 + λ 1/2 j-1 -λ 1/2 j 2 + λ -1/2 j-1 -λ -1/2 j 2
up to a multiple constant and using condition (C 2 ) together with (A.1) we get |ε 2,n | = o(θ -1 n ). Now using Itô's Lemma and the substitution λ j = λ 0 (1t j ) 4β one replaces

J ′ 1,n with J 2,n = m 2 j=m 1 λ -1/2 j-1 S t j-1 ϕ j-1 |κ j |∆λ j := m 2 j=m 1 ζ j , κ j = ̺ -1 σ(y t j-1 )Z 1,j + q j-1 , (7.23)
where q is defined in (2.10). We will determine the limit of J n throughout the Doob's decomposition w.r.t. the filtration F j m 1 ≤j≤m 2 of J 2,n . To this end, note that

E(ζ j |F j-1 ) = λ -1/2 j-1 S t j-1 ϕ j-1 ∆λ j E(|κ j ||F j-1 ), where E( κ j |F j-1 ) = ̺ -1 σ(y t j-1 )G(p j-1 ) := D j-1 and G(p) is defined in (7.7). Let B(λ, x, y) = λ -1/2 x ϕ(λ, x)D(λ, x, y) and J 3,n = m 2 j=m 1 B j-1 ∆λ j . (7.24)
We observe that J 2,n = J 3,n + U 2,m 2 , where

U 2,k = k j=m 1 λ -1/2 j-1 S t j-1 ϕ j-1 κ j ∆λ j and κ j := κ j -D j-1 . (7.25)
Making use of the substitution Št j-1 by Š1 everywhere in J 3,n gives J 3,n = J 4,n +J 5,n , where

J 4,n = m 2 j=m 1 B(λ j-1 , Š1 )∆λ j , J 5,n = m 2 j=m 1
B * j-1 ∆λ j and B * j-1 = B(λ j-1 , Št j-1 ) -B(λ j-1 , Š1 ). Observe that the sum J 4,n converges a.s. to J(S 1 , y 1 , ̺) at rate θ n by Lemma A.2. Now, Itô's Lemma applied for B * j-1 leads to the stochastic integrals with respect to the Wiener processes. Aproximation technique in Proposition 7.1 allows to approximate the obtained stochastic integrals by U 3,m 2 , where U

3,k = ̺ -1 2 i=1 k j=m 1 Q i,j-1 Z i,j ∆λ j with Q 1 = ∞ λ (xσ(y)∂ x B + rF 2 (t(λ), y)∂ y B)dz and Q 2 = √ 1 -r 2 F 2 (t(λ), y) ∞ λ ∂ y Bdz with t(λ) = 1 -(λ/λ 0 ) 4β .
The asymptotic form of J n is summarized in the following. Proposition 7.7. For any fixed ̺ > 0, the total trading volume J n admits the following asymptotic form

P -lim n-→∞ θ n J n -J(S 1 , y 1 , ̺) -(U 2,m 2 + U 3,m 2 ) = 0.
Now, the martingale part M m 2 of the hedging error is given by

M k = 1 2 U 1,k -κ * (U 2,k + U 3,k ) = ̺ -1 k j=m 1 3 i=1 A i,j-1 Z i,j ∆λ j ,
where

A 1 = -σ(y) x Ȟ/2, A 2 = κ * Q 2 and A 3 = -κ * σ(y)λ -1/2 x ϕ(λ, x).
It is easy to see that the assumption of Proposition 7.2 is fulfilled for these functions A i , i = 1, 2, 3 and hence, the sequence n β M m 2 n≥1 converges in law to a mixed Gaussian variable by Proposition 7.2 and Theorem 3.1 is proved. .

Proof of Theorem 3.2

When ̺ → ∞ under condition (C 2 ) the approximation of J n is slightly different since the dependence of volatility on the limits can be now removed completely. Observing that E |aZ + b| may be approximated by b(2Φ(b/a)-1) as a → 0, we replace J 3,n in (7.24) with the sum J 3,n = m 2 j=m 1 B j-1 ∆λ j , where B(λ, x) = λ -1/2 x ϕ(λ, x)q(λ, x) Φ(̺q(λ, x)), with Φ(q) = 2Φ(̺ q) -1 and q(λ, x) defined in (2.10). Putting J 4,n = m 2 j=m 1 B(λ j-1 , S 1 ) ∆λ j and J 5,n = J 3,n -J 4,n , we present J 5,n = m 2 j=m 1 B * j-1 ∆λ j , where B * j-1 = B(λ j-1 , S t j-1 ) -B(λ j-1 , S 1 ). Now, using Lemma A.2 we can show directly that | J 4,n -J * (S 1 )| = o(θ -1 n ). Furthermore, Itô's formula allows to replace B * j-1 by

1 t j-1 ∂ x B(λ j-1 , S u )dS u . Direct cal- culations give ∂ x B = λ -1/2 ϕ(λ, x)[-2q 2 (λ, x) Φ(λ, x) + 1 2λ Φ(λ, x) + ̺ λ ϕ(̺ q(λ, x))].
Clearly, Φ(̺q) → sign(q) and ϕ(̺ q) → 0 as ̺ → ∞. Now, using the technique in Proposition 7.1, we can approximate J 5,n by U 3,m 2 defined by

U 3,k = ̺ -1 k j=m 1 σ(y t j-1 )S t j-1 N j-1 Z 1,j ∆λ j ,
where N (λ, x) = +∞ λ z -1/2 ϕ(z, x) -2q 2 (z, x) + 1/(2z) sign(q(z, x))dz. The asymptotic representation of trading costs is summarized in the following.

Proof of Theorem 3.3

The key technique in Proposition 7.1 is used to obtain a smart martingale approximation for the sum i≥1 ∆S t i σ(y t j-1 )S t j-1 Y j-1 Z 1,j ∆λ j .

Proof. The proof follows from the substitution ∆S t j by ̺ -1 σ(y t j-1 )S t j-1 ∆λ t j as in Proposition 7.1.

Let us now study the trading volume J n following the procedure in the approximation of J n . In particular, Itô's Lemma leads to

γ t i -γ t i-1 = t i t i-1 C xx (u, S u )dS u + 1 2 t i t i-1
C xxx (u, S u )σ 2 (y u )S 2 u du, where the time-correction which involves the term q j-1 in the formula of κ j defined by (7.23) has been removed. We now approximate J n by J 1,n , where

J 1,n = ̺ -1 m 2 j=m 1
B j-1 Z 1,j ∆λ j and B(λ, x, y) = σ(y) x λ -1/2 ϕ(λ, x).

Since E|Z| = 2/π, for Z ∼ N (0, 1), the Dood' decomposition of J 1,n is given by J 2,n + Ū2,m 2 , where J 2,n = ̺ -1 2/π m 2 j=m 1 B j-1 ∆λ j and Ū2,m 2 = ̺ -1 m 2 j=m 1 B j-1 Z 4,j ∆λ j .

Again, the substitution Št j-1 by Š1 in J 2,n gives J 2,n = J 4,n + J 3,n where J 4,n = ̺ -1 2/π and B * j-1 = B(λ j-1 , Št j-1 )-B(λ j-1 , Š1 ). Observe that J 4,n converges a.s. to η min(S 1 , K) by Lemma A.2 and (2.12). We now find the suitable martingale approximation for J 3,n . By Itô's formula once again, B * j-1 can be replaced by 2 i=1 1 t Q i (λ j-1 , Šu )dW (i) u where Q 1 = σ(y)x∂ x B + rF 2 (t(λ), y)∂ y B and Q 2 = √ 1r 2 F 2 (t(λ), y)∂ y B. Direct calculations show that ∂ x B = σ(y)(2 -1 λ -1/2λ -3/2 ln(X/K)) ϕ(λ, x) and ∂ y B = σ ′ (y)λ -1/2 x ϕ(λ, x). Now, Proposition 7.1 is applied to approximate J 3,n by the martingale U 3,m 2 defined as U 3,k = ̺ -1 k j=m 1 (A 1,j-1 Z 1,j + A 2,j-1 Z 2,j )∆λ j , for explicit functions A i , i = 1, 2. The final asymptotic form of J n is given below. (A 1,j-1 Z 1,j + A 4,j Z 4,j-1 + A 2,j-1 Z 2,j )∆λ j for explicit functions A i holding the assumption of Proposition 7.3. Then, the convergence in law to a mixed Gaussian variable of the sequence n β M m 2 n≥1 is guaranteed by Proposition 7.3 and hence, Theorem 3.3 is proved.

.

Conclusion

We studied the option replication in Leland's spirit for general stochastic volatility settings using a new form of enlarged volatility, which is simpler than the ones used in the previous works. We established the limit theorems for both Leland' strategy and Lépinette's one, which proved that the influence of transaction costs can be approximately controlled.

The setting of model (3.1) is general enough for practice purposes since it includes many famous SV models. A connection of the present framework to high frequency markets with proportional transaction costs was also discussed. In fact, the approach is still applicable for more general settings where the friction rule admits a representation of separatevariable kind [START_REF] Nguyen | Option replication with general transaction costs in stochastic volatility markets[END_REF], which also includes the case where trading costs are based on the number of traded shares instead of trading volume in dollar value. 4 We pointed out that increasing volatility can compensate trading costs and the option price is now expensive and rapidly approaches to the buy-and-hold super-hedging price. This undesirable property can be relatively released in the spirit of quantile hedging. Lastly, in the accompanying paper, we extended the method to multidimensional frameworks for European options with general payoff written on several assets [START_REF] Nguyen | Approximate hedging with proportional transaction costs for multiasset options[END_REF].

obtain lim n→∞ θ w 2 (t)dt ≤ Cθ 2 n n -2 (ε n ) -(4β+1)/4β λ -1 0 , which tends to 0. The convergence for w 3 (t) can be shown in the same way.

C Moments of Orstein-Uhlenbeck's processes

Lemma C.1. Suppose that σ(z) ≤ γ(1 + |z|) for all z with some constant γ > 0 and let y t be an Orstein-Uhlenbeck process defined by dy t = (aby t )dt + dZ t with some constants a and b > 0. Put X α = exp 2αγ

Corollary 3 . 1 .

 31 Assume that ̺ → ∞ under condition (C 2 ) and condition (C 1 ) holds. Then, the wealth sequence V n 4 Applications for pricing problems

Proposition 4 . 1 .

 41 Under conditions (C 1 ) and (C 2 ), Plim n→∞ V n 1 ≥ h(S 1 ). The same property holds for Lépinette's strategy.

2 µ+1 λ b µ j- 1 2 µ+1 λ b µ j- 1 L- 1 ) 2 j=m 1 E(υ * 2 j

 21211212 a). On the other hand, ∆λ j = n -2β (1 + o(1))μ ̺ by Lemma A.1. So, n 2β E(υ * 2 j |F j-1 ) = (1 + o(1))μ ̺ (λ j-1 , Š * t j∆λ j . By Lemma A.5, the sum n 2β m |F j-1 ) converges in probability to ς * 2 (L). Thus, n β M * m 2 weakly converges to N (0, ς * 2 (L)) throughout Theorem 7.1. Moreover, the property (7.12) implies sup δ>0 lim L→∞ lim n→∞ P(n β |M m 2 -M * m 2 | > δ) = 0 .

Proposition 7 . 4 . 1 µ+1

 741 Under condition (C 2 ), the sequence n β ̺ -Mm 2 weakly converges to a mixed Gaussian variable with mean zero and variance ς2 = μ +∞ 0 λ b µ Ľ(λ, S 1 )dλ,

Proposition 7 . 6 .

 76 If ̺ either is a positive constant or satisfies condition (C 2 ) then θ n I 2,n converges to 0 in probability as n → ∞.Proof. See the Appendix B.

Proposition 7 . 8 . 1 ( 1 µ+1 Mm 2 ,

 78112 Under conditions (C 1 ) and (C 2 ), the trading volume J n admits the following asymptotic formPlim n-→∞ θ n |J n -J * (S 1 ) -(U 2,m 2 + U 3,m 2 )| = 0. Now, the martingale part ̺ -1Mm 2 of the hedging error is determined byMk = ̺ 2 U 1,kκ * ̺(U 2,k + U 3,k ) = k j=m Ǎ1,j-1 Z 1,j + Ǎ3,j-1 Z 3,j )∆λ j ,with two functions Ǎi , i = 1, 2 explicitly determined and satisfying the assumption of Proposition 7.4. Since θ n ̺ -1 Mm 2 = n β ̺ -Theorem 3.2 is proved throughout Proposition 7.4.

t i-1 0 C

 0 xt (u, S u )du. Proposition 7.9. If ̺ either is a positive constant or satisfies condition(C 2 ), then |I 2,n -U 1,m 2 | = o(θ -1 n ), where Y (λ, x) = ∞ λ z -3/2 ln(x/K) ϕ(z, x)dz and U 1,k = ̺ -1 k j=m 1

m 2 j=m 1 B j- 1 2 j=m 1 B

 1121 ∆λ j , J 3,n = ̺ -1 2/π m * j-1 ∆λ j

Proposition 7 . 10 .

 710 If ̺ is a positive constant independent of n then,Plim n→∞ θ n |J nη min(S 1 , K) -(U 2,m 2 + U 3,m 2 )| = 0.Hence, the martingale part of the hedging error for Lépinette's strategy is determined byM m 2 = U 1,m 2 + U 1,m 2κ * (U 2,m 2 + U 3,m 2 ), which can be represented in the form M k = ̺ -1 k j=m 1

2 1 0 y 2 s

 2 ds and α * = b 2 (2γ 2 (2b + a 2 )) -1 . Then, EX α < ∞ for 0 < α < α * .Proof. Remark that (aby)y ≤ a 2 /(2b)by 2 /2. Then, by adapting Proposition 1.1.5 in[START_REF] Kabanov | Two-scale stochastic systems, Asymptotic Analysis and Control[END_REF], p.24, we can show thatE |y t | 2m ≤ m! 2/b + a 2 /b 2 m , m ≥ 1. It follows that EX α ≤ m=0 (α2γ 2 ) m (m!) -1 E |y t | 2m ≤ m=0 2/b + a 2 /b 2 m (α2γ 2 ) m < ∞ for 0 < α < α * . Ify t is mean-reverting then b takes very big values and it is possible to choose α > 3/2 + √ 2 as discussed in Remark 9.

  C 1 ) holds if the volatility function σ and its derivative satisfy the condition of polynomial growth |σ(y)| ≤ C(1 + |y| m ) for some positive constant C and m ≥ 1.Hull-White models: Consider the case where y t follows a geometric Brownian motion dS t = (y t + σ min )S t dW t and dy t = y t (adt + bdZ t ), (5.2)where σ min > 0, a and b are some constants and Z is a standard Brownian motion correlated to W t . Put y

* = sup 0≤t≤1 |y t |. Then, by Theorem 5.1 one has

E (y * ) 2m ≤ C(1 + E|y 0 | 2m ) < ∞ as long as E|y 0 | 2m < ∞. Therefore, condition (C 1 ) is clearly fulfilled.

Uniform Elliptic Volatility models: Consider the case where volatility is driven by an Orstein-Uhlenbeck process of mean-reverting dS t = (y 2 t + σ min )S t dW t and dy t = (aby t )dt + dZ. (

Table 1 :

 1 Convergence for Lépinette's strategy with κ

	n	gain/loss	error	lower bound upper bound	price	strategy
	10	0.1523845 -0.2225988 -0.2363122	-0.2088854	0.7914033 0.9013901
	50	0.2966983 -0.0596194 -0.0670452	-0.0521936	0.9399330 0.9706068
	100	0.3086120 -0.0288526 -0.0350141	-0.0226911	0.9746527 0.9875094
	500	0.2955755 0.0032387	-0.0005821	0.0070594	0.9991733 0.9995891
	1000 0.2851002 0.0012409	-0.0021596	0.0046415	0.9999300 0.9999652

* = 0.01, ̺ = 2.

Table 2 :

 2 Convergence for Lépinette's strategy with κ * = 0.001, ̺ = 4.

	n	gain/loss	error	lower bound upper bound	price	strategy
	10	0.2859197 -0.0744180 -0.0813544	-0.0674816	0.9246420 0.9659700
	50	0.3172523 -0.0069238 -0.0115426	-0.0023049	0.9921661 0.9962377
	100	0.3033519 0.0007474	-0.0030916	0.0045864	0.9984346 0.9992385
	500	0.3618707 0.0001296	-0.0024741	0.0027333	0.9999977 0.9999989
	1000 0.3334375 0.0003996	-0.0020559	0.0028550	1	1

  ̟ 2 n (t)dt = 0 . Now put t j = min(t j , 1ε n ). It then remainsE(γ n t -C x (t, S t )) 2 dt = o(θ -2 n ). Let us introduce the discrete sums w 1 (t) = ) 2 ξ j (t), w 2 (t) =) 2 ξ j (t), where ξ j (t) = 1 ( b t j-1 , b t j ] (t) and x t = ln(S t /K). Clearly,|γ n t -C x (t, S t )| 2 ≤w 1 (t) + w 2 (t) + w 3 (t). Taking into account that sup (t)dt ≤ Cn 2β-3/2 ̺ 4β-1 , which converges to 0 by (C 2 ). The particular choice of ε ensures that θ 2 n E

			2 n E	1 1-ε n
	to prove that w 3 (t) = n j=1 (λ	n j=1 n j=1 λ t b t j b t j-1 -1 (x t -x b t j-1 1/2 t 1/2 -λ b t j-1	n j=1 x 2 t (λ	-1/2 t	-λ b t j-1 -1/2	) 2 ξ j (t) and
	n, 1≤j≤n	n sup 0≤t≤1	E(x t -x b t j-1	) 2 ξ j (t) < ∞ and	sup 0≤t≤1	E x 2 t < ∞,
	one gets θ 2 n E	1-ε n 0	w 1 1-ε n
						0

Seemingly, mean-square replication may not contain much useful information since gains and losses have different meaning in practice. Clearly, if α = 1/2 the modified volatility is independent of n

converges in probability to h(S 1 ) + min(S 1 , K) = S 1 . Note that no improved-convergence version of Theorem 3.3 is obtained since κ * J n converges to 0 at order of ̺.

It is important to know that the classical Black-Scholes strategy is not finite variation.

We would like to thank the anonymous referee for pointing out the correspondence of the case α = 1/2 to this setting.

This extension was presented by the first author at the 7th Colloquium Bachelier on Mathematical Finance and Stochastic Calculus in Metabief, January 2013.

Acknowledgment: The authors would like to thank the two referees and the editor for their remarks and suggestions that are largely useful for the paper's improvement. The first author wishes to express his gratitude to the Vietnam Overseas Scholarship Program (project 322) for financial support.

The second author is partially supported by the National Research Tomsk State University and the Normandy Regional research project "Réseaux d'interaction et systèmes complexes"(RISC).

where θ n = n β ̺ 2β , A j = A(λ j , Št j ) and A(λ, x, y) = ∞ λ A(z, x, y)dz.

Proof. Making use of the stochastic Fubini theorem one gets

Changing the variables v = λ t for the inner integral, we obtain

In other words, I n = I 1,n -I 2,n , where I 1,n = 1 0 Ǎu dW (i) u , Ǎu = A(λ u , Šu ) and I 2,n = 1 0 A(λ 0 , Šu ) dW (i) u . Moreover, we have

Let ε > 0 and b > 0. One observes that P(

In view of (H), one has

Using the Chebychev inequality one gets

Hence, due to condition (H), the integral I 2,n = o(θ -1 n ) as n → ∞. Similarly, taking into account that l * ≤ λ u ≤ λ 0 for 0 ≤ u ≤ t * , we get R 1,n = o(θ -1 n ). Next, let us show the same behavior for the last term in (7.11). Indeed, for some fixed η > 0 and L > 0, one has

where

Then, taking into account Lemma A.3 and the integrability condition (C 1 ), one gets lim η→0 lim L→∞ P(Γ c 1,η,L ) = 0. On Γ 1,η,L , we have Ǎ = Ǎ * and

By the Chebychev inequality again on obtains

Appendix A Auxiliary Lemmas

Lemma A.1. There exist two positive constants C 1 , C 2 such that

where ν 0 (x) = x (µ-1)/(µ+1) . Moreover,

Proof. It follows directly from the relation (7.3).

A technical condition (H 0 ): A : R + → R is a continuously differentiable function having absolutely integrable derivative A ′ and

The following result is straightforward to check.

Lemma A.2. If ̺ either is a positive constant or satisfies condition (C 2 ) then, for any function A satisfying condition

In particular, lim n→∞ θ n m 2 j=m 1

A(λ j-1 )∆λ j -∞ 0 A(λ)dλ = 0.

Lemma A.3. For any ε > 0, lim sup v→1 P(inf v≤u≤1 | ln(S u /K)| ≤ ε) = 0.

Proof. It follows from the explicit form of S t and the fact that conditioning on σ-field generated by the Wiener process driving y, the log-price process ln S t has Gaussian distribution.

Lemma A.4. Suppose that A 0 = A 0 (λ, x, y) and its derivatives

x, y)dz and define

By condition (H), there exists γ > 0 such that

where U is some function verifying sup 0≤t≤1 E U ( Š * t ) < ∞. For η > 0 and N > 0, let

By Lemma A.3, lim n→∞ P(Γ c 1,ε ) → 0 for any ε > 0 given. Thanks to the continuity of the functions S t and y t one gets lim n→∞ P sup (r,d)∈B | U (r, d) -U ( Š * 1 )| ≥ η = 0. Moreover, the integrability of U ( Š * 1 ) implies that P(| U ( Š * 1 )| > N ) converges to 0 as N → ∞. By (7.12), P(τ * < 1) converges to 0 as L → ∞ and the proof is completed.

Lemma A.5. Let A(λ, x, y) = λ A 0 (z, x, y) ϕ(z, x)dz, A = A 2 , where A 0 = A 0 i (λ, x, y) is a function having property (H). Then, for any γ > 0,

where Št = (S t , y t ). The same property still holds if A(λ, x, y) = A 0 (λ, x, y) ϕ(x, y) or the product of these above kinds.

Proof. We just prove for the first case A(λ, x, y) = λ A 0 (z, x, y) ϕ(z, x)dz since the same argument can be made for the other cases. First, we split the expression under the absolute sign as m 2 j=m 1 λ γ j-1 A(λ j-1 , Š1 )∆λ j + m 2 j=m 1 ∆ j,n ∆λ j , where ∆ j,n = A(λ j-1 , Št j-1 ) -A(λ j-1 , Š1 ) and A(λ, x, y) = λ γ A(λ, x, y). It is clear that for any (x, y), the function A(•, x, y) satisfies condition (H 0 ) hence, the sum m 2 j=m 1 A(λ j-1 , Š1 )∆λ j converges a.s. to ∞ 0 A(λ, Š1 )dλ = 0 by Lemma A.2. It remains to show that P(|∆ n | > ε) → 0 for any ε > 0 given but it can be showed by the same way as in Lemma A.3.

B Proof of Proposition 7.6

The singularity of C at the maturity T = 1 requires a separate treatment. Let ε n = n -2β ̺ -4β l * . We then represent I 2,n as I 2,n =