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, approximate hedging, high frequency markets Mathematics Subject Classification (2010): 91G20, 60G44, 60H07 JEL Classification G11, G13

. The asymptotic results obtained not only recover the existing results, but also enable us to fix the under-hedging property pointed out by Kabanov and Safarian in [18]. We also discuss possible methods to improve the convergence rate and to reduce the option price inclusive of transaction costs.

Introduction

Leland [START_REF] Leland | Option pricing and replication with transactions costs[END_REF] suggests a simple method for pricing standard European options in markets with proportional transaction costs. He argues that transaction costs can be accounted for in the option price by increasing volatility parameter in the classical Black-Scholes model [START_REF] Black | The pricing of options and corporate liabilities[END_REF]. Leland then claims, without giving a mathematically rigorous argument, that the replicating portfolio of the corresponding discrete delta strategy converges to the option payoff as the number of revisions n goes to infinity, if the transaction cost rate is a constant independent of n, or decreases to zero at rate n -1/2 . The latter statement is proved by Lott in his PhD thesis [START_REF] Lott | Ein verfahren zur eplikation von optionen unter transaktionkosten in stetiger Zeit[END_REF]. In fact, this property still holds if the transaction cost coefficient converges to zero at any power rate [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF].

However, a careful analysis shows that the replicating portfolio does not converge to the option payoff when the cost rate is a constant independent of n. Kabanov and Safarian [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF] find an explicit limit for the hedging error, which is negative, showing that the replication problem is not completely solved in Leland's framework. Pergamenshchikov [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF] obtains a weak convergence for the normalized hedging error and points out that, for the case of constant transaction cost, the rate of convergence in Kabanov-Safarian's result is n 1/4 . This limit theorem is of practical importance because it provides the asymptotic distribution of the hedging error. Note that the rate of convergence can be improved using non-uniform revisions [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]. In these papers, Lépinette and his co-authors suggest a modification to Leland's strategy to solve the discrepancy indicated in Kabanov-Safarian's result. For a recent account of the theory, we refer the reader to Section 2 and [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF][START_REF] Lépinette | Marché avec côuts de transaction: approximation de Leland et arbitrage[END_REF][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][START_REF] Darses | Limit theorem for a modified Leland hedging strategy under constant transaction costs rate[END_REF][START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF].

In this study, we examine the problem of approximate hedging of European style options with constant transaction costs in stochastic volatility (SV) markets (the reader is referred to e.g. [START_REF] Fouque | Derivatives in Financial Markets with Stochastic Volatility[END_REF] and the references therein for motivations and detailed discussions related to SV models). In particular, we establish a weak convergence for the normalized hedging error of Leland's strategy using a simple form of adjusted volatility in a general SV setting. The results obtained not only recover the existing results, but also provide a method of improving the rate of convergence. Furthermore, it turns out that superhedging can be attained by controlling a model parameter.

Let us emphasize that the classic form for adjusted volatility 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] may be no longer practically applicable for SV models. The reason is that option pricing and hedging in SV markets are intrinsically different from those in the classical Black-Scholes framework. In particular, the option price is now substantially dependent on future realizations of the volatility process. In general cases, this information may not be statistically available for all investors. To treat this issue, we suggest a new specification for adjusted volatility in Leland's algorithm. Although we employ an artificially modified volatility simpler than the well-known version used in the previous works, the same asymptotic results are obtained for SV contexts. In addition, the rate of convergence can be improved by controlling a model parameter. Note that in the above-mentioned papers, approximation procedures are mainly based on moment estimates. This essential technique no longer works in general SV models, unless some intrinsic conditions are imposed on the model parameters [START_REF] Andersen | Moment explosions in stochastic volatility models[END_REF][START_REF] Lions | Correlations and bounds for stochastic volatility models[END_REF]. It is useful to remember that our goal is to establish a weak convergence for the normalized replicating error which only requires convergence in probability of the approximation terms. Thus, in the approximation procedure, the integrability issue can be avoided in order to keep our model setting as general as possible.

As discussed in [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF], the option price (inclusive of transaction costs) in Leland's algorithm may be high (it, in fact, approaches the buy-and-hold price), even for small values of the revision number. Another practical advantage of our method is that the option price can be reduced as long as the option seller is willing to take a risk in option replication. This approach is inspired by 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 give a brief review of Leland's approach. Section 3 is devoted to formulating the problem and presenting our main results. Section 4 presents some direct applications to pricing and hedging. Section 5 discusses common SV models that fulfill our condition on volatility function. A numerical result for Hull-White's model is also provided for illustration. Section 6 connects our results to high-frequency markets with proportional transaction costs. The proofs of our main results are reported in Section 7. Auxiliary lemmas can be found in the Appendix.

2 Approximate hedging with transaction costs: A review of Leland's 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. The option price, defined as the replication cost, is the initial capital that the investor must introduce into his portfolio to obtain a complete hedge. In fact, the option price can be computed as the expectation of the discounted claim under the unique equivalent martingale measure. This principle plays a central role in the well-known Black-Scholes model. For simplicity, let us consider a continuous time model of two-asset financial market on the time interval [0, 1], where the bond price is equal to 1 over time. The stock price dynamics follow the stochastic differential equation

dS t = σ 0 S t dW t , (2.1)
where S 0 and σ 0 are positive constants and (W t ) 0≤t≤1 is a standard Wiener process. As usual, let F t = σ{W u , 0 ≤ u ≤ t}. We recall that a financial strategy (β t , γ t ) 0≤t≤1 is an admissible self-financing strategy if it is bounded from below, (F t ) -adapted 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 classic hedging problem is to find an admissible self-financing strategy (β t , γ t ) whose terminal portfolio value exceeds the payoff h(S 1 ) = max(S 1 -K, 0), or

V 1 = V 0 + 1 0 γ u dS u ≥ h(S 1 ) a.s.,
where K is the strike price. By the pricing principle, the option price C(t, S t ) is given by the well-known formula [START_REF] Black | The pricing of options and corporate liabilities[END_REF]

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 following, we denote by ϕ the N (0, 1) density: 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)
By assuming that continuous portfolio adjustments are possible with zero transaction costs, Black and Scholes [START_REF] Black | The pricing of options and corporate liabilities[END_REF] argue that the option payoff can be dynamically replicated using the delta strategy (i.e., the partial derivative of the option price with respect to the stock price).

It is clear that the assumption of continuous portfolio revision is not realistic. Moreover, continuous trading would be ruinously expensive in the case of nonzero constant proportional transaction costs because the delta strategy has infinite variation. This simple intuition contradicts the argument of Black and Scholes that if trading takes places reasonably frequently, then hedging errors are relatively small. Therefore, option pricing and replication with nonzero trading costs are intrinsically different from those in the Black-Scholes setting. Note that it may be very costly to assure a given degree of accuracy in replication with transaction costs. In what follow, we show that Leland's increasing volatility principle [START_REF] Leland | Option pricing and replication with transactions costs[END_REF] is practically helpful in such contexts.

Constant volatility case

Leland's approach [START_REF] Leland | Option pricing and replication with transactions costs[END_REF] provides an efficient technique to deal with transaction costs. This method is simply based on the intuition that transaction costs can be accounted for in the option price as a reasonable extra fee, necessary for the option seller to cover the option return. It means that in the presence of transaction costs, the option becomes more expensive than in the classic Black-Scholes framework. This would be intuitively equivalent to an increase in the volatility parameter in the Black-Scholes formula. Let us shortly describe the Leland's approach [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. We assume that the transaction cost rate is given by the law κ * n -α , where n is the number of revisions. Here, 0 ≤ α ≤ 1/2 and κ * > 0 are two fixed parameters. The basic idea of Leland's method is to replace the true volatility parameter in the Black-Scholes model by σ, artificially modified as

σ 2 = σ 2 0 + n 1/2-α with = κ * σ 0 8/π . (2.5)
In this case, the option price is given by C(t, x) = C(t, x, σ) by the Black-Scholes's formula.

For the problem of option replication, Leland suggests the following discrete strategy, known as 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), t i = i n , i ∈ {1, 2, .., n}. (2.6)
Here, 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 replicating 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 is given by

J n = n i=1 S t i |γ n t i -γ n t i-1
| (measured in dollar value). Recall that the option price C(t, x) is the solution 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 tracking error, V n 1 -h(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
| can be replaced 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 defined 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] conjectures that the replication error converges in probability to zero as n → ∞ for the case of constant proportional transaction cost (i.e., α = 0). He also gives a remark, without giving a rigorous proof, that this property is also true for the case α = 1/2. In fact, Leland's second conjecture for the case α = 1/2 is correct and is proved by Lott in his PhD thesis [START_REF] Lott | Ein verfahren zur eplikation von optionen unter transaktionkosten in stetiger Zeit[END_REF].

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

P -lim n→∞ V n 1 = h(S 1 ) .
This result is 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] observe that Leland-Lott's theorem remains true as long as the cost rate converges to zero as n → ∞.

Theorem 2.2 (Kabanov-Safarian [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF]). For any 0 < α ≤ 1/2, P -lim n→∞ V n 1 = h(S 1 ) . 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], the authors study Leland-Lott's approximation in the sense of L 2 convergence for the case α = 1/2 1 . Theorem 2.3 (Kabanov-Lépinette). Let α = 1/2. The mean-square approximation error of Leland's strategy, with defined in (2.5), satisfies the following asymptotic equality

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

Theorem 2.3 suggests that the normalized replication error 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 n - h(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.

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

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 important to note that Leland's approximation in Remark 1 is not mathematically correct and thus, his first conjecture is not valid for the case of constant transaction costs. In fact, as n → ∞, the trading volume J n can be approximated by the following sum (which converges in probability to J(S 1 , ) defined 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)

A careful study confirms that there is a non trivial discrepancy between the limit of the replicating portfolio and the payoff for the case α = 0.

Theorem 2.5 (Kabanov-Safarian [START_REF] Kabanov | On Leland's strategy of option pricing with transaction costs[END_REF]). For α = 0, 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 replication is not completely solved in the case of constant transaction costs. Indeed, considering 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

+∞ 0 λ -1/2 ϕ(λ, x) (E | Z| -E | Z + q(λ, x)|)
dλ. Now, Anderson'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, thus, the option is asymptotically under-hedged in this case.

In approximation procedures, one should also pay attention to the fact that C and its derivatives substantially depend on the number of revisions when 0 ≤ α < 1/2. In addition, the coefficient appearing in (2.5) can be chosen in an arbitrary way. Pergamenshchikov [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF] shows that the rate of convergence in Kabanov-Safarian's theorem is n 1/4 and provides a weak convergence for the normalized replication error.

Theorem 2.6 (Pergamenshchikov [START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[END_REF]). Consider Leland's 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 → ∞.

Theorem 2.6 is of practical importance because it not only gives the asymptotic information of the hedging error, but also provides a reasonable way to fix the under-hedging issue (see Section 4). Darses and Lépinette [START_REF] Darses | Limit theorem for a modified Leland hedging strategy under constant transaction costs rate[END_REF] modify Leland's strategy in order to improve the convergence rate in Theorem 2.6 by applying a non-uniform revision policy (t i ) 1≤i≤n , defined by

t i = g (i/n) , g(t) = 1 -(1 -t) µ for some µ ≥ 1. (2.14)
The adjusted volatility is then taken as σ 2 t = σ 2 0 +κ * σ 0 8/π nf (t), where f is the inverse function of g. Furthermore, the discrepancy in Theorems 2.5 and 2.6 can be removed by employing the following modified strategy, known as Lépinette's strategy [START_REF] Lépinette | Modified Leland's strategy for constant transaction costs rate[END_REF],

γn 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 1 -h(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-dependent volatility case

Assume now that σ is a positive non-random function and the payoff H is a continuous function with continuous derivatives, except at a finite number of points. Under the non-uniform rebalancing plan (2.14), the enlarged volatility should take the form

σ 2 t = σ 2 (t) + κ * σ(t)n 1/2-α f (t)8/π.
(2.17)

Theorem 2.8 (Lépinette [START_REF] Lépinette | Marché avec côuts de transaction: approximation de Leland et arbitrage[END_REF]). Let σ be a strictly positive Lipschitz and bounded function. Moreover, suppose that H(•) is a piecewise twice differentiable function and there exist x * ≥ 0 and δ ≥ 3/2, such that

sup x≥x * x δ |H (x)| < ∞.
Then, for any 1/2 ≥ α > 0, the replicating portfolio of Leland's strategy converges in probability to the payoff H(S 1 ) as n → ∞. Moreover, for α = 0,

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 that depend on the payoff H.

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

Discussion

From Remark 1, the modified volatility defined by (2.5) would give an appropriate approximation to account for transaction costs. However, this is not always the case because the option price inclusive of transaction costs now intrinsically depends on the rebalancing number. In more general models, this specific choice may generate technical issues. For example, in local volatility models [START_REF] Lépinette | Marché avec côuts de transaction: approximation de Leland et arbitrage[END_REF], proving the existence of the solution to (2.8) requires patience and effort, because σ depends on the stock price. On the other hand, it is interesting to observe that the true volatility plays no role in the approximation procedure from a mathematical point of view. In fact, all the results for the case α = 0 can be obtained 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 generally, we can take the following form

σ 2 t = nf (t), (2.18)
for some positive constant , which will be specified later. Of course, the limiting value of transaction costs will change slightly. Let us emphasize that using the simple form (2.18) has two folds of importance. First, it allows us to carry out a far simpler approximation than is used in the existing literature. Second, Leland's strategy with σ defined in (2.5) may no longer works in stochastic volatility (SV) markets. Indeed, in the latter context, option prices are substantially dependent on volatility future realizations, which are not statistically available. We show in the remainder of the paper that the simple form (2.18) (a deterministic function of t) is helpful for approximate hedging in a very general SV setting. It should be noted that the approximation methodology developed here still work well for the classical form (2.5) if the volatility risk premium depends only on the current value of the volatility process [START_REF] Pham | Equilibrium state prices in a stochastic volatility model[END_REF][START_REF] Touzi | Option hedging and implicit volatilities[END_REF]. We conclude the section by mentioning that Leland's algorithm is of practical importance owing to its easy and practical implementation. The case of constant transaction costs α = 0 should be investigated in more general situations, for instance, where volatility depends on external random factors, or jumps in stock prices are considered. Note that the method of modern approximate hedging theory requires a delicate treatment, which is seemingly impossible in general SV models.

Model and main results

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

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 is the correlation coefficient. It is well known in the literature of SDEs that if F 1 (t, y) and F 2 (t, y) are measurable in (t, y) ∈ [0, T ] × R, linearly bounded and locally Lipschitz, there exists a unique solution y to the last equation of system (3.1). For this fundamental result, see Theorem 5.1 and [START_REF] Friedman | Stochastic differential equations and applications[END_REF][START_REF] Liptser | Statistics of random processes I: General Theory[END_REF]. For simplicity, we assume that the bond interest rate equals zero. In other words, the non-risky asset is chosen as the numéraire.

In this section, we consider the problem of approximate hedging with constant proportional costs using the principle of increasing volatility for model (3.1). As discussed in Subsection 2.3, the adjusted volatility is chosen as

σ 2 t = nf (t) = µ -1/2 √ n(1 -t) 1-µ 2µ , 1 ≤ µ < 2. (3.2)
The replicating portfolio is revised at (t i ), as defined by (2.14). Note that the parameter > 0 plays an important role in controlling the rate of convergence and is specified later. As shown below, limiting value of the total trading volume is essentially related to the dependence of on the number of revisions. Remark 4. Intuitively, using an independent adjusted volatility is not too natural because it does not take into account the market information. However, our techniques developed in this note are well adapted to the case when the adjusted volatility depends on the volatility process that is driven by an independent Brownian motion. In such a context, if the volatility risk premium depends only on the current volatility process, then the no-arbitrage option price (without transaction costs) is given as the average of the Black-Scholes prices over the future paths of the volatility process [START_REF] Pham | Equilibrium state prices in a stochastic volatility model[END_REF][START_REF] Touzi | Option hedging and implicit volatilities[END_REF]. The detail will be reported in a next publication.

For the reader's convenience, we recall that C(t, x) is the solution of the Cauchy problem (2.8) with two first derivatives, as described 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) .

(3.3)

Remark 5. We see in Section 4 that the under-hedging 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 make use of the following condition on the volatility function.

(C 1 ) Assume that σ is a C 2 function and there exists 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 restrictive and is fulfilled in many popular SV models (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 study the replication error for Leland's strategy γ n t defined in (2.6). The replicating portfolio V n 1 is defined by (2.7). 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 replication error as 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 as in (2.7). Let us first emphasize that complete replication in SV models is far from obvious. In our setting, I 2,n converges to zero faster than n β , with β defined as in (2.16). The gamma error I 1,n approaches 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 . Our goal is to study the convergence of the normalized replication error corrected by these explicit limiting values by applying the theory of limit theorems for martingales [START_REF] Hall | Martingale limit theory and its applications[END_REF]. To do so, we search for the martingale part in the approximation of the above terms by developing a special discretization procedure in Section 7.

Theorem 3.1. Suppose that condition (C 1 ) holds and > 0 is a fixed positive constant. Then,

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 6. This theorem is a generalization that includes 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 7. Remark that h(x) + min(x, K) = x, where h(x) = (x -K) + is the payoff of a classical European call option. Then, from Theorem 3.1, the wealth process V n 1 approaches S 1 -κ * J(S 1 , y 1 , ) as n → ∞. This can be explained by the fact that the option is now sold at a higher price because C(0, S 0 , σ) → S 0 as σ → ∞. In other words, Leland's strategy now converges to the well-known buy-and-hold [START_REF] Karatzas | Methods of mathematical finance[END_REF]: buy a stock share at time t = 0 for price S 0 and keep it until expiry.

We now present a method of improving the rate of convergence in Theorem 3.1. To this end, 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 . Theorem 3.2. Under conditions (C 1 ) -(C 2 ), 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 8. The asymptotic distributions in Theorems 3.1 and 3.2 are explicitly determined in their proofs in Section 7.

Asymptotic result for Lépinette's strategy

Let us study the replication error of Lépinette's strategy γ n t , as defined in (2.15). As before, the replicating portfolio is given 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 | . (3.8)
Now, by Itô's formula, the tracking error is

V n 1 -h(S 1 ) = 1 2 I 1,n + I 2,n -κ * J n , (3.9)
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.
Then, we have the following strengthening of Theorem 2.7.

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 9. Theorem 2.7 can be established from Theorem 3.3 with = κ * σ 8/π when the volatility is a constant. In addition, in our model, the parameter µ takes its values in the interval [1 , 2), which is slightly more general than the condition imposed in Theorem 2.7. Moreover, if the classical form of adjusted volatility is applied for Lépinette's strategy, then complete replication can be reached by taking = κ * 8/π, and we again have the result established in [START_REF] Darses | Limit theorem for a modified Leland hedging strategy under constant transaction costs rate[END_REF].

Corollary 3.1. Under conditions (C 1 ) -(C 2 ), the wealth sequence V n 1 converges in prob- ability to h(S 1 ) + min(S 1 , K) = S 1 .
Note that we do not obtain an improved convergence version of Theorem 3.3 because κ * J n converges to zero at order of .

Application to the pricing problem

In this section, we present an application for the problem of option pricing with transaction costs. We first emphasize that it is impossible to obtain a non-trivial perfect hedge in the presence of transaction costs, even in constant volatility models. In fact, the seller can take the buy-and-hold strategy, but this leads to a high option price. We show below that option price can be reduced in certain ways so that the payoff is covered with a given probability.

Super-hedging with transaction costs

To be on the safe side, the investor search for strategies that provide the terminal value greater than the payoff. Such strategies usually concern solutions to dynamic optimization problems. More precisely, let H be a general contingent claim and denote A(x) and V π,x T as the set of all admissible strategies π with initial capital x and the terminal value of strategy π, respectively. Then, the super-replication cost for 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 for more detail). 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-and-hold strategy is the unique choice for super-replication, and then S 0 is the super-replication price. In this section, we show that this property still holds for approximate super-hedging. The following observation is a direct consequence of Theorem 3.2 when is used as a function of n.

Proposition 4.1. Under conditions (C 1 ) -(C 2 ), P -lim n→∞ V n 1 ≥ h(S 1
). The same property holds for Lépinette's strategy.

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). (4.2)
The left-hand side is obviously non-negative since κ * < 1. The conclusion for Lépinette's strategy directly follows from Theorem 3.3.

Asymptotic quantile pricing

As seen ealier, super-hedging in the presence of transaction costs leads to a high option price. Practically, one can ask that how much the initial capital can be reduced by accepting a shortfall probability at the terminal moment. More precisely, for a given significance level 0 ≤ ε ≤ 1, the seller may look for hedges with a minimal initial cost defined by inf x ∈ R, ∃π ∈ A(x) :

P V π,x T ≥ H ≥ 1 -ε .
This construction is motivated by quantile hedging theory, which goes back to [START_REF] Föllmer | Quantile hedging[END_REF][START_REF] Novikov | Hedging of options with a given probability[END_REF].

For related discussions, we refer to [START_REF] Föllmer | Quantile hedging[END_REF][START_REF] Novikov | Hedging of options with a given probability[END_REF][START_REF] Pergamenshchikov | Limit theorem for Leland's strategy[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]. Here, we adapt this idea to the hedging problem. Recall that the super-hedging price of Leland's algorithm is S 0 . On the seller's side, we propose a price δS 0 < S 0 for the option, for a properly chosen 0 < δ < 1.

We then follow Leland's strategy for replication. To be safe at the terminal moment, we need to choose such that the probability of the terminal portfolio exceeding the sum of the real objective (i.e., the payoff) and the additional amount (1 -δ)S 0 is greater than 1 -ε. Here, ε is a significance level predetermined by the seller. By Proposition 4.1, this aim can be achieved for sufficiently large . To determine the option price, it now remains to choose the smallest value of δ. Motivated by (4.2), we define this by

δ ε = inf {a > 0 : Υ(a) ≥ 1 -ε} , Υ(a) = P ((1 -κ * ) min(S 1 , K) > (1 -a)S 0 ) . (4.3)
Thus, the reduction in the option price is given by (1 -δ ε )S 0 . Clearly, smaller values of δ ε yield cheaper options.

Next, we show that the option price is significantly reduced, compared with powers of parameter ε. Proof. We first observe that 0 < δ ε ≤ 1 and δ ε tends to 1 as ε → 0. Set b = 1 -κ * . 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 ) , (4.5)
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 [29, Lemma 4.11, p.130]). Set R(υ) = 2υσ 2 max . 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 sufficiently small ε > 0, we have

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) . Then, 1 -a ≥ b e -ι ε (υ) , where ι ε (υ) = |ln ε(1 -R(υ))| /υ + σ 2 max /2. Letting a → δ ε , we get 1 -δ ε ≥ be -ι ε (υ) , which implies (4.4).
The boundedness of the volatility function is essential for the above comparison result. If we wish to relax this assumption, the price reduction will be 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 . (4.6)
Proof. For any positive constant L we set

τ = τ L = inf t > 0 : t 0 σ 2 (y s )ds ≥ L ∧ 1, (4.7)
which is understood to be the first time that the log-price's variance passes level L. Then, from (4.5),

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 , (4.8)
where

E t (σ) = e t 0 σ(ys)dW (1) s -1 2 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 stopped process χ t = E τ ∧t (-pσ) is a martingale and Eχ t = 1. Therefore, the first probability on the right side of (4.8) can be estimated as

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

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 . By 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). Then, including in the last inequality p = √ 2α we obtain property (4.6).

Remark 10. It is clear that r α < 1 for α > 3/2 + √ 2.
The condition used in Proposition 4.3 holds for such α when σ is linearly bounded and y t follows an Orstein-Uhlenbeck process (see the Appendix C). The same quantile pricing results can be established for the Lépinette strategy.

Examples and numerical results

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

dy t = F 1 (t, y t )dt + F 2 (t, y t )dZ t , y(0) = y 0 , (5.1)
where Z is a standard Wiener process and F 1 , F 2 are two smooth functions. We first recall the well-known result in SDE theory ( see for example [ 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] proposes a SV model where volatility is driven by a CIR process, which also known as squared root process. This model can be used in our context. Indeed, assume now that the price dynamics are given by the following: dS t = y t + σ min S t dW t and dy t = (a -by t )dt + √ y t dZ t , y 0 ≥ 0.

(5.5)

For any a and b > 0, the last equation admits a unique strong solution y t > 0. Note that the Lipschitz condition in Theorem 5.1 is violated, but by using the stopping times technique, we can directly show that E y * < ∞. Hence, this implies that condition (C 1 ) is satisfied for model (5.5).

Similarly, we can check 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, and M such that yF 1 (t, y) ≤ a -by 2 and |F 2 (t, y)| ≤ M, for all t > 0, y ∈ R. Proposition 5.1. Under condition (A), there exists α > 0 such that Ee αy * 2 < ∞.

Proof. The proof uses the same method as in Proposition 1.1.2 in [START_REF] Kabanov | Two-scale stochastic systems, Asymptotic Analysis and Control[END_REF]. Scott models: Suppose that volatility follows an Orstein-Uhlenbeck, as in Stein-Stein's models, and the function σ takes the exponential form dS t = (e δy t + σ min )S t dW

(1) t and dy t = (a -by t )dt + dZ t , (

where a, b and σ min > 0 are constants. Here, δ > 0 is chosen such that 2δ ≤ α, defined as in Proposition 5.1. Clearly, σ(y) = e δy + σ min and 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 the approximation result of Lépinette's strategy in Hull-White's model (5.2). By Theorem 3.3, the corrected replication error is given by of strategy γn . For a numerical view, we simulate N = 500 trajectories in a crude Monte-Carlo method, where correlation coefficient of two Brownian motions is 0.05 and the other initial values are given by S 0 = K = 1, y 0 = 2, σ min = 2, a = -2 and b = 1. For each value of n, we estimate the average value of the corrected error and give the corresponding 95% intervals defined by lower and upper bounds. Initial numbers of shares to hold are given in the last column of Table 1 and Table 2. It turns out that strategy γn t converges to the buy-and-hold and the option prices approach the super-hedging price S 0 . We also see that the convergence of the corrected replication error to zero is somehow slow. In fact, increasing values of can provide a better convergence, but this unexpectedly leads to super-replication more rapidly.

V n 1 -max(S 1 -K, 0) -η min(S 1 , K), where η = 1-κ * σ(y 1 ) -1 8/π. The difference V n 1 -max(S 1 -K, 0) can
We now provide a numerical illustration for the quantile hedging result of Proposition 4.2. For simplicity, we now suppose that σ(y) = sin 2 (y) + 0.1 and that y follows a geometric Brownian motion as above. To compare the reduction factor 1 -δ ε with powers of significance level ε, we compute (1 -δ ε )ε -r for 0.001 ≤ ε ≤ 0.1 and 0 ≤ r ≤ 0.1, with κ * = 0.001. Then, (4.4) is confirmed by the simulation result (see Figure 2a). The simulation also shows that the option price inclusive of transaction costs is 1 -0.385 = 0.615, which is cheaper than the super-hedging price S 0 = 1, for a shortfall probability less than 0, 1%. Of course it is reasonable to replace S 0 by the option price inclusive of transaction costs C(0, S 0 ). Then, simulation result for the reduction in the option price (1 -δ ε ) C(0, S 0 ) is given in Figure 2b. 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 t -ε t . Here 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 of finite variation ψ t , 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 ψ t . Observe that the first two terms are the classic components in frictionless frameworks, and respectively describe the initial capital and gains from trading. The last integral in (6.1) accounts for transaction costs incurred from the trading activities by weighting the total variation2 of the strategy with the halfwidth of the spread.

For 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 zero 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 Leland's spirit. Assume that for his replication aim, the option seller applies a discrete hedging strategy ψ n,ε t , revised 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)
To treat the risk of transaction costs, we again apply the increasing volatility principle. Note that in high frequency markets, the bid-ask spread is, in general, of the same order of magnitude as price jumps 3 . Hence, ε t should be of the form κ * n -1/2 S t , for some positive constant κ * . Then, this case corresponds to Leland-Lott's framework with α = 1/2. In our context, the method in Section 3 is still helpful when ψ n,ε t is replaced by Leland's or Lépinette's strategy. Proposition 6.1. Let ε t = κ * n -1/2 S t , and assume that the enlarged volatility is of the form σ 2 = nf (t) as in (3.2). For both Leland's and Lépinette's strategies, the sequence of replicating 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 a direct consequence of Theorem 3.1, because the total transaction cost now converges to zero.

Note that the case α = 0 studied in Section 3 corresponds to the assumption ε t = κ * S t , for some constant κ * . This specific form means that the market is more illiquid and the bid-ask spread is now proportional to the spot asset price in every trade. Therefore, approximate hedging results for this case are the same as those in Section 3.

We conclude the section by supposing that the stock spreads remain constant all the time, regardless of the current stock price. In other words, ε t = κ * for some positive constant κ * . Intuitively, transaction costs are now based on the volume of traded shares, instead of on the traded amount of money as in the literature and Section 3. It is interesting to see that our methodology still works in this case. The following result is just an analog of Theorem 3.1, with a small modification to the limiting value 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, for Lépinette's strategy, n β V n,ε 1 -h(S 1 ) -(1 -η 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 11. When → ∞ under condition (C 2 ), one obtains an improved-rate version of Proposition 6.2, as in Theorem 3.2.

Proofs

Our main results are proved in the following generic procedure.

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

Step 2: Represent the residual terms as discrete martingales. To this end, stochastic integrals will be discretized by following a special procedure set up in Subsection 7.2.

Step 3: Determine the limit distribution of normalized replication error by applying Theorem 7.1. 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 first 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 = µ √ n. (7.1)
Moreover, the function ϕ(λ, x), which appears in all k-th (k ≥ 2) degree derivatives of C with respect to x and derivatives in time via the relation (2.8), is exponentially decreasing to zero when λ tends to zero or infinity. 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 to index j / ∈ [m 1 , m 2 ] can be ignored at a certain order which depends on the choice of l * and l * . For our purpose, the desired order is θ n ∼ λ 2β 0 . Therefore, we take, for example, l * = 1/ ln 3 n, l * = ln 3 n and define

m 1 = n -n (l * /λ 0 ) 2/(µ+1) + 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 real 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)
Figure 3: The sequences (λ j ) and (t j ) defined by (7.3).

Note that t j is an increasing sequence taking 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, we make use the notations ∆t j = t j -t j-1 , whereas ∆λ j = λ j-1 -λ j , for m 1 ≤ j ≤ m 2 , to avoid recopying the negative sign in discrete sums. Below, Itô stochastic 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 style of reduced notation is abusively applied for functions appearing in the approximation procedure. Define

   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 our approximation terms. In order to represent the limit of transaction costs, we introduce

   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 P -lim 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 corresponding stopped processes. We provide an approximation procedure for Itô's stochastic integrals throughout the sequences (Z 1,j ) and (Z 2,j ). 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 for any x ≥ 0, y ∈ R,

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 12. We can check directly that for k ≥ 2,

∂ 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 0 appearing in the below 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 of its two first derivatives σ , σ . In bounded volatility settings, it can be shown with some computational effort (see 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)
The latter property is, however, not always fulfilled for SV models with unbounded volatility. In fact, it has been demonstrated in [START_REF] Andersen | Moment explosions in stochastic volatility models[END_REF][START_REF] Lions | Correlations and bounds for stochastic volatility models[END_REF] that the stock price does not admit integrable moments in general SV markets, unless some natural conditions necessarily imposed on the correlation and the volatility dynamics coefficients. Thus, asymptotic analysis using L 2 estimates as in the existing works may be impossible in general SV frameworks. Nevertheless, note that (7.9) is true for processes stopped by τ * . Below, the approximation analysis will be established in the sense of convergence in probability in order to avoid this integrability issue.

For simplicity, we use the notation Š = (S, y). The following technique is frequently applied 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)
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. By making use of the stochastic Fubini theorem, we gets

I n = 1 0 σ 2 t 1 t A(λ t , Šu )dW (i) u dt = 1 0 u 0 σ 2 t A(λ t , Šu )dt dW (i) u .
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 via 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, we introduce the 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) + n , where n = 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 {| A u,j |≤b(|λu-λ j-1 |+|S * u -S * t j-1 |+|y * u -y * t j-1 |)} . Then, n = 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 Cheby- chev's inequality. Clearly, E B 2 j,n is bounded by 3b 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, we get lim b→∞ lim n→∞ P(Γ c 2,b ) = 0. The proof is completed.

Limit theorem for approximations

We first recall the following result in [START_REF] Hall | Martingale limit theory and its applications[END_REF], which is 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 both 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.

In this subsection, we establish some special versions of Theorem 7.1. In fact, our aim is to study the asymptotic distribution of discrete martingales resulting from approximation (7.10) in Proposition 7.1. First, we define

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 defined as in (7.4) and in (7.6). To describe the asymptotic variance of (M), we 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 ) = μ 2 µ+1
+∞ 0 λ 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 (υ j ). To overcome this issue, we consider their "stopped version" (υ * j ), which are obtained by substituting Št j-1 by Š * t j-1 in A i , i.e., υ * j =

3 i=1 A i (λ j , Š * t j ) Z i,j ∆λ j . Let 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 ). To this end, setting Γ 1,η = {inf t * ≤u≤1 | ln(S * u /K)| > η} and a * j = E (υ * 2 j 1 {|υ * j |>δ} |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 that the first probability on the right side of (7.17) converges to zero. Indeed, from the proof of Proposition 7.1, one observes that on the set Γ 1,η , max i=1,2,3

A i (λ u , Š * u ) ≤ U ( Š * u )(1 + λ -γ u ), t * ≤ u ≤ t * , (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 {| υ * j |>δ} |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 have

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. By using Chebychev's inequality and then again Markov's inequality, we observe that

E a * j ≤ 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 .
Note that Z i,j have bounded moments. Then, by using (7.18), we obtain

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 zero by Lemma A.1.

We now 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 ).
Now, observe that for Z ∼ N (0, 1) and some constant a, E(Z |Z + a|) = 2Φ(a) -1 and

E (Z + a) 2 -(E|Z + a|) 2 = Λ(a). On the other hand, ∆λ j = n -2β (1 + o(1))μ 2 µ+1 λ µ j-1 by Lemma A.1. Therefore, n 2β E(υ * 2 j |F j-1 ) = (1 + o(1))μ 2 µ+1 λ µ j-1 L(λ j-1 , Š * t j-1
)∆λ j .

By Lemma A.5, n 2β m 2 j=m 1 E(υ * 2 j |F j-1 ) converges in probability to ς * 2 (L). Thus, n β M * m 2 weakly converges to N (0, ς * 2 (L)) by Theorem 7.1. Moreover, property (7.12) implies that for any δ > 0, lim

L→∞ lim n→∞ P(n β |M m 2 -M * m 2 | > δ) = 0 .
Therefore, by 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 ). This completes the proof.

Next, we study the asymptotic property of the following martingale

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)
The limiting variance will be 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)
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 µ L(λ, Š1 )dλ. The same property still holds if some (or all) 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 .

In the rest of the subsection, we establish a limit theorem for a martingale of the following form

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 , m 1 ≤ k ≤ m 2 ,
where A i (λ, x, y) = A 0 i (λ, x, y) ϕ(λ, x) and A 0 i , i = 1, 3 are functions having property (H). The following result is helpful for the case when diverges to infinity as in Theorem 3.2. λ µ Ľ(λ, S 1 )dλ, 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) A i are of the form

∞ λ A 0 i (z, x, y) ϕ(z, x)dz.
Proof. We 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 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) . Moreover, it can be checked 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), hence, sup a∈R |Λ(a)| < ∞. Note also that Q → Ľ a.s. as n → ∞ because p(λ, x, y) → ∞ as = (n) → ∞, for any x > 0 and λ = 2 ln(x/K). Using Lemma A.5, we claim that the sum of the terms on the right-hand side of (7.21) converges in probability to ς2 . The proof is completed by running again the argument in the proof of Proposition 7.2.

Proof of Theorem 3.1

We first observe that I 1,n approaches 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 = 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 0 σ 2 t 1 t σ(y u )S u H(λ t , S u )dW (1) u dt, where H = (2 -1 λ -1/2 -λ -3/2 ln(x/K)) ϕ(λ, x).
Discretization technique of Proposition 7.1 can be applied to replace the latter double integral by U 1,m 2 , defined by

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 /2 -z -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 I 2,n = o(θ -1 n ). Proposition 7.6. If either is a positive constant or satisfies condition (C 2 ), then θ n I 2,n converges to zero in probability as n → ∞.

Proof. See the Appendix B.

Let us study the trading volume J n . First, it is easy to check that for any v > 0, 1 -

Φ(v) ≤ v -1 ϕ(v). Now, observe that |γ n t i -γ n t i-1 | ≤ |1 -γ n t i | + |1 -γ n t i-1 |
, which almost surely converges to zero more rapidly than any power of n when inf 1≤i≤n λ i ≥ l * ⇐⇒ i ≤ m 1 . The same property can be deduced for the case sup i λ ≤ l * ⇐⇒ i ≥ m 2 . To see this, we note that for λ

u ≤ l * , S u (ω) = S 1-(λu/λ 0 ) 4β (ω) converges to S 1 (ω) as n → ∞ uniformly in λ u ∈ [0, l * ],
for any ω outside the zero probability set {S 1 = K}. 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 ζ j , κ j = -1 σ(y t j-1 )Z 1,j + q j-1 , (7. [START_REF] Leland | Option pricing and replication with transactions costs[END_REF] where q is defined in (2.10). We need to determine the limit of J n throughout the Doob's decomposition of J 2,n w.r.t. the filtration F j m 1 ≤j≤m 2 . 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 ), we can directly show that | J 4,n -J * (S 1 )| = o(θ -1 n ). Furthermore, owing to Itô's formula, we replace B * j-1 by

J 1,n = m 2 j=m 1 S t j ∆Φ j . Putting b j = ∆Φ j -ϕ j-1 ∆v j , we can represent 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
1 t j-1
∂ x B(λ j-1 , S u )dS u . Direct calculations yield that ∂ 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 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 volume is summarized in the following.

Proof of Theorem 3.3

The key technique in Proposition 7.1 can be 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 , m 1 ≤ k ≤ m 2 .

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 by following the procedure in the approximation of J n . First, by Itô's lemma,

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 = -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 . Now, putting B * j-1 = B(λ j-1 , Št j-1 ) -B(λ j-1 , Š1 ), we write J 2,n = J 4,n + J 3,n , where 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, B * j-1 can be replaced by

J 4,n = -1 2/π
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 = √ 1 -r 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 can be applied to approximate J 3,n by the martingale U 3,m 2 , defined by

U 3,k = -1 k j=m 1 (A 1,j-1 Z 1,j + A 2,j-1 Z 2,j )∆λ j , m 1 ≤ k ≤ m 2 ,
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 , m 1 ≤ k ≤ m 2 , for explicit functions A i holding the assumption of Proposition 7.3. Then, n β M m 2 n≥1 converges in law to a mixed Gaussian variable, which completes the proof.

Conclusion

We studied the problem of approximate option replication in SV settings using a new specification for adjusted volatility. Although our model employed an adjusted volatility simpler than those of previous works, we obtain the same asymptotic results for both Leland' and Lépinette's strategies in general SV markets. A possible connection with high frequency markets with proportional transaction costs was also discussed. As an application, we showed that the option price inclusive of transaction costs can be reduced adapting the theory of quantile hedging. Note that our approach is still helpful for more general settings, for example, when the friction rule admits a separate-variable representation [START_REF] Nguyen | Option replication with general transaction costs in stochastic volatility markets[END_REF]. This generalization includes the case where trading costs are based on the physical number of traded shares. Lastly, in the accompanying paper, we extended the method to multidimensional frameworks for European options with a general payoff written on several assets [START_REF] Nguyen | Approximate hedging with proportional transaction costs for multiasset options[END_REF].

Figure 1 :

 1 Figure 1: min(S 1 , K) -κ * J(S 1 ) on the left and J(S 1 ) on the right with K = 5.

  taking their values in R. Our financial market consists of one risky asset governed by the following equations on the time interval [0 , 1]

Proposition 4 . 2 .( 1 -

 421 Assume that σ max = sup y∈R σ(y) < ∞ . Then, for any r > 0 and δ ε defined by (4.3), lim ε→0 δ ε )ε -r = +∞ .(4.4)

  (a) Reduction factor 1 -δ ε and powers of ε (b) Reduction amount and ε 6 High-frequency markets

Proposition 7 . 4 .

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

j=m 1 S 2 up 2 j=m 1 λ- 1 / 2 j- 1 S

 1221121 t j-1 b j . In view of (A.1) and condition (C 2 ),ε 1,n = o(θ -1 n ) as n → ∞.Furthermore, by using the Taylor expansion, we obtain ε 2,n ≤ S sup m 2 j=m 1 ∆v j 2 up to a multiple constant, where S sup = sup 0≤t≤1 S t . Now, by taking into account that E v j-1 -v j 2 to a multiple constant and using condition (C 2 ) together with (A.1), we get |ε 2,n | = o(θ -1 n ). Next, by using Itô's Lemma and the substitution λ j = λ 0 (1 -t j ) 4β , we replace J 1,n by J 2,n = m t j-1 ϕ j-1 |κ j |∆λ j := m 2 j=m 1

Proposition 7 . 8 . 1 ( 1 Mm 2 = n β - 1 µ+1 Mm 2 ,

 7811212 Under conditions (C 1 ) -(C 2 ), P -lim 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,k -κ * (U 2,k + U 3,k ) = k j=m Ǎ1,j-1 Z 1,j + Ǎ3,j-1 Z 3,j )∆λ j ,where Ǎi , i = 1, 2 are explicitly determined and satisfy the assumption of Proposition 7.4. Then, in view of θ n -Theorem 3.2 is proved throughout Proposition 7.4.

t i-1 0 CProposition 7 . 9 .

 079 xt (u, S u )du. 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 2 j=m 1 B * j- 1

 1211 j-1 ∆λ j , J 3,n = -1 2/π m ∆λ j .

Proposition 7 . 10 .

 710 If is a positive constant independent of n then,P -lim n→∞ θ n |J n -η 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 2 -κ * (U 2,m 2 + U 3,m 2 ). The latter martingale sum can be represented in the formM k = -1 k j=m 1

  12, Theorem 2.3, p.107]). 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

	Theorem 5.1.

0≤s≤t |y s | 2m < M (1 + E |y 0 | 2m ),

where α, M are positive constants dependent on t, m.

In the context of Theorem 5.1, condition (C 1 ) holds if volatility function 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: Assume that 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, we have

E (y * ) 2m ≤ C(1 + E|y 0 | 2m ) < ∞, m ≥ 1,

as long as E|y 0 | 2m < ∞. Therefore, condition (C 1 ) is fulfilled in (5.2).

Uniform elliptic volatility models: Suppose that 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 = (a -by t )dt + dZ.

(5.3)

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

Stein-Stein models: Assume that dS t = y 2 t + σ min S t dW t and dy t = (a -by t )dt + dZ t .

(5.4)

Table 1 :

 1 be seen as the gain/loss Convergence for Lépinette's strategy with κ * = 0.01, = 2.

	n	gain/loss corrected 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
	n	gain/loss corrected 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

Table 2 :

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

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.

Acknowledgment: The authors would like to thank the two referees and editor for their remarks and suggestions which 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 Russian Science Foundation (research project No. 14-49-00079) and by the National Research Tomsk State University.

Then, changing the variables v = λ t for the inner integral yields u 0 σ 2 t A(λ t , Šu )dt = λ 0 λ u A(v, Šu )dv = A(λ u , Šu ) -A(λ 0 , Šu ).

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 In view of (H), one deduces A(λ 0 , x, y)| ≤ C √ K U (x, y)e -λ 0 /8 , where U (x, y) = x -1/2 U (x, y). Now, putting Ǎ * u = Ǎu∧τ * and I * 2,n =

Using the Chebychev inequality, we obtain

Hence, due to condition (H),

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, by taking into account Lemma A.3 and the integrability condition (C 1 ), one gets lim η→0 lim n→∞ lim L→∞ P(Γ c 1,η,L ) = 0.

On Γ 1,η,L , we have Ǎ = Ǎ * and

. By Chebychev's inequality, we obtain

where E( κ j |F j-1 ) = -1 σ(y t j-1 )G(p j-1 ) := D j-1 and G(p) defined in (7.7). Let B(λ, x, y) = λ -1/2 x ϕ(λ, x)D(λ, x, y) and

We observe that J 2,n = J 3,n + U 2,m 2 , where

j-1 S t j-1 ϕ j-1 κ j ∆λ j and κ j := κ j -D j-1 .

(7.25)

By substituting Št j-1 by Š1 everywhere in J 3,n , we write J 3,n = J 4,n + J 5,n , where

). Then, by Lemma A.2, we can check that J 4,n converges a.s. to J(S 1 , y 1 , ) at rate θ n . Now, an application of Itô's Lemma for B * j-1 yields stochastic integrals with respect to the Wiener processes. Owing to Proposition 7.1, the sum of these integrals can be approximated by U 3,m 2 , defined by

The asymptotic form of J n is summarized in the following. Proposition 7.7. For any fixed > 0,

Now, the martingale part of the hedging error is given by M m 2 , defined by

where

It is easy to see that the assumption of Proposition 7.2 is fulfilled for A i , i = 1, 2, 3. Hence, the sequence

converges in law to a mixed Gaussian variable by Proposition 7.2, which proves Theorem 3.1.

Proof of Theorem 3.2

When → ∞ under condition (C 2 ), the approximation for J n is slightly different. In particular, observe first that for any b ∈ R, E |aZ + b| can be approximated by b(2Φ(b/a)-1) as a → 0. Therefore, we can replace J 3,n in (7.24) by the sum

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). Puting

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, for any m 1 ≤ j ≤ m 2 , The following result is straightforward to check.

Lemma A.2. Let either be a positive constant or satisfy condition (C 2 ). Then, for any function A satisfying condition (H 0 ),

Proof. It follows from the fact that conditioning on σ-field generated by the volatility process, the log-price process ln S t has Gaussian distribution.

Lemma A.4. Suppose that A 0 and its derivatives Proof. Let ε > 0. On the set Γ

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

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

By Lemma A.3, lim ε→0 lim n→∞ P(Γ c 1,ε ) → 0. Thanks to the continuity of the functions S t and y t , one gets lim n→∞ P sup

Moreover, the integrability of U ( Š * 1 ) implies that P(| U ( Š * 1 )| > N ) converges to zero as N → ∞. By (7.12), P(τ * < 1) converges to 0 as L → ∞, which completes the proof.

, where A 0 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 is a product of these above kinds.

Proof. We only 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, m 2 j=m 1 A(λ j-1 , Š1 )∆λ j converges a.s. to zero by Lemma A.2. It remains to show that P(|∆ n | > ε) → 0 for any given ε > 0, but it can be done 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

) 2 ξ j (t) and

) 2 ξ j (t), where ξ j (t) = 1 ( t j-1 , t j ] (t) and x t = ln(S t /K). Clearly,

, which converges to zero by (C 2 ). Now, the particular choice of ε ensures that θ 4β+1)/4β λ -1 0 , which tends to zero. 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 ∈ R, for some constant γ > 0. Let y t be an Orstein-Uhlenbeck process defined by dy t = (a-by t )dt+dZ t with some constants a and b > 0. Put X α = exp 2αγ