Mean Square Error and Limit Theorem for the Modi fied Leland Hedging Strategy with a Constant Transaction Costs Coefficient

We study the Leland model for hedging portfolios in the presence of a constant proportional transaction costs coefficient. The modi fied Leland's strategy defi ned in [2], contrarily to the classical one, ensures the asymptotic replication of a large class of payoff . In this setting, we prove a limit theorem for the deviation between the real portfolio and the payoff . As Pergamenshchikov did in the framework of the usual Leland's strategy [11], we identify the rate of convergence and the associated limit distribution. This rate turns out to be improved using the modi fied strategy and non periodic revision dates.


Introduction
The present paper is concerned with the study of asymptotic hedging in the presence of transaction costs. The asymptotic replication of a given payoff is performed via a modified Leland's strategy recently introduced in [2].
Let us briefly recall the history and the main known results about Leland's strategy.
In 1985 Leland suggested an approach to price contingent claims under proportional transaction costs. His main idea was to use the classical Black-Scholes formula with a suitably adjusted volatility for a periodically revised portfolio whose terminal value approximates the payoff. The intuition behind this practical method is to compensate for transaction cost by increasing the volatility in the following way: where n is the number of the portfolio revision dates and kn = k 0 n −α , α ∈ [0, 1 2 ] is the transaction costs coefficient generally depending of n; f is an increasing and smooth function whose inverse g := f −1 defines the revision dates t n i := g( i n ), 1 ≤ i ≤ n. The principal results on convergence for models with transaction costs can be described as follows. First consider the case of approximate hedging of the European call option using the strategy with periodic portfolio revisions (i.e g(t) = t). We know the following results with T = 1: (a) For α = 1 2 , Lott gave the first rigorous result on the convergence of the approximating portfolio value V n 1 to the payoff V 1 = (S 1 − K) + . The sequence V n 1 − V 1 tends to zero in probability [10], and a stronger result holds: n E (V n 1 − V 1 ) 2 converges to a constant A 1 > 0 [5]; (b) For α ∈ (0, 1 2 ), the sequence V n 1 − V 1 tends to zero in probability (see [8]), and it is shown in [1] that n pα E (V n 1 − V 1 ) 2 → 0 as n → ∞, with pα < α. (c) For α = 0, the terminal values of portfolios do not converge to the European call as shown by Kabanov and Safarian [8]. Namely, there is a negative σ{S 1 }-measurable random variable ξ such that V n 1 − V 1 → ξ in probability. Pergamenshchikov [11] then analyzed the rate of convergence and proved a limit theorem: the sequence n 1 4 (V n 1 − V 1 − ξ) converges in law to a mixture of Gaussian distributions [11]. He noticed that one can increase the modified volatility to obtain the asymptotic replication. To do so, he utilizes the explicit form of the systematic hedging error for the European call option. For related results see also [6] and [12].
For models including uniform and non-uniform revision intervals one needs to impose certain conditions on the scale transform g. Generalizations of some of the above results to this more technical case as well as extensions to contingent claims of the form h(S 1 ) can be found in [12], [5], [1]. In particular, n 1/2+α E (V n 1 − V 1 ) 2 converges to a constant in the case α > 0. Moreover, for α = 1 2 , the distributions of the process Y n t := n 1 2 (V n t − V t ) 3 in the Skorohod space D[0, 1] converges weakly to the distribution of a two-dimensional Markov diffusion process component (see [4]). Notice that the asymptotic replication still does not hold for α = 0 in this more general setting. For more details we refer to [1], [3], [4] and references therein.
We solve the case α = 0 for a large class of payoff and with specific revision dates (including uniform dates) by means of the modified strategy introduced in [2].
This one makes the portfolio's terminal value converge to the contingent claim as n tends to infinity, that is the approximation error vanishes. The analysis we performed here suggests that it might be difficult to obtain a better convergence rate regarding uniform revision dates. In the framework of the non uniform grid we use, concentrating the revision dates near the maturity T = 1 accelerates the convergence rate.
The asymptotic behavior of the hedging error is a practical important issue. Since traders obviously prefer gains than losses, measuring the L 2 -norm of hedging errors is strongly criticized. Of course, the limiting distribution of the hedging error is much more informative. Our present work also aims at tackling this issue: we prove that where the law of Z is explicitly identified, EZ = 0 and p > 0 depends on the choosen grid.
The paper is organized as follows. In Section 2, we introduce the basic notations, models and assumptions of our study; In particular we recall the modified Leland's strategy defined in [2]. In Section 3, we state our main result: a limit theorem for the renormalized asymptotic hedging error. In Section 4, we establish two lemmas concerning, on one hand, random variables constructed from the geometric Brownian motion, and on the other hand, some change of variables for the revision dates. These auxiliary results will be used repeatedly throughout the paper. In Section 5, we prove the main result. An appendix finally recalls all the known technical results we need for the various proofs.

Black-Scholes model and hedging strategy
We are given a filtered probability space (Ω, F, (F t ) t∈[0,1] , P) on which a standard one-dimensional (F t )-adapted Brownian motion W is defined. As usual, we denote by We consider the classical Black-Scholes model composed of two assets without transaction costs, i.e. k 0 = 0 and σ = σ. The first one is riskless (bond) with the interest rate r = 0 and the second asset is S = (S t ), t ∈ [0, 1], a geometric Brownian motion that is S t = S 0 e σWt− 1 2 σ 2 t . It satisfies the SDE dS t = σS t dW t , with positive constants S 0 , σ. It means that the risky asset is seen under the martingale measure.
The well-known Black and Scholes problem without transaction costs is to hedge a payoff h(S 1 ), h being a continuous function of polynomial growth. The pricing function solves the terminal valued Cauchy problem Its solution can be written as where ρ 2 t = (1 − t)σ 2 and ϕ is the standard Gaussian density.
Without transaction costs (σ = σ) the self-financed portfolio process reads In the Itô formula for C(t, S t ) the integral over dt vanishes and, therefore, for all t ∈ [0, 1]. In particular, V 1 = h(S 1 ): At maturity the portfolio V replicates the terminal payoff of the option. Modeling assumptions of the above formulation include frictionless market and continuous trading for instance.
However, an investor revises the portfolio at a finite set of dates and keeps Cx(t i , S ti ) units of the stock until the next revision date t i+1 . It is well known that this discretized model converges to the Black-Scholes one in the sense that the corresponding portfolio terminal value converges to the payoff as the number of revision dates tends to infinity.

Reminder about Leland's strategy
We are now concerned with transaction costs. We directly work in a discrete time setting. Leland suggested to replace σ in the Cauchy problem above by a suitable modified volatility σ. In the case where σ does not depend on t, the solution C satisfies C(t, x) = C(t, x, σ), i.e. practitioners do not need to rectify their algorithms to compute the strategy. Leland obtained an explicit expression of σ by equalizing the transaction costs of the portfolio and the drift term generated by the additional term σ − σ 2 > 0 in the Ito expansion of the payoff h(S 1 ) = C(1, S 1 ). In the general case, the pricing function can be written as ϕ is the Gaussian density and g = f −1 is the revision date function.

A possible modification of Leland's strategy
The practically interesting case α = 0 (i.e., k 0 is constant), where is a systematic error attracted a lot of attention. Limit theorems were obtained by Granditz and Schachinger [6] and Pergamenshchikov [11]. Zhao and Ziemba [13], [14] provides a numerical study of the limiting error for practical values of parameters. Sekine and Yano, [12] suggested some scheme to reduce it. In the paper [11] a modification of the Leland strategy was suggested for the call option eliminating the limiting error. Unfortunately, the approach is based on the explicit formulae and, seemingly, cannot be easily generalized for more general pay-off functions. Our modification of the Leland strategy has the following features: 1) we use the same enlarged volatility; 2) the initial value of the portfolio V n 0 is exactly the same than for the initial Leland strategy (see [11] where the behavior of V n 0 is studied as n → ∞ and a method is suggested to lower the option price); 3) the only difference is at the revision dates t i ; We apply not the modified "delta" of the Black-Scholes formula with enlarged volatility, but correct it on the basis of previous revisions, see the formula (2.7). This is a technical modification of Leland's strategy which is difficult to economically interpret but has the advantage to release the limiting error.
In the model with proportional transaction costs and a finite number of revision dates the current value of the portfolio process at time t is described as where D n is a piecewise-constant process with D n = D n i on the interval (t i−1 , t i ], t i = t n i , i ≤ n, are the revision dates, and D n i are F ti−1 -measurable random variables. Recall that the transaction costs coefficient is a constant k 0 > 0 (that is α = 0 in the Leland model) and the dates t i are defined by a function g, namely t i = g( i n ). Let us denote by f the inverse of g. Set for all i 0 < n and let us define the dates The "enlarged volatility", depending on n, is given by the formula (2.5). We modify the usual Leland strategy (see [2]) by considering the process D n with Moreover, let us define K n t := i∈J n 1 (t) ∆K n

Assumptions and notational conventions
Throughout the paper, we adopt the following rules: (i) we will often omit the indexes n and the variable t (especially in the appendix) when there is no ambiguity; (ii) the constants C appearing in the various inequalities is independent of n and may change from one line to the next; (iii) we use the classical Landau notations O and o. These quantities will be always deterministic.
As shown in [3], recall that the solution C of the Cauchy problem we consider is infinitely differentiable on [0, T ) × (0, ∞). We use the abbreviations δ t := Cx(t, S t ), γ t := Cxx(t, S t ). We denote by (δ n t ) t the process equal to δ t n i on the interval [t n i , t n i+1 ) and (γ n t ) t is defined similarly. For an arbitrary process H, we set ∆ H ti := H ti −H ti−1 .
We will work under the following assumptions: (A1) The function g has the following form: (A2) h is a convex and continuous function on [0, ∞) which is twice differentiable except the points K 1 < · · · < Kp h where h and h admit right and left limits; Assumption (A1) is not too restrictive. A trader can in particular choose µ = 1 to balance its portfolio periodically. However, as we will see, it is more preferable to increase µ to obtain a better rate of convergence.
Note that f (t) = 1 − (1 − t) 1/µ , hence the derivative f for µ > 1 explodes at the maturity date and so does the enlarged volatility. We define the increasing function Under Assumption (A1), we have 0 ≤ p < 1/16.
In the sequel, will frequently appear the quantity

Main Result
In [2], it is proven that V n 1 converges in probability to h(S 1 ). We recall this result: Our main result here provides the rate of convergence for a specific family of revision dates funtions including the uniform grid (i.e. g(t) = t) and identifies the associated limit distribution of the deviation: where the law of Z is a mixed gaussian distribution, i.e. Z = ηN where N is a standard normal independent of η and Observe that EZ = 0. In the proof, Z will be identified by its characteristic function As we can see, concentrating the revision dates near the horizon date (p > 0) improves the convergence rate. Actually, we can observe that near T = 1, the derivative f explodes if p > 0 and so increases the modified volatility, which confirms the main Leland idea; Artificially increase the volatility to compensate for transaction costs. The proof of the theorem above is given in Section 5; To do so, we decompose the difference n 1 4 +p (V n t − C(t, S t )) into a martingale which converges in L 2 and a residual term tending to 0 at T = 1. We conclude with h(S 1 ) = C(1, S 1 ).

Geometric Brownian motion and related quantities
In the sequel, we shall use the decomposition given by Ito formula where The process M n is a square integrable martingale on [0, 1] by virtue of [2].
We set for u < v In the sequel, we will use several times the following basic results.
provided that f is no decreasing. Note that there is a constant C independent of n such that for all i ≤ n − 1, We shall often use the inequality where C is a constant independent of n.
There exists a constant C > 0 such that Moreover, for a given x, Proof Let us write In a similar way, we have Hence there is a constant c satisfying Since 0 ≤ (µ − 1)/(1 + µ) < 1, we also find a constant c such that which concludes the proof.
We now stress an important remark regarding a slight abuse of notation repeatedly used along the paper.

Remark 4.3
Throughout the sequel, we shall often use the change of variable x = ρ 2 t with dx = − σ 2 t dt. For ease of notation, we will use the abuse of notation t instead of t(n, x) := (ρ 2 ) −1 (x) when applying this change of variable in an integral.
Similarly, a direct computation yields the following lemma.
Lemma 4.4 Set y > 0 and v := v(n, y) such that y = ρ 2 v . There exists a constant Moreover, for a given y,

Proof of the limit theorem
The proof is divided into three parts. In Step 1 we split the hedging error into a martingale part M and a residual part . In Step 2 we show that the residual terms tend to 0 in L 2 (Ω) as n tends to infinity. The convergence rate n where for all n ∈ N, M n is a martingale of terminal value The residual term can be splited as

5.2
Step 2: The mean square residue tends to 0 with rate n The most technical part of this paper is the following. The deviation of the approximating portfolio from the payoff has been written in an integral form by virtue of the Ito formula. The "real world" portfolio may be interpreted as a discrete-time approximation of the theoretical portfolio C(t, S t ) yielding the residual terms above.
Consequently, the following analysis is mainly based on Taylor approximations involving the successive derivatives of C and so heavily utilizes estimates of the appendix.
Standard tools from stochastic calculus are also frequently used.
The following convergence holds: To prove this theorem, we show the suitable convergence to 0 concerning the From Corollary 6.6 and Inequalities (4.16-4.19), we deduce the following A Taylor formula suggests to write the following splitting: Proof The Doob inequality yields n 1 2 +2p E sup t (R n 10 (t)) 2 ≤ 4n 1 2 +2p E (R n 10 (1)) 2 where the r.h.s tends to 0 as shown below. Indeed, by the independence of the increments of the Wiener process, we write: It is easy to check the following asymptotic We then deduce Let us remark the abuse of notations ρ 2 0 = ρ 2 0,n and t i = t(n, x i ) as previously mentioned. First, let us show that fn satisfies the dominated convergence bound condition.
Applying the Lebesgue theorem, we deduce that Λ ti−1 converges to is integrable if µ = 1. We then apply the Lebesgue theorem to conclude the following Proof Using the Doob inequality, we obtain that E (sup t R n 11 (t)) 2 ≤ 4E (R n 11 (1)) 2 . By independence of the increments of the Wiener process, we deduce that It follows that since Corollary 6.13 gives where nf (t i−1 )∆t i is bounded. We then conclude.
Proof As previously, we have the Doob inequality E (sup t R n 12 (t)) 2 ≤ 4E (R n 12 (1)) 2 and the equality From (6.68), there exists a constant C such that: Using the Cauchy-Schwarz inequality and (4.15) with m = 8, we deduce that which proves the desired convergence to 0.
Proof We still consider the Doob inequality E (sup t R n 13 (t)) 2 ≤ 4E (R n 13 (1)) 2 and Moreover, using Lemma 6.18 and the Cauchy-Schwarz inequality, we deduce that Then, we obtain The conclusion follows.
Proof We use the Doob inequality E (sup t R n 14 (t)) 2 ≤ 4E (R n 14 (1)) 2 and the equality Then, Thus, we can conclude.
Let us now study the residual term R n 2 . Again, a Taylor formula suggests to write the following splitting R n 2 = R n 20 + · · · + R n 24 where R n 20 (t) := σk 0 2 π n Proof We have We use the Cauchy-Schwarz inequality, Inequalities (6.6) and (4.18). From the explicit formula of f , we thus obtain so that we can conclude.
We write n From (6.63), there exists a constant C such that: Using Assumption ( A1), we claim that there exists a constant c such that Thus, using (6.62-6.67), we obtain some constant C such that the following inequality holds: (5.29) By means of the stochastic Fubini theorem, we obtain that Since the Doob inequality E (sup t A n t ) 2 ≤ 4E (A n 1 ) 2 holds, it suffices to estimate E (A n 1 ) 2 . From the boundedness of (t i − u)/(1 − u) and f (u)(t i − u)n on u ∈ [t i−1 , t i ), we deduce the following estimates: Then, we conclude that E (sup t A n t ) 2 − −−− → n→∞ 0.
Secondly, we write: Then, sup t |B n t | ≤ cn 3/4+p Thus there exists a constant c such that E sup t |B n t | 2 ≤ c n 3 2 +2p Υ n where Using the Cauchy-Schwarz inequality and (5.29), we can then bound Υ n : c log n n 1+3/(4µ) . (5.30) In a same way, we obtain the following inequalities c log n n 7/2+1/(4µ) . Proof We write −R n 22 (t) = kn i∈J n We deduce that for n large enough, 0 ≤ E {E ti ti−1 } 2 s ≤ c(∆t i ) 3 2 . Using the Doob inequality E (sup t U n (t)) 2 ≤ 4E (U n (1)) 2 , it suffices to estimate E (U n (1)) 2 . The independence of the increments of the Brownian motion implies the equality Then, there exists a constant C such that At last, for n large enough, E {E ti ti−1 } 2 s ≥ 0. Hence, 0 ≤ sup t V n (t) ≤ N n (1). In order to prove that n 1 2 +2p E V n (1) 2 − −−− → n→∞ 0, we first analyze the following sum Using the Cauchy-Schwarz inequality, we also have We deduce that n Applying Taylor's formula to the difference Cx(t i , S ti ) − Cx(t i−1 , S ti−1 ) it is sufficient to estimate the following sums (5.33), · · · ,(5.36). The first one satisfies Indeed, from Corollary 6.13, we deduce that: The second one verifies Thirdly, from (6.69), we deduce that and it follows that Finally, from Lemma 6.18, we get that From above, we can conclude about that n As for R n 232 (t), we use the inequality Θ i − Θ 1 i ≤ |∆ K n ti | and we deduce from Using the Cauchy-Schwarz inequality, the boundedness of Moreover, .

24
By virtue of the Bienaymé-Tchebytchev inequality P( We deduce that E sup i (∆S ti ) 4 ≤ C n − 3 2 and finally E sup t (R n 232 (t)) 2 ≤ Cn −3/4 log 2 (n) so that we can conclude the lemma.
Proof Let us notice that sup t |R n 24 (t)| is bounded by the random variable Using the Ito formula for the increments We first prove that T 1 n − −−− → n→∞ 0. Using the Taylor Formula, we get that Using the suitable estimations from the Appendix, we then obtain The last estimate follows from Corollary 6.67. Indeed, the proof is the same since We then prove that T 2 n − −−− → n→∞ 0. We deduce from Appendix the following inequality: It suffices to obtain the convergence to conclude the lemma. This last lemma completes the proof of Theorem 5.1.

Step 3: Asymptotic distribution
From the previous subsection, it turns out that the deviation between the "real world" terminal portfolio and the payoff h(S 1 ) is essentially composed of a martingale as n → ∞. To study the asymptotic distribution of n 1 4 +p M n 1 , we consider it as terminal values of the following sequence of martingales (N n j ) j=0,··· ,n with respect to the filtration F n = (F ti ) i : We achieve the proof of this theorem by means of results in [7] recalled by Theorem 6.1 in the Appendix. We thus need to prove the following lemmas.
Using the bound (5.42), we obtain i E χ 2 We finally conclude the lemma. Inspecting the proof above, we deduce the following: The sequence of martingales (N n i ) i=0,··· ,n satisfies Proof Indeed, by virtue of Inequalities (5.41) and (5.43), for a given ε > 0 Lemma 5.16 The sequence of martingales (M n i ) i=0,··· ,n satisfies the following convergence Proof First, let us study the term ξ ϑ n : . Hence, using Lemma 4.1 and the change of variable y = ρ 2 u and x i = ρ 2 ti , We then deduce that In particular, Hence, a.s.
Therefore, using (4.21), But, due to Hölder's inequality, We can thus apply Lebesgue's theorem using Corollary 6.21 and (4.20): Second, let us study the term ξ χ n = i E χ 2 i |F ti−1 . By independence, we obtain (1)). We then deduce that Let us obtain a suitable bound for z χ n (x), integrable in x. Recall that Due to Inequality (6.60), we claim that a.s.(ω) for n large enough, there is a constant cω which does not depend on n such that Indeed, this is obvious for x ≥ 1. Otherwise, 1 ≥ x = ρ 2 u ≥ c n 1/2 (1 − u n (x)) implies that u = u n (x) is close to 1 uniformly in x ≤ 1 as soon as n is large enough. It then suffices to choose S 1 out of the null-set {S 1 = K 1 , · · · , Kp h } to obtain by continuity that S u n (x) is also far enough from the points K 1 , · · · , Kp h if x ≥ 1.
We conclude that for all j, log 2 (K j /S u n (x) ) ≥ c ω,j for some constants c ω,j > 0. Therefore, We can then apply the dominated convergence theorem using the limit (4.21). We obtain ξ χ n a.s.
Finally, let us study the term i E χ i ϑ i |F ti−1 .
Due to (4.20), we obtain (∆t i ) From the bounds (4.21), (5.48), (5.52) and by applying again Lebesgue's theorem, we then deduce the following Proof Due to the independence of the increments of the Wiener process, we have E (χ i + ϑ i )(χ j + ϑ j ) = 0 whenever i = j. We thus obtain Let us show that ξn := ξ χ n + ξ ϑ n is uniformly integrable. First let us note that ξn is bounded in L 1 (Ω). Indeed, from Corollary 6.6, Inequalities (5.49) and (4.20), we obtain for all n Now, using the Cauchy-Schwarz inequality and then the Markov inequality, we have Recall that Therefore, applying successively the Cauchy-Schwarz inequality, (4.20), Corollary 6.15, and the Markov inequality, we obtain Therefore, ξn is uniformly integrable, and so is i E (χ i + ϑ i ) 2 |F ti−1 , which moreover converges to η a.s. This yields the conclusion of the Lemma.
We easily deduce the following: These last lemmas and corollaries complete the proof of Theorem 5.13.

Conclusion
Let us summarize the results of the previous theorems: n The proof of the limit theorem is then complete.

Estimates
To study the residual terms generated by the discretization of the theoretical portfolio C(T, S T ), we use Taylor approximations. We then need to estimate some bounds of the successive derivatives of C. where Lemma 6.11 Assume that Assumption A1 holds. Then there exists a constant c such for n large enough.
Proof We have obviously We deduce that: Indeed, we use a first order Taylor expansion to estimate the difference f (g(θ i ) − h i ) − f (g(θ i )). We conclude by using the explicit expression of f , g but also the inequality (1 − t i−1 )/(1 − t i ) ≤ c for i ≤ n − 1.
The following lemma is of first importance to get estimations of expectations we need in some of our proofs.
Lemma 6.12 Suppose that t ≤ u < 1, m ∈ R, q ∈ 2N and K > 0. There exists a constant c = c(m, q) such that E S m u log q Su K exp − log 2 (Su/K) where P 0 (ρ t ) := ρ t , P 2 (ρ t ) := ρ 3 t + ρ 5 t , From now on, we can deduce the following results.
Corollary 6.13 If m ∈ R and u ≥ t, then there exists a constant cm > 0 such that Proof Indeed, it suffices to use Lemma 6.8 and apply the previous lemma.
In a similar way, we have: xx (t, S t ) Cxx(t, S t ) and we apply Cauchy-Schwarz' inequality with p = 4/3 and q = 4 such that p −1 + q −1 = 1. We obtain where the last inequality is deduced from Corollary 6.62. The conclusion follows. (1 − t i ) 4 .
Proof We have S m ti−1 ≤ S m ti−1 + S m ti , and ρ ti−1 ≥ ρ ti . Furthermore, in virtue of Lemma 6.8, recall that we have Then, the conclusion follows.
In the same way, we can prove the following results: Lemma 6.18 There exists a constant C such that Proof The arguments are similar to the previous ones but we also use the inequality: Lemma 6.19 There exists a constant C such that (6.70)

Technical Lemmas
Recall the two following lemmas (see [2]). These results ensures the convergence of the Leland scheme without any hedging error when using the modified Leland strategy.
The change of variable x = ρ 2 u appears to be as essential in the following proofs and points out the significative role of the revision dates near the maturity. Proof We apply Lemma 6.20 with the change of variable x = ρ 2 u . Recall that 0 ≤ 1 − u ≤ c x n −1/2 so that u → 1 as n → ∞ for a given x ≥ 0. We can apply the Lebesgue theorem by dominating the function G 1 (x, Su) whether x ≤ 1 or not because x ≤ 1 implies that u is sufficiently near from 1 independently of x for n ≥ n 0 . Indeed, outside of the null-set ∪ i {S 1 = K i }, we have 0 < a ≤ | log(Su/K j )| ≤ b for some constants a, b (depending on ω) provided that u is sufficiently near one.