On Leland s strategy of option pricing with transactions costs

Transcription

1 Finance Stochast., c Springer-Verlag 997 On Leland s strategy of option pricing with transactions costs Yuri M. Kabanov,, Mher M. Safarian 2 Central Economics and Mathematics Institute of the Russian Academy of Sciences, and Laboratoire de Mathématiques, Université de Franche-Comté, 6 Route de Gray, F-253 Besançon Cedex, France 2 Humboldt University, Unter den Linden, 6, D-7 Berlin, Germany Abstract. We compute the limiting hedging error of the Leland strategy for the approximate pricing of the European call option in a market with transactions costs. It is not equal to zero in the case when the level of transactions costs is a constant, in contradiction with the claim in Leland 985. Key words: Transactions costs, asymptotic hedging, call option, Black-Scholes formula JEL classification: G3 Mathematics Subject Classification 99: 9A9, 6H5. Introduction In his paper devoted to the problem of option pricing in the presence of transactions costs, Heyne Leland 985 suggested a trading strategy based on the nice idea of a periodic revision of a hedging portfolio using modified Black Scholes betas. He assumed that the level k of transactions costs is a constant and claimed that the terminal value of the portfolio approximates the payoff as the length of a revision interval tends to zero. In a footnote remark he also mentioned that the same holds also when the level is k n /2, n being the number of revision intervals. Both of these results are considered very helpful for practitioners, and the paper is widely quoted in the literature. However, Leland s arguments were on a heuristic level and his conclusions have to be considered only as conjectures. The research for this paper was carried out during a visit to Humboldt University at Berlin supported by the Volkswagenstiftung. The authors express their thanks to Uwe Küchler for fruitful discussions and to anonymous referees for important remarks and corrections. Manuscript received: February 996; final version received: October 996

2 24 Yu.M. Kabanov, M.M. Safarian Recently, Lott 993 provided a rigorous mathematical proof of the footnote remark together with a study of another approximating strategy. In the present note we show that, unfortunately, the main conjecture of Leland for the case of constant level of the transactions costs fails, and we calculate the hedging error. We also prove that the approximation result still holds in the case where the level is k n α, α ], /2[, k >. 2. Description of the model The stock price dynamics is given by the geometric Brownian motion S t = S exp{µ σ 2 /2t + σw t }, where W is the Wiener process. The bond price is constant over time and equal to one certainly, this is not a restriction. In the absence of transactions costs the fair price at time t of the European call option maturing at T = with the striking price K, i.e. with the terminal payoff H = hs =S K +, is given by the Black Scholes formula V t = C t, S t where C t, x =Ct,x,σ:=xΦd KΦd σ t, Φ is the standard normal distribution function with the density ϕ, and d = dx,σ:= lnx/k σ t + 2 σ t. The terminal payoff is replicated by the value at maturity of the self-financing portfolio which has initial endowment C, S and at time t contains φ t := C x t, S t units of the stock and hence V t C x t, S t S t units of the bond. Assume that in the stock market the cost of a single transaction is a fixed fraction of its trading volume and the corresponding coefficient is k = k n our definition corresponds to one half of Leland s round trip coefficient. Let us consider the self-financing trading strategy with initial endowment Ĉ, S and the portfolio containing at time t a number ξt n of shares of the stock given by the formula ξ n t := φ ti I ]ti,t i ]t = Ĉ x, S ti I ]ti,t i ]t where t i := i/n, Ĉ t, x :=Ct,x, σ, φ t := Ĉ x t, S t, σ 2 := σ 2 + γ 2 2, γ := 2 σ π k n =2 π k n /2 α 2 to simplify formulae we omit the dependence on n in obvious cases. The value process now has the form

3 On Leland s strategy of option pricing with transactions costs 24 t V t ξ n =Ĉ, S + ξuds n u k S ti ξt n i ξt n i. 3 t i t Remarks. We follow here the definition adopted by Lott 993. Leland 985 considered instead of self-financing an H -admissible strategy but the problem is essentially the same. 2 A reader may have some trouble with boundary effects. There is a transaction at time t = when the investor enters the market. The last transaction at t =is also special: the contract may admit different specification for the final settlement, e.g., to deliver or not to deliver the stock. We exclude these particular transactions from our considerations. Theorem Assume that k = k n = k n α where α ], /2], k >. Then P lim V ξ n =H. 4 n Theorem 2 Let k = k > be a constant. Then where J 2 = J 2 k := 4 P lim n V ξ n =H+J J 2 5 J := min{k, S }, 6 { S GS,v,k exp v lns /K + } 2 dv, 7 v 2 v 2 GS,v,k := x 2k lns /K + k e x 2 /2 dx 8 2π 2πv 2π Remark. The integral in 8 can be calculated explicitly, in particular, GS,v, = 2/ 2π. From the other hand, it is easy to check that { J := min{k, S } = 2 S exp v lns /K + } 2 dv = J 2. 2π v 2 v 2 It follows that J 2 J Bk where the constant B depends on S and K. Thus, the option is always underpriced in the limit though the hedging error is small for small values of k see Fig.. 3. Conclusion We have shown that the limiting error in Leland s hedging strategy for the approximate pricing of the European call is equal to zero only when the level of transaction costs decreases to zero as the revision interval tends to zero. In the case when the level of transaction costs is a constant the limiting hedging error is given in Theorem 2 and, in general, is not equal to zero.

4 242 Yu.M. Kabanov, M.M. Safarian -.2 J-J S lg k Fig.. Dependence of J J 2 on k = k and S = S for K = 5 Appendix A. Proof of Theorem We start with calculations that are common for both theorems, assuming also without loss of generality that P is the risk-neutral measure i.e., µ =. The case of α =/2 when σ and hence Ĉ do not depend on n has been considered in Lott 993. So we suppose from now on that α [, /2[. This implies that σ 2 = On /2 α as n. However, some ideas from Lott s study work well here and we use them in several places below, e.g., in Lemma 4. By the Ito formula we have that where M n t := A n t := Ĉ x t, S t =Ĉ x, S +M n t t t Ĉ xx u, S u ds u = t +A n t σs u Ĉ xx u, S u dw u, ] du. [ Ĉ xt u, S u + 2 σ2 S 2 uĉxxx u, S u The process M n is a square integrable martingale on [, ] with M n t = { t σ 2 2π σ 2 s exp lnss /K σ s + } 2 2 σ s ds. Following Lott we represent the difference V H in the form convenient for a further study.

5 On Leland s strategy of option pricing with transactions costs 243 Lemma We have V H = F ξ n +F 2 ξ n where F ξ n := F 2 ξ n := 2 γσ ξ n t φ t ds t, 9 St 2 Ĉ xx t, S t dt k ξt n i ξt n i S ti. Proof. According to the Black Scholes theorem the claim H admits the representation H = C, S + Comparing 3 and we get that φ u ds u. V H = k ξ n t φ t ds t + ξt n i ξt n i S ti. φ t φ t ds t + Ĉ, S C, S We have left to check that Ĉ, S C, S = 2 γσ St 2 Ĉ xx t, S t dt φ t φ t ds t. This identity follows easily from the Ito formula and the observation that C t, x is the solution of the parabolic equation /2σ 2 x 2 C xx +C t = with the boundary condition C, x =hx while Ĉ t, x is the solution of /2 σ 2 x 2 Ĉ xx + Ĉ t = with the same boundary condition. Lemma 2 For any α [, /2[ P lim n F ξ n =. 2 Proof. By the Lenglart inequality see, e.g., Jacod and Shiryayev 987 it is sufficient to show that P lim n ξ n t φ t 2 σ 2 S 2 t dt =. 3 Here the integrand is bounded by σ 2 sup t St 2. But for all t < we have that ξt n and φ t asn. The study of F 2 ξ n is more delicate. Put t := /n. It is easily seen that F 2 ξ n = 5 Ln i where

6 244 Yu.M. Kabanov, M.M. Safarian L n := σ γ 2 σ γ 2 L n 2 := σ γ 2 kσ L n 3 := kσ L n 4 := k L n 5 := k k St 2 Ĉ xx t, S t dt S 2 Ĉ xx, S ti I ]ti,t i ]tdt, S 2 Ĉ xx, S ti t S 2 Ĉ xx, S ti w ti w ti, S 2 Ĉ xx, S ti w ti w ti S ti M n t i S ti M n t i M n, M n k S ti S ti ξt n i ξt n i. Lemma 3 For any α [, /2[ we have σ γ 2 σ γ 2 S ti ξt n i ξt n i, St 2 Ĉ xx t, S t dt J a.s., S 2 Ĉ xx, S ti I ]ti,t i ]tdt J a.s., and, hence, L n a.s. when n. Proof. After the substitution v = σ 2 t the first integral can be written as σ σ γ 2 S 2 v/ σ lns v/ σ 2 2 σ 2 exp 2π v v /K v 2 dv and the second one as σ σ γ 2 2 σ 2 2π exp v i 2 S vi / σ 2 vi 2 lns vi / σ /K 2 + v i 2 I ]v i,v i ]vdv where v i := σ 2 t i. Clearly, both expressions tends a.s. to the integral

7 On Leland s strategy of option pricing with transactions costs 245 { 2 S exp v lns /K + } 2 dv 2π v 2 v 2 which is equal to J = min{k, S } since γ/ σ 2 /σ and the integrands above when S ω /= K are dominated by the function of the form c ω[v /2 e c2/v I [,ε] v+i ]ε,n ] v+e c3/v I [N, [ v] 4 and we get the result see Remark after Theorem 2. Lemma 4 For any α [, /2[ we have L n 2 in probability. Proof. The sequence of independent random variables w ti w ti n /2 2/π is a martingale difference with respect to the discrete filtration F ti, E w ti w ti n /2 2 2/π = 2/πn = 2/π t. By the Lenglart inequality we need to check only the convergence to zero in probability of the sequence σ 2 2/πk 2 S 4 Ĉ 2 xx, S ti t which for almost all ω is of the order k 2 / σ 2 = On /2 α. Lemma 5 For t [, [ we have { } ESt 2 Ĉxx 2 t, S t = 2π σ 2 t 2a2 + exp b2 2a 2 + where σ t a := σ t, b := lns /K σ 2 t/2 σ + t 2 σ t. Proof. Let η be a standard normal random variable. Then for any a and b Since the result follows. E exp{ aη + b 2 } = St 2 Ĉxx 2 t, S t = 2π { exp { 2a2 + exp σ 2 t b2 2a 2 + }. σwt σ t + lns /K σ 2 t/2 σ + t 2 σ t 2 }

8 246 Yu.M. Kabanov, M.M. Safarian As a corollary we get that for t [/2, [ ESt 2 Ĉxx 2 c t, S t σ t. 5 We shall use below some simple bounds for higher order derivatives of the function Ĉ x t, x =Φ d where d := dx, σ. We have: Ĉ xx t, x = x σ t ϕ d, Ĉ xxx t, x = x 2 σ d + t σ ϕ d, t [ 2 Ĉ xxxx t, x = x 3 σ d + t σ t ] d d + σ 2 + t σ ϕ d, t Ĉ xt t, x = lnx/k 2 σ t + σ ϕ d, 3/2 4 t [ Ĉ xxt t, x = 2x σ t 3/2 + x σ lnx/k t 2 σ t + ] σ ϕ d. 3/2 4 t It follows that for t < we have Ĉxxx 2 t, x c x 4 σ 2 t + σ 4 t 2, 6 Ĉxxxx 2 t, x c x 4 σ 2 t + σ 4 t 2 + σ 6 t 3, 7 σ Ĉ xt t, x = c +, 8 t c Ĉ xxt t, x = x σ t /2 t Lemma 6 For any α [, /2[ the sequence L n 3 in probability. Proof. Using the inequality a b a b we get that L n 3 k ti S ti σ[s ti Ĉ xx, S ti S t Ĉ xx t, S t ]dw t and it sufficient to show that the sequence ti [S ti Ĉ xx, S ti S t Ĉ 2 xx t, S t ]dw t

9 On Leland s strategy of option pricing with transactions costs 247 tends to zero in probability. But by the Burkholder and Jensen inequalities E ti [S ti Ĉ xx, S ti S t Ĉ xx t, S t ]dw t c /2 ti E [S ti Ĉ xx, S ti S t Ĉ xx t, S t ] 2 dt /2 ti c E[S ti Ĉ xx, S ti S t Ĉ xx t, S t ] 2 dt. It follows from 5 that the last summand in the right-hand side of the above inequality is finite and tends to zero. By the Ito formula where d[s t Ĉ xx t, S t ] = f t dw t + g t dt f t := σs t Ĉ xx t, S t +σ 2 St 2 Ĉ xxx t, S t, g t := S t Ĉ xxt t, S t + 2 σ2 S 3 t Ĉ xxxx t, S t +σ 2 St 2 Ĉ xxx t, S t and we can estimate all other summands as follows: ti E[S ti Ĉ xx, S ti S t Ĉ xx t, S t ] 2 dt ti [ t t ] /2 2dt E f u dw u + g u du 2 t /2 ti c t Ef 2 u du + t σ 2 t i + σ 4 t i 2 ti Eg 2 udu /2 /2 /2 +c t 3/2 σ t i + + 3/2 t i σ 2 t i /2 + σ 4 t i 2 + σ 6 t i 3 c t σ t i + /2 σ 2 t i +c t 3/2 σ /2 t i 3/4 + t i + /2 σ t i + /2 σ 2 t i +. σ 3 t i 3/2

10 248 Yu.M. Kabanov, M.M. Safarian It is clear that n n t σ σ t i /2 t σ 2 t i σ 2 ln n, n t /2 t /2 n /2 σ σ /2 t i 3/4 n t /2 t n /2 t i /4 n t /2 t σ t i dt, t /2 n /2 ln n, dt, t 3/4 dt, t /4 n t /2 t σ 3 t i 3/2 n /2 σ 3 n /2, and the result follows. Lemma 7 For any α ], /2[ the sequence L n 4 in probability and bounded in probability for α =. Proof. Using again the inequality a b a b we get that L n 4 k S ti A ti A ti kcω +kcω ti Ĉ xt u, S u du ti It follows from 6 that n ti n σ 2 Su 2 Ĉ xxx u, S u du c σ 2 S 2 u Ĉ xxx u, S u du. 2 t σ t i /2 + t σ 2 t i c σ + σ 2 ln n. But the first sum for any α [, /2[ converges to a finite limit cωk 2π lns /K + { 2 v 4 v exp v lns /K + } 2 dv 2 v 3/2 2 where k := lim k n which is to zero when α> and to k when α =. The convergence to zero of the last summand in 2 follows from 5.

11 On Leland s strategy of option pricing with transactions costs 249 Lemma 8 For any α [, /2[ the sequence L n 5 in probability. Proof. It is sufficient to show that the sequence k ξt n i ξt n i = k Ĉ x t i, S ti Ĉ x, S ti is bounded in probability. But this fact follows easily from Lemma 7 since we proved above that k n S Mt n i Mt n i converges in probability to J. Thus, we established that for α ], /2[ the sequence F 2 ξ n converges to zero in probability and Theorem is proved. Appendix B. Proof of Theorem 2 In view of Lemmas, 2, 8, and 3, 4, 6 it remains to show only that S ti ξt n i ξt n i J 2. k Put Zi n := w nwti ti lnst /K i 2σ n + σ2 4σ n, Z i n := Zi n EZi n F ti Evidently, S ti ξt n i ξt n i k J 2 =I n +I2 n +I3 n where I n := S ti ξt n i ξt n i lns /K 2σ t + 4σ σ2 t, σst 2 i C xx, S ti w ti w ti I n 2 := I n 3 := σs 2 Ĉ xx, S ti Z n i n /2, σst 2 i Ĉ xx, S ti EZi n F ti n /2 k J 2. Since σ 2 / n 2 2/πk σ we get using the definition 8 that I n 3 = 4 S ti σ GS ti, σ 2 ϕ ds ti, σ σ 2 t +o k J 2 a.s.

12 25 Yu.M. Kabanov, M.M. Safarian By the same considerations as in Lemma 4 we can show that I2 n in probability. Indeed, for any n the sequence Z i n is a martingale difference with respect to the discrete filtration F ti and for a certain easily calculated function R we have σ 2 St 4 i Ĉxx 2, S ti E Z i n 2 F ti n cωn /2 RS ti, σ 2 implying by the Lenglart inequality the convergence I2 n Finally, I n a.s. in probability. S ti M ti M ti σs ti Ĉ xx, S ti w ti w ti + Ati S ti A ti σs ti Ĉ xx, S ti lns /K 2σ + 4σ σ2 t. It follows from Lemma 6 that the first sum in the right-hand side of the above inequality converges to zero. The second sum is equivalent to the sum ti S ti Ĉ xt t, S t dt Ĉ xt, S ti t which tends to zero as n. References. Jacod, J., Shiryayev A.N.: Limit Theorems for Stochastic Processes. Berlin Heidelberg New York: Springer Leland, H.E.: Option pricing and replication with transactions costs. J. Finance 45, Lott, K.: Ein Verfahren zur Replikation von Optionen unter Transaktionkosten in stetiger Zeit, Dissertation. Universität der Bundeswehr München. Institut für Mathematik und Datenverarbeitung, 993