Option replication with transaction costs: general diffusion limits

Hyungsok Ahn; Mohit Dayal; Eric Grannan; Glen Swindle

doi:10.1214/aoap/1028903447

August 1998 Option replication with transaction costs: general diffusion limits

Hyungsok Ahn, Mohit Dayal, Eric Grannan, Glen Swindle

Ann. Appl. Probab. 8(3): 676-707 (August 1998). DOI: 10.1214/aoap/1028903447

Abstract

Transaction costs preclude the construction of hedging strategies for general contingent claims. Leland introduced the concept of diffusion limits of hedging strategies in a small transaction cost limit. This paper establishes such diffusion limits for very general hedging strategies and also establishes leading order asymptotic expressions for the replication error. In addition to subsuming previously considered temporal strategies, the results in this paper yield new results, namely expressions for replication errors of stock price strategies and a variety of "renewal" strategies. Most importantly, this paper provides a unified methodology for calculating hedging strategies and replication errors in the small transaction cost limit. This is an essential component of optimization methods, when, for example one is trying to minimize replication error for a given initial portfolio value.

References

1.

BENSAID, B., LESNE, J. P., PAGES, H. and SCHEINKMAN, J. 1992. Derivative asset pricing with transaction costs. Math. Finance 2 63 86. Z. 0900.90100BENSAID, B., LESNE, J. P., PAGES, H. and SCHEINKMAN, J. 1992. Derivative asset pricing with transaction costs. Math. Finance 2 63 86. Z. 0900.90100

2.

BLACK, F. and SCHOLES, M. 1974. The pricing of options and corporate liabilities. Journal of Political Economy 81 637 654. Z. BLACK, F. and SCHOLES, M. 1974. The pricing of options and corporate liabilities. Journal of Political Economy 81 637 654. Z.

3.

BOy LE, P. P. and VORST, T. 1992. Option replication in discrete time with transaction costs. Journal of Finance 47 271 294. Z. BOy LE, P. P. and VORST, T. 1992. Option replication in discrete time with transaction costs. Journal of Finance 47 271 294. Z.

4.

CONSTANTINIDES, G. M. and ZARIPHOPOULOU, T. 1995. Universal bounds on option prices with proportional transaction costs. Preprint. Z. CONSTANTINIDES, G. M. and ZARIPHOPOULOU, T. 1995. Universal bounds on option prices with proportional transaction costs. Preprint. Z.

5.

COX, J. C., ROSS, S. A. and RUBINSTEIN, M. 1979. Option pricing: a simplified approach. Journal of Financial Economics 7 229 263. Z. COX, J. C., ROSS, S. A. and RUBINSTEIN, M. 1979. Option pricing: a simplified approach. Journal of Financial Economics 7 229 263. Z.

6.

DAVIS, M. H. A. and NORMAN, A. R. 1990. Portfolio selection with transaction costs. Math. Oper. Res. 15 676 713. Z. 0717.90007 MR1080472 10.1287/moor.15.4.676 0364-765X%28199011%2915%3A4%3C676%3APSWTC%3E2.0.CO%3B2-BDAVIS, M. H. A. and NORMAN, A. R. 1990. Portfolio selection with transaction costs. Math. Oper. Res. 15 676 713. Z. 0717.90007 MR1080472 10.1287/moor.15.4.676 0364-765X%28199011%2915%3A4%3C676%3APSWTC%3E2.0.CO%3B2-B

7.

DAVIS, M. H. A., PANAS, V. G. and ZARIPHOPOULU, T. 1993. European option pricing with transaction costs. SIAM J. Control Optim. 31 470 493. Z. 0779.90011 MR1205985 10.1137/0331022DAVIS, M. H. A., PANAS, V. G. and ZARIPHOPOULU, T. 1993. European option pricing with transaction costs. SIAM J. Control Optim. 31 470 493. Z. 0779.90011 MR1205985 10.1137/0331022

8.

DONG, G. 1991. Nonlinear partial differential equations of second order. Trans. Math. Monogr. 95. Amer. Math. Soc., Providence, RI. Z. MR1134129 0759.35001DONG, G. 1991. Nonlinear partial differential equations of second order. Trans. Math. Monogr. 95. Amer. Math. Soc., Providence, RI. Z. MR1134129 0759.35001

9.

DURRETT, R. 1984. Brownian Motion and Martingales in Analy sis. Wadsworth, Belmont, CA. Z. MR87a:60054 0554.60075DURRETT, R. 1984. Brownian Motion and Martingales in Analy sis. Wadsworth, Belmont, CA. Z. MR87a:60054 0554.60075

10.

ETHIER, S. N. and KURTZ, T. G. 1985. Markov Processes. Wiley, New York. Z. MR838085ETHIER, S. N. and KURTZ, T. G. 1985. Markov Processes. Wiley, New York. Z. MR838085

11.

FELLER, W. 1971. An Introduction to Probability Theory and Its Applications 2. Wiley, New York. Z. MR42:5292FELLER, W. 1971. An Introduction to Probability Theory and Its Applications 2. Wiley, New York. Z. MR42:5292

12.

FRIEDMAN, A. 1964. Partial Differential Equations of Parabolic Ty pe. Prentice-Hall, Englewood Cliffs, NJ. Z. MR181836FRIEDMAN, A. 1964. Partial Differential Equations of Parabolic Ty pe. Prentice-Hall, Englewood Cliffs, NJ. Z. MR181836

13.

GRANNAN, E. R. and SWINDLE, G. H. 1994. Minimizing transaction costs of option hedging strategies. Math Finance. To appear. Z. GRANNAN, E. R. and SWINDLE, G. H. 1994. Minimizing transaction costs of option hedging strategies. Math Finance. To appear. Z.

14.

HENROTTE, P. 1991. Transactions costs and duplication strategies. Preprint. Z. HENROTTE, P. 1991. Transactions costs and duplication strategies. Preprint. Z.

15.

KARATZAS, I. and SHREVE, S. 1991. Brownian Motion and Stochastic Calculus. Springer, New York. Z. MR92h:60127KARATZAS, I. and SHREVE, S. 1991. Brownian Motion and Stochastic Calculus. Springer, New York. Z. MR92h:60127

16.

KARLIN, S. and TAy LOR, H. M. 1981. A Second Course in Stochastic Processes. Academic Press, New York. Z. MR82j:60003 0469.60001KARLIN, S. and TAy LOR, H. M. 1981. A Second Course in Stochastic Processes. Academic Press, New York. Z. MR82j:60003 0469.60001

17.

LELAND, H. E. 1985. Option pricing and replication with transactions costs. Journal of Finance 40 1283 1301. Z. LELAND, H. E. 1985. Option pricing and replication with transactions costs. Journal of Finance 40 1283 1301. Z.

18.

MERTON, R. C. 1971. Optimum consumption and portfolio rules in a continuous time model. J. Econom. Theory 3 373 413. Z. MR56:14599 10.1016/0022-0531(71)90038-XMERTON, R. C. 1971. Optimum consumption and portfolio rules in a continuous time model. J. Econom. Theory 3 373 413. Z. MR56:14599 10.1016/0022-0531(71)90038-X

19.

MEy ER, P. A. 1976. Un cours sur les integrales stochastiques. Seminaire de Probab. X. Lecture ´ ´ Notes in Math 511 246 400. Springer, Berlin. Z. MEy ER, P. A. 1976. Un cours sur les integrales stochastiques. Seminaire de Probab. X. Lecture ´ ´ Notes in Math 511 246 400. Springer, Berlin. Z.

20.

PROTTER, P. 1990. Stochastic Integration and Differential Equations. Springer, New York. MR91i:60148PROTTER, P. 1990. Stochastic Integration and Differential Equations. Springer, New York. MR91i:60148

21.

SANTA BARBARA, CALIFORNIA 93160-3110SANTA BARBARA, CALIFORNIA 93160-3110

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Hyungsok Ahn, Mohit Dayal, Eric Grannan, and Glen Swindle "Option replication with transaction costs: general diffusion limits," The Annals of Applied Probability 8(3), 676-707, (August 1998). https://doi.org/10.1214/aoap/1028903447

Published: August 1998

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