Abstract
We consider the dynamic hedging of a European option under a general local volatility model with small proportional transaction costs. Extending the approach of Leland, we introduce a class of continuous strategies of finite cost that asymptotically (super-)replicate the payoff. An associated central limit theorem for the hedging error is proved. We also obtain an explicit trading strategy minimizing the asymptotic error variance.
Similar content being viewed by others
References
Ahn, H., Dayal, M., Grannan, E., Swindle, G.: Option replication with transaction costs: general diffusion limits. Ann. Appl. Probab. 8, 676–707 (1998)
Barles, G., Soner, H.M.: Option pricing with transaction costs and a nonlinear Black–Scholes equation. Finance Stoch. 2, 369–397 (1998)
Bichuch, M.: Pricing a contingent claim liability with transaction costs using asymptotic analysis for optimal investment. Finance Stoch. 18, 651–694 (2014)
Davis, M., Panas, V.G., Zariphopoulou, T.: European option pricing with transaction costs. SIAM J. Control Optim. 31, 470–493 (1993)
Denis, E., Kabanov, Y.: Mean square error for the Leland–Lott hedging strategy: convex pay-offs. Finance Stoch. 14, 625–667 (2010)
El Karoui, N., Jeanblanc-Picqué, M., Shreve, S.E.: Robustness of the Black and Scholes formula. Math. Finance 8, 93–126 (1998)
Fukasawa, M.: Conservative delta hedging under transaction costs. In: Takahashi, A., et al. (eds.) Recent Advances in Financial Engineering 2011, pp. 55–72. World Scientific, Singapore (2012)
Glynn, P.W., Wang, R.J.: Central limit theorems and large deviations for additive functionals of reflecting diffusion processes. In: Dawson, D., et al. (eds.) Asymptotic Laws and Methods in Stochastics. Fields Institute Communications, vol. 76, pp. 329–345. Springer, Berlin (2015)
Grannan, E., Swindle, G.: Minimizing transaction costs of option hedging strategies. Math. Finance 6, 341–364 (1996)
Hodges, S.D., Neuberger, A.: Optimal replication of contingent claims under transaction costs. Rev. Futures Mark. 8, 222–239 (1989)
Jacod, J., Shiryaev, A.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)
Kabanov, Y., Safarian, M.: Markets with Transaction Costs. Mathematical Theory. Springer, Berlin (2009)
Kallsen, J., Li, S.: Portfolio optimization under small transaction costs: a convex duality approach. Preprint, available online at arXiv:1309.3479 (2013)
Leland, H.: Option pricing and replication with transaction costs. J. Finance 40, 1283–1301 (1985)
Levental, S., Skorokhod, A.V.: On the possibility of hedging options in the presence of transaction costs. Ann. Appl. Probab. 7, 410–443 (1997)
Papavasiliou, A., Pavliotis, G.A., Stuart, A.M.: Maximum likelihood drift estimation for multiscale diffusions. Stoch. Process. Appl. 119, 3173–3210 (2009)
Peskir, G.: A change-of-variable formula with local time on surfaces. In: Donati-Martin, C., et al. (eds.) Séminaire de Probabilités XL. Lecture Notes in Mathematics, vol. 1899, pp. 70–96. Springer, Berlin (2007)
Skorokhod, A.V.: Asymptotic Methods in the Theory of Stochastic Differential Equations. Am. Math. Soc., Providence (1987)
Soner, H.M., Touzi, N.: Homogenization and asymptotics for small transaction costs. SIAM J. Control Optim. 51, 2893–2921 (2013)
Soner, H.M., Shreve, S.E., Cvitanić, J.: There is no nontrivial hedging portfolio for option pricing with transaction costs. Ann. Appl. Probab. 5, 327–355 (1995)
Tanaka, H.: Stochastic differential equations with reflecting boundary conditions in convex regions. Hiroshima Math. J. 9, 163–177 (1979)
Toft, K.: On the mean-variance tradeoff in option replication with transaction costs. J. Financ. Quant. Anal. 31, 233–263 (1996)
Whalley, A.E., Wilmott, P.: An asymptotic analysis of an option hedging model for option pricing with transaction costs. Math. Finance 7, 307–324 (1997)
Acknowledgements
M. Fukasawa is grateful to Professors Chiaki Hara, Toshiki Honda, Masaaki Kijima, Shigeo Kusuoka and Jun Sekine for their helpful comments and suggestions. J. Cai thanks his advisors, Professors Mathieu Rosenbaum and Peter Tankov, for making this collaboration possible.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work is supported by Japan Society for the Promotion of Science, KAKENHI Grant Numbers 24684006, 25245046 and 25245034, and by Kyoto University Joint Usage and Research Center at the Institute of Economic Research.
Appendix: Note on the computation for (5.21)
Appendix: Note on the computation for (5.21)
The computation for (5.21) is straightforward but lengthy. Here we present a Maple worksheet to check it. Maple is a trademark of Waterloo Maple Inc.
> latex(factor(%));
Rights and permissions
About this article
Cite this article
Cai, J., Fukasawa, M. Asymptotic replication with modified volatility under small transaction costs. Finance Stoch 20, 381–431 (2016). https://doi.org/10.1007/s00780-016-0294-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00780-016-0294-2