Asia-Pacific Financial Markets

, Volume 20, Issue 2, pp 147–182

# Pricing Exotic Options and American Options: A Multidimensional Asymptotic Expansion Approach

Article

## Abstract

This paper introduces a new method for pricing exotic options whose payoff functions depend on several stochastic indices and American options in multidimensional models. This method is based on two ideas. One is an application of the asymptotic expansion method for the law of a multidimensional diffusion process. The other is the combination of the asymptotic expansion method and the method called backward induction. The author applies the method to the problems of pricing call options on the maximum of two assets in the CEV model, average strike options in the Black–Scholes model and American options in the Heston model. Numerical examples show practical effectiveness of the proposed method.

## Keywords

American option Asymptotic expansion Average strike option Black–Scholes model CEV model Call option on the maximum of two assets Heston model Monte Carlo Quasi-Monte Carlo Simulation

## References

1. Broadie, M., & Glasserman, P. (2004). A stochastic mesh method for pricing high-dimensional american options. Journal of Computational Finance, 7, 35–72.Google Scholar
2. Clarke, N., & Parrott, K. (1999). Multigrid for american option pricing with stochastic volatility. Applied Mathematical Finance, 6(3), 177–195.
3. Duffy, D. (2006). Finite difference methods in financial engineering: A partial differential equation approach. New York: Wiley.
4. Glasserman, P. (2003). Monte Carlo methods in financial engineering (Vol. 53). Berlin: Springer.
5. Hull, J. (2006). Options, futures, and other derivatives. London: Pearson.Google Scholar
6. Ikonen, S., & Toivanen, J. (2007). Efficient numerical methods for pricing american options under stochastic volatility. Numerical Methods for Partial Differential Equations, 24(1), 104–126.
7. Kunitomo, N., & Takahashi, A. (2001). The asymptotic expansion approach to the valuation of interest rate contingent claims. Mathematical Finance, 11(1), 117–151.
8. Kunitomo, N., & Takahashi, A. (2004). Applications of the asymptotic expansion approach based on malliavin-watanabe calculus in financial problems. Stochastic Processes and Applications to Mathematical Finance, (pp. 195–232). Proceedings of the Ritsumeikan Intern. Symposium: World ScientificGoogle Scholar
9. L’Ecuyer, P. (2009). Quasi-monte carlo methods with applications in finance. Finance and Stochastics, 13(3), 307–349.
10. Longstaff, F., & Schwartz, E. (2001). Valuing american options by simulation: A simple least-squares approach. Review of Financial studies, 14(1), 113–147.
11. Matsuoka, R., Takahashi, A., & Uchida, Y. (2004). A new computational scheme for computing greeks by the asymptotic expansion approach. Asia-Pacific Financial Markets, 11(4), 393–430.
12. Muroi, Y. (2012). Pricing credit derivatives using an asymptotic expansion approach. Journal of Computational Finance, 15(3), 135.Google Scholar
13. Ninomiya, M., & Ninomiya, S. (2009). A new higher-order weak approximation scheme for stochastic differential equations and the runge-kutta method. Finance and Stochastics, 13(3), 415–443.
14. Ninomiya, S., & Tezuka, S. (1996). Toward real-time pricing of complex financial derivatives. Applied Mathematical Finance, 3(1), 1–20.
15. Ninomiya, S., & Victoir, N. (2008). Weak approximation of stochastic differential equations and application to derivative pricing. Applied Mathematical Finance, 15(2), 107–121.
16. Oosterlee, C. (2003). On multigrid for linear complementarity problems with application to american-style options. Electronic Transactions on Numerical Analysis, 15, 165–185.Google Scholar
17. Press, W., Teukolsky, S., Vetterling, W., & Flannery, B. (2007). Numerical recipes 3rd edition: The art of scientific computing. Cambridge: Cambridge University Press.Google Scholar
18. Rogers, L. (2002). Monte Carlo valuation of american options. Mathematical Finance, 12(3), 271–286.
19. Takahashi, A. (1999). An asymptotic expansion approach to pricing financial contingent claims. Asia-Pacific Financial Markets, 6(2), 115–151.
20. Takahashi, A. (2007). An asymptotic expansion approach in finance. CIRJE Discussion Papers.Google Scholar
21. Takahashi, A., & Takehara, K. (2007). An asymptotic expansion approach to currency options with a market model of interest rates under stochastic volatility processes of spot exchange rates. Asia-Pacific Financial Markets, 14(1), 69–121.
22. Takahashi, A., Takehara, K., & Toda, M. (2009). Computation in an asymptotic expansion method. University of Tokyo working paper CIRJEF-621.Google Scholar
23. Takahashi, A., & Yoshida, N. (2004). An asymptotic expansion scheme for optimal investment problems. Statistical Inference for Stochastic Processes, 7(2), 153–188.
24. Takahashi, A., & Yoshida, N. (2005). Monte Carlo simulation with asymptotic method. Journal of The Japan Statistical Society, 35, 171–203.Google Scholar
25. Vellekoop, M., & Nieuwenhuis, H. (2009). A tree-based method to price american options in the heston model. Journal of Computational Finance, 13(1), 1.Google Scholar
26. Watanabe, S. (1987). Analysis of wiener functionals (malliavin calculus) and its applications to heat kernels. The Annals of Probability 15(1), 1–39.Google Scholar
27. Zvan, R., Forsyth, P., & Vetzal, K. (1998). Penalty methods for american options with stochastic volatility. Journal of Computational and Applied Mathematics, 91(2), 199–218.