Closed-Form Expansion, Conditional Expectation, and Option Valuation

Abstract

Enlightened by the theory of Watanabe [Watanabe S (1987) Analysis of Wiener functionals (Malliavin calculus) and its applications to heat kernels. Ann. Probab. 15:1–39] for analyzing generalized random variables and its further development in Yoshida [Yoshida N (1992a) Asymptotic expansions for statistics related to small diffusions. J. Japan Statist. Soc. 22:139–159], Takahashi [Takahashi A (1995) Essays on the valuation problems of contingent claims. Ph.D. thesis, Haas School of Business, University of California, Berkeley, Takahashi A (1999) An asymptotic expansion approach to pricing contingent claims. Asia-Pacific Financial Markets 6:115–151] as well as Kunitomo and Takahashi [Kunitomo N, Takahashi A (2001) The asymptotic expansion approach to the valuation of interest rate contingent claims. Math. Finance 11(1):117–151, Kunitomo N, Takahashi A (2003) On validity of the asymptotic expansion approach in contingent claim analysis. Ann. Appl. Probab. 13(3):914–952] etc., we focus on a wide range of multivariate diffusion models and propose a general probabilistic method of small-time asymptotic expansions for approximating option price in simple closed-form up to an arbitrary order. To explicitly construct correction terms, we introduce an efficient algorithm and novel closed-form formulas for calculating conditional expectation of multiplication of iterated stochastic integrals, which are potentially useful in a wider range of topics in applied probability and stochastic modeling for operations research. The performance of our method is illustrated through various models nested in constant elasticity of variance type processes. With an application in pricing options on VIX under GARCH diffusion and its multifactor generalization to the Gatheral double lognormal stochastic volatility models, we demonstrate the versatility of our method in dealing with analytically intractable non-Lévy and non-affine models. The robustness of the method is theoretically supported by justifying uniform convergence of the expansion over the whole set of parameters.