An Asymptotic Expansion Approach to Numerical Problems in Finance

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1.We developed mathematical validity of the asymptotic expansion approach to valuation of contingent claims in Markovian and non-Markovian setting. ("On Validity of the Asymptotic Expansion Approach in Contingent Claim Analysis," Annals of Applied Probability(2003).)2.We derived numerically tractable formulas of optimal portfolio in the dynamic investment problems when state variables follow general diffusion processes. Moreover, we provided analytic approximation formulas based on the asymptotic expansion approach and showed their effectiveness through numerical examples. ("An Asymptotic Expansion Scheme for the Optimal Investment Problems," forthcoming in Statistical Inference for Stochastic Processes.)3.We showed a decomposition of the value of an American option into the corresponding European value and the early exercise premium when the underlying price follow a general diffusion process. Then, we applied the asymptotic expansion approach to develop a semi-analytic computational scheme. ("An Asymptotic Expansion Approach to American Options," Monetary and Economic Studies(2003).)4.We proposed a new computational scheme with the asymptotic method to achieve variance reduction of Monte Carlo simulation for numerical analysis in finance. We provided general scheme and mathematical validity of our method. The examples of the application include pricing options under jump-diffusion processes, pricing interest rate derivatives in HJM framework and computing optimal portfolios in the dynamic investment problems. ("Applications of the Asymptotic Expansion Approach based on Malliavin-Watanabe Calculus in Financial Problems," forthcoming in Stochastic Processes and Applications to Mathematical Finance.)5.We published a book on the asymptotic expansion approach to finance.("Foundation of Mathematical Finance-Application of Malliavin Calculus and Asymptotic Expansion Method-," (2003).)