An Expansion Approach for the Pricing of Exotic Options

Prof. Masaaki Kijima

Tokyo Metropolitan University

About the Speaker

Masaaki Kijima is a Professor of Finance at Graduate School of Social Sciences, Tokyo Metropolitan University (TMU) since 2006. Before joining TMU, he was a Professor of Financial Engineering at Graduate School of Economics, Kyoto University.

He graduated from the Department of Information Sciences, Tokyo Institute of Technology in 1980, and received a Ph.D. in Business Administration from the William E. Simon Graduate School of Business Administration, University of Rochester, in 1986. He returned to Tokyo Institute of Technology to become an Assistant Professor in 1986 and started his research career of applied probability and finance.

He is the author of two books, “Markov Processes for Stochastic Modeling” in 1997 and “Stochastic Processes with Applications to Finance” in 2002, both published by Chapman & Hall, London. He has published more than 100 research papers in international journals specializing in applied probability and mathematical finance. He is currently an Associate Editor of the SIAM Journal on Financial Mathematics.

Abstract

As the distribution of the sum of log-normal distributions is not known, the pricing of basket and average options becomes non-trivial. Accordingly, a large number of numerical methods have been proposed in the literature for the pricing of such options. In this lecture, I first introduce an expansion approach and explain why this approach is so powerful for pricing average options. More precisely, in the first lecture, I propose an approximation method based on the chaos expansion for the pricing of European-style contingent claims, in particular generalized Asian options. In the second lecture, I apply the approach for the pricing of such exotic options as reciprocal options, volume-weighted-average-price (VWAP) options, and barrier options. The application of the expansion approach is non-trivial, because the payoffs of these options are nonlinear. In the third lecture, we consider a fractional volatility model and explain how the expansion approach can be applied. Through numerical examples, I show that a two-factor fractional volatility model can resolve the fractional puzzle, when the correlations between the underlying asset process and the factors of rough volatility and persistence belong to a certain range.