On pricing derivatives under GARCH models: a dynamic Gerber-Shiu's approach

Abstract

This paper proposes a method for pricing derivatives under the GARCH assumption for underlying assets in the context of a “dynamic” version of Gerber-Shiu’s optionpricing model. Instead of adopting the notion of local risk-neutral valuation relationship (LRNVR) introduced by Duan (1995), we employ the concept of conditional Esscher Transforms introduced by Buhlmann et al. (1996) to identify a martingale measure under the incomplete market setting. One advantage of our model is that it provides a unified and convenient approach to deal with different parametric models for the innovation of the GARCH stock-price process. Under the conditional normality assumption for the stock innovation, our pricing result is consistent with that of Duan (1995). In line with the ∗Tak Kuen Siu, Ph.D., is a Research Fellow in the Liu Bie Ju Centre for Mathematical Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, e-mail: [e-mail] †Howell Tong, Hon. F.I.A., Ph.D., is the Chair Professor of the Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, e-mail: [e-mail]. He is also a Chair Professor of the Department of Statistics, London School of Economics, U.K. ‡Hailiang Yang, A.S.A., Ph.D., is an Associate Professor in the Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong, e-mail: [e-mail].