On Pricing Derivatives under GARCH Models: A Dynamic Gerber-Shiu’s Approach

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 Bühlmann et al. (1996) to identify a martingale measure under the incomplete market setting. One advantage of our model is that it provides an 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 celebrated Gerber-Shiu’s option pricing model, we can justify our pricing result within the dynamic framework of utility maximization problems which makes the economic intuition of our pricing result more appealing. In fact, the use of the Esscher Transformation for option valuation can also be justified by the minimization of the relative entropy between the statistical probability and the risk-neutral pricing probability. Numerical results for the comparsion of our model with the Black-Scholes option pricing model are presented.