Modular Pricing of Options

Abstract

In this paper, we introduce a unified pricing framework for options by applying the Fourier analysis where stochastic volatility, stochastic interest rate and random jump are independently specified. The modeling of volatility and interest rate falls into four different alternatives: constant, mean-reverting Ornstein-Uhlenbeck process, mean-reverting square root process and mean-reverting double square root process while random jumps can be specified as pure jumps, lognormal jumps and Pareto jumps. This framework called Modular Pricing of Options includes most of the existing options pricing formulas as special cases, for example, the Black-Scholes (1973), the Merton (1976), the Heston (1993), the Bakshi-Cao-Chen (1997) as well as Schoebel-Zhu's (1999) formulas. Modular Pricing of Options is not only of theoretical importance but also provides practitioners a powerful and convenient tool to implement complex and flexible option pricing model to meet the challenge in steadily changing financial markets.