Option Pricing with Piecewise-Constant Parameters, Discrete Jumps and Regime-Switching

Abstract

In this paper, I address systematically how to enhance the most existing option models with piecewise-constant parameters, and how to derive the corresponding closed-form characteristic function under the risk-neutral measure. As long as the characteristic function with piecewise-constant parameters is analytical known, the pricing formula for a European call is then given by inverse transform of the derived characteristic function. The method developed here is based on iterated expectations by applying the tower rule, and is applicable for all existing option models with closed-form characteristic function. Particularly, the most popular option pricing models are extended with time-dependent parameters, including stochastic volatility models, jump-diffusion models and Levy pure jump models. Additionally, to demonstrate the potential power of iterated expectation, I develop a discrete jump model that allows for jumps at discrete time points. A concrete use of this discrete jump model is a discrete dividend model. Furthermore, a regime-switched option pricing model can also be easily dealt with where the underlying asset follows different types of stochastic process in the course of time. For instance, an asset can be governed first by stochastic volatility dynamics, and then by Levy pure jump dynamics. Finally, two stochastic volatility models of Heston (1993) as well as of Schobel and Zhu (1999) are examined in details. The time-dependent variants of these two models are calibrated to FX option markets, then compared to the conventional stochastic volatility models in terms of market data fitting and product pricing.