Valuing Exotic Derivatives with Jump Diffusions: The Case of Basket Options

Abstract

Exotic options are complicated derivatives instruments whose structure does not allow, in general, for closed-form analytic solutions, thus, making their pricing and hedging a difficult task. To overcome additional complexity such products are, as a rule, priced within a Black-Scholes framework, assuming that the underlying asset follows a Geometric Brownian Motion (GBM) stochastic process. This paper develops a more realistic framework for the pricing of exotic derivatives; and derives closed-form analytic solutions for the pricing and hedging of basket options. We relax the simplistic assumption of the GBM, by introducing the Bernoulli Jump Diffusion process (BJD) and approximate the terminal distribution of the underlying asset with a log-normal distribution. Potential extension of the model with the use of the Edgeworth Series Expansion (ESE) is also discussed. Monte Carlo simulation confirms the validity of the proposed BJD model.