Option Pricing Under a Mixed-Exponential Jump Diffusion Model

• Published in: Management science 2011-11, Vol.57 (11), p.2067-2081
• Main Authors: Cai, Ning, Kou, S. G.
• Format: Article
• Language: English
• Subjects: Algorithms; Applications; Applied sciences; Approximation; Asset pricing; barrier options; Calibration; Call options; Diffusion index; Distribution theory; Economic models; European option; Exact sciences and technology; Experimental economics; first passage times; Insurance, economics, finance; jump diffusion; Jump diffusion model; Laplace transformation; Laplace transforms; lookback options; Management science; Mathematical economics; Mathematical models; Mathematics; Merton's normal jump diffusion model; Microelectromechanical systems; mixed-exponential distributions; Modeling; Monte Carlo methods; Normal distribution; Operational research and scientific management; Operational research. Management science; Option pricing; Options (Finance); Portfolio theory; Probability and statistics; Probability theory and stochastic processes; Sciences and techniques of general use; Securities prices; Size distribution; Statistics; Stochastic models; Studies
• ISSN: 0025-1909; 1526-5501
• DOI: 10.1287/mnsc.1110.1393

This paper aims to extend the analytical tractability of the Black-Scholes model to alternative models with arbitrary jump size distributions. More precisely, we propose a jump diffusion model for asset prices whose jump sizes have a mixed-exponential distribution, which is a weighted average of exponential distributions but with possibly negative weights. The new model extends existing models, such as hyperexponential and double-exponential jump diffusion models, because the mixed-exponential distribution can approximate any distribution as closely as possible, including the normal distribution and various heavy-tailed distributions. The mixed-exponential jump diffusion model can lead to analytical solutions for Laplace transforms of prices and sensitivity parameters for path-dependent options such as lookback and barrier options. The Laplace transforms can be inverted via the Euler inversion algorithm. Numerical experiments indicate that the formulae are easy to implement and accurate. The analytical solutions are made possible mainly because we solve a high-order integro-differential equation explicitly. A calibration example for SPY options shows that the model can provide a reasonable fit even for options with very short maturity, such as one day. This paper was accepted by Michael Fu, stochastic models and simulation.