First passage times and option pricing under a mixed-exponential jump diffusion model (abstract only)

Abstract

This paper aims at extending 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 hyper-exponential and double-exponential jump diffusion models, as the mixed-exponential distribution can approximate any distribution as closely as possible, including the normal distribution and various heavy-tailed distributions. The mixedexponential jump diffusion model can lead to analytical solutions for Laplace transforms of prices and sensitivity parameters for pathdependent 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 related to first passage times explicitly. A calibratrion example for SPY options shows that the model can provide a reasonable fit even for options with very short maturiy, such as one day. This is a joint work with Ning Cai at Hong Kong Univ. of Science and Technology.