Analytical Pricing of Double-Barrier Options under a Double-Exponential Jump Diffusion Process: Applications of Laplace Transform

Abstract

We derive explicit formulas for pricing double (single) barrier and touch options with time-dependent rebates assuming that the asset price follows a double-exponential jump diffusion process. We also consider incorporating time-dependent volatility. Assuming risk-neutrality, the value of a barrier option satisfies the generalized Black-Scholes equation with the appropriate boundary conditions. We take the Laplace transform of this equation in time and solve it explicitly. Option price and risk parameters are computed via the numerical inversion of the corresponding solution. Numerical examples reveal that the pricing formulas are easy to implement and they result in accurate prices and risk parameters. Proposed formulas allow fast computing of smile-consistent prices of barrier and touch options.

Keywords: jump diffusion processes, exponential jumps, volatility smile, option pricing, path-dependent options, double barrier options, double touch options, Laplace transform

JEL Classification: C00, G00