The first passage time problem for mixed-exponential jump processes with applications in insurance and finance

TL;DR

This paper derives explicit Laplace transform solutions for first passage times in mixed-exponential jump processes and applies these results to insurance risk, option pricing, and credit risk models.

Contribution

It provides the first explicit Laplace transform solutions for first passage times in mixed-exponential jump diffusion processes with practical applications.

Findings

Explicit Laplace transform solutions for first passage times.

Closed-form expressions for Gerber-Shiu functions.

Analytical solutions for barrier and lookback options.

Abstract

This paper stidies the first passage times to constant boundaries for mixed-exponential jump diffusion processes. Explicit solutions of the Laplace transforms of the distribution of the first passage times, the joint distribution of the first passage times and undershoot (overshoot) are obtained. As applications, we present explicit expression of the Gerber-Shiu functions for surplus processes with two-sided jumps, present the analytical solutions for popular path-dependent options such as lookback and barrier options in terms of Laplace transforms and give a closed-form expression on the price of the zero-coupon bond under a structural credit risk model with jumps.