A Wiener–Hopf based approach to numerical computations in fluctuation theory for Lévy processes

Abstract

This paper focuses on numerical evaluation techniques related to fluctuation theory for Lévy processes; they can be applied in various domains, e.g., in finance in the pricing of so-called barrier options. More specifically, with $\bar{X}_t:= \sup _{0\le s\le t} X_s$ denoting the running maximum of the Lévy process $X_t$, the aim is to evaluate $\mathbb{P }(\bar{X}_t \in \mathrm{d}x)$ for $t,x>0$. The starting point is the Wiener–Hopf factorization, which yields an expression for the transform $\mathbb E e^{-\alpha \bar{X}_{e(\vartheta )}}$ of the running maximum at an exponential epoch (with $\vartheta ^{-1}$ the mean of this exponential random variable). This expression is first rewritten in a more convenient form, and then it is pointed out how to use Laplace inversion techniques to numerically evaluate $\mathbb{P }(\bar{X}_t\in \mathrm{d}x).$ In our experiments we rely on the efficient and accurate algorithm developed in den Iseger (Probab Eng Inf Sci 20:1–44, 2006). We illustrate the performance of the algorithm with various examples: Brownian motion (with drift), a compound Poisson process, and a jump diffusion process. In models with jumps, we are also able to compute the density of the first time a specific threshold is exceeded, jointly with the corresponding overshoot. The paper is concluded by pointing out how our algorithm can be used in order to analyze the Lévy process’ concave majorant.

Appendix: Numerical output

See Tables 1, 2 and  3.

Table 1 Probability ${X_t}$ exceeds $x$ before time $1$, for the standard Brownian model of Example 1, with absolute value of the error

Table 2 Probability ${X_t}$ exceeds $x$ before time 1, for the compound poisson process with Normally distributed jumps of Example 2, with absolute value of the error

Table 3 Probability $e^{X_t}$ exceeds $H=1.2$ before $t$, for the jump diffusion model