An analytic recursive method for optimal multiple stopping: Canadization and phase-type fitting

TL;DR

This paper introduces a recursive analytic method for solving optimal multiple stopping problems driven by spectrally negative Levy processes, using Canadization and phase-type fitting techniques to approximate complex distributions.

Contribution

It develops a novel recursive algorithm that combines Canadization and phase-type fitting to compute optimal stopping strategies analytically.

Findings

The method provides closed-form solutions for the value function.

Numerical examples demonstrate accuracy compared to Monte Carlo simulations.

The approach handles positive and negative discount rates effectively.

Abstract

We study an optimal multiple stopping problem for call-type payoff driven by a spectrally negative Levy process. The stopping times are separated by constant refraction times, and the discount rate can be positive or negative. The computation involves a distribution of the Levy process at a constant horizon and hence the solutions in general cannot be attained analytically. Motivated by the maturity randomization (Canadization) technique by Carr (1998), we approximate the refraction times by independent, identically distributed Erlang random variables. In addition, fitting random jumps to phase-type distributions, our method involves repeated integrations with respect to the resolvent measure written in terms of the scale function of the underlying Levy process. We derive a recursive algorithm to compute the value function in closed form, and sequentially determine the optimal exercise thresholds. A series of numerical examples are provided to compare our analytic formula to results from Monte Carlo simulation.