A General Approach for Parisian Stopping Times under Markov Processes

TL;DR

This paper introduces a versatile method using continuous-time Markov chain approximation to accurately compute distributions and prices of Parisian options under various one-dimensional Markov processes, including complex models.

Contribution

The authors develop a general, convergent approach for Parisian stopping times applicable to diverse Markov models, with explicit convergence rates and extensions to advanced financial instruments.

Findings

Method achieves high accuracy in diffusion and jump models.

Convergence rate estimates improve computational efficiency.

Extensions enable modeling of complex financial derivatives.

Abstract

We propose a method based on continuous time Markov chain approximation to compute the distribution of Parisian stopping times and price Parisian options under general one-dimensional Markov processes. We prove the convergence of the method under a general setting and obtain sharp estimate of the convergence rate for diffusion models. Our theoretical analysis reveals how to design the grid of the CTMC to achieve faster convergence. Numerical experiments are conducted to demonstrate the accuracy and efficiency of our method for both diffusion and jump models. To show the versality of our approach, we develop extensions for multi-sided Parisian stopping times, the joint distribution of Parisian stopping times and first passage times, Parisian bonds and for more sophisticated models like regime-switching and stochastic volatility models.