Occupation Times for Time-changed Processes with Applications to Parisian Options

TL;DR

This paper develops a theoretical framework for time-changed stochastic processes, particularly inverse subordinators, and applies it to price Parisian options, revealing new connections between occupation measures, excursion theory, and fractional PDEs.

Contribution

It introduces an excursion theory for time-changed reflected Brownian motion and links it to option pricing and occupation measure distributions, advancing mathematical finance models.

Findings

Expressed Parisian option prices via solutions of time-fractional PDEs.

Proved a Ray-Knight theorem for occupation measures of time-changed Brownian motion.

Showed excursion durations follow a Poisson-Dirichlet distribution under inverse stable subordinator.

Abstract

Stochastic processes time-changed by an inverse subordinator have been suggested as a way to model the price of assets in illiquid markets, where the jumps of the subordinator correspond to periods of time where one is unable to sell an asset. We develop an excursion theory for time-changed reflected Brownian motion and use this to express the price of certain European options with Parisian barrier condition in terms of solutions of a time-fractional PDE. We provide a general description of the occupation measures of time-changed processes and use this to prove a Ray-Knight theorem for the occupation measure of a time-changed Brownian motion with negative drift. We also show that the duration of the excursions on finite time intervals obey a Poisson-Dirichlet distribution when a reflected Brownian motion is time-changed by an inverse stable subordinator.