Pricing Derivatives in a Regime Switching Market with Time Inhomogeneous Volatility

TL;DR

This paper develops a model for pricing derivatives in a market with regime switching and time-dependent volatility, providing PDE and integral equation solutions for European options.

Contribution

It introduces a semi-Markov regime switching model with explicit time-dependent volatility and derives novel PDE and integral equation methods for pricing and hedging.

Findings

Existence and uniqueness of classical solutions to the PDE.

Efficient computational methods via Volterra integral equations.

Explicit expression for quadratic residual risk.

Abstract

This paper studies pricing derivatives in an age-dependent semi-Markov modulated market. We consider a financial market where the asset price dynamics follow a regime switching geometric Brownian motion model in which the coefficients depend on finitely many age-dependent semi-Markov processes. We further allow the volatility coefficient to depend on time explicitly. Under these market assumptions, we study locally risk minimizing pricing of a class of European options. It is shown that the price function can be obtained by solving a non-local B-S-M type PDE. We establish existence and uniqueness of a classical solution of the Cauchy problem. We also find another characterization of price function via a system of Volterra integral equation of second kind. This alternative representation leads to computationally efficient methods for finding price and hedging. Finally, we analyze the PDE to establish continuous dependence of the solution on the instantaneous transition rates of semi-Markov processes. An explicit expression of quadratic residual risk is also obtained.