Volterra equation for pricing and hedging in a regime switching market

Abstract

It is known that the risk minimizing price of European options in Markov-modulated market satisfies a system of coupled PDE, known as generalized B–S–M PDE. In this paper, another system of equations, which can be categorized as a Volterra integral equations of second kind, are considered. It is shown that this system of integral equations has smooth solution and the solution solves the generalized B–S–M PDE. Apart from showing existence and uniqueness of the PDE, this IE representation helps to develop a new computational method. It enables to compute the European option price and corresponding optimal hedging strategy by using quadrature method.

Keywords

• Markov modulated market
• locally risk minimizing option price
• Black–Scholes–Merton equations
• Volterra equation
• quadrature method