Inhomogeneous Terminal Value Problems Related to the Option Price in a Regime Switching Market

Abstract

This project attempts to study an inhomogeneous terminal-value problem related to the option pricing PDE in a MMGBM model. This problem has not been studied before, to the best of our knowledge. To begin with, the problem of pricing European Call option contracts in a MMGBM market model is re-examined, and several approaches to solve the option-pricing PDE are presented, including a complete proof for the existence of a unique solution via the classical theory for parabolic PDE, which has not been explicitly presented in existing literature. Some new results have been developed regarding the smoothness properties of the option price function, using an equivalent integral equation. Coming to the related inhomogeneous PDE - the presence of an inhomogeneous term is a challenge. The aim is to establish the existence of a unique classical solution in the class of functions having at most quadratic growth, which is achieved in two steps: in the first step, a mild solution is constructed using semi-group theory, and, in the second step, it is shown that the mild solution is sufficiently smooth, thereby making it a classical solution. The motivation for studying this particular terminal value problem becomes apparent later, when a potential application is presented. The procedure to compute the implied values of the volatility vector of a risky asset is sketched briefly. Repeatedly solving particular forms of the general inhomogeneous terminal-value problem from before is a promising approach towards proving local existence of the implied volatility vector.