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.
