| We investigate a method of solution to pricing problems in quantitative finance that arise when a function of both deterministic and stochastic variables is modified to consider additional random factors. In particular, we consider several extensions of the Black-Scholes paradigm for option pricing, which assumes constant volatility of asset returns and constant interest rates. We model the interest rate processes using the Heath-Jarrow-Morton paradigm for forward rates, and we also model the instantaneous volatility of the spot process by various diffusions. The method we employ uses both analytical and numerical methods to arrive at a solution, and hence may be referred to as a semi-analytical method. The method is grounded in the theory of asymptotic analysis, in which the solution of a partial differential equation that is subjected to a perturbation is related to the solution of the unperturbed problem. |
