Multiscale Models and Dimensionality reduction in the pricing and hedging of Path Dependent Financial Options

Multiscale models address the main drawback in a one-factor stochastic volatility framework by providing a better fit to short maturity European-style options. In this project we focus on the complex considerations that a model needs to fulfill to prove practical. We build on the theoretical work of Fouque, Papanicolaou and Sircar, implementing calibrations for both first-order and second order perturbation expansions. This allows us to investigate properties such as goodness of fit, parameter stability and computational tractability of calibrations. We find that the first-order expansion with time-dependency shows decent fits and parameter stability for contracts that are not very close to expiry. The second order quadratic fits provide more flexibility and improve results, but only at a high computational cost. We develop a hybrid calibration using first order expansion results and a parameter reduction technique which both prove successfull in reducing the dimension of the non-convex optimization problem. A combination of the two techniques allows us to derive a quasi-closed form solution to the second-order calibration, reducing its computational cost.