Chiheb Ben Hammouda, "Numerical Methods For Uncertainty Quantification In Option Pricing", Graduation Project Report, Tunisia Polytechnic School, Hosting Institution: KAUST, June 2013.
Model or/and parametric uncertainty, in the context of derivative pricing, results in mis- pricing of contingent claims due to uncertainty on the choice of the pricing models or/and the values of parameters within these models. In this thesis, we introduce a quantitative framework for investigating the impact of parametric uncertainty on option pricing under Black-Scholes framework. We start with one dimensional case by quantifying the impact of the volatility parameter on the price of European put. From historical data, we fit the sample density of the volatility by a parametric distribution. We then use Monte Carlo Sampling (MCS) and construct Polynomial Chaos (PC) approximation to compute statistical infor- mation (i.e, mean quantities, α level confidence bounds (α bounds), and sample densities) of option's price. We show that both methods give the same results but PC approximation method performs significantly better. In the second part of our project, we extend the work to multidimensional case by quantifying the impact of the covariance matrix on the price of European basket option. After modeling the randomness in the covariance matrix with a Wishart distribution, we developed three ways to compute mean quantities and sample densities of basket put price: a nested MC simulation and two methods, based on Monte Carlo (MC) and Sparse Grid Quadrature (SGQ) techniques, that use an approximation of basket option price. We show that the three methods give the same results, those using ap- proximation of basket option price are more eficient and the performance of SGQ decreases for high dimensional problems.