Function approximation for option pricing and risk management Methods, theory and applications.

Abstract

This thesis investigates the application of function approximation techniques for computationally demanding problems in nance. We focus on the use of Chebyshev interpolation and its multivariate extensions. The main contribution of this thesis is the development of a new pricing method for path-dependent options. In each step of the dynamic programming time-stepping we approximate the value function with Chebyshev polynomials. A key advantage of this approach is that it allows us to shift all modeldependent computations into a pre-computation step. For each time step the method delivers a closed form approximation of the price function along with the options' delta and gamma. We provide a theoretical error analysis and nd conditions that imply explicit error bounds. Numerical experiments con rm the fast convergence of prices and sensitivities. We use the new method to calculate credit exposures of European and path-dependent options for pricing and risk management. The simple structure of the Chebyshev interpolation allows for a highly e cient evaluation of the exposures. We validate the accuracy of the computed exposure pro les numerically for di erent equity products and a Bermudan swaption. Benchmarking against the least-squares Monte Carlo approach shows that our method delivers a higher accuracy in a faster runtime. We extend the method to e ciently price early-exercise options depending on several risk-factors. As an example, we consider the pricing of callable bonds in a hybrid twofactor model. We develop an e cient and stable calibration routine for the model based on our new pricing method. Moreover, we consider the pricing of early-exercise basket options in a multivariate Black-Scholes model. We propose a numerical smoothing in the dynamic programming time-stepping using the smoothing property of a Gaussian kernel. An extensive numerical convergence analysis con rms the e ciency.