ORIE Colloquium on 4/23/2013 - Marie Chazal: Option Pricing in a One-Dimensional Affine Term Structure Model Via Spectral Representations

ORIE Colloquium: Marie Chazal (Université libre de Bruxelles) - Option Pricing in a One-Dimensional Affine Term Structure Model Via Spectral Representations Tuesday, April 23, 2013 at 4:15pm Frank H. T. Rhodes Hall, 253 Affine processes have proved to be useful from both theoretical and applied perspectives. In finance, they offer some interesting abilities for modeling the dynamics of interest rates or of stochastic volatility of asset prices. One remarkable feature of these models is that the Laplace transform of the pricing semigroup is an exponential affine function of the short rate present value, identified in terms of some solutions to generalized Riccati equations. In this work, we are interested in the pricing of interest rate derivatives in one factor affine term structure models. Classical numerical methods for solving this type of problems rely on Laplace/Fourier inversion procedures. However, this turns out to be a difficult task since one has to resort to numerical methods to solve the aforementioned Riccati equations. We offer a significant analytical simplication to the valuation problem by showing that the pricing semigroup admits a discrete spectral representation based on eigenfunctions and eigenmeasures that are directly characterized in terms of the CBI mechanisms of the short rate. Our results rely on a paper by Ogura (1970) which is notable in the framework of non self-adjoint semigroups. This representation allows us to get series expansions for the price of European options on the short rate as well as bond and yield options. We illustrate our approach on two examples. The first one treats a class of self-similar CBI-processes. The second one discusses a CIR process with jumps. Numerical experiments show that the spectral approach leads to an extremely fast method for computing option prices.