Pitfalls of the Fourier Transform Method in Affine Models, and Remedies

Abstract

We study sources of potentially serious errors of popular numerical realizations of the Fourier method in affine models, and explain that, in many cases, a calibration procedure based on such a realization will be able to find a correct parameter set only in a rather small region of the parameter space, with a blind spot: an interval of strikes depending on the model and time to maturity, where accurate calculations are extremely time consuming. We explain how to construct more accurate and faster pricing and calibration procedures. Efficient realizations require analytic continuation of the characteristic function not only into a strip around the real axis (the problem of finding the maximal strip of analyticity has been extensively studied in the literature under the name moment explosion) but into much wider regions of the complex plane, and an appropriate conformal deformation of the contour of integration. In the paper, we study analytic continuation of the characteristic functions in wide classes of affine models, models with jumps including. An important ingredient of our method is the study of the associated system of generalized Riccati equations. As a byproduct, we show that the straightforward application of the Runge-Kutta method may lead to sizable errors, and suggest certain remedies. The methodology of the paper can be applied to other models (e.g. quadratic term structure models, Wishart dynamics, 3/2-model), where European options and other options can be priced using the Fourier transform techniques.