A Highly Efficient Shannon Wavelet Inverse Fourier Technique for Pricing European Options

Abstract

Abstract. In the search for robust, accurate, and highly efficient financial option valuation techniques, we here present the SWIFT method (Shannon wavelets inverse Fourier technique), based on Shannon wavelets. SWIFT comes with control over approximation errors made by means of sharp quantitative error bounds. The nature of the local Shannon wavelets basis enables us to adaptively determine the proper size of the computational interval. Numerical experiments on European-style options show exponential convergence and confirm the bounds, robustness, and efficiency.

SWIFT Calibration of the Heston Model

“…Typically, the ChF of a random variable with density function f is defined asf (u) = R f (x)e iux dx. However, to be consistent with [1], it is defined in this work asf (u) = R f (x)e −iux dx. We can see that there is a sign difference in all the u-dependent equations and expressions in [5].…”

“…In this section, we give a brief overview on the SWIFT method, originally developed for pricing European options in [1]. In this work, the method will be extended to European options' calibration.…”

“…It encompasses machinery for option pricing as well as an optimization procedure, aiming at minimizing the differences between the market option prices and the prices given by the valuation method. Regarding the pricing, we use the SWIFT method presented in [1] for European options. This belongs to the class of Fourier inversion methods and has already been used for pricing Bermudan, barrier and Asian options (see [2,3]).…”

“…We extend the SWIFT method to the calibration problem by deriving the option price gradient; • We implement and test the speeding-up techniques mentioned in [1] based on multiple strike valuation; • We propose a novel method for calibrating the Heston model with a set of options with certain fixed strikes that can be later used for arbitrary strikes by interpolation; • We develop and implement speeding up techniques for the option price gradient.…”

Calibration and simulation of Heston model

“…We would like to close this section by mentioning also the approximation by the usage of fast Fourier transform (FFT) as was suggested by Carr and Madan [29]. Although the FFT methods, including the fractional FFT modification [30,31], cosine FFT method [32] or methods based on wavelets [33], are fast in calculating an approximation of the inverse Fourier transform integral in many discrete points at once, however, in option pricing problem many values are calculated redundantly and, moreover, with relatively low precision that should be in modern financial applications considered unsatisfactory. Heston model in particular was studied from the FFT perspectives by Zhylyevskyy [34,35].…”