Pricing early-exercise and discrete barrier options by Shannon wavelet expansions

Abstract

We present a pricing method based on Shannon wavelet expansions for early-exercise and discretely-monitored barrier options under exponential Lévy asset dynamics. Shannon wavelets are smooth, and thus approximate the densities that occur in finance well, resulting in exponential convergence. Application of the Fast Fourier Transform yields an efficient implementation and since wavelets give local approximations, the domain boundary errors can be naturally resolved, which is the main improvement over existing methods. Mathematics Subject Classification

On the wavelet-based SWIFT method for backward stochastic differential equations

“…For the computation of the expectations appearing in the discrete FBSDEs (2.3), we will use the wavelet-based SWIFT method. In this section, we first provide an alternative derivation for the SWIFT formula used in [14] and [13]. Instead of using an approximation space based on Shannon wavelets on the whole real line, we construct a Shannon wavelet scaling function on a finite domain and derive our formula with this scaling function.…”

“…Furthermore, we mitigate the problem of errors near the computation boundaries by means of an antireflective boundary technique, giving an improved approximation. We test our algorithm with different numerical experiments.In this article, we shall adopt the quick variant of the SWIFT formula proposed in [13].The reason behind this is left for the error section, but this gives an approximation for E x p [v(t p+1 , X ∆ t p+1 )], which we see in the discrete-time approximation of FBSDE,…”

A Shannon Wavelet Method for Pricing American Options under Two-Factor Stochastic Volatilities and Stochastic Interest Rate

“…where J ≔ log 2 (πN) . How to compute the vector of coefficients D m,k (x) ≔ 〈f(• | x), ϕ m,k (•)〉 efficiently using the fast Fourier transform has been showed in by Maree [19]. To sum up, by means of the SWIFT method, the resulting option value at time t can be written as follows:…”

“…where G m,k (y 1 , y 2 ) means the payoff coefficients, and it can be efficiently computed using the fast Fourier transform as Oosterlee mentioned in [19]. Once a formula for the value coefficients at maturity is obtained, we can calculate the coefficients at any time t n , for n � N − 1, .…”

A Shannon wavelet method for pricing foreign exchange options under the Heston multi-factor CIR model

“…This paper aims to further extend the applicabilities of these state-of-the-art numerical integration methods to the above-mentioned general jump-diffusion FX model. We use the SWIFT method, due to the established robustness of Shannon wavelets in option pricing, as demonstrated in a number of works, such as Colldeforns-Papiol et al (2017); Maree et al (2017); Ortiz-Gracia and Oosterlee (2016). The proposed SWIFT-based method is developed within the hybrid MC-PDE computational framework put forward in Dang et al (2015bDang et al ( , 2017.…”