Notes on the BENCHOP implementations for the COS method

This text describes the COS method and its implementation for the BENCHOP-project. 1 Fourier cosine expansion formula (COS formula) We explain the COS method to approximate the European option value u(x, t0) = e −r∆tE [u(XT , T )|Xt0 = x] , (1) with ∆t = T − t0. Here Xt is the state process, which can be any monotone function of the underlying asset price St, for example, the scaled log-asset price, Xt = ln(St/K), where K is the options strike price. We assume a continuous transitional density, which is denoted by p(y|x). In other words, ∫ B p(y|x)dy = P(XT ∈ B|Xt0 = x), ∀ Borel subsets B ∈ R. We omit the dependence on ∆t for notational convenience. We rewrite u(x, t0) = e −r∆t ∫ R u(y, T )p(y|x)dy. (2) The numerical method is based on Fourier cosine series expansions of the option value at time level T and the density function, as we will show below. The resulting equation is called the COS formula, due to the use of Fourier cosine series expansions. In the derivation of the COS formula, we distinguish three different approximation steps. Step 1: For the problems we work on, the integrand decays to zero as y → ±∞. Because of that, we can truncate the infinite integration range of the ∗Centrum Wiskunde & Informatica, Amsterdam, the Netherlands, email: [email protected]. †Centrum Wiskunde & Informatica, Amsterdam, the Netherlands, email: [email protected], and Delft University of Technology, Delft, the Netherlands.