Andrea Perchiazzo

Università degli Studi di Milano (UNIMI)

Title

Pricing European Options using the Gauss-Laguerre Quadrature

Authors

Andrea Perchiazzo

Abstract

In this work we propose a novel method for pricing European options numerically in an efficient manner using the Gauss-Laguerre quadrature. Instead of employing the Carr-Madan formula [1], which needs an appropriate choice for the damping factor, or the COS method of Fang and Oosterlee [2] in which three approximation errors are introduced (e.g., truncation of the integration range in the risk-neutral valuation formula) as discussed in [3, Chapter 6.2.3], the pricing of European options using the characteristic function is based on the Gauss-Laguerre quadrature. The approach does not necessitate the truncation of the integration range in the risk-neutral valuation formula and the approximation error term is controlled by the order of Laguerre polynomials. The new methodology is initially tested on the Black-Scholes model in order to confirm its efficacy and then applied to several models (e.g., the Merton, Kou, Variance Gamma, and Heston models). In addition, a comparison between the novel approach based on the Gauss-Laguerre quadrature, the COS method, and the Carr-Madan formula is performed.

[1] Peter Carr and Dilip Madan. “Option valuation using the fast Fourier transform”. In: Journal of computational finance 2.4 (1999), pp. 61–73.
[2] Fang Fang and Cornelis W Oosterlee. “A Novel Pricing Method for European Options Based on Fourier-Cosine Series Expansions”. In: SIAM Journal on Scientific Computing 31.2 (2009), pp. 826–848. doi: 10.1137/080718061.
[3] Cornelis W Oosterlee and Lech A Grzelak. Mathematical modeling and computation in finance: with exercises