Closed-form approximations for basket option pricing under normal tempered stable Lévy model

Abstract

In this paper, we study the pricing problems of basket options and spread options under the Normal Tempered Stable Lévy model, which is a general model for financial assets and covers many well-known models as special cases such as the Variance Gamma model, Normal Inverse Gaussian model etc. Our approach draws inspiration from the lower bound approximation strategy used in Gaussian models in Bjerksund and Stensland (2014). The approximation formula we derived involves some one-dimensional integrations. We calculate these integrals using the generalized Gauss–Laguerre quadrature rule and Taylor expansion methods. In particular, we derive an analytical approximation formula under the Variance Gamma model for some exchange options. Moreover, we extend the approximation formulas proposed by Kirk (1995) and Carmona and Durrleman (2003b) to the Normal Tempered Stable Lévy model. Numerical tests show that our approximation formulas are highly accurate. Furthermore, we show that our approximation formulas outperform the Fourier inversion method introduced by Caldana et al. (2016) in accuracy, especially for low prices cases.