Pricing and Hedging of Lookback Options in Hyper-exponential Jump Diffusion Models

ABSTRACT

In this article, we consider the problem of pricing lookback options in certain exponential Lévy market models. While in the classic Black-Scholes models the price of such options can be calculated in closed form, for more general asset price model, one typically has to rely on (rather time-intense) Monte-Carlo or partial (integro)-differential equation (P(I)DE) methods. However, for Lévy processes with double exponentially distributed jumps, the lookback option price can be expressed as one-dimensional Laplace transform (cf. Kou, S. G., Petrella, G., & Wang, H. (2005). Pricing path-dependent options with jump risk via Laplace transforms. The Kyoto Economic Review, 74(9), 1–23.). The key ingredient to derive this representation is the explicit availability of the first passage time distribution for this particular Lévy process, which is well-known also for the more general class of hyper-exponential jump diffusions (HEJDs). In fact, Jeannin and Pistorius (Jeannin, M., & Pistorius, M. (2010). A transform approach to calculate prices and Greeks of barrier options driven by a class of Lévy processes. Quntitative Finance, 10(6), 629–644.) were able to derive formulae for the Laplace transformed price of certain barrier options in market models described by HEJD processes. Here, we similarly derive the Laplace transforms of floating and fixed strike lookback option prices and propose a numerical inversion scheme, which allows, like Fourier inversion methods for European vanilla options, the calculation of lookback options with different strikes in one shot. Additionally, we give semi-analytical formulae for several Greeks of the option price and discuss a method of extending the proposed method to generalized hyper-exponential (as e.g. NIG or CGMY) models by fitting a suitable HEJD process. Finally, we illustrate the theoretical findings by some numerical experiments.