Abstracts

In this paper, we study the maturity randomization (Canadization) technique for pricing of European and American-style Parisian options. We consider a hyper-exponential jump-diffusion process which can approximate any Levy process with completely monotone density. The model is general and flexible enough to capture the asymmetric leptokurtic feature and the volatility smile, and one of its main advantages is the analytical tractability for pricing of path-dependent options. We follow the Gaver-Stehfest Canadization approach for solving a system of two-dimensional partial integro-differential equations describing the option price dynamics. In the first step, we analytically solve the pricing problem for Canadized European and American-style Parisian options by taking the double Laplace transform with respect to the option maturity and the Parisian window. Subsequently, we utilize the recursive algorithm for Gaver-Stehfest inversion, and present both theoretical and numerical results for the computation of option prices, Greeks, and early exercise boundaries. Our approach is presented for both up-and-out and down-and-out Parisian call options with and without already started excursion. In order to obtain prices of other types of options we provide necessary parity and symmetry relations in the hyper-exponential jump-diffusion setting. Furthermore, we examine the convergence of the obtained results for European (American) Parisian options to the European (American) plain vanilla or barrier options depending on the length of the Parisian window. Finally, the impact of jumps on the option prices, the Greeks and the early exercise boundary is discussed in the paper. Therefore, the Gaver-Stehfest Canadization method for Parisian options combines the mathematical appeal inherent to analytical approach with the ease of implementation of the Laplace numerical inversion, and provides important economic insight for pricing and hedging in the presence of jumps.