Levy Process And Application In Finance

Modeling financial assert with exponential Levy process is quite popular in recent years. But the density functions of Levy process can not be expressed analytically. So how to price the exotic derivatives under Exp-Levy model is a very interesting question. In this paper, we will focus on pricing two classes of exotic products: discretely monitored Barrier Options and discretely moni-tored Lookback Options. The main technique involves a sequential evaluation of Hilbert transforms of expressions involving the characteristic function of the Levy process. An efficient computational algorithm based on the FFT-based Toeplitz matrix-vector multiplication is used to calculate the Hilbert Transform. The dif-ference is that a backward recursion is used for pricing Barrier Options, and a forward recursion is used for Lookback Options. For Barrier Options, a new cri-terion is proposed to maintain the robustness of the algorithm. Numerical tests are implemented to demonstrate the robustness.