Recombined multinomial tree based on saddle-point approximation and its application to Levy models options pricing

Abstract

This paper studies the constructing methods of a recombined multinomial tree based on saddle-point approximation and its application to Levy models options pricing. Firstly, the Levy process and the European option pricing are introduced. Then, we used the characteristic function of Levy process to generate density function value at discrete point in a certain range, based on the saddle-point approximation method. Then, we provide the method to construct recombined multinomial tree and give the pricing formula of European option and path-dependent option pricing based on the backward iteration. Finally, we used the CGMY process to demonstrate its application to European option, American option and American barrier option pricing. It proves that saddle-point approximation turns the inverse Fourier integral transform problem to several function value calculation. Comparing to IFFT, the speed of calculation using this method is faster, and it avoid the negative probability density function value based on IFFT. Because of the linear growth of the number of nodes, we can extend the saddle-point approximation method to the calculation of path-dependent option.