High-order Compact Finite Difference Scheme For Pricing American Options Under The Stochastic Volatility Model With Jump

Efficient pricing American options derivatives is one of the important topics in quantitative finance. As one of the typical stochastic volatility models with jump, the Bates model not only inherits the advantages of the Heston stochastic volatility model and the Merton jump model, such as having the "heavy tail, high peak and skewed" characteristic for asset return and "smile and smirk" for the implied volatility, but also can obtain the analytic solution for pricing European options. The model has been one of the most popular references in the real financial world.In the past years, a plentiful of progresses have been made to the finite difference method (FDM) for American option pricing under the B-S model. However, there are still two big challenges that are more complex than B-S for existing methods. First, the volatility term will add one more dimension to the pricing PDE discretization. Second, the PDE becomes the PIDE caused by the non-locality of the jump term, which brings us a new challenge of pricing American options under the Bates model. Until now, there are few papers solving the two problems at the same time.In this paper I present an efficient high-order numerical method, HOCJ-CF, for pricing American put options under the Bates model, based on the Jain’s high-order compact finite difference, convolution integral and Fast Fourier Transformation. The critical idea behind HOCJ-CF is:to avoid the inversion operation of full matrix introduced by the non-locality of the jump term, we put aside temporarily the integral term resulting from the jump, thus deriving a nine-point compact scheme for discretizing partial derivatives, and then reconsider the integral term to develop the final whole valuation scheme.For the differential terms of pricing PIDE (the same as PDE under the Heston model), we continue to split them into three sub-PDEs with coefficients (to be determined later), and then apply the Nemerov discretization approach to deal with these sub-PDEs to form a compact stencil as a whole. The HOCJ has been approved conditional convergent later. Besides, numerical illustration demonstrates that, on the same space grids, our HOCJ scheme has a better accuracy, faster convergence rate and higher efficiency than the HOCS scheme under the same Heston model settings.For the integral term, we evaluate it independently using the multiplication as a Convolution integral, and then employ FFTs (hereinafter, CF). Finally, we combine both to form a whole valuation scheme. To make a fair comparison between our HOCJ-CF method and the IMEX scheme in American option pricing, the LU decomposition along with the operator splitting (OS) method are also embedded in HOCJ-CF. Numerical illustration demonstrates that, on the same space grids, our HOCJ-CF scheme has a better accuracy, faster convergence rate and higher efficiency than the IMEX scheme under the same model settings.The HOCJ-CF method has a few creativities for option pricing in practice. Firstly, the method has extended the application of Numerov discretization for one-dimensional second-order ODEs to two-dimensional second-order quasi-linear PDE under asset price random model option pricing. Secondly, compared with the HOCJ and IMEX scheme, the form of HOCJ-CF scheme seems much simpler. By splitting the pricing PDE into three sub-PDEs, we successfully avoid the complicated manipulations such as drilling the finite difference approximation of high-order derivatives from truncation, like in the HOCJ scheme. Also, by temporarily dropping the jump terms, we obtain a linear compact valuation scheme for American options. Thirdly, the advantages of the high-order compact discretization of Jain are inherited by the HOCJ-CF method. Thus, the accuracy of HOCJ-CF in American option pricing can be guaranteed. Lastly, recognizing that PDEs under the Heston stochastic volatility model are terms under Bates model without jump, I deal with these terms through new HOCJ and terms without jump through combing the Convolution integral with FFTs. Thus, high order accuracy is promised and complex calculation on full matrix inversion is avoided.This paper enriches the theory of option pricing. Besides, we can understand the role of Options in financial markets better, recognize the importance of rational pricing, and invest with good reason. In practice, HOCJ-CF is more accurate, direct and fast. It can be universally applied:on one hand, it can be applied to characterize other PIDEs like stochastic volatility model with jumps; on the other hand, it can be used in other types of options besides the American type options. In addition, the method provides a reliable reference for quants, enables investors to make better use of options to hedge, and produces effective risk indicators for the risk managers.