High-performance computation for pricing financial options

Abstract

Options are widely traded financial instruments and computing their fair prices is interesting as it requires one to solve challenging problems from various fields, including Mathematics, Computer science and Finance. Option pricing models available in the literature use stochastic processes to model the market dynamics that includes underlying assets, volatilities and interest rates. Depending on the model’s assumption, under the risk-neutral framework, the corresponding governing equation could be a partial differential equation (PDE) or a partial integro-differential equation (PIDE) for a European option. On the other hand, one has to solve a complementarity problem to price an American option. Often these pricing problems do not have analytical solutions, thus approximate solutions are computed numerically. Method-of-lines is a very popular approach for numerically solving multi-dimensional option pricing problems by using approximation of derivatives, such as finite difference (FD). It involves storing and handling huge discretization matrices when the FD grid is sufficiently refined. Moreover, for the problem of type PIDE, the integral approximation matrix is dense in nature. Hence, on a desktop with a limited amount of memory, the applied technique suffers severely for speed and memory, even after using efficient numerical methods, such as the alternating direction implicit (ADI) schemes. This thesis deals with memory-efficient and fast numerical pricing of options, be it European or American, which has been achieved by designing state-of-the-art parallel algorithms and employing them with customization for optimal performance. In spite of robust numerical schemes with well-known theoretical stability and convergence results are available to price options, their parallelization is seriously underdeveloped. Unlike approximating derivatives with central FD scheme that yields block diagonal discretization matrices with tridiagonal blocks, option pricing problems often demand the use of different FD schemes, for instance higher order or one sided divided difference. In these cases one obtains blocks of discretization matrices with different band structures, such as pentadiagonal. In this thesis, we have developed a parallel cyclic reduction (PCR) algorithm, PCR-Penta, for pentadiagonal systems which is highly efficient and has fine-grained parallelism. The novel algorithm is implemented on a graphics processing unit (GPU) using CUDA and studied for its performance. PCR-Penta is employed to solve a convection-dominated Heston PDE on a GPU for pricing European options in parallel, and overall significant speed-ups are noted compared to efficient sequential technique as well as other contemporary parallel solvers. Two-factor forward variance model that tries to capture forward variance of the underlying asset with two stochastic processes are popular for pricing foreign exchange options. In this thesis, one pertinent model is considered for pricing European options, which yields a three-dimensional PDE. The central FD scheme is applied for spatial discretization, and a new nonuniform mesh generating technique is proposed in the asset direction. Since the corresponding discretization matrices are block diagonal with tridiagonal blocks, in the parallel implementation of the employed ADI schemes, the unidirectional implicit steps are solved by using the existing PCR algorithm for tridiagonal systems. As expected, the parallel solutions using CUDA are sufficiently accurate and fast compared to solutions that are computed sequentially in MATLAB. Jump-diffusion models are popular for modeling asset prices, especially when the market is highly volatile. Bates model is one such model and it has an additional benefit of incorporating stochastic volatility. Pricing a European option under this model requires one to solve the two-dimensional Bates PIDE that has a one-dimensional non-local integral term. Numerical solutions of this PIDE are computed in parallel in thesis. Note that in addition to the computeintensive implicit steps of the ADI schemes, execution of their explicit steps also becomes challenging mainly due to handling the integral term. Significant speed-ups are observed when parallel solutions computed using CUDA on a GPU are compared with those calculated using OpenMP on multi-core CPU and in MATLAB by using an efficient sequential technique. In the last and most important part of the thesis, pricing of a two-asset American option under the Merton jump-diffusion model is considered, which gives rise to a two-dimensional partial integro-differential complementarity problem (PIDCP) that has a nonlocal two-dimensional integral term. Numerically solving this PIDCP requires a humongous amount of memory as the huge approximation matrix corresponding to the double integral term is fully dense. However, exploiting its block Toeplitz with Toeplitz block structure, which involves usage of fast Fourier transformations and bilinear interpolations, the memory requirement is reduced drastically. The semidiscretized PIDCP is solved using an ADI-IT scheme, which is a combination of ADI scheme with Ikonen-Toivanen (IT) splitting technique. The employed ADI-IT scheme is parallelized on a GPU using CUDA and associated bottlenecks are cleverly resolved. As a result, in comparison with an efficient sequential solution, a substantially faster parallel solution is achieved. Some interesting future work and open problems are included at the end.