Implicit-explicit predictor-corrector methods combined with improved spectral methods for pricing European style vanilla and exotic options

In this paper we present a robust numerical method to solve several types of European style option pricing problems. The governing equations are described by variants of Black-Scholes partial differential equations (BS-PDEs) of the reaction-diffusion-advection type. To discretise these BS-PDEs numerically, we use the spectral methods in the asset (spatial) direction and couple them with a third-order implicit-explicit predictor-corrector (IMEX-PC) method for the discretisation in the time direction. The use of this high-order time integration scheme sustains the better accuracy of the spectral methods for which they are well-known. Our spectral method consists of a pseudospectral formulation of the BS-PDEs by means of an improved Lagrange formula. On the other hand, in the IMEX-PC methods, we integrate the diffusion terms implicitly whereas the reaction and advection terms are integrated explicitly. Using this combined approach, we first solve the equations for standard European options and then extend this approach to digital options, butterfly spread options, and European calls in the Heston model. Numerical experiments illustrate that our approach is highly accurate and very efficient for pricing financial options such as those described above

Pricing European and American Options under Heston Model using Discontinuous Galerkin Finite Elements

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments

Robust Spectral Methods for Solving Option Pricing Problems

Doctor Scientiae - DScRobust Spectral Methods for Solving Option Pricing Problems by Edson Pindza PhD thesis, Department of Mathematics and Applied Mathematics, Faculty of Natural Sciences, University of the Western Cape Ever since the invention of the classical Black-Scholes formula to price the financial derivatives, a number of mathematical models have been proposed by numerous researchers in this direction. Many of these models are in general very complex, thus closed form analytical solutions are rarely obtainable. In view of this, we present a class of efficient spectral methods to numerically solve several mathematical models of pricing options. We begin with solving European options. Then we move to solve their American counterparts which involve a free boundary and therefore normally difficult to price by other conventional numerical methods. We obtain very promising results for the above two types of options and therefore we extend this approach to solve some more difficult problems for pricing options, viz., jump-diffusion models and local volatility models. The numerical methods involve solving partial differential equations, partial integro-differential equations and associated complementary problems which are used to model the financial derivatives. In order to retain their exponential accuracy, we discuss the necessary modification of the spectral methods. Finally, we present several comparative numerical results showing the superiority of our spectral methods

Finite element methods for Bellman and Isaacs Equations

This work concerns the numerical analysis of the Partial Differential Equations (PDEs) with a particular focus on fully nonlinear PDEs. More specifically, the main goal is to provide a finite element method to approximate solutions of Isaacs equations, which come from game theory and can be thought of as generalisation of Hamilton-Jacobi-Bellman (HJB) equations. Both of these classes of problems arise from the stochastic optimal control problems. Is is widely known that nonlinear PDEs do not in general admit classical solutions. A way to circumvent this issue is to use a relaxed definition of derivative leading to the notion of a generalised solution. One such notion is that of viscosity solution introduced in 1980s by Crandall and Lions. The main idea is to regularise non-smooth functions by using comparison principles and subtractive testing. The theory of viscosity solutions gave rise to novel numerical methods. A general framework of formulating convergent numerical schemes for (possibly degenerate) elliptic PDEs was formulated by Barles and Souganidis in 1991. The main result states that, given a comparison principle depending on the application at hand, a monotone, stable and consistent numerical scheme converges to the unique viscosity solution of a fully nonlinear problem. This framework is used throughout this work to formulate convergent numerical schemes. The main three contributions of the thesis are as follows. First we present a Finite Element Method to approximate solutions of isotropic parabolic problems of Isaacs type with possibly degenerate diffusions. Second we design a method of numerically approximating isotropic parabolic Hamilton-Jacobi-Bellman equations with nonlinear, mixed boundary conditions where Robin type boundary conditions are imposed via one-sided Dini derivatives. In both cases we prove the convergence of the numerical solution to the unique viscosity solution. The uniqueness of numerical solution is guaranteed by Howard’s algorithm. The analysis of the HJB equations with mixed boundary conditions is motivated by option pricing in a financial setting, which leads to our third contribution. We extend the Heston model of mathematical finance to permit the uncertain market price of volatility risk and we interpret it as an HJB equation. Finally, we present a case study investigating the effects of the market price of volatility risk on the option value and its derivatives