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

Application of Operator Splitting Methods in Finance

Financial derivatives pricing aims to find the fair value of a financial contract on an underlying asset. Here we consider option pricing in the partial differential equations framework. The contemporary models lead to one-dimensional or multidimensional parabolic problems of the convection-diffusion type and generalizations thereof. An overview of various operator splitting methods is presented for the efficient numerical solution of these problems. Splitting schemes of the Alternating Direction Implicit (ADI) type are discussed for multidimensional problems, e.g. given by stochastic volatility (SV) models. For jump models Implicit-Explicit (IMEX) methods are considered which efficiently treat the nonlocal jump operator. For American options an easy-to-implement operator splitting method is described for the resulting linear complementarity problems. Numerical experiments are presented to illustrate the actual stability and convergence of the splitting schemes. Here European and American put options are considered under four asset price models: the classical Black-Scholes model, the Merton jump-diffusion model, the Heston SV model, and the Bates SV model with jumps

Pricing Options under Heston’s Stochastic Volatility Model via Accelerated Explicit Finite Differencing Methods

We present an acceleration technique, effective for explicit finite difference schemes describing diffusive processes with nearly symmetric operators, called Super-Time-Stepping (STS). The technique is applied to the two-factor problem of option pricing under stochastic volatility. It is shown to significantly reduce the severity of the stability constraint known as the Courant-Friedrichs-Lewy condition whilst retaining the simplicity of the chosen underlying explicit method. For European and American put options under Heston’s stochastic volatility model we demonstrate degrees of acceleration over standard explicit methods sufficient to achieve comparable, or superior, efficiencies to a benchmark implicit scheme. We conclude that STS is a powerful tool for the numerical pricing of options and propose them as the method-of-choice for exotic financial instruments in two and multi-factor models.

Avrupa Tipi Satış Opsiyonu Modeli için Nümerik bir Değerlendirme

The Black-Scholes equations have been increasingly popular over the last three decades since they provide more practical information for optional behaviours. Therefore, effective methods have been needed to analyse these models. This study will mainly focus on investigating the behaviour of the Black-Scholes equation for the European put option pricing model. To achieve this, numerical solutions of the Black-Scholes European option pricing model are produced by three combined methods. Spatial discretization of the Black-Scholes model is performed using a fourth-order finite difference (FD4) scheme that allows a highly accurate approximation of the solutions. For the time discretization, three numerical techniques are proposed: a strong-stability preserving Runge Kutta (SSPRK3), a fourth-order Runge Kutta (RK4) and a one-step method. The results produced by the combined methods have been compared with available literature and the exact solution. It has seen that the results with minimal computational effort are sufficiently accurate.Black-Scholes denklemleri opsiyon davranışlarında pratik bilgiler sağladığından son otuz yılda daha popüler hale gelmiştir. Bu nedenle, bu modelleri analiz etmek için etkili yöntemlere ihtiyaç duyulmaktadır. Bu çalışma temel olarak Avrupa tipi satış opsiyonu fiyatlama modeli için Black-Scholes denkleminin davranışını araştırmaya odaklanmıştır. Bunun için, Black-Scholes Avrupa tipi opsiyon fiyatlama modelinin sayısal çözümleri üç birleştirilmiş yöntem ile üretilmiştir. Black-Scholes modelinin uzaysal ayrıklaştırması, çözümlerin yüksek hassasiyetli yaklaşımlarına izin veren dördüncü mertebeden bir sonlu fark (FD4) şeması kullanılarak yapılmıştır. Zaman ayrıklaştırması için üç sayısal teknik kullanılmıştır: Kuvvetli kararlılık koruyan RungeKutta (SSPRK3), dördüncü mertebe Runge Kutta (RK4) ve tek adımlı bir yöntem. Birleştirilmiş yöntemlerle üretilen sonuçlar literatürde mevcut olan çözüm ve tam çözüm ile karşılaştırılmıştır. Sonuçların minimum hesaplama çabasıyla yeterince hassas olduğu görülmüştür

High-order full discretization for anisotropic wave equations

Two-dimensional linear wave equation in anisotropic media, on a rectangular domain with initial conditions and periodic boundary conditions, is considered. The energy of the problem is contemplated. The space discretization is reached by means of finite differences on a uniform grid, paying attention to the mixed derivative of the equation. The discrete energy of the semi-discrete problem is introduced. For the time integration of the system of ordinary differential equations obtained, a fourth order exponential splitting method, which is a geometric integrator, is proposed. This time integrator is efficient and easy to implement. The stability condition for time step and space step ratio is deduced. Numerical experiments displaying the good behavior in the long time integration and the efficiency of the numerical solution are provided.MTM2015-66837-P del Ministerio de Economía y Competitivida