This paper presents a convergence analysis of Crank-Nicolson and Rannacher time-marching methods which are often used in finite difference discretisations of the Black-Scholes equations. Particular attention is paid to the important role of Rannacher's startup procedure, in which one or more initial timesteps use Backward Euler timestepping, to achieve second order convergence for approximations of the first and second derivatives. Numerical results confirm the sharpness of the error analysis which is based on asymptotic analysis of the behaviour of the Fourier transform. The relevance to Black-Scholes applications is discussed in detail, with numerical results supporting recommendations on how to maximise the accuracy for a given computational cost.

/pdf/convergence-analysis-of-crank-nicolson-and-rannacher-time-4bxbtqnqm2.pdf

Convergence analysis of Crank-Nicolson and Rannacher time-marching

The Mathematics of Financial Derivatives: Preface

Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model, which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential equation (PIDE), while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids, and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy-to-implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. For American options two ways to solve the LCPs are described: an operator slitting method and a penalty method. Numerical experiments confirm that the developed methods are very efficient, as fairly accurate option prices can be computed in a few milliseconds on a PC.

/pdf/numerical-valuation-of-european-and-american-options-under-13mhjpv57x.pdf

Numerical Valuation of European and American Options under Kou's Jump-Diffusion Model

Preface 1. Introduction 2. Storables modeling basics 3. Continuously produced commodity price dynamics 4. Empirical performance of the two-factor model 5. Stochastic volatility 6. Seasonal commodities 7. Carbon markets 8. Non-storable commodities.

Commodity Price Dynamics: A Structural Approach

A unified approach to Bermudan and barrier options under stochastic volatility models with jumps

Operator splitting methods for american option pricing

Stochastic volatility models lead to more realistic option prices than the Black-Scholes model which uses a constant volatility. Based on such models a two-dimensional parabolic partial differential equation can derived for option prices. Due to the early exercise possibility of American option contracts the arising pricing problems are free boundary problems. In this paper we consider the numerical pricing of American options when the volatility follows a stochastic process. We propose operator splitting methods for performing time stepping after a finite difference space discretization is done. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach we can use any efficient numerical method to solve the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank-Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting does not increase essentially the error. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.

http://users.jyu.fi/~tene/papers/reportB11-04.pdf

Operator Splitting Methods for Pricing American Options with Stochastic Volatility

Pricing American options using partial (integro-)differential equation based methods leads to linear complementarity problems (LCPs) The numerical solution of these problems resulting from the Black-Scholes model, Kou’s jump-diffusion model, and Heston’s stochastic volatility model are considered The finite difference discretization is described The solutions of the discrete LCPs are approximated using an operator splitting method which separates the linear problem and the early exercise constraint to two fractional steps The numerical experiments demonstrate that the prices of options can be computed in a few milliseconds on a PC

http://users.jyu.fi/~tene/papers/OP60.pdf

An Operator Splitting Method for Pricing American Options

Option pricing models with a stochastic volatility are more realistic than the Black-Scholesmodelwhichusesaconstantvolatility. Thepricesofoptionsbasedonsuch models can be obtained by solving a parabolic partial difierential equation. Particularly, weconsiderthemodelpresentedbyHeston. Thevariablesintheseproblemsarethetime, the underlying asset value, and the volatility. Due to the early exercise possibility of American options the arising problems are free boundary problems. This paper considers the numerical solution of this type of option pricing models. We study operator splitting methods for performing time stepping after a flnite difier- encespacediscretizationisdone. Theideaistodecoupletheearlyexerciseconstraintfor thepricesatthegridpointsandthesolutionofthesystemoflinearequationstoseparate fractional time steps. With this approach we can use any e-cient numerical method to solve the systems of linear equations in the flrst fractional step and then make a simple update to satisfy the constraint in the second fractional step. This leads to more simple ande-cientsolutionprocedures. Thispaperstudiestheaccuracyoftheoperatorsplitting methods without considering e-cient solution procedures which is the topic of our future research. Numerical experiments compare the accuracy of our operator splitting methods with traditionalunsplitimplicittimediscretizations. Wedemonstratethattheadditionalerror due to the splitting does not essentially increase the time discretization error.

http://users.jyu.fi/~tene/papers/Ikonen.pdf

Operator splitting methods for american options with stochastic volatility

We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.

https://masterfinance.math.univ-paris-diderot.fr/attachments/article/50/ikonen_toivanen.pdf

Operator splitting methods for pricing American options under stochastic volatility

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