Article2006

Convergence analysis of Crank–Nicolson and Rannacher time-marching

124 citations0/0 files storedThe Journal of Computational Finance

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Stochastic processes and financial applications Advanced Numerical Methods in Computational Mathematics Differential Equations and Numerical Methods

Abstract

This paper presents a convergence analysis of Crank–Nicolson and Rannacher time-marching methods which are often used in finite difference discretizations 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 behavior of the Fourier transform. The relevance to Black–Scholes applications is discussed in detail, with numerical results supporting recommendations on how to maximize the accuracy for a given computational cost. 1

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References

1993A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options 1995The Mathematics of Financial Derivatives 1984The American Put Option Valued Analytically 1996American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods 2002Quadratic Convergence for Valuing American Options Using a Penalty Method 2006Finite Difference Methods in Financial Engineering: A Partial Differential Equation Approach 2003Numerical convergence properties of option pricing PDEs with uncertain volatility 2003Convergence remedies for non-smooth payoffs in option pricing

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