Bergische Universität Wuppertal Fachbereich Mathematik und Naturwissenschaften Angewandte Mathematik - Numerische Analysis

People Research Publications Teaching

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Exponentially fitted schemes and Box Methods for Exotic Options

Master's Thesis in Financial Mathematics, Halmstad University, Sweden

Supervisor:

• Prof. Dr. Matthias Ehrhardt

Description

Exotic options derivatives that are especially designed for customers needs. For these non-standard options the well-known transformations of the Black-Scholes equation to the heat equation do not work any more. Hence, one has to discretize the Black-Scholes equation directly. If one simply applies standard numerical methods for parabolic equations without taking special care of the financial meaning of the variables then one faces severe problems.
Here, in this thesis, we will show how to modify standard numerical schemes, e.g. by nonlocal discretization approaches, to remove unrealistic restrictions on the time step, preserve the positivity of the solution obeying a discrete maximum principle, adequate treatment of non-smooth initial data (pay-off function), etc. while keeping all the stability properties of the original scheme. As an illustrative example we consider a discretely monitored barrier option, a knock-and-out European Put option.

Keywords

positivity-preserving finite difference schemes, M-matrices, nonlocal discretization, nonstandard finite difference methods (NSFD), discrete maximum principle, spurious oscillations, discretely monitored Barrier options, Box scheme, exponentially-fitted method, error damping, non-smooth initial conditions

References:

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2. D.J. Duffy, A Critique of the Crank-Nicolson Strengths and Weaknesses for Financial Instrument Pricing, Technical Article, WILMOTT magazine, 2004, pp. 68-76
3. D.J. Duffy, Robust and Accurate Finite Difference Methods in Option Pricing - One Factor Models, Working Report, Datasim, 2001.
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