Solving complex PDE systems for pricing American options with regime‐switching by efficient exponential time differencing schemes |
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Authors: | AQM Khaliq B Kleefeld RH Liu |
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Institution: | 1. Department of Mathematical Sciences, Middle Tennessee State University, Murfreesboro, Tennessee 37132;2. Institut für Mathematik, Brandenburgische Technische Universit?t Cottbus, Postfach 101344, 03013 Cottbus, Germany;3. Department of Mathematics, University of Dayton, 300 College Park, Dayton, Ohio 45469‐2316 |
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Abstract: | In this article, we study the numerical solutions of a class of complex partial differential equation (PDE) systems with free boundary conditions. This problem arises naturally in pricing American options with regime‐switching, which adds significant complexity in the PDE systems due to regime coupling. Developing efficient numerical schemes will have important applications in computational finance. We propose a new method to solve the PDE systems by using a penalty method approach and an exponential time differencing scheme. First, the penalty method approach is applied to convert the free boundary value PDE system to a system of PDEs over a fixed rectangular region for the time and spatial variables. Then, a new exponential time differncing Crank–Nicolson (ETD‐CN) method is used to solve the resulting PDE system. This ETD‐CN scheme is shown to be second order convergent. We establish an upper bound condition for the time step size and prove that this ETD‐CN scheme satisfies a discrete version of the positivity constraint for American option values. The ETD‐CN scheme is compared numerically with a linearly implicit penalty method scheme and with a tree method. Numerical results are reported to illustrate the convergence of the new scheme. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013 |
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Keywords: | American option exponential time differencing free boundary value problem numerical PDE penalty method regime‐switching |
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