Positive Second Order Finite Difference Methods on Fokker-Planck Equations with Dirac Initial Data - Application in Finance
12 Pages Posted: 13 May 2015 Last revised: 5 Nov 2015
Date Written: November 2015
One approach to price financial derivatives in an arbitrage free manner is to work directly with the probability density function. For many models, it is possible to find an expansion of the density that follows a Fokker-Planck partial differential equation. In order to keep the arbitrage-free property numerically, it is particularly important to use a positivity preserving finite difference scheme. With a discontinuous initial condition like a Dirac, many schemes will produce oscillations or negative densities. This paper analyzes the behavior of a few simple schemes related to backward Euler, namely BDF2, Lawson-Morris and Lawson-Swaybe on the specific problem of a diffusion with Dirac initial condition. The case of the arbitrage free SABR model for interest rate derivatives is then evaluated.
Keywords: SABR, Fokker-Planck, Dirac, BDF2, Richardson extrapolation, Crank-Nicolson, finite difference
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