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Partial Differential Equation
  • Unconditionally Stable and Second-Order Accurate Explicit Finite Difference Schemes Using Domain Transformation: Part I One-Factor Equity Problems We introduce a class of stable and second-order accurate finite difference schemes that resolve a number of problems when approximating the solution of option pricing models in computational finance using the finite difference method (FDM). In particular, we show how to avoid having to apply truncation methods to the domain of integration as well as the resulting ad-hoc experimentation in choosing suitable numerical boundary conditions that we must apply on the boundary of the truncated domain. Instead, we transform the PDE option model (which is originally defiined on a semi-infinite interval) to a PDE that is defined on a bounded interval. We then apply the elegant Fichera theory to help us determine which boundary conditions to apply to the transformed PDE. We show that the new system is well-posed by proving energy inequalities in the space of square-integrable functions. Having done that, we adapt the Alternating Direction Explicit (ADE) (a method that dates from the first half of the last century) method to compute an approximation to the solution of the transformed PDE. We prove that the explicit scheme is unconditionally stable and second-order accurate. The scheme is very easy to model, to program and to parallelise and the generalization to n-factor problems is easily motivated. We examine equity problems and we compare the ADE approach (in terms of accuracy and speedup) with more traditional finite difference schemes and with the Monte Carlo method. Finally, we stress-test the scheme by varying critical parameters over a spectrum of values and the results are noted. The methods in this article - in particular the combination of pure and applied techniques - are relatively new in the computational finance literature in our opinion, in particular the realization that a PDE on a semi-infinite interval can be transformed to one on a bounded interval and that the new PDE can be approximated by schemes other than Crank-Nicolson and its workarounds. Finally, we shall report on n-factor models in future articles. , D.Duffy (2009)
  • Finite Difference Scheme for Heston Model , S.Lin (2008)
  • ADI finite difference schemes for option pricing in the Heston model with correlation , K.J.Hout, S.Foulon (2007)
  • Partial differential equations in finance , Y.Achdou, O.Bokanowski, T.Lelievre (2007)
  • Numerical Methods for Controlled Hamilton-Jacobi-Bellman PDEs in Finance , P.A.Forsyth, G.Labahn (2007)
  • Pricing Equity Derivatives under Stochastic Volatility : A Partial Differential Equation Approach , R. Sheppard (2007)
  • Finite Difference Methods for Option Valuation with Jump Processes , V.Surkov (2006)
  • Convergence analysis of Crank-Nicolson and Rannacher time-marching , M.B. Giles, R. Carter (2005)
  • Singularity Removing Transformations for Partial Differential Equations in Finance , P. Raahauge (2005)
  • Convergence analysis of Crank-Nicolson and Rannacher time-marching , M.B. Giles, R. Carter (2005)
  • Numerical Methods and Volatility Models for Valuing Cliquet Options , H.A.Windcliff, P.A.Forsyth, K.R.vetzal (2005)
  • Derivative Pricing, Numerical Methods , K.R. Vetzal (2003)
  • Numerical solution of the Black-Scholes equation with a small number of grid points , C.C.W. Leentvaar (2003)
  • A Penalty Method for American Options with Jump Diffusion Processes , Y. d Halluin, P.A. Forsyth, G. Labahn (2003)
  • Why Be Backward? Forward Equations for American Options , P.Carr, A.Hirsa (2002)
  • Robust Numerical Methods for PDE Models of Asian Options , R. Zvan, P.A. Forsyth, K.R. Vetzal (2000)

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