## Numerical pricing of options using high-order compact finite difference schemes.(English)Zbl 1146.91338

Summary: We consider high-order compact (HOC) schemes for quasilinear parabolic partial differential equations to discretise the Black-Scholes PDE for the numerical pricing of European and American options. We show that for the heat equation with smooth initial conditions, the HOC schemes attain clear fourth-order convergence but fail if non-smooth payoff conditions are used. To restore the fourth-order convergence, we use a grid stretching that concentrates grid nodes at the strike price for European options. For an American option, an efficient procedure is also described to compute the option price, Greeks and the optimal exercise curve. Comparisons with a fourth-order non-compact scheme are also done. However, fourth-order convergence is not experienced with this strategy. To improve the convergence rate for American options, we discuss the use of a front-fixing transformation with the HOC scheme. We also show that the HOC scheme with grid stretching along the asset price dimension gives accurate numerical solutions for European options under stochastic volatility.

### MSC:

 91B28 Finance etc. (MSC2000) 65N06 Finite difference methods for boundary value problems involving PDEs
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### References:

 [1] X. Chen, J. Chadam, R. Stamicar, The optimal exercise boundary for American put options: analytic and numerical approximations, Preprint, 2000.; X. Chen, J. Chadam, R. Stamicar, The optimal exercise boundary for American put options: analytic and numerical approximations, Preprint, 2000. [2] Fornberg, B., Calculation of weights in finite difference formulas, SIAM Rev., 40, 3, 685-691 (1998) · Zbl 0914.65010 [3] Goodman, J.; Ostrov, D. N., On the early exercise boundary of the American put option, SIAM J. Appl. Math., 62, 5, 1823-1835 (2002) · Zbl 1029.91028 [4] Han, H.; Wu, X., A fast numerical method for the Black-Scholes equation of American options, SIAM J. Numer. Anal., 41, 6, 2081-2095 (2003) · Zbl 1130.91336 [5] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financial Studies, 6, 327-343 (1993) · Zbl 1384.35131 [6] Jain, M. K.; Jain, R. K.; Mohanty, R. K., A fourth-order difference method for the one-dimensional general quasilinear parabolic differential equation, Numer. Methods Partial Differential Equations, 6, 311-319 (1990) · Zbl 0715.65067 [7] Leisen, D.; Reimer, M., Binomial models for option valuation-examining and improving convergence, Appl. Math. Finance, 3, 319-346 (1996) · Zbl 1097.91513 [8] McCartin, B. J.; Labadie, S. M., Accurate and efficient pricing of vanilla stock options via the Crandall-Douglas scheme, Appl. Math. Comput., 143, 39-60 (2003) · Zbl 1053.91062 [9] C.W. Oosterlee, C.C.W. Leentvaar, X. Huang, Accurate American option pricing by grid stretching and high-order finite differences, Working papers, DIAM, Delft University of Technology, the Netherlands, 2005.; C.W. Oosterlee, C.C.W. Leentvaar, X. Huang, Accurate American option pricing by grid stretching and high-order finite differences, Working papers, DIAM, Delft University of Technology, the Netherlands, 2005. [10] Seydel, R., Tools for Computational Finance (2002), Springer: Springer Berlin · Zbl 1044.91534 [11] W.F. Spotz, G.F. Carey, High-order compact finite difference methods, in: V. Andrew, V. Ilin, L. Ridgway Scott (Eds.), ICOSAHOM95, Proceedings of the Third International Conference on Spectral and High Order Methods, 1996, pp. 397-408.; W.F. Spotz, G.F. Carey, High-order compact finite difference methods, in: V. Andrew, V. Ilin, L. Ridgway Scott (Eds.), ICOSAHOM95, Proceedings of the Third International Conference on Spectral and High Order Methods, 1996, pp. 397-408. · Zbl 0866.65066 [12] Spotz, W. F.; Carey, G. F., A high-order compact formulation for the 3D Poisson equation, Numer. Methods Partial Differential Equations, 12, 235-243 (1996) · Zbl 0866.65066 [13] Spotz, W. F.; Carey, G. F., Extension of high order compact schemes to time dependent problems, Numer. Methods Partial Differential Equations, 17, 6, 657-672 (2001) · Zbl 0998.65101 [14] D.Y. Tangman, A. Gopaul, M. Bhuruth. A fast high-order finite difference algorithm for pricing American options, J. Comput. Appl. Math., submitted.; D.Y. Tangman, A. Gopaul, M. Bhuruth. A fast high-order finite difference algorithm for pricing American options, J. Comput. Appl. Math., submitted. · Zbl 1147.91032 [15] Tavella, D.; Randall, C., Pricing Financial Instruments: The Finite Difference Method (2000), Wiley: Wiley New York [16] Wu, L.; Kwok, Y. K., A front-fixing finite difference method for the valuation of American options, J. Financial Eng., 6, 2, 83-97 (1997) [17] Zhu, Y. L.; Wu, X.; Chern, I. L., Derivatives Securities and Difference Methods (2004), Springer: Springer New York · Zbl 1061.91036
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