##
**Compact finite difference method for American option pricing.**
*(English)*
Zbl 1151.91552

Summary: A compact finite difference method is designed to obtain quick and accurate solutions to partial differential equation problems. The problem of pricing an American option can be cast as a partial differential equation. Using the compact finite difference method this problem can be recast as an ordinary differential equation initial value problem. The complicating factor for American options is the existence of an optimal exercise boundary which is jointly determined with the value of the option. In this article we develop three ways of combining compact finite difference methods for American option price on a single asset with methods for dealing with this optimal exercise boundary. Compact finite difference method one uses the implicit condition that solutions of the transformed partial differential equation be nonnegative to detect the optimal exercise value. This method is very fast and accurate even when the spatial step size \(h\) is large \((h\geqslant 0.1)\). Compact difference method two must solve an algebraic nonlinear equation obtained by K. N. Pantazopoulos et al. [Comput. Econ. 12, No. 3, 255–273 (1998; Zbl 0913.90022)] at every time step. This method can obtain second order accuracy for space \(x\) and requires a moderate amount of time comparable with that required by the Crank Nicolson projected successive over relaxation method. Compact finite difference method three refines the free boundary value by a method developed by Barone-Adesi and Lugano [The saga of the American put (2003], and this method can obtain high accuracy for space \(x\). The last two of these three methods are convergent, moreover all the three methods work for both short term and long term options. Through comparison with existing popular methods by numerical experiments, our work shows that compact finite difference methods provide an exciting new tool for American option pricing.

### MSC:

91G60 | Numerical methods (including Monte Carlo methods) |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

91G20 | Derivative securities (option pricing, hedging, etc.) |

60G40 | Stopping times; optimal stopping problems; gambling theory |

### Keywords:

compact finite difference method; free boundary value; American option pricing; optimal exercise boundary; Black-Scholes equation### Citations:

Zbl 0913.90022
PDFBibTeX
XMLCite

\textit{J. Zhao} et al., J. Comput. Appl. Math. 206, No. 1, 306--321 (2007; Zbl 1151.91552)

Full Text:
DOI

### References:

[1] | M. Ahmed, An exploration of compact finite difference methods for the numerical solution of PDE, Ph.D. Thesis, the University of Western Ontario, 1997.; M. Ahmed, An exploration of compact finite difference methods for the numerical solution of PDE, Ph.D. Thesis, the University of Western Ontario, 1997. |

[2] | Barone-Adesi, G.; Elliott, R., Approximations for the values of American options, Stochast. Anal. Appl., 9, 2 (1991) · Zbl 0729.60056 |

[3] | G. Barone-Adesi, U. Lugano, The saga of the American put, 2003.; G. Barone-Adesi, U. Lugano, The saga of the American put, 2003. |

[4] | G. Barone-Adesi, R. Whaley, Efficient analytic approximation of American option values, J. Finance (1987) 301-320.; G. Barone-Adesi, R. Whaley, Efficient analytic approximation of American option values, J. Finance (1987) 301-320. |

[5] | Blum, E. K., Numerical Analysis and Computation: Theory and Practice (1972), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co. Reading, MA · Zbl 0273.65001 |

[6] | Boyle, P., A lattice framework for option pricing with two state variables, J. Financial Quant. Anal., 23, 1, 1-12 (1988) |

[7] | M.J. Brennan, E.S. Schwartz, Finite difference methods and jump processes arising in the pricing of contingent claims: a synthesis, J. Financial Quant. Anal. (1978) 461-474.; M.J. Brennan, E.S. Schwartz, Finite difference methods and jump processes arising in the pricing of contingent claims: a synthesis, J. Financial Quant. Anal. (1978) 461-474. |

[8] | P. Carr, D. Faguet, Fast accurate valuation of American options, Working Paper, Cornell University, 1994.; P. Carr, D. Faguet, Fast accurate valuation of American options, Working Paper, Cornell University, 1994. |

[9] | Carr, P.; Jarrow, R.; Myneni, R., Alternative characterizations of American put options, J. Math. Finance, 2, 87-106 (1992) · Zbl 0900.90004 |

[10] | R.M. Corless, J. Rokicki, J. Zhao, FINDIF: a routine for generation of finite difference formulae, share library package 1994, and upgraded to n dimensions for “iguana”, 2006.; R.M. Corless, J. Rokicki, J. Zhao, FINDIF: a routine for generation of finite difference formulae, share library package 1994, and upgraded to n dimensions for “iguana”, 2006. |

[11] | Courtadon, G., A more accurate finite difference approximations for the valuation of options, J. Financial Quant. Anal., 17, 5, 697-703 (1982) |

[12] | Cox, J. C.; Ross, S. A.; Rubinstein, M., Option pricing: a simplified approach, J. Financial Economics, 7, 229-263 (1979) · Zbl 1131.91333 |

[13] | Düring, B.; Fournié, M.; Jüngel, A., High order compact finite difference schemes for a nonlinear Black-Scholes equation, Int. J. Theor. Appl. Finance, 6, 7, 767-789 (2003) · Zbl 1070.91024 |

[14] | Düring, B.; Fournié, M.; Jüngel, A., Convergence of a high-order compact finite difference schemes for a nonlinear Black-Scholes equation, Math. Model. Numer. Anal., 38, 2, 359-369 (2004) · Zbl 1124.91031 |

[15] | Druskin, V.; Moshow, S., Three-point finite-difference schemes, Padé and the spectral Galerkin method. I. One-sided impedance approximation, Math. Comput., 71, 239, 995-1019 (2001) · Zbl 0997.65117 |

[16] | Gamet, L.; Ducros, F.; Nicoud, F., Compact finite difference schemes on non-uniform meshes, Application to direct numerical simulations of compressible flows, Int. J. Numer. Methods Fluids, 29, 159 (1999) · Zbl 0939.76060 |

[17] | Geske, R.; Johnson, H., The American put options valued analytically, J. Finance, 39, 1511-1524 (1984) |

[18] | Hirsh, R. S., Higher order accurate difference solutions of fluid mechanics problems by a compact differencing technique, J. Comput. Phys., 19, 90-109 (1975) · Zbl 0326.76024 |

[19] | R.S. Hirsh, Higher order approximations in fluid mechanics, Von Karman Institute for Fluid Dynamics Lecture Series, 1983-1984.; R.S. Hirsh, Higher order approximations in fluid mechanics, Von Karman Institute for Fluid Dynamics Lecture Series, 1983-1984. |

[20] | Jacka, S. D., Optimal stopping and the American put, Math. Finance, 1, 2, 1-14 (1991) · Zbl 0900.90109 |

[21] | Jarrow, R.; Rudd, A., Tests of an approximate option-valuation formula, (Brenner, M., Option Pricing Theory and Applications (1983), Lexington Books: Lexington Books Lexington, MA), 81-100 |

[22] | Jiang, Y.; Floryan, J. M., Finite-difference 4th-order compact scheme for the direct numerical simulation of instabilities of shear layers, Internat. J. Numer. Methods Fluids, 48, 1259-1281 (2005) · Zbl 1112.76421 |

[23] | Jin, Y.; Zhao, J.; Ma, L.; Corless, R. M., A new accurate algorithm for solving the partial differential equation in two dimensional cardiac tissue models, WSEAS Trans. Biol. Biomed., 3, 2, 63-68 (2006) |

[24] | John, C. H., Options, Futures, and Other Derivatives (1999), Prentice-Hall Inc.: Prentice-Hall Inc. Englewood Cliffs, NJ · Zbl 1087.91025 |

[25] | Kim, I. J., The analytic valuation of American options, Rev. Financial Stud., 3, 547-572 (1990) |

[26] | Lele, K. K., Compact finite-difference schemes with spectral-like resolution, J. Comput. Phys., 103, 16 (1992) · Zbl 0759.65006 |

[27] | Longstaff, F. A.; Schwartz, E. S., Valuing American options by simulation: a simple least-squares approach, Rev. Financial Stud., 14, 113-147 (2001) |

[28] | MacMillan, L. W., An analytical approximation for the American put prices, Adv. Futures Options Res., 1, 119-139 (1986) |

[29] | Meyer, G. H.; Van der Hoek, J., The valuation of American options with the method of lines, Adv. Futures Options Res., 9, 265-286 (1997) |

[30] | K.N. Pantazopoulos, Numerical methods and software for the pricing of American financial derivatives, Ph.D. Thesis, Computer Science Department, Purdue University, 1998.; K.N. Pantazopoulos, Numerical methods and software for the pricing of American financial derivatives, Ph.D. Thesis, Computer Science Department, Purdue University, 1998. |

[31] | M.F. Pettigrew, On compact finite difference schemes with applications to moving boundary problems, Ph.D. thesis, the University of Western Ontario, 1989.; M.F. Pettigrew, On compact finite difference schemes with applications to moving boundary problems, Ph.D. thesis, the University of Western Ontario, 1989. |

[32] | Rokicki, J.; Floryan, J. M., A compact finite-difference method for the Navier-Stokes equations—its implementation on parallel computers, (Proceedings of the XI Annual Conference on Fluid Mechanics. Proceedings of the XI Annual Conference on Fluid Mechanics, Warsaw, Poland, October 17-21 (1994)), 78-91 |

[33] | L.F. Shampine, M.W. Reichelt, The Matlab ODE suite, SIAM J. Sci. Comput., 2006, to appear.; L.F. Shampine, M.W. Reichelt, The Matlab ODE suite, SIAM J. Sci. Comput., 2006, to appear. · Zbl 0868.65040 |

[34] | Shampine, L. F.; Reichelt, M. W., The Matlab ODE suite, SISC, 18, 1 (1997) · Zbl 0868.65040 |

[35] | S.E. Sherer, J.N. Scott, High-order compact finite-difference methods on general overset grids, J. Comput. Phys., 2006, to appear.; S.E. Sherer, J.N. Scott, High-order compact finite-difference methods on general overset grids, J. Comput. Phys., 2006, to appear. · Zbl 1113.76068 |

[36] | Shukla, R. K.; Zhong, X. L., Derivation of high-order compact finite difference schemes for non-uniform grid using polynomial interpolation, J. Comput. Phys., 205, 404-429 (2004) · Zbl 1067.65088 |

[37] | R. Smith, Optimal and near-optimal advection-diffusion finite-difference schemes 3. Black-Scholes equation, Proc. Roy. Soc. Lond. A 1999.; R. Smith, Optimal and near-optimal advection-diffusion finite-difference schemes 3. Black-Scholes equation, Proc. Roy. Soc. Lond. A 1999. · Zbl 0937.65095 |

[38] | H. Sun, J. Zhang, A high order compact boundary value method for solving one dimensional heat equations, Technical Report No. 333-02, University of Kentucky, 2002.; H. Sun, J. Zhang, A high order compact boundary value method for solving one dimensional heat equations, Technical Report No. 333-02, University of Kentucky, 2002. |

[39] | Wilmott, P.; Dewynne, J.; Howison, S., Option Pricing: Mathematical Model and Computation (1995), Oxford Financial Press · Zbl 0844.90011 |

[40] | G.B. Wright, B. Fornberg, Scattered node compact finite difference-type formulas generated from radial basis functions, 2005, to appear.; G.B. Wright, B. Fornberg, Scattered node compact finite difference-type formulas generated from radial basis functions, 2005, to appear. · Zbl 1089.65020 |

[41] | Zhang, J., An explicit fourth-order compact finite difference scheme for three dimensional convection-diffusion equation, Commun. Numer. Methods Eng., 14, 263-280 (1998) · Zbl 0903.65080 |

[42] | Zhao, J.; Corless, R. M., Compact finite difference method for integro-differential equations, Appl. Math. Comput., 177, 271-288 (2006) · Zbl 1331.65180 |

[43] | J. Zhao, R.M. Corless, Compact finite difference method for high order integro-differential equations, Appl. Numer. Math., 2005, submitted for publication.; J. Zhao, R.M. Corless, Compact finite difference method for high order integro-differential equations, Appl. Numer. Math., 2005, submitted for publication. |

[44] | Zhao, J.; Corless, R. M.; Davison, M., Financial applications of symbolically generated compact finite difference formulae, (International Workshop on Symbolic-Numerical Computation Proceedings. International Workshop on Symbolic-Numerical Computation Proceedings, Xi’an, China (2005)), 220-234 |

[45] | Zhao, J.; Jin, Y.; Ma, L.; Corless, R. M., A highly efficient and accurate algorithm for solving the partial differential equation in cardiac tissue models, (Proceedings of the 2006 WSEAS International Conference on Mathematical Biology and Ecology. Proceedings of the 2006 WSEAS International Conference on Mathematical Biology and Ecology, Miami, FL, USA, January 18-20 (2006)), 81-86 |

[46] | J. Zhao, T. Zhang, R.M. Corless, Convergence of the compact finite difference method for 2nd order elliptic equations, Appl. Math. Comput., 2006, accepted.; J. Zhao, T. Zhang, R.M. Corless, Convergence of the compact finite difference method for 2nd order elliptic equations, Appl. Math. Comput., 2006, accepted. · Zbl 1111.65098 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.