## Pricing multi-asset option problems: a Chebyshev pseudo-spectral method.(English)Zbl 1419.91654

Summary: The aim of this paper is to contribute a new second-order pseudo-spectral method via a non-uniform distribution of the computational nodes for solving multi-asset option pricing problems. In such problems, the prices are required to be as accurately as possible around the strike price. Accordingly, the proposed modification of the Chebyshev-Gauss-Lobatto points would concentrate on this area. This adaptation is also fruitful for the non-smooth payoffs which cause discontinuities in the strike price. The proposed scheme competes well with the existing methods such as finite difference, meshfree, and adaptive finite difference methods on several benchmark problems.

### MSC:

 91G60 Numerical methods (including Monte Carlo methods) 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 91G20 Derivative securities (option pricing, hedging, etc.)

### Software:

BENCHOP; Mathematica; Eigtool
Full Text:

### References:

 [1] Abell, M.L., Braselton, J.P.: Mathematica by Example, 5th edn. Academic Press, Dordrecht (2017) · Zbl 1366.68001 [2] Borovkova, S.; Permana, F.; Weide, JV, American basket and spread option pricing by a simple binomial tree, J. Deriv., 19, 29-38, (2012) [3] Butcher, J.C.: Numerical Methods for Ordinary Differential Equations, 2nd edn. Wiley, England (2008) · Zbl 1167.65041 [4] Company, R.; Egorova, VN; Jódar, L.; Soleymani, F., A local radial basis function method for high-dimensional American option pricing problems, Math. Model. Anal., 23, 117-138, (2018) [5] Dang, DM; Christara, CC; Jackson, KR, An efficient graphics processing unit-based parallel algorithm for pricing multi-asset American options, Concurr. Computat. Pract. Exp., 24, 849-866, (2012) [6] d’Halluin, Y.; Forsyth, PA; Labahn, G., A penalty method for American options with jump diffusion processes, Numer. Math., 97, 321-352, (2004) · Zbl 1126.91036 [7] Ehrhardt, M., Günther, M., ter Maten, E.J.W.: Novel Methods in Computational Finance. Springer, Switzerland (2017) · Zbl 1390.91011 [8] Fornberg, B.: A Practical Guide to Pseudospectral Methods. Cambridge University Press, Cambridge (1996) · Zbl 0844.65084 [9] Gad, KST; Pedersen, JL, Rationality parameter for exercising American put, Risks, 3, 103-111, (2015) [10] Geske, R.; Shastri, K., Valuation by approximation: a comparison of alternative approaches, J. Func. Quant. Anal., 20, 45-71, (1985) [11] Giribone, PG; Ligato, S., Option pricing via radial basis functions: performance comparison with traditional numerical integration scheme and parameters choice for a reliable pricing, Int. J. Financ. Eng., 2, 1550018, (2015) [12] Glasserman, P.: Monte Carlo Methods in Financial Engineering. Springer, New York (2003) · Zbl 1038.91045 [13] Gzyl, H.; Milev, M.; Tagliani, A., Discontinuous payoff option pricing by Mellin transform: a probabilistic approach, Financ. Res. Lett., 20, 281-288, (2017) [14] Hout, KJI; Foulon, S., ADI finite difference schemes for option pricing in the Heston model with correlation, Int. J. Numer. Anal. Model., 7, 303-320, (2010) [15] Ikonen, S.; Toivanen, J., Operator splitting methods for American option pricing, Appl. Math. Lett., 17, 809-814, (2004) · Zbl 1063.65081 [16] Janson, S.; Tysk, J., Feynman-Kac formulas for Black-Scholes-type operators, Bull. Lond. Math. Soc., 38, 269-282, (2006) · Zbl 1110.35021 [17] Kangro, R.; Nicolaides, R., Far field boundary conditions for Black-Scholes equations, SIAM J. Numer. Anal., 38, 1357-1368, (2000) · Zbl 0990.35013 [18] Khaliq, AQM; Voss, DA; Kazmi, K., Adaptive $$\theta$$-methods for pricing American options, J. Comput. Appl. Math., 222, 210-227, (2008) · Zbl 1151.91521 [19] Knapp, R., A method of lines framework in Mathematica, J. Numer. Anal. Indust. Appl. Math. (JNAIAM), 3, 43-59, (2008) · Zbl 1154.65070 [20] Kovalov, P.; Linetsky, V.; Marcozzi, M., Pricing multi-asset American options: a finite element method-of-lines with smooth penalty, J. Sci. Comput., 33, 209-237, (2007) · Zbl 1210.91133 [21] Kressner, D.; Tobler, C., Krylov subspace methods for linear systems with tensor product structure, SIAM J. Matrix Anal. Appl., 31, 1688-1714, (2010) · Zbl 1208.65044 [22] Leentvaar, C.C.W.: Pricing multi-asset options with sparse grids. PhD Thesis, TU Delft, The Netherlands (2008) [23] Lötstedt, P.; Persson, J.; Sydow, L.; Tysk, J., Space-time adaptive finite difference method for European multi-asset options, Comput. Math. Appl., 53, 1159-1180, (2007) · Zbl 1154.91462 [24] Martín-Vaquero, J.; Khaliq, AQM; Kleefeld, B., Stabilized explicit Runge-Kutta methods for multi-asset American options, Comput. Math. Appl., 67, 1293-1308, (2014) · Zbl 1386.91165 [25] Milovanović, S.; Sydow, L., Radial basis function generated finite differences for option pricing problems, Comput. Math. Appl., 75, 1462-1481, (2017) [26] Milovanović, S., Shcherbakov, V.: Pricing derivatives under multiple stochastic factors by localized radial basis function methods (2018). arXiv:1711.09852 [27] Nielsen, BF; Skavhaug, O.; Tveito, A., Penalty methods for the numerical solution of American multi-asset option problems, J. Comput. Appl. Math., 222, 3-16, (2008) · Zbl 1152.91542 [28] Ramage, A.; Sydow, L., A multigrid preconditioner for an adaptive Black-Scholes solver, BIT, 51, 217-233, (2011) · Zbl 1211.91251 [29] Reddy, SC; Trefethen, LN, Stability of the method of lines, Numer. Math., 62, 235-267, (1992) · Zbl 0734.65077 [30] Shcherbakov, V., Radial basis function partition of unity operator splitting method for pricing multi-asset American options, BIT, 56, 1401-1423, (2016) · Zbl 1354.91169 [31] Shcherbakov, V.; Larsson, E., Radial basis function partition of unity methods for pricing vanilla basket options, Comput. Math. Appl., 71, 185-200, (2016) [32] Sofroniou, M., Knapp, R.: Advanced Numerical Differential Equation Solving in Mathematica. Wolfram Mathematica, Tutorial Collection, USA (2008) [33] Tavella, D., Randall, C.: Pricing Financial Instruments: The Finite Difference Method. Wiley, New York (2007) [34] Traub, J.F.: Iterative Methods for Solution of Equation. Prentice-Hall, Englewood Cliffs (1964) · Zbl 0121.11204 [35] Trefethen, L.N., Embree, M.: Spectra and Pseudospectra. The Behavior of Nonnormal Matrices and Operators. Princeton Unversity Press, Princeton and Oxford (2005) · Zbl 1085.15009 [36] Dorsselaer, JLM; Kraaijevanger, JFBM; Spijker, MN, Linear stability analysis in the numerical solution of initial value problems, Acta Numerica, 2, 99-237, (1993) · Zbl 0796.65091 [37] Sydow, L.; Höök, LJ; Larsson, E.; Lindström, E.; Milovanović, S.; Persson, J.; Shcherbakov, V.; Shpolyanskiy, Y.; Sirén, S.; Toivanen, J.; Waldén, J.; Wiktorsson, M.; Levesley, J.; Li, J.; Oosterlee, CW; Ruijter, MJ; Toropov, A.; Zhao, Y., BENCHOP—the BENCHmarking project in option pricing, Int. J. Comput. Math., 92, 2361-2379, (2015) · Zbl 1335.91113 [38] Yousuf, M.; Khaliq, AQM; Liu, R., Pricing American options under multi-state regime switching with an efficient $$L$$-stable method, Int. J. Comput. Math., 92, 2530-2550, (2015) · Zbl 1386.91168
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.