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Option pricing with Legendre polynomials. (English) Zbl 1414.91412

Summary: Here we develop an option pricing method based on Legendre series expansion of the density function. The key insight, relying on the close relation of the characteristic function with the series coefficients, allows to recover the density function rapidly and accurately. Based on this representation for the density function, approximations formulas for pricing European type options are derived. To obtain highly accurate result for European call option, the implementation involves integrating high degree Legendre polynomials against exponential function. Some numerical instabilities arise because of serious subtractive cancellations in its formulation (96) in Proposition A.1. To overcome this difficulty, we rewrite this quantity as solution of a second-order linear difference equation and solve it using a robust and stable algorithm from Olver. Derivation of the pricing method has been accompanied by an error analysis. Errors bounds have been derived and the study relies more on smoothness properties which are not provided by the payoff functions, but rather by the density function of the underlying stochastic models. This is particularly relevant for options pricing where the payoffs of the contract are generally not smooth functions. The numerical experiments on a class of models widely used in quantitative finance show exponential convergence.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
65T50 Numerical methods for discrete and fast Fourier transforms
91G20 Derivative securities (option pricing, hedging, etc.)
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)

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[1] Karatzas, I.; Shreve, S., Brownian Motion and Stochastic Calculus (1997), Springer
[2] Cont, R.; Tankov, P., Financial Modelling with Jump Processes (2004), Chapman and Hall/CRC Press · Zbl 1052.91043
[3] Heston, S., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6, Article 327343 pp. (1993) · Zbl 1126.91368
[4] Bakshi, G.; Chen, Z., An alternative valuation model for contingent claims, J. Financ. Econ., 44, Article 123165 pp. (1997)
[5] Carr, P.; Madan, D., Option valuation using the fast fourier transform, J. Comput. Finance, 2, 6173 (1999)
[6] Andricopoulos, A.; Widdicks, M.; Duck, P.; Newton, P., Universal option valuation using quadrature methods, J. Financ. Econ., 67, Article 447471 pp. (2003)
[7] Lord, R.; Fang, F.; Bervoets, F., A fast and accurate fft-based method for pricing early-exercise options under levy processes, SIAM J. Sci. Comput., 30, 16781705 (2008) · Zbl 1170.91389
[8] Feng, L.; Linetsky, V., Pricing discretely monitored barrier options and defaultable bonds in levy process models: a fast hilbert transform approach, Math. Finance, 18, Article 337384 pp. (2008) · Zbl 1141.91438
[9] Fang, F.; Oosterlee, C., A novel pricing method for European options based on fourier- cosine series expansions, SIAM J. Sci. Comput., 31, 826-848 (2008) · Zbl 1186.91214
[10] Hurn, K. L.A.; McClelland, A., On the efficacy of fourier series approximations for pricing European options, Appl. Math., 5, 2786-2807 (2014)
[11] Ding, D.; U, S., Efficient option pricing methods based on fourier series expansions, J. Math. Res. Exposition, 31, 1222 (2011)
[13] Boyd, J. P., Chebyshev and Fourier Spectral Methods (2000), Dover Publications, Inc.
[14] Evans, G.; Webster, J., A comparison of some methods for the evaluation of highly oscillatory integrals, J. Comput. Appl. Math., 112, 55-69 (1999) · Zbl 0947.65148
[15] Fishback, P., Taylor series are limits of legendre expansions, Missouri J. Math. Sci., 19, 29-34 (2007) · Zbl 1143.42030
[16] Cohena, M.; Tan, C., A polynomial approximation for arbitrary functions, Appl. Math. Lett., 25, 19471952 (2012)
[17] Dicke, R.; Wittke, J., Introduction to Quantum Mechanics (1960), Addison-Wesley Publishing Company Inc.: Addison-Wesley Publishing Company Inc. Reading, Massachusetts · Zbl 0093.20701
[18] Hollas, J., Modern Spectroscopy (1992), John Wiley and Sons: John Wiley and Sons Chichester
[19] Jackson, J., Classical Electrodynamics (1962), John Wiley and Sons: John Wiley and Sons New York · Zbl 0114.42903
[21] Pulch, R.; Emmerich, C., Polynomial chaos for simulating random volatilities, J. Math. Comput. Simul., 80, 245-255 (2009) · Zbl 1180.91312
[22] Ibsen, C.; Almeida, R., Affine processes, arbitrage-free term structure of legendre polynomial, and option pricing, Int. J. Theor. Appl. Finance, 08, 161-184 (2005) · Zbl 1100.91050
[23] Almeida, R.; Duarte, A.; Fernandes, C., Decomposing and simulating the movements of term structures of interest rates in emerging eurobond markets, J. Fixed Income, 1, 21-31 (1998)
[25] Lebedev, N., Special Functions and their Applications (1972), Dover Publications: Dover Publications New York · Zbl 0271.33001
[26] Davis, P., Interpolation and Approximation (1975), Dover Publications, Inc.
[27] Tolstov, G., Fourier Series (1962), Prentice-Hall, Inc.: Prentice-Hall, Inc. Englewood Cliffs, NJ, translated by R. A. Silverman
[28] Olver, F.; Lozier, D.; Boisvert, R.; Clark, C., NIST Handbook of Mathematical Functions (2010), Cambridge University Press · Zbl 1198.00002
[29] Nualart, D., The Malliavin Calculus and Related Topics (2006), Springer-Verlag · Zbl 1099.60003
[31] Nachman, D., Spanning and completeness with options, Rev. Financ. Stud., 1, 311-328 (1998)
[33] Avellaneda, M.; Laurence, P., Quantitative Modeling of Derivative Securities: From Theory to Practice (1999), Chapman and Hall/CRC
[34] Selezneva, I. A.; Ratis, Y. L.; Hernandez, E.; Perez-Quiles, J.; de Cordoba, P. F., A code to calculate high order legendre polynomials and function, Rev. Acad. Colombiana Cienc., 37, 145, 541-544 (2013) · Zbl 1323.65017
[35] Klemm, A. D.; Larsen, S., Some integrals involving legendre polynomials providing combinatorial identities, J. Aust. Math. Soc. Ser. B, 32, 304-310 (1990) · Zbl 0726.33005
[36] Haug, E. G., The Complete Guide to Option Pricing Formulas (2007), McGraw-Hill: McGraw-Hill New York
[37] Olver, F., Numerical solution of second-order linear difference equations, J. Res. Natl. Bur. Stand., 71B (1967) · Zbl 0171.36601
[38] Cash, J., A note on the iterative solution of recurrence relations, Numer. Math., 27, 165-170 (1977) · Zbl 0341.65084
[39] Gautschi, W., Computational aspects of three-term recurrence relations, SIAM Rev., 9, 27 (1967) · Zbl 0168.15004
[40] Wand, H.; Xiang, S., On the convergence rates of legendre approximation, Math. Comput., 81, 278 (2012)
[42] Bender, C.; Orszag, S., Advanced Mathematical Methods for Scientists and Engineers (1978), McGraw-Hill: McGraw-Hill New York · Zbl 0417.34001
[43] Glasserman, P., Monte Carlo Methods in Financial Engineering (2003), Springer
[44] Kloeden, P.; Platen, E., Numerical Solution of Stochastic Differential Equations (1992), Springer: Springer Berlin · Zbl 0752.60043
[45] Black, F.; Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ., 81, 637-654 (1973) · Zbl 1092.91524
[46] Gatheral, J., The Volatility Surface: A Practitioner’s Guide (2006), Wiley Finance
[47] Wilmott, P., Paul Wilmott on Quantitative Finance (2006), Wiley · Zbl 1127.91002
[49] Clark, I., Foreign Exchange Option Pricing: A Practitioners Guide (2011), Wiley Finance
[51] Merton, R., Option pricing when underlying stock returns are discontinuous, J. Financ. Econ., 3, 125-144 (1976) · Zbl 1131.91344
[52] Andersen, L.; Andreasen, J., Jump-diffusion processes: Volatility smile fitting and numerical methods for option pricing, Rev. Deriv. Res., 4, Article 231262 pp. (1998)
[53] Kou, S. G., A jump-diffusion model for option pricing, J. Manag. Sci., 48, 1086-1101 (2002) · Zbl 1216.91039
[54] Lord, R.; Kahl, C., Complex logarithms in heston-like models, Math. Finance, 20, Article 671694 pp. (2010) · Zbl 1232.91728
[56] der Cruyssen, P. V., A reformulation of olver’s algorithm for the numerical solution of second-order linear difference equations, Numer. Math., 32, Article 327343 pp. (1979) · Zbl 0386.65051
[58] Bompis, R.; Hok, J., Forward implied volatility expansion in time-dependent local volatility models, ESAIM: Proc., 45 (2014) · Zbl 1401.91556
[59] Pellser, A., Efficient Methods for Valuing Interest Rate Derivatives (2000), Springer: Springer Heidelberg
[61] Detlefsen, K.; Hardle, W., Calibration risk for exotic options, J. Deriv., 14, 4, 4763 (2007)
[62] Gilli, M.; Schumann, E., Calibrating option pricing models with heuristics, Nat. Comput. Comput. Finance, 4, 9-37 (2011), Springer
[63] Taormina, R.; Chau, K., Data-driven input variable selection for rainfall-runoff modeling using binary-coded particle swarm optimization and extreme learning machines, J. Hydrol., 529, 1617-1632 (2015)
[64] Zhang, J.; Chau, K., Multilayer ensemble pruning via novel multi-sub-swarm particle swarm optimization, J. Hydrol., 15, 4, 840-858 (2009)
[65] Zhang, S.; Chau, K., Dimension reduction using semi-supervised locally linear embedding for plant leaf classification, Lecture Notes in Comput. Sci., 5754, 948-955 (2009)
[66] May, R.; Dandy, G.; Maier, H., Review of Input Variable Selection Methods for Artificial Neural Networks (2011), INTECH Open Access Publisher
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