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Chebyshev interpolation for parametric option pricing. (English) Zbl 1402.91782
Summary: Recurrent tasks such as pricing, calibration and risk assessment need to be executed accurately and in real time. We concentrate on parametric option pricing (POP) as a generic instance of parametric conditional expectations and show that polynomial interpolation in the parameter space promises to considerably reduce run-times while maintaining accuracy. The attractive properties of Chebyshev interpolation and its tensorized extension enable us to identify broadly applicable criteria for (sub)exponential convergence and explicit error bounds. The method is most promising when the computation of the prices is most challenging. We therefore investigate its combination with Monte Carlo simulation and analyze the effect of (stochastic) approximations of the interpolation. For a wide and important range of problems, the Chebyshev method turns out to be more efficient than parametric multilevel Monte Carlo. We conclude with a numerical efficiency study.

91G20 Derivative securities (option pricing, hedging, etc.)
91G60 Numerical methods (including Monte Carlo methods)
41A10 Approximation by polynomials
Full Text: DOI
[1] Black, F.; Scholes, M., The pricing of options and corporate liabilities, J. Polit. Econ., 81, 637-654, (1973) · Zbl 1092.91524
[2] Boyarchenko, S.I., Levendorskiĭ, S.Z.: Non-Gaussian Merton-Black-Scholes Theory. World Scientific, Singapore (2002) · Zbl 0997.91031
[3] Brennan, M.J.; Schwartz, E.S., The valuation of American put options, J. Finance, 2, 449-462, (1977)
[4] Burkovska, O.; Haasdonk, B.; Salomon, J.; Wohlmuth, B., Reduced basis methods for pricing options with the Black-Scholes and Heston models, SIAM J. Financ. Math., 6, 685-712, (2015) · Zbl 1335.91099
[5] Canuto, C.; Quarteroni, A., Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comput., 38, 67-86, (1982) · Zbl 0567.41008
[6] Carr, P.; Madan, D.B., Option valuation and the fast Fourier transform, J. Comput. Finance, 2, 61-73, (1999)
[7] Cont, R.; Lantos, N.; Pironneau, O., A reduced basis for option pricing, SIAM J. Financ. Math., 2, 287-316, (2011) · Zbl 1227.91033
[8] Cuchiero, C.; Keller-Ressel, M.; Teichmann, J., Polynomial processes and their applications to mathematical finance, Finance Stoch., 4, 711-740, (2012) · Zbl 1270.60079
[9] Davis, P.J.: Interpolation and Approximation. (1975). Courier Corporation · Zbl 0329.41010
[10] Duffie, D.; Filipović, D.; Schachermayer, W., Affine processes and applications in finance, Ann. Appl. Probab., 13, 984-1053, (2003) · Zbl 1048.60059
[11] Eberlein, E.; Glau, K.; Papapantoleon, A., Analysis of Fourier transform valuation formulas and applications, Appl. Math. Finance, 17, 211-240, (2010) · Zbl 1233.91267
[12] Eberlein, E.; Keller, U.; Prause, K., New insights into smile, mispricing and value at risk: the hyperbolic model, J. Bus., 71, 371-405, (1998)
[13] Eberlein, E.; Özkan, F., The Lévy LIBOR model, Finance Stoch., 9, 327-348, (2005) · Zbl 1088.60074
[14] Feng, L.; Linetsky, V., Pricing discretely monitored barrier options and defaultable bonds in Lévy process models: a fast Hilbert transform approach, Math. Finance, 18, 337-384, (2008) · Zbl 1141.91438
[15] Filipović, D.; Larsson, M.; Trolle, A., Linear-rational term structure models, J. Finance, 72, 655-704, (2017)
[16] Gaß, M.: PIDE Methods and Concepts for Parametric Option Pricing. PhD thesis, Technical University of Munich (2016). Available online at https://mediatum.ub.tum.de/604993?query=pide+methods&show_id=1311705 · Zbl 1257.91043
[17] Gaß, M.; Glau, K.; Mair, M., Magic points in finance: empirical interpolation for parametric option pricing, SIAM J. Financ. Math., 8, 766-803, (2017) · Zbl 1429.91320
[18] Giles, M.B., Multilevel Monte Carlo methods, Acta Numer., 24, 259-328, (2015) · Zbl 1316.65010
[19] Glau, K., Hashemi, B., Mahlstedt, M., Pötz, C.: Spread option in 2D Black-Scholes. Example for Chebfun3 in the Chebfun toolbox. Available online at http://www.chebfun.org/examples/applics/BlackScholes2D.html
[20] Glau, K., Mahlstedt, M.: Improved error bound for multivariate Chebyshev polynomial interpolation. Preprint (2016). Available online at https://arxiv.org/abs/1611.08706 · Zbl 1199.65004
[21] Haasdonk, B.; Salomon, J.; Wohlmuth, B.; Cangiani, A. (ed.); etal., A reduced basis method for the simulation of American options, 9th European Conference on Numerical Mathematics and Advanced Applications, Leicester, Berlin · Zbl 1269.91086
[22] Heinrich, S., Monte Carlo complexity of global solution of integral equations, J. Complex., 14, 151-175, (1998) · Zbl 0920.65090
[23] Heinrich, S.; Margenov, S. (ed.); etal., Multilevel Monte Carlo methods, LSSC 2001, Sozopol, Bulgaria, June 6-10, 2001, Berlin · Zbl 1031.65005
[24] Heinrich, S.; Sindambiwe, E., Monte Carlo complexity of parametric integration, J. Complex., 15, 317-341, (1999) · Zbl 0958.68068
[25] Heston, S.L., A closed-form solution for options with stochastic volatility with applications to bond and currency options, Rev. Financ. Stud., 6, 327-343, (1993) · Zbl 1384.35131
[26] Keller-Ressel, M.; Papapantoleon, A.; Teichmann, J., The affine LIBOR models, Math. Finance, 23, 627-658, (2013) · Zbl 1275.91140
[27] Kolda, T.G.; Bader, B.W., Tensor decompositions and applications, SIAM Rev., 51, 455-500, (2009) · Zbl 1173.65029
[28] Kudryavtsev, O.; Levendorskiĭ, S.Z., Fast and accurate pricing of barrier options under Lévy processes, Finance Stoch., 13, 531-562, (2009) · Zbl 1194.91179
[29] L’Ecuyer, P., Quasi-Monte Carlo methods with applications in finance, Finance Stoch., 13, 307-349, (2009) · Zbl 1199.65004
[30] Lee, R.W., Option pricing by transform methods: extensions, unification, and error control, J. Comput. Finance, 7, 51-86, (2004)
[31] Lord, R.; Fang, F.; Bervoets, F.; Oosterlee, C.W., A fast and accurate FFT-based method for pricing early-exercise options under Lévy processes, SIAM J. Sci. Comput., 30, 1678-1705, (2008) · Zbl 1170.91389
[32] Mason, J.C., Handscomb, D.C.: Chebyshev Polynomials. CRC Press Taylor & Francis Group, Boca Raton (2003) · Zbl 1015.33001
[33] Merton, R.C., Theory of rational option pricing, Bell J. Econ. Manag. Sci., 4, 141-183, (1973) · Zbl 1257.91043
[34] Metropolis, N., The beginning of the Monte Carlo method, Los Alamos Sci., 15, 125-130, (1987)
[35] Pachon, R.: Numerical pricing of European options with arbitrary payoffs. Preprint (2016). Available online at SSRN: http://ssrn.com/abstract=2712402 · Zbl 1088.60074
[36] Pironneau, O., Reduced basis for vanilla and basket options, Risk Decis. Anal., 2, 185-194, (2011) · Zbl 1409.91280
[37] Pistorius, M.; Stolte, J., Fast computation of vanilla prices in time-changed models and implied volatilities, Int. J. Theor. Appl. Finance, 15, (2012) · Zbl 1246.91151
[38] Platte, R.B.; Trefethen, N.L.; Fitt, A. (ed.); etal., Chebfun: a new kind of numerical computing, No. 15, 69-86, (2008), Berlin · Zbl 1220.65100
[39] Raible, S.: Lévy Processes in Finance: Theory, Numerics, and Empirical Facts. Ph.D. thesis, Universität Freiburg (2000). Available online at https://freidok.uni-freiburg.de/data/51
[40] Runge, C.: Über empirische Funktionen und die Interpolation zwischen äquidistanten Ordinaten. Z. Angew. Math. Phys. 46, 224-243 (1901) · JFM 32.0272.02
[41] Sachs, E.W.; Schu, M., Reduced order models in PIDE constrained optimization, Control Cybern., 39, 661-675, (2010) · Zbl 1282.49022
[42] Sauter, S., Schwab, C.: Boundary Element Methods. Springer Series in Computational Mathematics, vol. 39. Springer, Berlin (2011) · Zbl 1215.65183
[43] Stein, E.M., Shakarchi, R.: Complex Analysis. Princeton University Press, Princeton (2003) · Zbl 1020.30001
[44] Tadmor, E., The exponential accuracy of Fourier and Chebyshev differencing methods, SIAM J. Numer. Anal., 23, 1-10, (1986) · Zbl 0613.65017
[45] Trefethen, L.N.: Talk: Six myths of polynomial interpolation and quadrature (2011). Available online at https://people.maths.ox.ac.uk/trefethen/mythstalk.pdf
[46] Trefethen, L.N.: Approximation Theory and Approximation Practice. SIAM Society for Industrial and Applied Mathematics (2013) · Zbl 1264.41001
[47] Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987) · Zbl 0623.35006
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