Generalized exponential basis for efficient solving of homogeneous diffusion free boundary problems: Russian option pricing. (English) Zbl 1532.35543
Summary: This paper develops a method for solving free boundary problems for time-homogeneous diffusions. We combine the complete exponential system of solutions for the heat equation, transmutation operators and recently discovered Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations to construct a complete system of solutions for the considered partial differential equations. The conceptual algorithm for the application of the method is presented. The valuation of Russian options with finite horizon is used as a numerical illustration. The solution under different horizons is computed and compared to the results that appear in the literature.
MSC:
| 35R35 | Free boundary problems for PDEs |
| 35K05 | Heat equation |
| 45G10 | Other nonlinear integral equations |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
free boundary problems; transmutation operators; Neumann series of Bessel functions; Russian optionsSoftware:
Regularization toolsReferences:
| [1] | Alexidze, M., 1991. Fundamental functions in approximate solutions of boundary value problems (in Russian). Moscow: Nauka. · Zbl 0752.65079 |
| [2] | Black, F., Scholes, M., 1973. The pricing of options and corporate liabilities. Journal of Political Economy 81, 637-654. · Zbl 1092.91524 |
| [3] | Carr, P., 1998. Randomization and the American put. Review of Financial Studies 11, 597-626. · Zbl 1386.91134 |
| [4] | Colton, D., 1976. Solution of boundary value problems by the method of integral operators. Pitman London. · Zbl 0332.35001 |
| [5] | Colton, D., Reemtsen, R., 1984. The numerical solution of the inverse Stefan problem in two space variables. SIAM Journal on Applied Mathematics 44, 996-1013. · Zbl 0576.65125 |
| [6] | Colton, D., Watzlawek, W., 1977. Complete families of solutions to the heat equation and generalized heat equation in \(\mathbb{R}^n \). Journal of Differential Equations \(25, 96 - 107\). · Zbl 0336.35049 |
| [7] | Colton, D.L., 1980. Analytic theory of partial differential equations. Pitman. · Zbl 0468.35001 |
| [8] | Davis, P.J., Rabinowitz, P., 1984. Methods of numerical integration. Second edition. Academic Press, San Diego, California. · Zbl 0537.65020 |
| [9] | Doicu, A., Eremin, Y.A., Wriedt, T., 2000. Acoustic and electromagnetic scattering analysis using discrete sources. · Zbl 0948.78007 |
| [10] | Duffie, J.D., Harrison, J.M., et al., 1993. Arbitrage pricing of Russian options and perpetual lookback options. The Annals of Applied Probability 3, 641-651. · Zbl 0783.90009 |
| [11] | Duistermaat, J., Kyprianou, A.E., van Schaik, K., 2005. Finite expiry Russian options. Stochastic Processes and their Applications 115, 609-638. · Zbl 1151.91505 |
| [12] | Ekström, E., 2004. Russian options with a finite time horizon. Journal of Applied Probability 41, 313-326. · Zbl 1062.60040 |
| [13] | Fairweather, G., Karageorghis, A., 1998. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics 9, 69. · Zbl 0922.65074 |
| [14] | Hansen, P.C., 1994. Regularization tools: A MATLAB package for analysis and solution of discrete ill-posed problems. Numerical Algorithms 6, 1-35. · Zbl 0789.65029 |
| [15] | Herrera-Gomez, A., Porter, R.M., 2017. Mixed linear-nonlinear least squares regression. arXiv preprint arXiv:1703.04181. |
| [16] | Jeon, J., Han, H., Kim, H., Kang, M., 2016. An integral equation representation approach for valuing Russian options with a finite time horizon. Communications in Nonlinear Science and Numerical Simulation 36, 496-516. · Zbl 1470.91280 |
| [17] | Kamenov, A.A., 2008. Bachelier-version of Russian option with a finite time horizon. Teoriya Veroyatnostei i ee Primeneniya 53, 576-587. · Zbl 1202.91320 |
| [18] | Kamenov, A.A., 2014. Non-additive problems about optimal stopping for stationary diffusions (in Russian). Ph.D. thesis. Lomonosov Moscow State University. Moscow. |
| [19] | Khmelnytskaya, K.V., Kravchenko, V.V., Torba, S.M., Tremblay, S., 2013. Wave polynomials, transmutations and Cauchy’s problem for the Klein-Gordon equation. Journal of Mathematical Analysis and Applications 399, 191-212. · Zbl 1256.35009 |
| [20] | Kimura, T., 2008. Valuing finite-lived Russian options. European Journal of Operational Research 189, 363-374. · Zbl 1148.90333 |
| [21] | Kravchenko, I.V., Kravchenko, V.V., Torba, S.M., 2019. Solution of parabolic free boundary problems using transmuted heat polynomials. Mathematical Methods in the Applied Sciences 42, 5094-5105. · Zbl 1423.35466 |
| [22] | Kravchenko, V.V., Morelos, S., Torba, S.M., 2016. Liouville transformation, analytic approximation of transmutation operators and solution of spectral problems. Applied Mathematics and Computation 273, 321-336. · Zbl 1410.34071 |
| [23] | Kravchenko, V.V., Navarro, L.J., Torba, S.M., 2017a. Representation of solutions to the one-dimensional Schrödinger equation in terms of Neumann series of Bessel functions. Applied Mathematics and Computation 314, 173-192. · Zbl 1426.34025 |
| [24] | Kravchenko, V.V., Otero, J.A., Torba, S.M., 2017b. Analytic approximation of solutions of parabolic partial differential equations with variable coefficients. Advances in Mathematical Physics 2017. · Zbl 1453.35108 |
| [25] | Kravchenko, V.V., Porter, R.M., 2010. Spectral parameter power series for Sturm-Liouville problems. Mathematical Methods in the Applied Sciences 33, 459-468. · Zbl 1202.34060 |
| [26] | Kravchenko, V.V., Torba, S.M., 2018. A Neumann series of Bessel functions representation for solutions of Sturm-Liouville equations. Calcolo 55, 11. · Zbl 1391.34060 |
| [27] | Kupradze, V.D., 1967. On the approximate solution of problems in mathematical physics. Russian Mathematical Surveys 22, 58-108. |
| [28] | Kuznetsov, A., 2013. On the convergence of the Gaver-Stehfest algorithm. SIAM Journal on Numerical Analysis 51, 2984-2998. · Zbl 1461.65258 |
| [29] | Lawson, C.L., Hanson, R.J., 1995. Solving least squares problems. volume 15 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Revised reprint of the 1974 original. · Zbl 0860.65029 |
| [30] | Madsen, K., Nielsen, H., 2010. Introduction to optimization and data fitting. Technical University of Denmark. |
| [31] | Merton, R.C., 1973. Theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141-183. · Zbl 1257.91043 |
| [32] | Nocedal, J., Wright, S.J., 2006. Numerical optimization. Springer, New York. · Zbl 1104.65059 |
| [33] | Peskir, G., 2005. The Russian option: finite horizon. Finance and Stochastics 9, 251-267. · Zbl 1092.91029 |
| [34] | Peskir, G., Shiryaev, A., 2006. Optimal stopping and free-boundary problems. Birkhäuser Verlag. · Zbl 1115.60001 |
| [35] | Polyanin, A.D., 2001. Handbook of linear partial differential equations for engineers and scientists. CRC Press. · Zbl 1027.35001 |
| [36] | Reemtsen, R., Lozano, C.J., 1982. An approximation technique for the numerical solution of a Stefan problem. Numerische Mathematik 38, 141-154. · Zbl 0463.65079 |
| [37] | Rose, M.E., 1960. A method for calculating solutions of parabolic equations with a free boundary. Mathematics of Computation, 249-256. · Zbl 0096.10102 |
| [38] | Rosenbloom, P., Widder, D., 1959. Expansions in terms of heat polynomials and associated functions. Transactions of the American Mathematical Society 92, 220-266. · Zbl 0086.27203 |
| [39] | Sarsengeldin, M., Arynov, A., Zhetibayeva, A., Guvercin, S., 2014. Analytical solutions of heat equation by heat polynomials. Bulletin of National Academy of Sciences of the Republic of Kazakhstan 5, 21-27. |
| [40] | Shepp, L., Shiryaev, A.N., 1993. The Russian option: reduced regret. The Annals of Applied Probability, 631-640. · Zbl 0783.90011 |
| [41] | Shepp, L.A., Shiryaev, A.N., 1995. A new look at pricing of the Russian option. Theory of Probability and Its Applications 39, 103-119. · Zbl 0834.60072 |
| [42] | Szegö, G., 1975. Orthogonal polynomials, 4th ed. American Mathematical Society. · Zbl 0305.42011 |
| [43] | Widder, D.V., 1962. Analytic solutions of the heat equation. Duke Math. J. 29, 497-503. · Zbl 0109.04604 |
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