Previous |  Up |  Next


option pricing; discontinuous Galerkin method; path-dependent option; basket option; floating strike
Option pricing models are an important part of financial markets worldwide. The PDE formulation of these models leads to analytical solutions only under very strong simplifications. For more general models the option price needs to be evaluated by numerical techniques. First, based on an ideal pure diffusion process for two risky asset prices with an additional path-dependent variable for continuous arithmetic average, we present a general form of PDE for pricing of Asian option contracts on two assets. Further, we focus only on one subclass---Asian options with floating strike---and introduce the concept of the dimensionality reduction with respect to the payoff leading to PDE with two spatial variables. Then the numerical option pricing scheme arising from the discontinuous Galerkin method is developed and some theoretical results are also mentioned. Finally, the aforementioned model is supplemented with numerical results on real market data.
[1] Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing. Frontiers in Applied Mathematics 30, Society for Industrial and Applied Mathematics, Philadelphia (2005). DOI 10.1137/1.9780898717495 | MR 2159611 | Zbl 1078.91008
[2] Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973), 637-654. DOI 10.1086/260062 | MR 3363443 | Zbl 1092.91524
[3] Boyle, P., Broadie, M., Glasserman, P.: Monte Carlo methods for security pricing. J. Econ. Dyn. Control 21 (1997), 1267-1321. DOI 10.1016/S0165-1889(97)00028-6 | MR 1470283 | Zbl 0901.90007
[4] Cen, Z., Xu, A., Le, A.: A hybrid finite difference scheme for pricing Asian options. Appl. Math. Comput. 252 (2015), 229-239. DOI 10.1016/j.amc.2014.12.007 | MR 3305101 | Zbl 1338.91150
[5] Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4, North-Holland Publishing Company, Amsterdam (1978). DOI 10.1016/s0168-2024(08)x7014-6 | MR 0520174 | Zbl 0383.65058
[6] Cont, R., Tankov, P.: Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series, Chapman & Hall/CRC, Boca Raton (2004). DOI 10.1201/9780203485217 | MR 2042661 | Zbl 1052.91043
[7] Cox, J. C., Ross, S. A., Rubinstein, M.: Option pricing: A simplified approach. J. Financ. Econ. 7 (1979), 229-263. DOI 10.1016/0304-405X(79)90015-1 | Zbl 1131.91333
[8] Dolejší, V., Feistauer, M.: Error estimates of the discontinuous Galerkin method for nonlinear nonstationary convection-diffusion problems. Numer. Funct. Anal. Optimization 26 (2005), 349-383. DOI 10.1081/NFA-200067298 | MR 2153838 | Zbl 1078.65078
[9] Dolejší, V., Feistauer, M.: Discontinuous Galerkin Method. Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics 48, Springer, Cham (2015). DOI 10.1007/978-3-319-19267-3 | MR 3363720 | Zbl 06467550
[10] Dubois, F., Lelièvre, T.: Efficient pricing of Asian options by the PDE approach. J. Comput. Finance 8 (2005), 55-63. DOI 10.21314/JCF.2005.138
[11] Feistauer, M., Felcman, J., Straškraba, I.: Mathematical and Computational Methods for Compressible Flow. Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford (2003). MR 2261900 | Zbl 1028.76001
[12] Hale, J. K.: Ordinary Differential Equations. Pure and Applied Mathematics 21, Wiley-Interscience a division of John Wiley & Sons, New York (1969). MR 0419901 | Zbl 0186.40901
[13] Harrison, J. M., Kreps, D. M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20 (1979), 381-408. DOI 10.1016/0022-0531(79)90043-7 | MR 0540823 | Zbl 0431.90019
[14] Haug, E. G.: The Complete Guide to Option Pricing Formulas McGraw-Hill, New York. (2006).
[15] Hecht, F.: New development in freefem++. J. Numer. Math. 20 (2012), 251-265. DOI 10.1515/jnum-2012-0013 | MR 3043640 | Zbl 1266.68090
[16] Hozman, J.: Discontinuous Galerkin method for the numerical solution of option pricing. Aplimat -- J. Appl. Math. 5 (2012), 197-206.
[17] Hozman, J.: Analysis of the discontinuous Galerkin method applied to the European option pricing problem. AIP Conference Proceedings 1570 (2013), 227-234. DOI 10.1063/1.4854760
[18] Hozman, J., Tichý, T.: Black-Scholes option pricing model: Comparison of $h$-convergence of the DG method with respect to boundary condition treatment. ECON -- Journal of Economics, Management and Business 24 (2014), 141-152 \bgroup\spaceskip.28em plus.16em minus.12em.
[19] Hozman, J., Tichý, T., Cvejnová, D.: A discontinuous Galerkin method for two-dimensional PDE models of Asian options. AIP Conference Proceedings 1738 (2016), Article no. 080011 \99999DOI99999 10.1063/1.4951846 \egroup.
[20] J. E. Ingersoll, Jr.: Theory of Financial Decision Making. Rowman & Littlefield Publishers, New Jersey (1987).
[21] Kufner, A.: Weighted Sobolev Spaces. Teubner-Texte zur Mathematik 31, BSB B. G. Teubner Verlagsgesellschaft, Leipzig (1980). MR 0664599 | Zbl 0455.46034
[22] Kumar, A., Waikos, A., Chakrabarty, S. P.: Pricing of average strike Asian call option using numerical PDE methods. Int. J. Pure Appl. Math. 76 (2012), 709-725.
[23] Merton, R. C.: Theory of rational option pricing. Bell J. Econ. Manage. 4 (1973), 141-183. DOI 10.2307/3003143 | MR 0496534 | Zbl 1257.91043
[24] Reed, W. H., Hill, T. R.: Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory (1973).
[25] Rivière, B.: Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations. Theory and Implementation. Frontiers in Applied Mathematics 35, Society for Industrial and Applied Mathematics, Philadelphia (2008). DOI 10.1137/1.9780898717440 | MR 2431403 | Zbl 1153.65112
[26] Shreve, S., Večeř, J.: Options on a traded account: Vacation calls, vacation puts and passport options. Finance Stoch. 4 (2000), 255-274. DOI 10.1007/s007800050073 | MR 1779579 | Zbl 0997.91020
[27] Večeř, J.: Unified pricing of Asian options. Risk 15 (2002), 113-116.
[28] Zeidler, E.: Nonlinear Functional Analysis and Its Applications. II/A: Linear Monotone Operators. Springer, New York (1990). DOI 10.1007/978-1-4612-0985-0 | MR 1033497 | Zbl 0684.47028
Partner of
EuDML logo