| Title:
|
DG method for pricing European options under Merton jump-diffusion model (English) |
| Author:
|
Hozman, Jiří |
| Author:
|
Tichý, Tomáš |
| Author:
|
Vlasák, Miloslav |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
64 |
| Issue:
|
5 |
| Year:
|
2019 |
| Pages:
|
501-530 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Under real market conditions, there exist many cases when it is inevitable to adopt numerical approximations of option prices due to non-existence of analytical formulae. Obviously, any numerical technique should be tested for the cases when the analytical solution is well known. The paper is devoted to the discontinuous Galerkin method applied to European option pricing under the Merton jump-diffusion model, when the evolution of the asset prices is driven by a Lévy process with finite activity. The valuation of options under such a model with lognormally distributed jumps requires solving a parabolic partial integro-differential equation which involves both the integrals and the derivatives of the unknown pricing function. The integral term related to jumps leads to new theoretical and numerical issues regarding the solving of the pricing equation in comparison with the standard approach for the Black-Scholes equation. Here we adopt the idea of the relatively modern technique that the integral terms in Merton-type models can be viewed as solutions of proper differential equations, which can be accurately solved in a simple way. For practical purposes of numerical pricing of options in such models we propose a two-stage implicit-explicit scheme arising from the discontinuous piecewise polynomial approximation, i.e., the discontinuous Galerkin method. This solution procedure is accompanied with theoretical results and discussed within the numerical results on reference benchmarks.\looseness 1 (English) |
| Keyword:
|
option pricing |
| Keyword:
|
Merton jump-diffusion model |
| Keyword:
|
integro-differential equation |
| Keyword:
|
discontinuous Galerkin method |
| Keyword:
|
semi-implicit discretization |
| Keyword:
|
a priori error estimates |
| MSC:
|
35Q91 |
| MSC:
|
65M15 |
| MSC:
|
65M60 |
| MSC:
|
91G60 |
| MSC:
|
91G80 |
| idZBL:
|
07144726 |
| idMR:
|
MR4022161 |
| DOI:
|
10.21136/AM.2019.0305-18 |
| . |
| Date available:
|
2019-10-16T10:58:46Z |
| Last updated:
|
2021-11-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147848 |
| . |
| Reference:
|
[1] Achdou, Y., Pironneau, O.: Computational Methods for Option Pricing.Frontiers in Applied Mathematics 30, Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2005). Zbl 1078.91008, MR 2159611, 10.1137/1.9780898717495 |
| Reference:
|
[2] Almendral, A., Oosterlee, C. W.: Numerical valuation of options with jumps in the underlying.Appl. Numer. Math. 53 (2005), 1-18. Zbl 1117.91028, MR 2122957, 10.1016/j.apnum.2004.08.037 |
| Reference:
|
[3] Andersen, L., Andreasen, J.: Jump-diffusion processes: volatility smile fitting and numerical methods for option pricing.Rev. Deriv. Res. 4 (2000), 231-262. Zbl 1274.91398, 10.1023/A:1011354913068 |
| Reference:
|
[4] Bensoussan, A., Lions, J.-L.: Impulse Control and Quasi-Variational Inequalities.\hbox{Gauthier}-Villars, Paris (1984). MR 0756234 |
| Reference:
|
[5] Black, F., Scholes, M.: The pricing of options and corporate liabilities.J. Polit. Econ. 81 (1973), 637-654. Zbl 1092.91524, MR 3363443, 10.1086/260062 |
| Reference:
|
[6] Boyle, P., Broadie, M., Glasserman, P.: Monte Carlo methods for security pricing.J. Econ. Dyn. Control 21 (1997), 1267-1321. Zbl 0901.90007, MR 1470283, 10.1016/S0165-1889(97)00028-6 |
| Reference:
|
[7] Carr, P., Mayo, A.: On the numerical evaluation of option prices in jump diffusion processes.Eur. J. Finance 13 (2007), 353-372. MR 2488026, 10.1080/13518470701201512 |
| Reference:
|
[8] Cont, R., Tankov, P.: Financial Modelling with Jump Processes.Chapman & Hall/CRC Financial Mathematics Series, Chapman and Hall/CRC, Boca Raton (2004). Zbl 1052.91043, MR 2042661, 10.1201/9780203485217 |
| Reference:
|
[9] Cont, R., Voltchkova, E.: A finite difference scheme for option pricing in jump diffusion and exponential Lévy models.SIAM J. Numer. Anal. 43 (2005), 1596-1626. Zbl 1101.47059, MR 2182141, 10.1137/S0036142903436186 |
| Reference:
|
[10] Cox, J. C., Ross, S. A., Rubinstein, M.: Option pricing: a simplified approach.J. Financ. Econ. 7 (1979), 229-263. Zbl 1131.91333, MR 3821653, 10.1016/0304-405X(79)90015-1 |
| Reference:
|
[11] d'Halluin, Y., Forsyth, P. A., Vetzal, K. R.: Robust numerical methods for contingent claims under jump diffusion processes.IMA J. Numer. Anal. 25 (2005), 87-112. Zbl 1134.91405, MR 2110236, 10.1093/imanum/drh011 |
| Reference:
|
[12] Dolejší, V., Feistauer, M.: Discontinuous Galerkin Method. Analysis and Applications to Compressible Flow.Springer Series in Computational Mathematics 48, Springer, Cham (2015). Zbl 06467550, MR 3363720, 10.1007/978-3-319-19267-3 |
| Reference:
|
[13] Dolejší, V., Vlasák, M.: Analysis of a BDF-DGFE scheme for nonlinear convection-diffusion problems.Numer. Math. 110 (2008), 405-447. Zbl 1158.65068, MR 2447245, 10.1007/s00211-008-0178-2 |
| Reference:
|
[14] Feistauer, M., Švadlenka, K.: Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems.J. Numer. Math. 12 (2004), 97-117. Zbl 1059.65083, MR 2062581, 10.1515/156939504323074504 |
| Reference:
|
[15] Feng, L., Linetsky, V.: Pricing options in jump-diffusion models: an extrapolation approach.Oper. Res. 56 (2008), 304-325. Zbl 1167.91367, MR 2410308, 10.1287/opre.1070.0419 |
| Reference:
|
[16] Haug, E. G.: The Complete Guide to Option Pricing Formulas.McGraw-Hill, New York (2006). |
| Reference:
|
[17] Hecht, F.: New development in freefem++.J. Numer. Math. 20 (2012), 251-265. Zbl 1266.68090, MR 3043640, 10.1515/jnum-2012-0013 |
| Reference:
|
[18] Hozman, J.: Analysis of the discontinuous Galerkin method applied to the European option pricing problem.AIP Conf. Proc. 1570 (2013), 227-234. 10.1063/1.4854760 |
| Reference:
|
[19] Hozman, J., Tichý, T.: On the impact of various formulations of the boundary condition within numerical option valuation by DG method.Filomat 30 (2016), 4253-4263. Zbl 06750048, MR 3601917, 10.2298/FIL1615253H |
| Reference:
|
[20] Hozman, J., Tichý, T.: DG method for numerical pricing of multi-asset Asian options---the case of options with floating strike.Appl. Math., Praha 62 (2017), 171-195. Zbl 06738487, MR 3649516, 10.21136/AM.2017.0273-16 |
| Reference:
|
[21] Hozman, J., Tichý, T.: DG method for the numerical pricing of two-asset European-style Asian options with fixed strike.Appl. Math., Praha 62 (2017), 607-632. Zbl 06861548, MR 3745743, 10.21136/AM.2017.0176-17 |
| Reference:
|
[22] Hozman, J., Tichý, T.: DG framework for pricing European options under one-factor stochastic volatility models.J. Comput. Appl. Math. 344 (2018), 585-600. Zbl 1394.65099, MR 3825537, 10.1016/j.cam.2018.05.064 |
| Reference:
|
[23] Itkin, A.: Pricing Derivatives Under Lévy Models. Modern Finite-Difference and Pseudo-Differential Operators Approach.Pseudo-Differential Operators. Theory and Applications 12, Birkhäuser/Springer, Basel (2017). Zbl 06658825, MR 3618292, 10.1007/978-1-4939-6792-6 |
| Reference:
|
[24] Kou, S. G.: A jump-diffusion model for option pricing.Manage. Sci. 48 (2002), 1086-1101. Zbl 1216.91039, 10.1287/mnsc.48.8.1086.166 |
| Reference:
|
[25] Kufner, A., John, O., Fučík, S.: Function Spaces. Monographs and Textsbooks on Mechanics of Solids and Fluids. Mechanics: Analysis.Noordhoff International Publishing, Leyden; Academia, Praha (1977). Zbl 0364.46022, MR 0482102 |
| Reference:
|
[26] Kwon, Y., Lee, Y.: A second-order finite difference method for option pricing under jump-diffusion models.SIAM J. Numer. Anal. 49 (2011), 2598-2617. Zbl 1232.91712, MR 2873249, 10.1137/090777529 |
| Reference:
|
[27] Marcozzi, M. D.: An adaptive extrapolation discontinuous Galerkin method for the valuation of Asian options.J. Comput. Appl. Math. 235 (2011), 3632-3645. Zbl 1215.60031, MR 2788114, 10.1016/j.cam.2011.02.024 |
| Reference:
|
[28] Matache, A.-M., Petersdorff, T. von, Schwab, C.: Fast deterministic pricing of options on Lévy driven assets.M2AN, Math. Model. Numer. Anal. 38 (2004), 37-71. Zbl 1072.60052, MR 2073930, 10.1051/m2an:2004003 |
| Reference:
|
[29] Merton, R. C.: Theory of rational option pricing.Bell J. Econ. Manage. Sci. 4 (1973), 141-183. Zbl 1257.91043, MR 0496534, 10.2307/3003143 |
| Reference:
|
[30] Merton, R. C.: Option pricing when underlying stock returns are discontinuous.J. Financ. Econ. 3 (1976), 125-144. Zbl 1131.91344, 10.1016/0304-405X(76)90022-2 |
| Reference:
|
[31] Nicholls, D. P., Sward, A.: A discontinuous Galerkin method for pricing American options under the constant elasticity of variance model.Commun. Comput. Phys. 17 (2015), 761-778. Zbl 1375.91243, MR 3371520, 10.4208/cicp.190513.131114a |
| Reference:
|
[32] Reed, W. H., Hill, T. R.: Triangular Mesh Methods for the Neutron Transport Equation.Technical Report LA-UR-73-479, Los Alamos Scientific Laboratory, New Mexico (1973), Available at https://www.osti.gov/servlets/purl/4491151. |
| Reference:
|
[33] 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 (SIAM), Philadelphia (2008). Zbl 1153.65112, MR 2431403, 10.1137/1.9780898717440 |
| Reference:
|
[34] Roos, H.-G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Convection-diffusion and Flow Problems.Springer Series in Computational Mathematics 24, Springer, Berlin (1996). Zbl 0844.65075, MR 1477665, 10.1007/978-3-662-03206-0 |
| Reference:
|
[35] Vlasák, M.: Time discretizations for evolution problems.Appl. Math., Praha 62 (2017), 135-169. Zbl 06738486, MR 3647039, 10.21136/AM.2017.0268-16 |
| Reference:
|
[36] Vlasák, M., Dolejší, V., Hájek, J.: A priori error estimates of an extrapolated space-time discontinuous Galerkin method for nonlinear convection-diffusion problems.Numer. Methods Partial Differ. Equations 27 (2011), 1456-1482. Zbl 1237.65105, MR 2838303, 10.1002/num.20591 |
| Reference:
|
[37] Wilmott, P., Dewynne, J., Howison, S.: Option Pricing: Mathematical Models and Computation.Financial Press, Oxford (1995). Zbl 0844.90011 |
| Reference:
|
[38] Zhang, K., Wang, S.: A computational scheme for options under jump diffusion processes.Int. J. Numer. Anal. Model. 6 (2009), 110-123. Zbl 1159.91402, MR 2574899 |
| . |
