Abstract
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.
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References
Y. Achdou, O. Pironneau: Computational Methods for Option Pricing. Frontiers in Applied Mathematics 30, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2005.
A. Almendral, C. W. Oosterlee: Numerical valuation of options with jumps in the underlying. Appl. Numer. Math. 53 (2005), 1–18.
L. Andersen, J. Andreasen: Jump-diffusion processes: volatility smile fitting and numerical methods for option pricing. Rev. Deriv. Res. 4 (2000), 231–262.
A. Bensoussan, J.-L. Lions: Impulse Control and Quasi-Variational Inequalities. Gauthier-Villars, Paris, 1984.
F. Black, M. Scholes: The pricing of options and corporate liabilities. J. Polit. Econ. 81 (1973), 637–654.
P. Boyle, M. Broadie, P. Glasserman: Monte Carlo methods for security pricing. J. Econ. Dyn. Control 21 (1997), 1267–1321.
P. Carr, A. Mayo: On the numerical evaluation of option prices in jump diffusion processes. Eur. J. Finance 13 (2007), 353–372.
R. Cont, P. Tankov: Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series, Chapman and Hall/CRC, Boca Raton, 2004.
R. Cont, E. Voltchkova: A finite difference scheme for option pricing in jump diffusion and exponential Lévy models. SIAM J. Numer. Anal. 43 (2005), 1596–1626.
J. C. Cox, S. A. Ross, M. Rubinstein: Option pricing: a simplified approach. J. Financ. Econ. 7 (1979), 229–263.
Y. d’Halluin, P. A. Forsyth, K. R. Vetzal: Robust numerical methods for contingent claims under jump diffusion processes. IMA J. Numer. Anal. 25 (2005), 87–112.
V. Dolejší, M. Feistauer: Discontinuous Galerkin Method. Analysis and Applications to Compressible Flow. Springer Series in Computational Mathematics 48, Springer, Cham, 2015.
V. Dolejší, M. Vlasák: Analysis of a BDF-DGFE scheme for nonlinear convection-diffusion problems. Numer. Math. 110 (2008), 405–447.
M. Feistauer, K. Švadlenka: Discontinuous Galerkin method of lines for solving nonstationary singularly perturbed linear problems. J. Numer. Math. 12 (2004), 97–117.
L. Feng, V. Linetsky: Pricing options in jump-diffusion models: an extrapolation approach. Oper. Res. 56 (2008), 304–325.
E. G. Haug: The Complete Guide to Option Pricing Formulas. McGraw-Hill, New York, 2006.
F. Hecht: New development in freefem++. J. Numer. Math. 20 (2012), 251–265.
J. Hozman: Analysis of the discontinuous Galerkin method applied to the European option pricing problem. AIP Conf. Proc. 1570 (2013), 227–234.
J. Hozman, T. Tichý: On the impact of various formulations of the boundary condition within numerical option valuation by DG method. Filomat 30 (2016), 4253–4263.
J. Hozman, T. Tichý: DG method for numerical pricing of multi-asset Asian options—the case of options with floating strike. Appl. Math., Praha 62 (2017), 171–195.
J. Hozman, T. Tichý: DG method for the numerical pricing of two-asset European-style Asian options with fixed strike. Appl. Math., Praha 62 (2017), 607–632.
J. Hozman, T. Tichý: DG framework for pricing European options under one-factor stochastic volatility models. J. Comput. Appl. Math. 344 (2018), 585–600.
A. Itkin: Pricing Derivatives Under Lévy Models. Modern Finite-Difference and Pseudo-Differential Operators Approach. Pseudo-Differential Operators. Theory and Applications 12, Birkhäuser/Springer, Basel, 2017.
S. G. Kou: A jump-diffusion model for option pricing. Manage. Sci. 48 (2002), 1086–1101.
A. Kufner, O. John, S. Fučík: Function Spaces. Monographs and Textsbooks on Mechanics of Solids and Fluids. Mechanics: Analysis. Noordhoff International Publishing, Leyden; Academia, Praha, 1977.
Y. Kwon, Y. Lee: A second-order finite difference method for option pricing under jump-diffusion models. SIAM J. Numer. Anal. 49 (2011), 2598–2617.
M. D. Marcozzi: An adaptive extrapolation discontinuous Galerkin method for the valuation of Asian options. J. Comput. Appl. Math. 235 (2011), 3632–3645.
A.-M. Matache, T. von Petersdorff, C. Schwab: Fast deterministic pricing of options on Lévy driven assets. M2AN, Math. Model. Numer. Anal. 38 (2004), 37–71.
R. C. Merton: Theory of rational option pricing. Bell J. Econ. Manage. Sci. 4 (1973), 141–183.
R. C. Merton: Option pricing when underlying stock returns are discontinuous. J. Financ. Econ. 3 (1976), 125–144.
D. P. Nicholls, A. Sward: A discontinuous Galerkin method for pricing American options under the constant elasticity of variance model. Commun. Comput. Phys. 17 (2015), 761–778.
W. H. Reed, T. R. Hill: 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/pur1/4491151 .
B. Rivière: 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.
H.-G. Roos, M. Stynes, L. Tobiska: Numerical Methods for Singularly Perturbed Differential Equations. Convection-diffusion and Flow Problems. Springer Series in Computational Mathematics 24, Springer, Berlin, 1996.
M. Vlasák: Time discretizations for evolution problems. Appl. Math., Praha 62 (2017), 135–169.
M. Vlasák, V. Dolejší, J. Hájek: 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.
P. Wilmott, J. Dewynne, S. Howison: Option Pricing: Mathematical Models and Computation. Financial Press, Oxford, 1995.
K. Zhang, S. Wang: A computational scheme for options under jump diffusion processes. Int. J. Numer. Anal. Model. 6 (2009), 110–123.
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The first two authors were supported through the Czech Science Foundation (GA ČR) under project 16-09541S. Furthermore, the second author also acknowledges the support provided within SP2019/5, an SGS research project of VŠB-TU Ostrava. The research of the third author was supported by grant 17-01747S of the Czech Science Foundation; he is a junior member of the university center for mathematical modeling, applied analysis and computational mathematics (MathMAC). The support is greatly acknowledged.
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Hozman, J., Tichý, T. & Vlasák, M. DG Method for Pricing European Options under Merton Jump-Diffusion Model. Appl Math 64, 501–530 (2019). https://doi.org/10.21136/AM.2019.0305-18
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DOI: https://doi.org/10.21136/AM.2019.0305-18
Keywords
- option pricing
- Merton jump-diffusion model
- integro-differential equation
- discontinuous Galerkin method
- semi-implicit discretization
- a priori error estimates