Abstract
In this paper we focus on qualitative properties of solutions to a nonlocal nonlinear partial integro-differential equation (PIDE). Using the theory of abstract semilinear parabolic equations we prove existence and uniqueness of a solution in the scale of Bessel potential spaces. Our aim is to generalize known existence results for a wide class of Lévy measures including with a strong singular kernel. As an application we consider a class of PIDEs arising in the financial mathematics. The classical linear Black–Scholes model relies on several restrictive assumptions such as liquidity and completeness of the market. Relaxing the complete market hypothesis and assuming a Lévy stochastic process dynamics for the underlying stock price process we obtain a model for pricing options by means of a PIDE. We investigate a model for pricing call and put options on underlying assets following a Lévy stochastic process with jumps. We prove existence and uniqueness of solutions to the penalized PIDE representing approximation of the linear complementarity problem arising in pricing American style of options under Lévy stochastic processes. We also present numerical results and comparison of option prices for various Lévy stochastic processes modelling underlying asset dynamics.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook of mathematical functions with formulas, graphs, and mathematical tables, volume 55 of National Bureau of Standards Applied Mathematics Series. U.S. Government Printing Office, Washington, D.C (1964)
Alvarez, O., Tourin, A.: Viscosity solutions of nonlinear integro-differential equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 13(3), 293–317 (1996)
Applebaum, D.: Lévy Processes and Stochastic Calculus volume 116 of Cambridge Studies in Advanced Mathematics, 2nd edn. Cambridge University Press, Cambridge (2009)
Arregui, I., Salvador, B., Ševčovič, D., Vázquéz, C.: Total value adjustment for european options with two stochastic factors. Mathematical model, analysis and numerical simulation. Comput. Math. Appl. 76(4), 725–740 (2018)
Arregui, I., Salvador, B., Ševčovič, D., Vázquéz, C.: Mathematical analysis of a nonlinear PDE model for European options with counterparty risk. Comptes Rendus Mathematique 357(3), 252–257 (2019)
Awatif, S.: Équations d’Hamilton–Jacobi du premier ordre avec termes intégro-différentiels. II. Existence de solutions de viscosité. Commun. Partial Differ. Equ. 16(6–7), 1075–1093 (1991)
Barles, G., Buckdahn, R., Pardoux, E.: Backward stochastic differential equations and integral-partial differential equations. Stoch. Stoch. Rep. 60(1–2), 57–83 (1997)
Barndorff-Nielsen, O.E., Levendorskiĭ, S.Z.: Feller processes of normal inverse Gaussian type. Quant. Finance 1, 318–331 (2001)
Bensoussan, A., Lions, J.-L.: Contrôle impulsionnel et inéquations quasi variationnelles, volume 11 of Méthodes Mathématiques de l’Informatique [Mathematical Methods of Information Science]. Gauthier-Villars, Paris (1982)
Carr, P., Geman, H., Madan, D.B., Yor, M.: The fine structure of asset returns: an empirical investigation. J. Bus. 75, 305–332 (2002)
Carr, P., Madan, D.B.: Option valuation using the fast Fourier transform. J. Comput. Finance 2(4), 61–73 (1999)
Cont, R., Tankov, P.: Financial Modeling with Jump Processes. Chapman & Hall/CRC, Boca Raton (2004)
Crandall, M.G., Ishii, H., Lions, P.-L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. (N.S.) 27(1), 1–67 (1992)
Cruz, J., Ševčovič, D.: Option pricing in illiquid markets with jumps. Appl. Math. Finance 25(4), 389–409 (2018)
d’Halluin, Y., Forsyth, P., Labahn, G.: A penalty method for American options with jump diffusion processes. Numerische Mathematik 97(2), 321–352 (2004)
Florescu, I., Liu, R., Mariani, M.C.: Solutions to a partial integro-differential parabolic system arising in the pricing of financial options in regime-switching jump diffusion models. Electron. J. Differ. Equ. 231, 12 (2012)
Garroni, M.G., Menaldi, J.L.: Second Order Elliptic Integro-differential Problems, volume 430 of Chapman & Hall/CRC Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton (2002)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)
Kou, S.: A jump-diffusion model for option pricing. Manag. Sci. 48, 1086–1101 (2002)
Kwok, Y.-K.: Mathematical Models of Financial Derivatives, 2nd edn. Springer Finance, Springer, Berlin (2008)
Lamberton, D., Lapeyre, B.: Introduction to Stochastic Calculus Applied to Finance. Financial Mathemathics Series. Chapman and Hall/CRC, New York (2007)
Lauko, M., Ševčovič, D.: Comparison of numerical and analytical approximations of the early exercise boundary of American put options. ANZIAM J. 51(4), 430–448 (2010)
Lesman, D.C., Wang, S.: Penalty approach to a nonlinear obstacle problem governing American put option valuation under transaction costs. Appl. Math. Comput. 251, 318–330 (2015)
Madan, D.B., Carr, P., Chang, E.C.: The Variance Gamma process and option pricing. Eur. Finance Rev. 2, 79–105 (1998)
Mariani, M.C., SenGupta, I.: Solutions to an integro-differential parabolic problem arising in the pricing of financial options in a Lévy market. Nonlinear Anal. Real World Appl. 12(6), 3103–3113 (2011)
Mariani, M.C., SenGupta, I., Salas, M.: Solutions to a gradient-dependent integro-differential parabolic problem arising in the pricing of financial options in a Lévy market. J. Math. Anal. Appl. 385(1), 36–48 (2012)
Merton, R.C.: Option pricing when underlying stock returns are discontinuous. J. Financial Econ. 3, 125–144 (1976)
Mikulevicius, R., Pragarauskas, G.: On the uniqueness of solutions of the martingale problem that is associated with degenerate Lévy operators. Liet. Mat. Rink. 33(4), 455–475 (1993)
Mikulevicius, R., Pragarauskas, H.: On the Cauchy problem for integro-differential operators in Hölder classes and the uniqueness of the martingale problem. Potential Anal. 40(4), 539–563 (2014a)
Mikulevicius, R., Pragarauskas, H.: On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem. J. Differ. Equ. 256(4), 1581–1626 (2014b)
Pham, H.: Optimal stopping of controlled jump diffusion processes: a viscosity solution approach. J. Math. Syst. Estim. Control 8(1), 1–27 (1998)
Sato, K.: Lévy Processes and Infinitely Divisible Distributions, volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1999). (Translated from the 1990 Japanese original, Revised by the author)
SenGupta, I., Mariani, M.C., Amster, P.: Solutions to integro-differential problems arising on pricing options in a Lévy market. Acta Appl. Math. 118, 237–249 (2012)
SenGupta, I., Wilson, W., Nganje, W.: Barndorff-Nielsen and Shephard model: oil hedging with variance swap and option. Math. Finance Econ. 13(2), 209–226 (2019)
Soner H.M.: Optimal control of jump-Markov processes and viscosity solutions. In: Stochastic differential systems, stochastic control theory and applications (Minneapolis, Minn., 1986), pp. 501–511, IMA Vol. Math. Appl., 10. Springer, New York (1988)
Stamicar, R., Ševčovič, D., Chadam, J.: The early exercise boundary for the American put near expiry: numerical approximation. Can. Appl. Math. Quart. 7(4), 427–444 (1999)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, no 30. Princeton University Press, Princeton (1970)
Wang, S.: An interior penalty method for a large-scale finite-dimensional nonlinear double obstacle problem. Appl. Math. Model. 58, 217–228 (2017)
Wang, S., Yang, X.Q., Teo, K.L.: Power penalty method for a linear complementarity problem arising from American option valuation. J. Opt. Theory Appl. 129(2), 227–254 (2006)
Zhu, S.-P.: A new analytical approximation formula for the optimal exercise boundary of American put options. Int. J. Theor. Appl. Finance 9(7), 1141–1177 (2006)
Zvan, R., Forsyth, P., Vetzal, K.: Penalty methods for American options with stochastic volatility. J. Comput. Appl. Math. 91(2), 199–218 (1998)
Acknowledgements
This research was supported by the project CEMAPRE MULTI/00491 financed by FCT/MEC through national funds and the Slovak research Agency Project VEGA 1/0062/18.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Cruz, J.M.T.S., Ševčovič, D. On solutions of a partial integro-differential equation in Bessel potential spaces with applications in option pricing models. Japan J. Indust. Appl. Math. 37, 697–721 (2020). https://doi.org/10.1007/s13160-020-00414-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13160-020-00414-2
Keywords
- Partial integro-differential equation
- Sectorial operator
- Analytic semigroup
- Bessel potential space
- Option pricing under Lévy stochastic process
- Lévy measure