Abstract
Prices of financial options in a market with liquidity risk are shown to be weak solutions of a class of semilinear parabolic partial differential equations with nonnegative characteristic form. We prove the existence and uniqueness of such solutions, and then show the solutions correspond to option prices as defined in terms of replication in a probabilistic setup. We obtain an asymptotic representation of the price and the hedging strategy as a liquidity parameter converges to zero.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, New York (2003)
Amann, H.: Existence and regularity for semilinear parabolic evolution equations. Ann. Scuola Norm. Sup. Pisa 11(4), 593–676 (1984)
Bank, P., Baum, D.: Hedging and portfolio optimization in financial markets with a large trader. Math. Finance 14(1), 1–18 (2004)
Barles, G., Burdeau, J.: The Dirichlet problem for semilinear second-order degenerate elliptic equations and applications to stochastic exit time control problems. Commun. PDE 20(1–2), 129–178 (1995)
Barles, G., Lesigne, E.: SDE, BSDE and PDE. In: El Karoui, N., Mazliak, M. (eds.) Backward Stochastic Differential Equations, Volume 364 of Pitman Research Notes in Mathematics Series, pp. 47–82. Longman, Harlow (1997)
Bensoussan, A., Lions, J.L.: Applications of Variational Inequalities in Stochastic Control. North-Holland, Amsterdam (1982)
Boccardo, L., Murat, F., Puel, J.-P.: Existence results for some quasilinear parabolic equations. Nonlinear Anal. 13(4), 373–392 (1989)
Boccardo, L., Segura de León, S., Trombetti, C.: Bounded and unbounded solutions for a class of quasi-linear elliptic problems with a quadratic gradient term. J. Math. Pures Appl. 80(9), 919–940 (2001)
Briand, P., Hu, Y.: BSDE with quadratic growth and unbounded terminal value. Probab. Theory Relat. Fields 136, 604–618 (2006)
Çetin, U., Jarrow, R.A., Protter, P.: Liquidity risk and arbitrage pricing theory. Finance Stoch. 8(3), 311–341 (2004)
Çetin, U., Soner, H.M., Touzi, N.: Option hedging for small investors under liquidity costs. Finance Stoch. 14(3), 317–341 (2010)
Choe, H.J.: On the regularity of parabolic equations and obstacle problems with quadratic growth nonlinearities. J. Differ. Equ. 102(1), 101–118 (1993)
Crandall, M.C., Ishii, H., Lions, P.L.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc 27, 1–67 (1992)
Darling, R.W.R., Pardoux, E.: Backwards SDE with random terminal time and applications to semilinear elliptic PDE. Ann. Probab. 25(3), 1135–1159 (1997)
Di Benedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993)
Düring, B., Jüngel, A.: Existence and uniqueness of solutions to a quasilinear parabolic equation with quadratic gradients in financial markets. Nonlinear Anal. 62, 519–544 (2005)
Evans, L.C.: Weak Convergence Methods for Nonlinear Partial Differential Equations. American Mathematical Society, Providence, RI (1990)
Evans, L.C.: Partial Differential Equations, 2nd edn. American Mathematical Society, Providence, RI (2010)
Feehan, P.M.N., Pop, C.A.: A Schauder approach to degenerate-parabolic partial differential equations with unbounded coefficients. J. Differ. Equ. 254(12), 4401–4445 (2013)
Feehan, P.M.N., Pop, C.A.: Degenerate-elliptic operators in mathematical finance and higher-order regularity for solutions to variational equations. Adv. Differ. Equ. 20(3–4), 361–432 (2015)
Ferone, V., Posteraro, R.M.: On a class of quasilinear elliptic equations with quadratic growth in the gradient. Nonlinear Anal. 20(6), 703–711 (1993)
Fichera, G.: On a unified theory of boundary value problems for elliptic-parabolic equations of second order. In: R.E. Langer (ed.) Boundary Problems in Differential Equations, pp. 97–120. University of Wisconsin Press, Madison (1960)
Fleming, W.H., Soner, H.M.: Controlled Markov Processes and Viscosity Solutions, Volume 25 of Stochastic Modelling and Applied Probability, 2nd edn. Springer, New York (2006)
French, K., Schwert, G.W., Stambaugh, R.F.: Expected stock returns and volatility. J. Finance Econ. 19(1), 3–29 (1987)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, NJ (1964)
Giaquinta, M., Struwe, M.: An optimal regularity result for a class of quasilinear parabolic systems. Manuscr. Math. 36(2), 223–239 (1981)
Giaquinta, M., Struwe, M.: On the partial regularity of weak solutions of nonlinear parabolic systems. Math. Z. 179(4), 437–451 (1982)
Ivanov, A.V.: Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of Second Order. American Mathematical Society, Providence, RI (1984)
Jarrow, R.A., Roch, A.F.: Liquidity risk and the term structure of interest rates. Math. Finance Econ. 9, 57–83 (2015)
Kobylanski, M.: Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28(2), 558–602 (2000)
Koshelev, A.: Regularity Problem for Quasilinear Elliptic and Parabolic Systems, Volume 1614 of Lecture Notes in Math. Springer, Berlin (1995)
Kufner, A.: Weighted Sobolev Spaces. Wiley, New York (1985)
Ladyzhenskaya, O.A., Solonnikov, V.A., Uraltseva, N.N.: Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society, Providence, RI (1968)
Lions, J.L.: Quelques méthodes de résolution des problemes aux limites non linéaires. Dunod, Paris (1969)
Lunardi, A.: Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 16. Birkhäuser, Basel (1995)
Oleinik, O.A., Radkevich, E.V.: Equations of Second Order with Nonnegative Characteristic Form. American Mathematical Society, Providence, RI (1973)
Pardoux, E., Peng, S.: Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations. Lecture Notes in Control Information Science, vol. 176, pp. 200–217 (1992)
Peng, S.: Probabilistic interpretation for systems of quasilinear parabolic partial differential equations. Stoch. Stoch. Rep. 37(1–2), 61–74 (1991)
Possamaï, D., Soner, H.M., Touzi, N.: Large liquidity expansion of super-hedging costs. Asymptot. Anal. 79, 45–64 (2012)
Ran, Q., Zhang, T.: Existence and uniqueness of bounded weak solutions of a semilinear parabolic PDE. J. Theor. Probab. 23(4), 951–971 (2010)
Showalter, R.E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. American Mathematical Society, Providence, RI (1997)
Stroock, D.W., Varadhan, S.R.S.: Multidimensional Diffusion Processes. Springer, Berlin (1979)
Struwe, M.: Some regularity results for quasilinear parabolic systems. Comment. Math. Univ. Carol. 26(1), 129–150 (1985)
Tolksdorf, P.: On some parabolic variational problems with quadratic growth. Ann. Sci. Norm. Super. Pisa Class. Sci. (4) 13(2), 193–223 (1986)
Veraar, M.C.: The stochastic Fubini theorem revisited. Stochastics 84, 543–551 (2012)
Zeidler, E.: Nonlinear Functional Analysis, vol. I. Springer, New York (1985)
Zeidler, E.: Nonlinear Functional Analysis and its Applications: Nonlinear Monotone Operators, vol. II/B. Springer, New York (1990)
Acknowledgements
The authors thank participants of the 6th Liquidity Risk Forum at the Institut Bachelier, the IME 2013 conference, QMF 2013 and the Bachelier Congress 2014 for constructive remarks on an earlier version of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
A. Roch’s research was supported by the Institute of Financial Mathematics of Montreal, and the Natural Sciences and Engineering Research Council of Canada.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Fahrenwaldt, M.A., Roch, A.F. Option prices under liquidity risk as weak solutions of semilinear diffusion equations. Nonlinear Differ. Equ. Appl. 24, 12 (2017). https://doi.org/10.1007/s00030-017-0435-0
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00030-017-0435-0