Abstract
This paper considers the American put option valuation in a jump-diffusion model and relates this optimal-stopping problem to a parabolic integro-differential free-boundary problem, with special attention to the behavior of the optimal-stopping boundary. We study the regularity of the American option value and obtain in particular a decomposition of the American put option price as the sum of its counterpart European price and the early exercise premium. Compared with the Black-Scholes (BS) [5] model, this premium has an additional term due to the presence of jumps. We prove the continuity of the free boundary and also give one estimate near maturity, generalizing a recent result of Barleset al. [3] for the BS model. Finally, we study the effect of the market price of jump risk and the intensity of jumps on the American put option price and its critical stock price.
Similar content being viewed by others
References
Aase, K. K. (1988), Contingent Claims Valuation When the Security Price is a Combination of an Itô Process and a Random Point Process, Stochastic Process. Appl., 28, 185–220.
Ahn, C. M., and H. E. Thompson (1992), The Impact of Jump Risks on Nominal Interest Rates and Foreign Exchange Rates, Rev. Quant. Finan. Account., 2, 17–31.
Barles, G., J. Burdeau, M. Romano, and N. Samsoen (1995), Critical Stock Price Near Expiration, Math. Finan., 5(2), 77–95.
Bensoussan, A., and J. L. Lions (1978), Applications des Inéquations Variationnelles en Contrôle stochastique, Dunod, Paris.
Black, F., and M. Scholes (1973), The Pricing of Options and Corporate Liabilities, J. Polit. Econ., 81, 637–659.
Carr, P., R. Jarrow, and R. Myneni (1992), Alternative Characterizations of American Put Options, Math. Finan., 2, 87–106.
Colwell, D. B., and R. J. Elliott (1993), Discontinuous Asset Prices and Non-Attainable Contingent Claims, Math. Finan., 3(3), 295–308.
El Karoui, N., A. Millet, and J. P. Lepeltier (1992), A Probabilistic Approach to the Réduite in Optimal Stopping, Probab. Math. Statist., 13, 97–121.
Friedman, A. (1975), Parabolic Variational Inequalities in One Space Dimension and Smoothness of the Free Boundary, J. Funct. Anal. 18, 151–176.
Friedman, A., and M. Robin (1978), The Free Boundary for Variational Inequalities with Nonlocal Operators, SIAM J. Control Optim., 16(2), 347–372.
Harrison, J. M., and D. M. Kreps (1979), Martingale and Arbitrage in Multiperiods Securities Markets, J. Econ. Theory, 20, 381–408.
Harrison, J. M., and S. R. Pliska (1981), Martingales and Stochastic Integrals in the Theory of Continuous Trading, Stochastic Process. Appl., 11, 215–260.
Jacka, S. (1991), Optimal Stopping and the American Put, Math. Finan., 1, 1–14.
Jacod, J. (1979), Calcul Stochastique et Problèmes de Martingales, Lectures Notes in Mathematics, vol. 714, Springer-Verlag, Berlin.
Jaillet, P., D. Lamberton, and B. Lapeyre (1990), Variational Inequalities and the Pricing of American Options, Acta Appl. Math., 21, 263–289.
Jorion, P. (1988), On Jump Processes in the Foreign Exchange and Stock Markets, Rev. Finan. Stud., 4, 427–445.
Kim, I. J. (1990), The Analytic Valuation of American Options, Rev. Finan. Stud., 3, 547–572.
Ladyzenskaja, O. A., V. A. Solonnikov, and N. N. Ural’ceva (1968), Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI.
Lamberton, D. (1994), Critical Price for an American Option near Maturity, Preprint, Université Marne la Vallée.
Lions, P. L. (1983), Optimal Control of Diffusion Processes and Hamilton-Jacobi-Bellman Equations. Part 1: The Dynamic Programming Principle and Applications and Part 2: Viscosity Solutions and Uniqueness, Comm. Partial Differential Equations, 8, 1101–1174 and 1229–1276.
Maingueneau, M. A. (1978), Temps d’Arrêts Optimaux et Théorie Générale, Séminaire de Probabilités XII, Lecture Notes in Mathematics, vol. 649, Springer-Verlag, Berlin, pp. 457–467.
McKean, H. P., Jr. (1965), Appendix: a Free Boundary Problem for the Heat Equation Arising from a Problem in Mathematical Economics, Indust. Manage. Rev., 6, 32–39.
Merton, R. (1973), Theory of Rational Option Pricing, Bell J. Econ. Manage. Sci., 4, 141–183.
Merton, R. (1976), Option Pricing when the Underlying Stock Returns are Discontinuous, J. Finan. Econ., 5, 125–144.
Meyer, P. A. (1976), Un Cours sur les Intégrales Stochastiques, Lecture Notes in Mathematics, vol. 511, Springer-Verlag, Berlin, 245–398.
Myneni, R. (1992), The Pricing of American Option, Ann. Appl. Probab., 2, 1–23.
Naik, V., and M. Lee (1990), General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns, Rev. Finan. Stud., 3, 493–521.
Pham, H. (1995), Applications des Methodes Probabilistes et de Contrôle Stochastique aux Mathematiques Financières, Part III, Doctoral dissertation, Université Paris IX Dauphine.
Pham, H. (1995), Optimal Stopping of Controlled Jump Diffusion Processes: a Viscosity Solution Approach, C. R. Acad. Sci. Sér. I, 320, 1113–1118, Forthcoming in J. Math. System Estim. Control.
Shiryaev, A. N. (1978), Optimal Stopping Rules, Springer-Verlag, New York.
Van Moerbeke, P. (1976), On Optimal Stopping and Free Boundary Problems, Arch. Rational Mech. Anal., 60, 101–148.
Zhang, X. (1994), Analyse Numérique des Options Américaines dans un Modèle de Diffusion avec des Sauts, Doctoral dissertation, Ecole Nationale des Ponts et Chaussées.
Author information
Authors and Affiliations
Additional information
Communicated by A. Bensoussan
Rights and permissions
About this article
Cite this article
Pham, H. Optimal stopping, free boundary, and American option in a jump-diffusion model. Appl Math Optim 35, 145–164 (1997). https://doi.org/10.1007/BF02683325
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02683325