Value function regularity in option pricing problems under a pure jump model. (English) Zbl 1406.91445
Summary: In this paper, we consider an option pricing problem in a pure jump model where the process \(X(t)\) models the logarithm of the stock price. By the Schauder fixed point theorem, we show the existence and uniqueness of the solutions in Hölder spaces for the European and American option pricing problems respectively. Due to the estimates of fractional heat kernel, we give the regularity of the value functions \(u_{E}(t,x)\) and \(u_{A}(t,x)\) of the European option and the American option respectively.
MSC:
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
| 45K05 | Integro-partial differential equations |
| 49L25 | Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games |
| 60G51 | Processes with independent increments; Lévy processes |
| 60J75 | Jump processes (MSC2010) |
| 91G80 | Financial applications of other theories |
| 47F05 | General theory of partial differential operators |
| 60G40 | Stopping times; optimal stopping problems; gambling theory |
Keywords:
option price; fractional Laplace operator; partial integro-differential operator; regularityReferences:
| [1] | Bayraktar, E., Emmerling, T., Menaldi, J.-L.: On the impulse control of jump diffusions. SIAM J. Control Optim. 51(3), 2612-2637 (2013) · Zbl 1272.49073 · doi:10.1137/120863836 |
| [2] | Bayraktar, E., Xing, H.: Regularity of the optimal stopping problem for jump diffusions. SIAM J. Control Optim. 50(3), 1337-1357 (2012) · Zbl 1255.60068 · doi:10.1137/100810915 |
| [3] | Bensoussan, A., Lions, J.-L.: Impulse Control and Quasivariational Inequalities. Gauthier-Villars, Montrouge (1984) |
| [4] | Caffarelli, L., Figalli, A.: Regularity of solutions to the parabolic fractional obstacle problem. J. Reine Angew. Math. 680, 191-233 (2013) · Zbl 1277.35088 |
| [5] | Caffarelli, L., Salsa, S., Silvestre, L.: Regularity estimates for the solution and the free boundary of the obstacke problem for the fractional Laplacian. Invent. Math. 171(2), 425-461 (2005) · Zbl 1148.35097 · doi:10.1007/s00222-007-0086-6 |
| [6] | Chen, Y.-S.A., Guo, X.: Impulse control of multidimensional jump diffusions in finite time horizon. SIAM J. Control Optim. 51(3), 2638-2663 (2013) · Zbl 1275.49062 · doi:10.1137/110854205 |
| [7] | Cont, R., Tankov, P.: Financial Modelling with Jump Processes. CRC Press, Boca Raton (2003) · Zbl 1052.91043 · doi:10.1201/9780203485217 |
| [8] | Cont, R., Voltchkova, E.: Integro-differential equations for option prices in exponential Lévy models. Finance Stoch. 9(3), 299-325 (2005) · Zbl 1096.91023 · doi:10.1007/s00780-005-0153-z |
| [9] | Davis, M.H.A., Guo, X., Wu, G.L.: Impulse control of multidimensional jump diffusions. SIAM J. Control Optim. 48(8), 5276-5293 (2010) · Zbl 1208.49045 · doi:10.1137/090780419 |
| [10] | Droniou, J., Gallou \[\ddot{e}\] e¨t, T., Vovelle, J.: Global Solution and Smoothing Effect for a Non-local Regularization of a Hyperbolic Equation. Nonlinear Evolution Equations and Related Topics, pp. 499-521. Birkhauser, Basel (2004) · Zbl 1036.35123 |
| [11] | Friedman, A.: Variational Principles and Free-boundary Problems. Courier Dover Publications, New York (2010) |
| [12] | Friedman, A.: Partial Differential Equations of Parabolic Type. Courier Dover Publications, New York (2010) |
| [13] | Garroni, M.G., Menaldi, J.-L.: Green Functions for Second Order Parabolic Integro-differential Problems. Longman Scientific and Technical, Harlow (1992) · Zbl 0806.45007 |
| [14] | Garroni, M.G., Menaldi, J.-L.: Second Order Elliptic Integro-differential Problems. CRC Press, Boca Raton (2010) · Zbl 0806.45007 |
| [15] | Guo, X., Wu, G.L.: Smooth fit principle for impulse control of multidimensional diffusion processes. SIAM J. Control Optim. 48(2), 594-617 (2009) · Zbl 1194.49016 · doi:10.1137/080716001 |
| [16] | Jiang, L.S.: Mathematical Modeling and Methods of Option Pricing. World Scientific, Singapore (2005) · Zbl 1140.91047 · doi:10.1142/5855 |
| [17] | Jiang, L.S., Bian, B.J., Yi, F.H.: A parabolic variational inequality arising from the valuation of fixed rate mortgages. Eur. J. Appl. Math. 16(3), 361-383 (2005) · Zbl 1133.91026 · doi:10.1017/S0956792505006297 |
| [18] | Krylov, N.V.: Lectures on Elliptic and Parabolic Equations in Hölder Spaces. American Mathematical Society, Providence (1996) · Zbl 0865.35001 · doi:10.1090/gsm/012 |
| [19] | Madych, W.R., Riviere, N.M.: Multipliers of the \[H\ddot{o}\] o¨lder classes. J. Funct. Anal. 21(4), 369-379 (1976) · Zbl 0338.46035 · doi:10.1016/0022-1236(76)90032-X |
| [20] | Miao, C.X., Yuan, B.Q., Zhang, B.: Well-posedness of the Cauchy problem for the fractional power dissipative equations. Nonlinear Anal. Theory Methods Appl. 68(3), 461-484 (2008) · Zbl 1132.35047 · doi:10.1016/j.na.2006.11.011 |
| [21] | Mikulevičius, Remigijus, Zhang, Changyong: On the rate of convergence of weak Euler approximation for nondegenerate SDEs driven by Lévy processes. Stoch. Process. Appl. 121(8), 1720-1748 (2011) · Zbl 1221.60101 · doi:10.1016/j.spa.2011.04.004 |
| [22] | Øksendal, B.K., Sulem, A.: Applied Stochastic Control of Jump Diffusions. Springer, Berlin (2005) · Zbl 1074.93009 |
| [23] | Pham, H.: Optimal stopping, free boundary, and American option in a jump-diffusion model. Appl. Math. Optim. 35(2), 145-164 (1997) · Zbl 0866.60038 · doi:10.1007/BF02683325 |
| [24] | Pham, H.: Continuous-time Stochastic Control and Optimization with Financial Applications. Springer, Berlin (2009) · Zbl 1165.93039 · doi:10.1007/978-3-540-89500-8 |
| [25] | Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun. Pure Appl. Math. 60, 67-112 (2007) · Zbl 1141.49035 · doi:10.1002/cpa.20153 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
