Corrected random walk approximations to free boundary problems in optimal stopping

Corrected random walk approximations to free boundary problems in optimal stopping

Tze Leung Lai, Yi-Ching Yao, and Farid Aitsahlia

Source: Adv. in Appl. Probab. Volume 39, Number 3 (2007), 753-775.

Abstract

Corrected random walk approximations to continuous-time optimal stopping boundaries for Brownian motion, first introduced by Chernoff and Petkau, have provided powerful computational tools in option pricing and sequential analysis. This paper develops the theory of these second-order approximations and describes some new applications.

First Page: Show

Primary Subjects: 60G40

Secondary Subjects: 60H30, 90C39

Keywords: Optimal stopping; free boundary partial differential equation; random walk; renewal theory; singular stochastic control

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aap/1189518637
Digital Object Identifier: doi:10.1239/aap/1189518637
Mathematical Reviews number (MathSciNet): MR2357380
Zentralblatt MATH identifier: 1127.60038

back to Table of Contents

References

AitSahlia, F. and Lai, T. L. (2001). Exercise boundaries and efficient approximations to American option prices and hedge parameters. J. Comput. Finance 4, 85--103.

Alvarez, L. H. R. (2001). Singular stochastic control, linear diffusions, and optimal stopping: a class of solvable problems. SIAM J. Control Optimization 39, 1697--1710.

Mathematical Reviews (MathSciNet): MR1825860

Digital Object Identifier: doi:10.1137/S0363012900367825

Zentralblatt MATH: 0997.93102

Baldursson, F. M. (1987). Singular stochastic control and optimal stopping. Stochastics 21, 1--40.

Mathematical Reviews (MathSciNet): MR899953

Zentralblatt MATH: 0615.60040

Bather, J. and Chernoff, H. (1967). Sequential decisions in the control of a spaceship. In Proc. Fifth Berkeley Symp. Math. Statist. Prob., Vol. III, University of California Press, Berkeley, pp. 181--207.

Mathematical Reviews (MathSciNet): MR224218

Benes, V. E., Shepp, L. A. and Witsenhausen, H. S. (1980). Some solvable stochastic control problems. Stochastics 4, 39--83.

Mathematical Reviews (MathSciNet): MR587428

Brezzi, M. and Lai, T. L. (2002). Optimal learning and experimentation in bandit problems. J. Econom. Dynamics Control 27, 87--108.

Mathematical Reviews (MathSciNet): MR1925627

Broadie, M., Glasserman, P. and Kou, S. (1997). A continuity correction for discrete barrier options. Math. Finance 7, 325--348.

Mathematical Reviews (MathSciNet): MR1482707

Digital Object Identifier: doi:10.1111/1467-9965.00035

Zentralblatt MATH: 1020.91020

Carr, P., Jarrow, R. and Myneni, R. (1992). Alternative characterizations of American put options. Math. Finance 2, 87--106.

Mathematical Reviews (MathSciNet): MR1143390

Digital Object Identifier: doi:10.1214/aoap/1177005768

Project Euclid: euclid.aoap/1177005768

Zentralblatt MATH: 0753.60040

Chen, X. and Chadam, J. (2007). A mathematical analysis of the optimal exercise boundary for American put options. SIAM J. Math. Anal. 38, 1613--1641.

Mathematical Reviews (MathSciNet): MR2286022

Digital Object Identifier: doi:10.1137/S0036141003437708

Zentralblatt MATH: 05194948

Chen, X., Chadam, J., Jiang, L. and Zhang, W. (2007). Convexity of the exercise boundary of the American put option on a zero dividend asset. To appear in Math. Finance.

Mathematical Reviews (MathSciNet): MR2380946

Chernoff, H. (1965). Sequential tests for the mean of a normal distribution. IV. (Discrete case). Ann. Math. Statist. 36, 55--68.

Mathematical Reviews (MathSciNet): MR170443

Digital Object Identifier: doi:10.1214/aoms/1177700270

Project Euclid: euclid.aoms/1177700270

Chernoff, H. (1972). Sequential Analysis and Optimal Design. Society for Industrial and Applied Mathematics, Philadelphia, PA.

Mathematical Reviews (MathSciNet): MR309249

Zentralblatt MATH: 0265.62024

Chernoff, H. and Petkau, A. J. (1976). An optimal stopping problem for sums of dichotomous random variables. Ann. Prob. 4, 875--889.

Mathematical Reviews (MathSciNet): MR431559

Digital Object Identifier: doi:10.1214/aop/1176995933

Zentralblatt MATH: 0347.62064

JSTOR: links.jstor.org

Chernoff, H. and Petkau, A. J. (1984). Numerical methods for Bayes sequential decision problems. Tech. Rep. ONR 34, MIT Statistics Center.

Chernoff, H. and Petkau, A. J. (1986). Numerical solutions for Bayes sequential decision problems. SIAM J. Sci. Statist. Comput. 7, 46--59.

Mathematical Reviews (MathSciNet): MR819456

Digital Object Identifier: doi:10.1137/0907003

Zentralblatt MATH: 0598.62102

Chow, Y. S. and Lai, T. L. (1979). Moments of ladder variables for driftless random walks. Z. Wahrscheinlichkeitsth 48, 253--257.

Mathematical Reviews (MathSciNet): MR537671

Chow, Y. S. and Robbins, H. (1965). On optimal stopping rules for $S_n/n$. Illinois J. Math. 9, 444--454.

Mathematical Reviews (MathSciNet): MR179897

Cox, J., Ross, S. and Rubinstein, M. (1979). Option pricing: a simplified approach. J. Financial Econom. 7, 229--263.

Dvoretzky, A. (1967). Existence and properties of certain optimal stopping rules. In Proc. Fifth Berkeley Symp. Math. Statist. Prob., Vol. I, University of California Press, Berkeley, pp. 441--452.

Mathematical Reviews (MathSciNet): MR214232

Zentralblatt MATH: 0255.60033

Ekström, E. (2004). Convexity of the optimal stopping boundary for the American put option. J. Math. Anal. Appl. 299, 147--156.

Mathematical Reviews (MathSciNet): MR2091277

Digital Object Identifier: doi:10.1016/j.jmaa.2004.06.018

Zentralblatt MATH: 1129.91326

Ekström, E. (2004). Properties of American option prices. Stoch. Process. Appl. 114, 265--278.

Mathematical Reviews (MathSciNet): MR2101244

Digital Object Identifier: doi:10.1016/j.spa.2004.05.002

Zentralblatt MATH: 1114.91050

Evans, J. D., Kuske, R. and Keller, J. B. (2002). American options on assets with dividends near expiry. Math. Finance 12, 219--237.

Mathematical Reviews (MathSciNet): MR1910594

Digital Object Identifier: doi:10.1111/1467-9965.02008

Fakeev, A. G. (1970). Optimal stopping rules for stochastic processes with continuous time. Theory Prob. Appl. 15, 324--331.

Mathematical Reviews (MathSciNet): MR267718

Feller, W. (1966). An Introduction to Probability Theory and Its Applications, Vol. 2. John Wiley, New York.

Mathematical Reviews (MathSciNet): MR210154

Friedman, A. (1979). Optimal stopping problems in stochastic control. SIAM Rev. 21, 71--80.

Mathematical Reviews (MathSciNet): MR516383

Digital Object Identifier: doi:10.1137/1021005

JSTOR: links.jstor.org

Zentralblatt MATH: 0402.93049

Friedman, A. (1979). Time dependent free boundary problems. SIAM Rev. 21, 213--221.

Mathematical Reviews (MathSciNet): MR524512

Digital Object Identifier: doi:10.1137/1021029

JSTOR: links.jstor.org

Zentralblatt MATH: 0417.76055

Friedman, A. (1981). Variational inequalities in sequential analysis. SIAM J. Math. Anal. 12, 385--397.

Mathematical Reviews (MathSciNet): MR613319

Digital Object Identifier: doi:10.1137/0512034

Zentralblatt MATH: 0462.62067

Friedman, A. (1982). Asymptotic behavior of the free boundary of parabolic variational inequalities and applications to sequential analysis. Illinois J. Math. 26, 653--697.

Mathematical Reviews (MathSciNet): MR674232

Friedman, A. (1982). Variational Principles and Free Boundary Problems. John Wiley, New York.

Mathematical Reviews (MathSciNet): MR679313

Zentralblatt MATH: 0564.49002

Gut, A. (1974). On the moments and limit distributions of some first passage times. Ann. Prob. 2, 277--308.

Mathematical Reviews (MathSciNet): MR394857

Digital Object Identifier: doi:10.1214/aop/1176996709

Zentralblatt MATH: 0278.60031

Hlawka, E. (1984). The Theory of Uniform Distribution. A B Academic Publishers, Berkhamsted.

Mathematical Reviews (MathSciNet): MR750652

Zentralblatt MATH: 0563.10001

Hogan, M. (1986). Comments on a problem of Chernoff and Petkau. Ann. Prob. 14, 1058--1063.

Mathematical Reviews (MathSciNet): MR841604

Digital Object Identifier: doi:10.1214/aop/1176992458

Project Euclid: euclid.aop/1176992458

Zentralblatt MATH: 0658.60070

Jacka, S. D. (1991). Optimal stopping and the American put. Math. Finance 1, 1--14.

Karatzas, I. and Shreve, S. E. (1984). Connections between optimal stopping and singular stochastic control. I. Monotone follower problems. SIAM J. Control Optimization 22, 856--877.

Mathematical Reviews (MathSciNet): MR762624

Digital Object Identifier: doi:10.1137/0322054

Zentralblatt MATH: 0551.93078

Karatzas, I. and Shreve, S. E. (1998). Methods of Mathematical Finance. Springer, New York.

Mathematical Reviews (MathSciNet): MR1640352

Zentralblatt MATH: 0941.91032

Krylov, N. V. (1980). Controlled Diffusion Processes. Springer, New York.

Mathematical Reviews (MathSciNet): MR601776

Lai, T. L. (1976). Uniform Tauberian theorems and their applications to renewal theory and first passage problems. Ann. Prob. 4, 628--643.

Mathematical Reviews (MathSciNet): MR410966

Digital Object Identifier: doi:10.1214/aop/1176996032

Zentralblatt MATH: 0365.60095

Lai, T. L. and Yao, Y. -C. (2006). The optimal stopping problem for $S_n/n$ and its ramifications. In Random Walk, Sequential Analysis and Related Topics, eds A. C. Hsiung, Z. Ying and C.-H. Zhang, World Scientific, Singapore, pp. 131--149.

Mathematical Reviews (MathSciNet): MR2367704

Lamberton, D. (1998). Error estimates for the binomial approximation of American put options. Ann. Appl. Prob. 8, 206--233.

Mathematical Reviews (MathSciNet): MR1620362

Digital Object Identifier: doi:10.1214/aoap/1027961041

Project Euclid: euclid.aoap/1027961041

Zentralblatt MATH: 0939.60022

Lamberton, D. (2002). Brownian optimal stopping and random walks. Appl. Math. Optimization 45, 283--324.

Mathematical Reviews (MathSciNet): MR1885822

Digital Object Identifier: doi:10.1007/s00245-001-0033-7

Zentralblatt MATH: 1040.60032

Lorden, G. (1970). On excess over the boundary. Ann. Math. Statist. 41, 520--527.

Mathematical Reviews (MathSciNet): MR254981

Digital Object Identifier: doi:10.1214/aoms/1177697092

Project Euclid: euclid.aoms/1177697092

McKean, H. P., Jr. (1965). A free boundary problem for the heat equation arising from a problem in mathematical economics. Industrial Management Rev. 6, 32--39.

Myneni, R. (1992). The pricing of the American option. Ann. Appl. Prob. 2, 1--23.

Mathematical Reviews (MathSciNet): MR1143390

Digital Object Identifier: doi:10.1214/aoap/1177005768

Project Euclid: euclid.aoap/1177005768

Zentralblatt MATH: 0753.60040

Peskir, G. (2005). On the American option problem. Math. Finance 15, 169--181.

Mathematical Reviews (MathSciNet): MR2116800

Digital Object Identifier: doi:10.1111/j.0960-1627.2005.00214.x

Zentralblatt MATH: 1109.91028

Peskir, G. and Shiryaev, A. (2006). Optimal Stopping and Free Boundary Problems. Birkhäuser, Basel.

Mathematical Reviews (MathSciNet): MR2256030

Zentralblatt MATH: 05016447

Shepp, L. A. (1969). Explicit solutions to some problems of optimal stopping. Ann. Math. Statist. 40, 993--1010.

Mathematical Reviews (MathSciNet): MR250415

Digital Object Identifier: doi:10.1214/aoms/1177697604

Project Euclid: euclid.aoms/1177697604

Simons, G. (1987). Easily determining which urns are `favorable'. Statist. Prob. Lett. 5, 43--48.

Mathematical Reviews (MathSciNet): MR873934

Van Moerbeke, P. (1974). An optimal stopping problem with linear reward. Acta Math. 132, 111--151.

Mathematical Reviews (MathSciNet): MR376225

Digital Object Identifier: doi:10.1007/BF02392110

Zentralblatt MATH: 0297.60027

Van Moerbeke, P. (1976). On optimal stopping and free boundary problems. Arch. Ration. Mech. Anal. 60, 101--148.

Mathematical Reviews (MathSciNet): MR413250

Walker, L. H. (1969). Regarding stopping rules for Brownian motion and random walks. Bull. Amer. Math. Soc. 75, 46--50.

Mathematical Reviews (MathSciNet): MR239630

Digital Object Identifier: doi:10.1090/S0002-9904-1969-12140-3

Zentralblatt MATH: 0185.45301

previous :: next