Optimal Multiple Stopping of Linear Diffusions

Motivated by the analysis of financial instruments with multiple exercise rights of American type and mean reverting underlyers, we formulate and solve the optimal multiple-stopping problem for a general linear regular diffusion process and a general reward function. Instead of relying on specific properties of geometric Brownian motion and call and put option payoffs as in most of the existing literature, we use general theory of optimal stopping for diffusions, and we illustrate the resulting optimal exercise policies by concrete examples and constructive recipes.

[1]  N. Meinshausen,et al.  MONTE CARLO METHODS FOR THE VALUATION OF MULTIPLE‐EXERCISE OPTIONS , 2004 .

[2]  R. Carmona,et al.  Pricing Asset Scheduling Flexibility using Optimal Switching , 2008 .

[3]  Savas Dayanik,et al.  On the optimal stopping problem for one-dimensional diffusions , 2003 .

[4]  Robert C. Dalang,et al.  Sequential Stochastic Optimization , 1996 .

[5]  Ronnie Sircar,et al.  A General Framework for Evaluating Executive Stock Options , 2004 .

[6]  Patrick Jaillet,et al.  Valuation of Commodity-Based Swing Options , 2004, Manag. Sci..

[7]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[8]  Tim Leung,et al.  ACCOUNTING FOR RISK AVERSION, VESTING, JOB TERMINATION RISK AND MULTIPLE EXERCISES IN VALUATION OF EMPLOYEE STOCK OPTIONS , 2007 .

[9]  Albert N. Shiryaev,et al.  Optimal Stopping Rules , 1980, International Encyclopedia of Statistical Science.

[10]  Robert J. Vanderbei,et al.  Optimal stopping and supermartingales over partially ordered sets , 1981 .

[11]  Mark Freidlin,et al.  The Dynkin Festschrift , 1994 .

[12]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[13]  R. Carmona,et al.  OPTIMAL MULTIPLE STOPPING AND VALUATION OF SWING OPTIONS , 2008 .

[14]  A. Borodin,et al.  Handbook of Brownian Motion - Facts and Formulae , 1996 .

[15]  Alʹbert Nikolaevich Shiri︠a︡ev,et al.  Optimal stopping rules , 1977 .

[16]  Alexander Yushkevich.  Optimal switching problem for countable Markov chains: average reward criterion , 2001, Math. Methods Oper. Res..

[17]  Luis H. R. Alvarez,et al.  Optimal harvesting of stochastically fluctuating populations , 1998 .

[18]  R. Carmona,et al.  Valuation of energy storage: an optimal switching approach , 2010 .

[19]  M. Barlow.  A DIFFUSION MODEL FOR ELECTRICITY PRICES , 2002 .

[20]  S. Karlin,et al.  A second course in stochastic processes , 1981 .

[21]  S. Shreve,et al.  Connections between Optimal Stopping and Singular Stochastic Control I. Monotone Follower Problems , 1984 .

[22]  Eduardo S. Schwartz.  The stochastic behavior of commodity prices: Implications for valuation and hedging , 1997 .

[23]  Eduardo S. Schwartz,et al.  Investment Under Uncertainty. , 1994 .

[24]  M. Kohlmann,et al.  Connections between optimal stopping and singular stochastic control , 1998 .

[25]  F. Boetius.  Bounded Variation Singular Stochastic Control and Associated Dynkin Game , 2001 .

[26]  S. Shreve,et al.  Connections Between Optimal Stopping and Singular Stochastic Control II. Reflected Follower Problems , 1985 .

[27]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[28]  Savas Dayanik,et al.  Optimal Stopping of Linear Diffusions with Random Discounting , 2008, Math. Oper. Res..

[29]  A. G. Fakeev.  Optimal Stopping of a Markov Process , 1971 .

[30]  S. Shreve,et al.  Methods of Mathematical Finance , 2010 .

[31]  A. A. Yushkevich,et al.  Optimal Switching Problem for Markov Chains , 2002 .

[32]  Robert J. Vanderbei,et al.  Optimal switching between a pair of Brownian motions , 1990 .

[33]  I. Karatzas.  On the pricing of American options , 1988 .

[34]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[35]  H. McKean,et al.  Diffusion processes and their sample paths , 1996 .