- Volume 48 Issue 4
This work is concerned with an optimal selling rule for a large position of stock in a market. Selling a large block of stock in a short period typically depresses the market, which would result in a poor filling price. In addition, the large selling intensity makes the regime more likely to be poor state in the market. In this paper, regime switching and depressing terms associated with selling intensity are considered on a set of geometric Brownian models to capture movements of underlying asset. We also consider the liquidation strategy to sell much smaller number of shares in a long period. The goal is to maximize the overall return under state constraints. The corresponding value function with the selling strategy is shown to be a unique viscosity solution to the associated HJB equations. Optimal liquidation rules are characterized by a finite difference method. A numerical example is given to illustrate the result.
regime switching;optimal selling rule;optimal control;HJB equation;viscosity solution
- W. H. Fleming and R. Rishel, Deterministic and Stochastic Optimal Control, Springer-Verlag, New York, 1975.
- W. H. Fleming and H. M. Soner, Controlled Markov Processes and Viscosity solutions, Springer-Verlag, New York, 2005.
- X. Guo and Q. Zhang, Closed-form solutions for perpetual american put options with regime switching, SIAM J. Appl. Math. 64 (2004), no. 6, 2034-2049. https://doi.org/10.1137/S0036139903426083
- E. R. Jakobsen and K. H. Karlsen, A "maximum principle for semicontinuous functions" applicable to integro-partial differential equations, NoDEA Nonlinear Differential Equations Appl. 13 (2006), no. 2, 137-165. https://doi.org/10.1007/s00030-005-0031-6
- B. Oksendal, Stochastic differential Equations, 6th ed., Springer-Verlag, New York, 2005.
- M. Pemy and Q. Zhang, Optimal stock liquidation in a regime switching model with finite time horizon, J. Math. Anal. Appl. 321 (2006), no. 2, 537-552. https://doi.org/10.1016/j.jmaa.2005.08.034
- M. Pemy, Q. Zhang, and G. Yin, Liquidation of a large block of stock, J. Bank Finance 31 (2007), 1295-1305. https://doi.org/10.1016/j.jbankfin.2006.10.014
- Q. Zhang, Stock trading: An optimal selling rule, SIAM J. Control Optim. 40 (2001), no. 1, 64-87. https://doi.org/10.1137/S0363012999356325
- Q. Zhang, G. Yin, and R. H. Liu, A near-optimal selling rule for a two-time-scale market model, Multiscale Model. Simul. 4 (2005), no. 1, 172-193. https://doi.org/10.1137/040606338
- X. Y. Zhou, Verification theorems within the framework of viscosity solutions, J. Math. Anal. Appl. 177 (1993), no. 1, 208-225. https://doi.org/10.1006/jmaa.1993.1253