When to Sell an Asset Where Its Drift Drops from a High Value to a Smaller One

Author(s)
Pham Van Khanh

ABSTRACT

To solve the selling problem which is resembled to the buying problem in [1], in this paper we solve the problem of determining the optimal time to sell a property in a location the drift of the asset drops from a high value to a smaller one at some random change-point. This change-point is not directly observable for the investor, but it is partially observable in the sense that it coincides with one of the jump times of some exogenous Poisson process representing external shocks, and these jump times are assumed to be observable. The asset price is modeled as a geometric Brownian motion with a drift that initially exceeds the discount rate, but with the opposite relation after an unobservable and exponentially distributed time and thus, we model the drift as a two-state Markov chain. Using filtering and martingale techniques, stochastic analysis transform measurement, we reduce the problem to a one-dimensional optimal stopping problem. We also establish the optimal boundary at which the investor should liquidate the asset when the price process hit the boundary at first time.

KEYWORDS

Optimal Stopping Time, Posterior Probability, Threshold, Markov Chain, Jump Times, Martingale, Brownian Motion

Optimal Stopping Time, Posterior Probability, Threshold, Markov Chain, Jump Times, Martingale, Brownian Motion

Cite this paper

Van Khanh, P. (2015) When to Sell an Asset Where Its Drift Drops from a High Value to a Smaller One.*American Journal of Operations Research*, **5**, 514-525. doi: 10.4236/ajor.2015.56040.

Van Khanh, P. (2015) When to Sell an Asset Where Its Drift Drops from a High Value to a Smaller One.

References

[1] Khanh, P. (2014) Optimal Stopping Time to Buy an Asset When Growth Rate Is a Two-State Markov Chain. American Journal of Operations Research, 4, 132-141.

http://dx.doi.org/10.4236/ajor.2014.43013

[2] Khanh, P. (2012) Optimal Stopping Time for Holding an Asset. American Journal of Operations Research, 4, 527-535.

http://dx.doi.org/10.4236/ajor.2012.24062

[3] Peskir, G. and Shiryaev, A.N. (2006) Optimal Stopping and Free-Boundary Problems (Lectures in Mathematics ETH Lectures in Mathematics. ETH Zürich (Closed)). Birkhäuser, Basel.

[4] Shiryaev, A.N., Xu, Z. and Zhou, X.Y. (2008) Thou Shalt Buy and Hold. Quantitative Finance, 8, 765-776.

http://dx.doi.org/10.1080/14697680802563732

[5] Guo, X. and Zhang, Q. (2005) Optimal Selling Rules in a Regime Switching Model. IEEE Transactions on Automatic Control, 9, 1450-1455.

http://dx.doi.org/10.1109/TAC.2005.854657

[6] Lipster, R.S. and Shiryaev, A.N. (2001) Statistics of Random Process: I. General Theory. Springer-Verlag, Berlin, Heidelberg.

[7] Shiryaev, A.N. (1978) Optimal Stopping Rules. Springer Verlag, Berlin, Heidelberg.

[1] Khanh, P. (2014) Optimal Stopping Time to Buy an Asset When Growth Rate Is a Two-State Markov Chain. American Journal of Operations Research, 4, 132-141.

http://dx.doi.org/10.4236/ajor.2014.43013

[2] Khanh, P. (2012) Optimal Stopping Time for Holding an Asset. American Journal of Operations Research, 4, 527-535.

http://dx.doi.org/10.4236/ajor.2012.24062

[3] Peskir, G. and Shiryaev, A.N. (2006) Optimal Stopping and Free-Boundary Problems (Lectures in Mathematics ETH Lectures in Mathematics. ETH Zürich (Closed)). Birkhäuser, Basel.

[4] Shiryaev, A.N., Xu, Z. and Zhou, X.Y. (2008) Thou Shalt Buy and Hold. Quantitative Finance, 8, 765-776.

http://dx.doi.org/10.1080/14697680802563732

[5] Guo, X. and Zhang, Q. (2005) Optimal Selling Rules in a Regime Switching Model. IEEE Transactions on Automatic Control, 9, 1450-1455.

http://dx.doi.org/10.1109/TAC.2005.854657

[6] Lipster, R.S. and Shiryaev, A.N. (2001) Statistics of Random Process: I. General Theory. Springer-Verlag, Berlin, Heidelberg.

[7] Shiryaev, A.N. (1978) Optimal Stopping Rules. Springer Verlag, Berlin, Heidelberg.