_{1}

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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.

In this paper we consider the following problem: How to find the optimal stopping time to sell a stock (or an asset) when the expected return of a stock is assumed to be a constant larger than the discount rate up until some random, and unobservable, time τ, at which it drops to a constant smaller than the discount rate.

An investor wants to hold the position as long as the inertia is present by taking advantage of the drift which is exceeding the discounted rate (or interest rate). On the other hand, when the inertia disappears the investor would like to exit the position by selling the asset.

The under study problem in this paper was also addressed in [

The author of [

Some classical optimal stopping time problem has been considered in [

For related studies of stock selling problems, see [

In this paper, the asset price is modeled as a linear Brownian motion with a drift that drops from one constant to a smaller constant at some unobservable time. This drift is modeled as a Markov chain with two states which are denoted by 0 and 1 where 0 is denoted for price decrease and 1 is denoted for price increase.

We define the asset price model in Section 2, and the optimal selling problem is set up. In Section 3, we study the simulation to examine our studies and finally, Section 4 is conclusion.

We take as given a complete probability space

where λ is the intensity of the transition from state 1 to state 0 and assume that λ is positive and that belongs to [0; 1). Denote the drift of the price process a_{t}, t ≥ 0, can be modeled as a Markov chain with two states a_{l} denoted by state 0 and a_{h} denoted by state 1 such that_{l} < r < a_{h} where r is discounted rate which is a given constant and process a_{t}, t ≥ 0 can only transit from state 1 to state 0

with transition density matrix as follows

dent of τ. The asset price process X is modeled by a geometric Brownian motion with a drift that drops from a_{h} to a_{l} at time τ. More precisely,

and

At the time of

where

Find

Similar the buying problem, posterior probability process

or

where

Moreover, in terms of

Processes

Put

We define new process

and a new measure

By Girsanov theorem,

where

The price process

or in term of

The solution of this stochastic equation is

Now we consider the process:

then

Let

Put

From this we have

Denote

then

To solve the problem (2.1) we solve the following auxiliary problem:

Put

The optimal stopping time is the first hitting time of the process

where

Differential equation in (2.3) has the general solution as follows:

Changing variables and using some analytic transformations we obtain:

then

We also have

as

We have

since

Moreover

since

These mean that the function

But

According to (2.3) we have

So B is the solution of the following equation:

Lemma 2.1. The free boundary Equation (2.4) has unique positive solution B.

Proof: The Equation (2.4) is equivalent to

Denote:

The graph of

We have

It follows that

Because

and

we obtain

We will prove that

since

Consequently,

solution on

Theorem 2.2. Stopping time

Proof: Let

and we will prove that

Now, we examine the function

Take the derivative we obtain

This follows

Using the Dynkin’s formula to the process

Because B satisfying

By

We will show that B satisfy the condition:

with positive value will be in continuation area

The optimal stopping time is the first hitting time of

Thus function G satisfy the following condition:

We define the function:

Now, we assume

This contradicts to the existence of

The optimal stopping time

But

by this, we have finished the provement.

To make visual for the above theory we simulate the asset price process, the posterior probability process

process

parameters is used in our simulating are

As can be seen in the figures from 4 to 8 if the price is increasing then the

Another simulation is shown in

In

The same scenario with the simulation in

This research considers the problem of how to find the optimal time to liquidate an asset when the asset price is modeled by the geometric Brownian motion which has a change point. In particular, the drift of the process drops from a high value to a smaller one and this drift process can be modeled as two-state Markov process. The results of this research indicate that a optimal selling decision is made when the probability of downtrend surpassed some certain threshold. We also simulate the price process with a number of parameters and conduct

numerical solution to the experimental selling threshold. In next studies, we will consider problems in which the price growth rate is a Markov process which has more than 2 states and establish some properties as well as distribution of stopping time.

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 10103-2012.17.

PhamVan Khanh, (2015) When to Sell an Asset Where Its Drift Drops from a High Value to a Smaller One. American Journal of Operations Research,05,514-525. doi: 10.4236/ajor.2015.56040