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.
Highlights
In this paper we consider the following problem: How to find the optimal stopping time to sell a stock 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
Denote the drift of the price process at, t ≥ 0, can be modeled as a Markov chain with two states al denoted by state 0 and ah denoted by state 1 such that P= π0 ; P (a = al )= 1− π0 at time 0, al < r < ah where r is discounted rate which is a given constant and process at, t ≥ 0 can only transit from state 1 to state 0 with transition density matrix as follows
The asset price process X is modeled by a geometric Brownian motion with a drift that drops from ah to al at time τ
Summary
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). Optimal Stopping Time, Posterior Probability, Threshold, Markov Chain, Jump Times, Martingale, Brownian Motion
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