Abstract
In this paper we consider the problem of determining the optimal time to buy an asset in a position of an uptrend or downtrend in the financial market and currency market as well as other markets. Asset price is modeled as a geometric Brownian motion with drift being a two-state Markov chain. Based on observations of asset prices, investors want to detect the change points of price trends as accurately as possible, so that they can make the decision to buy. Using filtering techniques and stochastic analysis, we will develop the optimal boundary at which investors implement their decisions when the posterior probability process reaches a certain threshold.
Highlights
In [1], the authors consider the problem of determining the optimal time to sell a property while price growth rate is a random variable that takes the value of the given set
Under the assumptions of the problem considered in [1], growth rate only gets one of the possible values that do not change from this value to other values, which means that transition probability density is 0; but at a time I do not know the accuracy of price growth rate and the probability of receiving a certain value of growth rate changes over time
The method we use to study in this paper is the martingale theory, change of measure and the optimal stopping time is referred in the literature [2] [5] and [6]
Summary
In [1], the authors consider the problem of determining the optimal time to sell a property while price growth rate is a random variable that takes the value of the given set. The authors in [3] consider the optimal stopping time problem when the growth rate of price process is not a random variable but in many cases. In [4] the authors consider the problem to find the optimal time to sell when the price growth rate is Markov chain, their approach is different from our method in this paper and the results are different. If a person wants to buy an asset, he will wait for the appearance of an opportunity of going down in price and wait for prices to go down further (in momentum) until no longer falling price, he decides to buy According to this way of investment, investors are expected to buy the property at the bottom of market and sell the property at the highest point of the market. The method we use to study in this paper is the martingale theory, change of measure and the optimal stopping time is referred in the literature [2] [5] and [6]
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