The right time to sell a stock whose price is driven by Markovian noise

We consider the problem of seeking the optimal time to sell a stock, subject to a fixed sale cost and an exponential discounting rate ρ. We assume that the price of the stock fluctuates according to the equation dYt = Yt (μdt+σ ξ(t) dt), where (ξ(t)) is an alternating Markov renewal process with values in {±1}, with an exponential renewal time. We determine the critical value of ρ under which the value function is finite. We examine the validity of the “principle of smooth fit,” and use this to give a complete and essentially explicit solution to the problem, which exhibits a surprisingly rich structure. The corresponding result when the stock price evolves according to the Black and Scholes model is obtained as a limit case. Abbreviated title: The right time to sell a stock MSC 2000 Subject Classifications. Primary 60G40; Secondary 90A09, 60J27.