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

Abstract

We consider the problem of finding the optimal time to sell a stock, subject to a fixed sales cost and an exponential discounting rate ρ. We assume that the price of the stock fluctuates according to the equation dY_t=Y_t(μ dt+σξ(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.