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

Robert C. Dalang and M. -O. Hongler

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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 \rho. We assume that the price of the stock fluctuates according to the equation dY_t=Y_t(\mu dt+\sigma\xi(t) dt), where (\xi(t)) is an alternating Markov renewal process with values in {\pm1}, with an exponential renewal time. We determine the critical value of \rho 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.

Date: 2005-03
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Published in Annals of Applied Probability 2004, Vol. 14, No. 4, 2176-2201

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