Abstract
We study the problem of selling an illiquid asset in a market in which unfavorable bid prices are readily available and favorable offers enter the market less frequently. We setup a liquidity model in which the liquidity premium implicit in the bid prices evolves stochastically and the rate of arrival of favorable offers is governed by a regime-shifting Markov process. The objective is to maximize the expected utility of the proceeds received from the sale of the illiquid asset. We formulate this problem as a multidimensional optimal stopping time problem with random maturity. We characterize the objective function as the unique viscosity solution of the associated system of variational Hamilton–Jacobi–Bellman inequalities. We derive explicit solutions in the case of power and logarithmic utility functions when the liquidity premium factor follows a mean-reverting CIR process.
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