  This paper studies the risk-adjusted optimal timing to liquidate an option at
the prevailing market price. In addition to maximizing the expected discounted
return from option sale, we incorporate a path-dependent risk penalty based on
shortfall or quadratic variation of the option price up to the liquidation
time. We establish the conditions under which it is optimal to immediately
liquidate or hold the option position through expiration. Furthermore, we study
the variational inequality associated with the optimal stopping problem, and
prove the existence and uniqueness of a strong solution. A series of analytical
and numerical results are provided to illustrate the non-trivial optimal
liquidation strategies under geometric Brownian motion (GBM) and exponential
Ornstein-Uhlenbeck models. We examine the combined effects of price dynamics
and risk penalty on the sell and delay regions for various options. In
addition, we obtain an explicit closed-form solution for the liquidation of a
stock with quadratic penalty under the GBM model.
