Optimal Derivative Liquidation Timing Under Path-Dependent Risk Penalties

TL;DR

This paper investigates the optimal timing for liquidating options considering path-dependent risk penalties, providing analytical and numerical insights into strategies under different stochastic models.

Contribution

It introduces a framework incorporating path-dependent risk penalties into optimal liquidation timing and derives explicit solutions and conditions for various models.

Findings

Optimal liquidation strategies depend on price dynamics and risk penalties.

Explicit solutions are obtained for stock liquidation with quadratic penalties under GBM.

Numerical results illustrate complex sell and delay regions influenced by risk considerations.

Abstract

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.