Optimal Liquidation of Derivative Portfolios

We consider the problem facing a risk averse agent who seeks to liquidate or exercise a portfolio of (infinitely divisible) perpetual American style options on a single underlying asset. The optimal liquidation strategy is of threshold form and can be characterised explicitly as the solution of a calculus of variations problem. Apart from a possible initial exercise of a tranche of options, the optimal behaviour involves liquidating the portfolio in infinitesimal amounts, but at times which are singular with respect to calendar time. We consider a number of illustrative examples involving CRRA and CARA utility, stocks and portfolios of options with different strikes, and a model where the act of exercising has an impact on the underlying asset price.