Hedging of American Options in Illiquid Markets with Price Impacts

We consider a setup in which a large trader has sold a number of American-style derivatives and can have an impact on prices by trading the underlying asset for hedging purposes. The price impacts are assumed to be temporary and decay exponentially with time. Due to the impact of trading on prices, the large trader may also be tempted to minimize the payoff of the derivative by manipulating the underlying asset. Since the option holders have the right to exercise the option at any time before expiry, we consider a robust optimization problem for the large trader, in which the underlying uncertainty is the exercise time. It is shown that the solution of this optimization problem can be described as the solution of a double obstacle variational inequality. The optimal strategy for the large trader and the worst-case exercise time for the option holder are obtained explicitly in terms of the value function. We conclude with a sensitivity analysis in which we compare the timing and size of trades by the large trader as well as the exercise region for the options holders for different levels of liquidity, and identify situations that may lead to potential price manipulation