Optimal strategies for options on target volatility funds

We deal with a stochastic optimal control problem rising from hedging the risky securities underlying a target volatility strategy, a portfolio whose asset-allocation is adjusted to maintain the realized volatility of the portfolio at a certain level. We consider the point of view of a derivative writer selling an option contract as protection to a portfolio manager on the invested capital. The uncertainty in the risky portfolio composition along with the difference in hedging costs of its components requires to adjust the protection price to include these costs in the worst-case scenario for the seller. We derive an analytical solution of the problem in a Black and Scholes scenario. Then, we use Reinforcement Learning techniques to determine the fund composition leading to the optimal policy under the local volatility model, for which an a priori solution is not available. We show how the performances of the numerical solution are compatible with those obtained by applying path-wise the analytical solution previously derived.