Partial Hedging in Rough Volatility Models

This paper studies the problem of partial hedging within the framework of rough volatility models in an incomplete market setting. We employ a stochastic control problem formulation to minimize the discrepancy between a stochastic target and the terminal value of a hedging portfolio. As rough volatility models are neither Markovian nor semimartingales, stochastic control problems associated with rough models are quite complex to solve. Therefore, we propose a multifactor approximation of the rough volatility model and introduce the associated Markov stochastic control problem. We establish the convergence of the optimal solution for the Markov partial hedging problem to the optimal solution of the original problem as the number of factors tends to infinity. Furthermore, the optimal solution of the Markov problem can be derived by solving a Hamilton–Jacobi–Bellman equation and more precisely a nonlinear partial differential equation (PDE). Due to the inherent complexity of this nonlinear PDE, an explicit formula for the optimal solution is generally unattainable. By introducing the dual solution of the Markov problem and expressing the primal solution as a function of the dual solution, we derive approximate solutions to the Markov problem using a dual control method. This method allows for suboptimal choices of dual control to deduce lower and upper bounds on the optimal solution as well as suboptimal hedging ratios. In particular, explicit formulas for partial hedging strategies in a rough Heston model are derived.