Aspects of quadratic utility: mean-variance hedging in rough volatility models, and CAPM-type equilibria

Abstract

The first main topic of this thesis, considered in Chapters I and II, is the study of the mean-variance hedging problem in the rough Heston model. Rough volatility models have become quite popular recently, but the question of hedging in such models is still underexplored. Previous work has focused on perfect hedging in a complete market and on approximate hedging under the risk-neutral measure. We use instead a mean-variance hedging approach under the historical measure, which is more natural for the purposes of risk management. Because the rough volatility process is neither Markovian nor a semimartingale, the rough Heston model poses difficulties to classical techniques in stochastic optimal control. By using the affine structure of the model, we obtain explicit formulas for the optimal mean-variance hedging strategies for a wide class of European-type payoffs, including vanilla call and put options, that can be implemented in practice. We then use those results to find optimal semistatic trading strategies in the underlying asset and a basket of derivatives. The second part of the thesis, developed in Chapters III and IV, pertains to quadratic market equilibria in continuous time. Many classical results on the existence and uniqueness of Radner equilibria such as the capital asset pricing model (CAPM) require the assumption of a complete market. The study of equilibria in incomplete setups is more challenging due to the absence of Pareto optimality. We obtain an explicit equilibrium in an incomplete semimartingale setup with quadratic utilities by using the linearity properties of mean-variance hedging. We then extend our results to mean-variance preferences and find an explicit solution in the linear case. More generally, we show the stability of the mean-variance hedging problem with respect to the quadratic equilibrium price process by using a novel result on the stability of quadratic backward stochastic differential equations under a BMO condition on the stochastic driver and in a continuous filtration. This yields sufficient conditions for the existence of an equilibrium for general mean-variance utility functions via a fixed-point argument.