Hedging in affine stochastic volatility models

Typically an investor incurs risk by issuing a contingent claim. She can try to reduce this risk by trading in the underlying asset according to a strategy which is in some sense appropriate. In an incomplete financial market there usually are several meaningful choices for the determination of such a hedging strategy. We discuss two of them based on optimization criteria: asymptotic exponential utility-based hedging and variance-optimal hedging. We investigate these approaches under the assumption that the underlying can be described by subclasses of affine processes in each case. These classes contain some popular affine stochastic volatility models, e.g. the Barndorff-Nielsen & Shephard model and the Heston model. Furthermore they allow for models which exhibit jumps in the return process as well as in the volatility component. For both hedging approaches we obtain semi-explicit expressions for the objects of interest (i.e. first order approximations of the optimal hedging strategy and the associated utility indifference price for asymptotic utility-based hedging as well as the optimal hedging strategy, optimal initial endowment and the corresponding variance of the hedging error for quadratic hedging). We mainly use general structural results, a Laplace transform technique and semimartingale calculus in order to derive these findings. For the exponential utility-based hedging approach we additionally require a characterization of the the density process of the minimal entropy martingale measure and the associated solution to the exponential utility maximization problem without additional claims. We provide these in the more general framework of multivariate affine processes. Finally, some numerical examples illustrate the results of the two hedging approaches by computing the relevant quantities in concrete stochastic volatility models.