Introduction

At present there is much uncertainty in the choice of the pricing measure for the hedging of derivatives in incomplete markets. Incompleteness can arise for instance in the presence of stochastic volatility, as will be studied in the following. This chapter provides comparative numerical results for two important hedging methodologies, namely local risk -minimisation and global mean-variance hedging.

We first describe the theoretical framework that underpins these two approaches. Some comparative studies are then presented on expected squared total costs and the asymptotics of these costs, differences in prices and optimal hedge ratios. In addition, the density functions for squared total costs and proportional transaction costs are estimated as well as mean transaction costs as a function of hedging frequency. Numerical results are obtained for variations of the Heston and the Stein–Stein stochastic volatility models.

To produce accurate and reliable estimates, combinations of partial differential equation and simulation techniques have been developed that are of independent interest. Some explicit solutions for certain key quantities required for mean-variance hedging are also described. It turns out that mean-variance hedging is far more difficult to implement than what has been attempted so far for most stochastic volatility models. In particular the mean -variance pricing measure is in many cases difficult to identify and to characterise. Furthermore, the corresponding optimal hedge, due to its global optimality properties, no longer appears as a simple combination of partial derivatives with respect to state variables. It has more the character of an optimal control strategy.

The importance of this chapter is that it documents for some typical stochastic volatility models some of the quantitative differences that arise for two major hedging approaches.