Pricing and Hedging Variable Annuity Guarantees with Multiasset Stochastic Investment Models

Abstract

Variable annuities are often sold with guarantees to protect investors from downside investment risk. The majority of variable annuity guarantees are written on more than one asset, but in practice, single-asset (univariate) stochastic investment models are mostly used for pricing and hedging these guarantees. This practical shortcut may lead to problems such as basis risk. In this article, we contribute a multivariate framework for pricing and hedging variable annuity guarantees. We explain how to transform multivariate stochastic investment models into their risk-neutral counterparts, which can then be used for pricing purposes. We also demonstrate how dynamic hedging can be implemented in a multivariate framework and how the potential hedging error can be quantified by stochastic simulations.

Acknowledgments

The authors would like to thank Professor Sheldon Lin and two anonymous referees for their comments on earlier drafts of this article. This work is supported by a Discovery Grant from the Natural Science and Engineering Research Council of Canada.

Notes

By natural filtration we mean the information up to and including time t.

It would be numerically very difficult to ensure that H _t is a covariance matrix if its structure is specified arbitrarily. It is typical to use an eigenvalue–eigenvector decomposition to ensure the positive definiteness of a matrix. As the dimension of the matrix increases, such a procedure becomes increasingly time-consuming and unstable.

We refer readers to Appendix A of Panjer et al. (1998) for definitions of a Radon-Nikodym derivative and a measurable event.

Siu et al. (2004) used similar arguments in deriving a univariate risk-neutral GARCH model.

We use crude Monte Carlo in our calculations. However, variance reduction methods described in Glasserman (2003) may be used.

A symmetric first difference approximation is used in our calculations.

The value of the hedge is calculated from the risk-neutral CCC-GARCH model, while the sample paths of the log returns are simulated from the CCC-GARCH model under the real-world probability measure.