Robustly Hedging Variable Annuities with Guarantees Under Jump and Volatility Risks

Accurately quantifying and robustly hedging options embedded in the guarantees of variable annuities is a crucial task for insurance companies in preventing excessive liabilities. Due to sensitivities of the benefits to tails of the account value distribution, a simple Black-Scholes model is inadequate. A model which realistically describes the real world price dynamics over a long time horizon is essential for the risk management of the variable annuities. In this paper, both jump risk and volatility risk are considered for risk management of lookback options embedded in guarantees with a ratchet feature. We evaluate relative performances of delta hedging and dynamic discrete risk minimization hedging strategies. Using the underlying as the hedging instrument, we show that, under a Black-Scholes model, local risk minimization hedging is significantly better than delta hedging. In addition, we compare risk minimization hedging using the underlying with that of using standard options. We demonstrate that, under a Merton’s jump diffusion model, hedging using standard options is superior to hedging using the underlying in terms of the risk reduction. Finally we consider a market model for volatility risks in which the at-the-money implied volatility is a state variable. We compute risk minimization hedging by modeling at-the-money Black-Scholes implied volatility explicitly; the hedging effectiveness is evaluated, however, under a joint underlying and implied volatility model which also includes instantaneous volatility risk. Our computational results suggest that, when implied volatility risk is suitably modeled, risk minimization hedging using standard options, compared to hedging using the underlying, can potentially be more effective in risk reduction under both jump and volatility risks. ∗The authors would like to thank their colleagues Peter Mansfield and Shirish Chinchalkar for many fruitful discussions.