I study dynamic hedging for variable annuities under basis risk. Basis risk, which arises from the imperfect correlation between the underlying fund and the proxy asset used for hedging, has a highly negative impact on the hedging performance. I investigate whether the choice of a suitable hedging strategy can help to reduce the risk for the insurance company. Comparing several cross-hedging strategies that only trade in the proxy asset, I observe very similar hedging performances. Particularly, I find that well-established but complex strategies from mathematical finance do not outperform simple and naive approaches. Nonetheless, a naive delta-gamma hedge, which additionally uses plain vanilla options on the proxy asset, performs better than all tested one-instrument strategies. A more substantial risk reduction could, however, be achieved by diversification.
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