Multi-period minimax hedging strategies

Option pricing and hedging under transaction costs are of major importance to marketmakers and investors. In this paper we present minimax strategy which determines the optimum number of shares that minimizes the worst-case potential hedging error under transaction costs for the next period. We present two extensions of this strategy. The first extension is the two-period minimax where the worst case is defined over a two-period setting. The objective function of the basic minimax strategy is augmented to include the hedging error for the second period. The second extension is the variable minimax strategy where early rebalancing is triggered by the minimax hedging error. Simulation results suggest that the basic minimax strategy and its two extensions are superior in performance to delta hedging and that the variable minimax strategy is superior to both the basic and the two-period strategies. This result is due to the opportunity provided by the variable minimax strategy to rebalance early. The greatest amount of business for traded options is done for at-the-money options; in this paper, we have concluded that the minimax strategies are particularly suitable for managing the risk of such options. In the Appendix, we present the minimax algorithm used for the implementation of these strategies.