The Risk of Optimal, Continuously Rebalanced Hedging Strategies and It's Efficient Evaluation via Fourier Transform

Abstract

This paper derives a closed-form formula for the hedging error of optimal and continuously rebalanced hedging strategies in a model with leptokurtic IID returns and, in contrast to the standard Black-Scholes result, shows that continuous hedging is far from riskless even in the absence of transaction costs. Our result can be seen as an extension of the Capital Asset Pricing Model and the Arbitrage Pricing Theory, allowing for intertemporal risk diversification. The paper provides an efficient implementation of the optimal hedging strategy and of the hedging error formula via fast Fourier transform and demonstrates their speed and accuracy. We compute the size of hedging errors for individual options based on the historical distribution of returns on FT100 equity index as a function of moneyness and time to maturity. The resulting option price bounds are found to be non-trivial, and largely insensitive to model parameters, while the optimal hedging strategy remains virtually identical to the standard Black-Scholes delta hedge. Thus, with leptokurtic returns Black-Scholes price is the right value to hedge towards, but not the right value to price at.