Discrete Hedging Under Piecewise Linear Risk Management

Abstract

In an incomplete market it is usually impossible to eliminate the intrinsic risk of an option. In this case quadratic risk-minimization is often used to determine a hedging strategy. However, it may be more natural to use piecewise linear risk-minimization since in this case the risk is measured in actual dollars (not dollars squared). We investigate hedging strategies using piecewise linear risk-minimization. We illustrate that piecewise linear risk-minimization often leads to smaller expected total hedging cost and significantly different, possibly more desirable, hedging strategies from those of quadratic risk minimization. The distributions of the total hedging cost and risk show that hedging strategies obtained by piecewise linear risk-minimization have a larger probability of small cost and risk, though they also have a very small probability of larger cost and risk. Comparative numerical results are provided.