Hedging Portfolios Of Derivatives Securities With Maximin Strategies

In this paper I introduce maxmin portfolios as hedging strategies in contingent claims pricing models. I show that portfolios of derivatives securities can be better hedged with maxmin strategies, and I examine the relation among maxmin strategies, {\it Greeks} hedging, and value-at-risk techniques. Maxmin portfolios guarantee the highest value of any portfolio for a given hedging period and for a given set of changes into state variables. I show that maxmin portfolios solve a continuous linear semi-infinite program that can be computed using a simplex algorithm. When the hedging period shortens, maxmin portfolios converge to standard {\it Greeks} hedging strategies, but when the hedging period lengthens, the two strategies exhibit significant differences. This is illustrated hedging a "plain vanilla" European call option. In the standard Black-Scholes-Merton model, the maxmin strategy is close to {\it Delta(-Gamma)} hedging, but in a model with stochastic volatility, maxmin hedging takes into account both sources of uncertainty whereas {\it Greeks} hedging has no rationale to choose between standard {\it Delta-Gamma} and {\it Delta-Vega}. Consequently, maxmin portfolios introduce a rationale for hedging and take into account the hedging period. Maxmin strategies can be used to hedge complex portfolios that include derivative securities. Finally, since maxmin strategies focus on worst-case scenarios, they are intuitively related to optimal value-at-risk techniques.