When are Static Superhedging Strategies Optimal?

This paper deals with the superhedging of derivatives on incomplete markets, i.e.with portfolio strategies which generate payoffs at least as high as that of a givencontingent claim. The simplest solution to this problem is in many cases a staticsuperhedge, i.e. a buy-and-hold strategy generating an affine-linear payoff.We study whether a superhedge can be achieved with less initial capital if we alsoallow for dynamic trading strategies. The answer to this question depends on thekind of the non-traded risk factors. Our main findings for a stochastic volatilitymodel with unbounded volatility show that there is always an optimal static su-perhedge. Additionally, there may be infinitely many optimal dynamic superhedgeswhich require the same initial capital. In a model with stochastic jumps, it is alwayseither a dynamic or a static strategy which is optimal, but never both. In a modelwith a stochastic short rate the properties of the interest rate process are also rele-vant. When there are no bounds for the interest rate optimal superhedges (if theyexist) are always static, since the strategy will never contain an investment in themoney market account. On the other hand, when interest rates are either boundedor non-negative either a static or a dynamic strategy is optimal, depending on therespective contingent claim.Our results have important implications for the design of superhedges as they showunder which conditions we can restrict the analysis to static strategies. There is nosuch thing as the incomplete market when it comes to superhedging. Although incontinuous-time models the class of possible trading strategies contains much moreelements than just static strategies, there is a number of cases where buy-and-holdis as good as or even superior to dynamic strategies.