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 given contingent claim. The simplest solution to this problem is in many cases a static superhedge, 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 also allow for dynamic trading strategies. The answer to this question depends on the kind of the non-traded risk factors. Our main findings for a stochastic volatility model with unbounded volatility show that there is always an optimal static superhedge. Additionally, there may be infinitely many optimal dynamic superhedges which require the same initial capital. In a model with stochastic jumps, it is always either a dynamic or a static strategy which is optimal, but never both. In a model with a stochastic short rate the properties of the interest rate process are also relevant. When there are no bounds for the interest rate optimal superhedges (if they exist) are always static, since the strategy will never contain an investment in the money market account. On the other hand, when interest rates are either bounded or non-negative either a static or a dynamic strategy is optimal, depending on the respective contingent claim. Our results have important implications for the design of superhedges as they show under which conditions we can restrict the analysis to static strategies. There is no such thing as the incomplete market when it comes to superhedging. Although in continuous-time models the class of possible trading strategies contains much more elements than just static strategies, there is a number of cases where buy-and-hold is as good as or even superior to dynamic strategies