Static Benefits Breaking Barriers

as a way to hedge barrier options when the underlying is a futures price with no drift. Derman, Ergener & Hani (1994) relaxed this drift restriction by introducing an algorithm for hedging barrier options in a bi-nomial model, using options with a single strike but multiple expiries. By contrast, this article provides explicit formulas for static hedges in the standard Black-Scholes (1973) model, using options with the same expiry but multiple strikes. Exotic options are very sophisticated instruments , and the techniques used to hedge and value them are fairly complex. The most popular are barrier options, which were introduced in the US over-the-counter markets years before vanilla options were listed (see Snyder, 1969). Barrier options are special cases of barrier securities, which may involve single or multiple barriers. Examples of the latter include double barrier options, rolldown calls and lookback options. For simplicity, we will focus on single barrier securities in this article. These allow for an arbitrary payout at maturity provided that the barrier has been touched (in-barrier securities) or not touched (out-barrier securities). They are usually further classified into " down securities " (barrier below spot) and " up securities " (barrier above spot). The standard methodology for hedging and valuing barrier securities applies dynamic replication strategies in the underlying assets. We hope to add insight into these structures by looking at them in another way. In particular , we show how barrier securities can be broken up into more fundamental securities, which, in turn, can be created out of vanilla European options. This allows us to hedge path-dependent barrier securities with path-independent vanilla options, with trading in the vanilla options occurring only at the initiation and expiry 1 of the hedge. Due to the relative infrequency of trading, such hedges are commonly termed " static ". In common with dynamic hedging, static hedging provides valuation formulas for barrier securities. When both types of hedging strategies are cast in the same economic model, the formulas result in identical values. We will show how valuation formulas based on dynamic replication can be used to uncover the static hedge. Since barrier option formulas have been around since Merton's seminal 1973 paper, they are now widely available (see Nelken, 1995, and Zhang, 1995, for surveys on exotic options). These formulas can be used to determine static hedges for a host of exotic options beyond those explicitly presented in this paper, …