Dealing with Dangerous Digitals

Options with discontinuous payoffs are generally traded above their theoretical Black–Scholes prices because of the hedging difficulties created by their large delta and gamma values. A theoretical method for pricing these options is to constrain the hedging portfolio and incorporate this constraint into the pricing by computing the smallest initial capital which permits superreplication of the option. We develop this idea for exotic options, in which case the pricing problem becomes one of stochastic control. The high cost of exact super-replication coincides with market price quotations for dangerous derivatives such as reverse knock-out barrier options, which are often higher than their risk-neutral expected payoff (theoretical value). This paper illustrates how the theory of leverage constrained pricing can be successfully applied to compute close-to-market option values and serves as a practitioner’s guide to derive explicit formulae and compute prices by finite difference methods. ∗This article first appeared in Foreign Exchange Risk, Risk Publications, London 2001 †Research supported by Credit Suisse, Swiss Re and UBS through RiskLab, Switzerland. ‡Work supported by grant DMS-01-03814 of the U.S. National Science Foundation. Dealing with Dangerous Digitals 2