Exact Superreplication Strategies for a Class of Derivative Assets

A superreplicating hedging strategy is commonly used when delta hedging is infeasible or is too expensive. This article provides an exact analytical solution to the superreplication problem for a class of derivative asset payoffs. The class contains common payoffs that are neither uniformly convex nor concave. A digital option, a bull spread, a bear spread, and some portfolios of bull spreads or bear spreads, are all included as special cases. The problem is approached by first solving for the transition density of a process that has a two-valued volatility. Using this process to model the underlying asset and identifying the two volatility values as σmin and σmax, the value function for any derivative asset in the class is shown to solve the Black-Scholes-Barenblatt equation. The subreplication problem and several related extensions, such as option pricing with transaction costs, calculating superreplicating bounds, and superreplication with multiple risky assets, are also addressed.