HEDGING UNDER PORTFOLIO CONSTRAINTS BY FACE-LIFTING AND OPTIMAL STOPPING

Nizar Touzi
(joint work with Mete Soner)

Abstract

The problem of super-hedging under portfolio constraints has a well-known beautiful solution in the Black-Scholes framework: the optimal hedging strategy consists in the classical (unconstrained) hedging strategy of a conveniently face-lifted payoff. In the singular control literature the face-lift effect is known as the boundary layer. We show that the solution of the problem of super-hedging under an upper bound constraint on the gamma is of similar type. Considering a lower bound constraint on the gamma turns out to lead to a different type of solution: the value function reduces to an optimal stopping problem, and the hedging strategy is a succession of classical unconstrained hedging and buy-and-hold strategies.