When a stock’s price follows a local volatility model, Peter Carr observed the convex duality relationship between the price of a contingent claim and the value of the delta-hedging portfolio when the payoff of the contingent claim is convex, which leads to significant additional insight in the relationship among the stock dynamics, the delta-hedging process, the value process of the contingent claim and the cash borrowed for constructing the hedging portfolio.

The financial innovations in the past several decades have lead to the creation of many new types of financial derivatives. They become increasingly liquid and, thus, can also be used as hedging devices. It turns out, when hedging a target contingent claim using another hedging claim, a similar relationship also emerges between the value of the target contingent claim and value of the hedging portfolio in terms of generalized duality.

In this talk we explore the ramification of this generalized duality. In particular, the preservation of the generalized convexity of the target contingent claim’s payoff and its various applications.

This is a joint research with Dr. Peter Carr.