Super-Hedging

Super-hedging is a strategy that hedges positions with a self-financing trading plan. It utilizes the lowest price that can be paid for a portfolio such that its worth will be greater or equal at a set future time.

A hedging transaction limits investment risk of an underlying asset by using options or futures. These are bought in opposing positions to the underlying asset in order to lock in a certain amount of gain. The super-hedging price of a Portfolio A is equivalent to the smallest amount necessary to be paid for an admissible Portfolio B at the current time so that at some specified point in the future the value of B is at least as great as A. In a complete market, the super-hedging price is equivalent to the price for hedging the initial portfolio. In an incomplete market, such as options, the cost of such a strategy may prove too high. The idea of super-hedging has been studied by academics; however, it is a theoretical ideal and is difficult to implement in the real world.

The sub-hedging price is the greatest value that can be paid so that in any possible situation at a specified point in the future, you have a second portfolio worth less or equal to the initial one. The upper and lower bounds created by the sub-hedging and super-hedging prices are the no-arbitrage bounds, an example of good-deal bounds.

The acceptance set (set of acceptable future net worth) for the super-hedging price is the negative of the set of values of a self-financing portfolio at the terminal time.

A self-financing portfolio is an important concept in financial mathematics. A portfolio is self-financing if there is no external infusion or withdrawal of money. In other words, the purchase of a new asset must be financed by the sale of an old one.

A self-financing portfolio is a replicating portfolio. In mathematical finance, a replicating portfolio for a given asset or series of cash flows is a portfolio of assets with the same properties.

Given an asset or liability, an offsetting replicating portfolio is called a hedge (it can be static or dynamic), and constructing such a portfolio (by selling or purchasing) is called hedging (static or dynamic). In practice, replicating portfolios are seldom, if ever, exact replications. Also, there is credit risk, and dynamic replication is imperfect, since actual price movements are not infinitesimal, and transaction costs to change the hedge are not zero.