Hedging in its broadest sense means the reduction of risk by exploiting relationships or correlation (or lack of correlation ) between various risky investments. The purpose behind hedging is that it can lead to an improved risk/return. In the classical Modern Portfolio Theory framework, for example, it is usually possible to construct many portfolios having the same expected return but with different variance of returns (risk ). Clearly, if you have two portfolios with the same expected return the one with the lower risk is the better investment.

To help understand why one hedges it is useful to look at the different types of hedging.

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The two main classifications

Probably the most important distinction between types of hedging is between model-independent and model-dependent hedging strategies.

• Model-independent hedging: An example of such hedging is put-call parity. There is a simple relationship between calls and puts on an asset (when they are both European and with the same strikes and expiries ), the underlying stock and a zero-coupon bond with the same maturity. This relationship is completely independent of how the underlying asset changes in value. Another example is spot-forward parity. In neither case do we have to specify the dynamics of the asset, not even its volatility. to find a possible hedge. Such model-independent hedges are few and far between.

• Model-dependent hedging: Most sophisticated finance hedging strategies depend on a model for the underlying asset. The obvious example is the hedging used in the Black-Scholes analysis that leads to a whole theory for the value of derivatives. In pricing derivatives we typically need to at least know the volatility of the underlying asset. If the model is wrong then the option value and any hedging strategy could also be wrong.

Delta hedging

One of the building blocks of derivatives theory is delta hedging. This is the theoretically perfect elimination of all risk by using a very clever hedge between the option and its underlying. Delta hedging exploits the perfect correlation between the changes in the option value and the changes in the stock price. This is an example of dynamic hedging ; the hedge must be continually monitored and frequently adjusted by the sale or purchase of the underlying asset. Because of the frequent rehedging, any dynamic hedging strategy is going to result in losses due to transaction costs. In some markets this can be very important. If a portfolio is being delta hedged on a frequent basis, then it is said to be delta-neutral, in other words the portfolio will not lose money whatever the sign or magnitude of the delta.

Gamma hedging

To reduce the size of each rehedge and/or to increase the time between rehedges, and thus reduce costs, the technique of gamma hedging is often employed. A portfolio that is delta hedged is insensitive to movements in the underlying as long as those movements are quite small. There is a small error in this due to the convexity of the portfolio with respect to the underlying. Gamma hedging is a more accurate form of hedging that theoretically eliminates these second-order effects. If a portfolio is being delta hedged on a frequent basis, then it is said to be gamma-neutral, in other words the portfolio will not lose money whatever the sign or magnitude of the gamma.

Vega hedging

The prices and hedging strategies are only as good as the model for the underlying. The key parameter that determines the value of a contract is the volatility of the underlying asset. Unfortunately, this is a very difficult parameter to measure. Nor is it usually a constant as assumed in the simple theories. Obviously, the value of a contract depends on this parameter, and so to ensure that a portfolio value is insensitive to this parameter we can vega hedge. This means that we hedge one option with both the underlying and another option in such a way that both the delta and the vega. the sensitivity of the portfolio value to volatility, are zero.

Static hedging

There are quite a few problems with delta hedging. on both the practical and the theoretical side. In practice, hedging must be done at discrete times and is costly. Sometimes one has to buy or sell a prohibitively large number of the underlying in order to follow the theory. This is a problem with barrier options and options with discontinuous payoff. On the theoretical side, the model for the underlying is not perfect, at the very least we do not know parameter values accurately. Delta hedging alone leaves us very exposed to the model, this is model risk. Many of these problems can be reduced or eliminated if we follow a strategy of static hedging as well as delta hedging ; buy or sell more liquid traded contracts to reduce the cashflows in the original contract. The static hedge is put into place now, and left until expiry. In the extreme case where an exotic contract has all of its cashflows matched by cashflows from traded options then its value is given by the cost of setting up the static hedge; a model is not needed.

Superhedging

In incomplete markets you cannot eliminate all risk by classical dynamic delta hedging. But sometimes you can superhedge meaning that you construct a portfolio that has a positive payoff whatever happens to the market.

Margin hedging

Often what causes banks, and other institutions, to suffer during volatile markets is not the change in the paper value of their assets but the requirement to suddenly come up with a large amount of cash to cover an unexpected margin call. Examples where margin has caused significant damage are Metallgesellschaft and Long Term Capital Management. To prevent this situation it is possible to margin hedge. That is, set up a portfolio such that a margin calls on one part of the portfolio are balanced by refunds from other parts. Usually over-the-counter contracts have no associated margin requirements and so won’t appear in the calculation.

Crash (Platinum) hedging

The final variety of hedging is specific to extreme markets. Market crashes have at least two obvious effects on our hedging. First of all, the moves are so large and rapid that they cannot be traditionally delta hedged. The convexity effect is not small. Second, normal market correlations become meaningless. Typically all correlations become one (or minus one). Crash or Platinum hedging exploits the latter effect in such a way as to minimize the worst possible outcome for the portfolio. The method, called CrashMetrics. does not rely on parameters such as volatilities and so is a very robust hedge. Platinum hedging comes in two types: hedging the paper value of the portfolio and hedging the margin calls.