WHY HEDGE
'Hedging' in its broadest sense means the reduction of risk by exploiting relationships or correlation between various risky investments. The reason for 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 there are two portfolios with the same expected return the one with the lower risk is the better investment.
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 both are 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 assets change in value. Another example is spot-forward parity. In neither case do the dynamics of the asset have to be specified, 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 the model for the underlying asset. The obvious example is the hedging used in the Black-Scholes analysis that leads to whole theory for the value of derivatives. In pricing derivatives, typically it is necessary to at least know the volatility of the underlying asset. If the model is wrong then the option value and any hedging strategy will 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 of 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.
GAMMA HEDGING
To reduce the size of each rehedge and/or to increase the time between hedges, 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. Typically, one hedges one exotic contract with a vanilla contract and the underlying. The quantities of the underlying are chosen so as to make both the portfolio delta and the portfolio gamma zero.
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 or even estimate. 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 the portfolio value is insensitive to this parameter it is possible to 'vega hedge.' This means hedging 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. This is often quite satisfactory in practice but is usually theoretically inconsistent; it is important not to use a constant volatility (basic Black-Scholes) model to calculate sensitivities to parameters (volatility, dividend yield, interest rate) that are assumed not to vary. The distinction between variables (underlying asset price and time) and parameters is extremely important here. It is justifiable to rely on sensitivities of prices to variables, but usually not sensitivity to parameters. To get around this problem it is possible to independently model volatility etc. as variables themselves. In such a way it is possible to build up a consistent theory.
STATIC HEDGING
There are quite a few problems with delta hedging, on both the practical and theoretical side. In practice, hedging must be done at discrete times and is costly. Sometimes one has to buy and 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 the parameter values are not known accurately. Delta hedging alone leaves the portfolio very exposed to the model, this is model risk. Many of these problems can be reduced or eliminated through a strategy of 'static hedging' as well as delta hedging: buy or sell more liquid traded contracts to reduce the cashflows in the original contracts. 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, and a model is not needed.
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. Recent examples where margin has caused significant damage are Metallgesellschaft and Long-Term Capital Management. Writing options is very risky. The downside of buying an option is just the initial premium, the upside may be unlimited. The upside of writing an option is limited, but the downside could be huge. For this reason, to cover the risk of default in the event of an unfavorable outcome, the clearing houses that register and settle options insist on the deposit of a margin by the writers of options. Margin comes in two forms, the initial margin and the maintenance margin. The initial margin is the amount deposited at the initiation of the contract. The total amount held as margin must stay above a prescibed maintenance margin. If it ever falls below this level then more money (or equivalent in bonds, stocks etc.) must be deposited. The amount of margin that must be deposited depends on the particular contract. A dramatic market move could result in a sudden large margin call that may be difficult to meet. 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 associateed margin requirements and so won't appear in the calculation.
CRASH (PLATINUM) HEDGING
The final variety of hedging to discuss 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 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 difficult-to-measure 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.
This week's Learning Curve was written by Philip Hua,v.p. corporate portfolio management at Bankers Trustin London and Paul Wilmott, a Royal Society Universityresearch fellow at Oxford UniversityandImperial College, London.
