It is better to be approximately right rather than be precisely wrong.
Beta has plenty of trading applications. To construct a portfolio of any financial instrument you must know its volatility with respect to the market.
Otherwise, it is simply impossible to understand its likely performance and risk profiles. More practically, in order to construct a portfolio that will produce a desired risk/reward profile, we need to understand beta.(See box: The anatomy of risk variables)
When we're relating vanilla equity performance to a market Index, it is easy enough in theory. (In practice of course, equity betas can change with surprising rapidity.) A high-beta stock carries more risk; a positive beta stock moves in the same direction as the market.
We can play all sorts of games to try and reduce risk without reducing returns. However, matters become more complex when we're relating a portfolio that has derivative elements to a market Index.
In basic terms, a derivative has sensitivity to the underlying instrument. The underlying may have some sort of beta relationship to the Index. We must understand both these relationships in order to exploit beta while using derivatives.
The sensitivity of a derivative to the underlying is usually described in terms of two Greek letters, delta and gamma. Delta is the derivative equivalent of beta, the direct linear change imposed on a derivative's value by every change in the underlying. (We are referring to the premium here when we say "derivative's value" although the position may be "in-the-money").
Suppose that the underlying instrument is a Satyam stock. Let us suppose that the delta is 2. This means that for small changes in Satyam prices, the derivative return is about the same as two Satyam shares. The portfolio in question could actually be Satyam stock, or a derivative position in Satyam, or some combination of stock and derivative.
Notice that we are talking of small changes. The profit curves of derivatives are frequently bounded within a narrow range and change drastically outside that. Delta tells us the direction of return in just the same fashion that the beta does. Gamma tells us the shape of the return.
Does the return increase, flatten, or decrease as the underlying rises? In terms of calculus, delta is the first derivative of a portfolio value with respect to the underlying and gamma is the second derivative with respect to the underlying.
If we map derivative returns versus the underlying, an upward-sloping curve (where the derivative return increases as the underlying increases) is positive gamma while a downward slope (where the return flattens at a maximum value and then declines) is negative gamma. Short positions are often negative gamma.
There are other important measures of derivative sensitivity. Vega is a measure of the changes in the premium compared to changes in implied volatility. In general, long positions benefit from a rise in implied volatility. And we must bear in mind the impact of interest rate changes (rho) and the time decay element (theta) that eventually destroy portfolio values.
So much for mathematics. Let's revert to the practical trader who is trying to extract maximum returns. How can he use his knowledge of beta and derivatives to achieve this aim? The disadvantage of using a derivative portfolio is time-decay.
However, there is the obvious advantage of great leverage and also flexibility. For example, let's assume a trader is looking for a high-beta return. The simplest derivative route is to pick high-beta underlyings and then construct a portfolio containing high-delta, positive-gamma derivatives of those underlyings. Such a portfolio will offer very high returns if the trader has correctly judged the direction.
Vega can also be exploited to add to returns. If the implied volatility is low and expected to rise, the trader should buy high-vega options to create a long position. Or else, he should be creating short positions, if he expects implied volatility to fall.
The derivative trader can also look for a high-beta return through other routes. He does not necessarily need to pick high-beta underlyings. Instead, he could look for high-delta positions in low-beta underlyings.
Or even negative-delta positions in a negative beta underlying. By cleverly using an understanding of the Greeks, beta and margining systems, it is sometimes possible to construct portfolios that could always yield positive returns regardless of market direction - these are called market-neutral strategies.
Typically, this can involve selling an option and using the realised premium to buy an opposing position in something else.
A rigorous formal mathematical relationship for these positions can be very difficult to develop. However, in these circumstances it is better to be approximately right rather than be precisely wrong. The trader is always working with historical patterns of premium movements to the underlying when he is trying to calculate these relationships.
Those patterns could always change and it is silly to try and work things out to the second decimal.
