Bollinger bands

Used in technical analysis to determine areas of support for and resistance to price changes. On a chart these plot the standard deviation of the moving average of a price. So when they are plotted above and below the moving average, the bands widen and narrow according to the underlying volatility of the average. The longer the period of low volatility, the closer together the lines become and the greater is the likelihood that there will be a break-out from the established price pattern.

Black scholes option pricing model

A pricing model that ranks among the most influential. It was devised by Fischer Black and Myron Scholes, two Chicago academics, in 1973, the year that formalized options trading began on the Chicago board of trade. The Black-Scholes model, or adaptations of it, has gained universal acceptance for pricing options because its results are almost as good as those achieved by other options pricing models without the complexity.

Behind the model is the assumption that asset prices must adjust to prevent arbitrage between various combinations of options and cash on the one hand and the actual asset on the other. Additionally, there are specific minimum and maximum values for an option which are easily observable. Assuming, for example, that it is a call option then its maximum value must be the share price. Even if the exercise price is zero, no one will pay more than the share price simply to acquire the right to buy the shares. The minimum value, meanwhile, will be the difference between the share's price and the option's exercise price adjusted to its present value.

The model puts these fairly easy assumptions into a formula and then adjusts it to account for other relevant factors.

• The cost of money, because buying an option instead of the underlying stock saves money and, therefore, makes the option increasingly valuable the higher interest rates go.
• The time until the option expires, because the longer the period, the more valuable the option becomes since the option holder has more time in which to make a profit.
• The volatility of the underlying share price, because the more it is likely to bounce around, the greater chance the option holder has to make a profit.

Of these, volatility, as measured by the standard deviation of share returns, is the most significant factor. Yet it was the factor over which Black and Scholes struggled because it is not intuitively obvious that greater volatility should equal greater value. That it is so is because of the peculiar nature of options: they peg losses to the amount paid for the option, yet they offer unlimited potential for profit.

Note that the basic Black-Scholes model is for pricing a call option, but it can be readily adapted for pricing a put option. It also ignores the effect on the price of the option of any dividends that are paid on the shares during the period until the option expires. This is remedied either by deducting the likely present value of any dividend from the share price that is input into the model, or by using a refinement of the Black-Scholes model which writes off the effect of the dividend evenly over the period until it is paid.

Beta

A widely used statistic which measures the sensitivity of the price of an investment to movements in an underlying market. In other words, beta measures an investment's price volatility, which is a substitute for its risk. The important point is that beta is a relative, not an absolute, measure of risk. In stock market terms, it defines the relationship between the returns on a share relative to the market's returns (the most commonly used absolute measure of risk is standard deviation). But in so far as much of portfolio theory says that a share's returns will be driven by its sensitivity to market returns, then beta is a key determinant of value in price models for share or portfolio returns.

An investment's beta is expressed as a ratio of the market's beta, which is always 1.0. Therefore a share with a beta of 1.5 would be expected to rise 15% when the market goes up 10% and fall 15% when the market drops 10%. In technical terms, beta is calculated using a least-squared regression equation and it is the coefficient that defines the slope of the regression line on a chart measuring, say, the relative returns of a share and its underlying market. However, the beta values derived from the regression calculation can vary tremendously depending on the data used. A share's beta generated from weekly returns over, say, one year might be very different from the beta produced from monthly returns over five years.

This highlights a major weakness of beta: that it is not good at predicting future price volatility based on past performance. This is certainly true of individual shares. For portfolios of shares beta works far better, basically because the effects of erratically changing betas on individual shares generally cancel each other out in a portfolio. Also, to the extent that portfolio theory is all about reducing risk through aggregating investments, beta remains a useful tool in price modeling.

Basis

In a futures market, basis is defined as the cash price (or spot price) of whatever is being traded minus its futures price for the contract in question. It is important because changes in the relationship between cash and futures prices affect the value of using futures as a hedge. A hedge, however, will always reduce risk as long as the volatility of the basis is less than the volatility of the price of whatever is being hedged.