Binomial option pricing model

The basic principle behind this and other option pricing models is that an option to buy or sell a specific stock can be replicated by holding a combination of the underlying stock and cash borrowed or lent. The idea is that the cash and security combined can be fairly accurately estimated and their combined value must equal the value of the option. This has to be so, otherwise there would be the opportunity to make risk-free profits by switching between the two.

Take a simple example, the aim of which is to find the value today of a call option on a common stock that expires in one year's time. The current stock price is $100, as is the call's exercise price. To maintain clarity and avoid the complicating effect of an option's delta on the arithmetic involved, imagine that an investor holds just half of this stock (that is, $50-worth) in his portfolio. The portfolio's only other component is a short position in a zero-coupon bond currently worth $42.45, which has to be repaid at $45 in a year's time.

Next assume that the value of the stock in a year's time will be either $110 or $90. From these two postulated outcomes several conclusions arise. First, we can value the call option in a year's time. It will be either $10 or zero. Second, we can value the portfolio. It too will be either $10 or zero. This must be so, since the value of the portfolio is the stock's value minus the debt on the zero-coupon bond. So it is either $55 minus $45, or $45 minus $45. The future value of the stock may be uncertain, but the value of the debt on the bond is not. Third, the alternative values for both the call option and the portfolio at the year end are the same. If this is so, then their start value must be the same as well. The start value for the portfolio can be easily calculated. It is $50 minus $42.45; that is, $7.55. So this must also be the present value of the call option.

From this basic building block of the binomial model comes the formula that the value of a call will be the current value of the stock in question multiplied by the option's delta (which, in effect, was 0.5 in our example) minus the borrowing needed to replicate the option. Using our example, the linear representation would be:

Call value = ($100 x 0.5) - $42.45 = $7.55

This is the single-period binomial model, so called because the starting point is to take two permitted outcomes for the stock price and then work back to find what this means for the present value of the option.

In the real world, however, a single-period model is not practical, hence the development of the multi-period binomial model where each period used to estimate the price of the option can be as short as computer power will allow. As the number of price outcomes rises by 2 to the power of the number of periods under review, the model is computer-intensive; a model using 20 periods, for example, would need over 1m calculations. Additionally, rather than using arbitrary stock-price outcomes from which to estimate the value of the option, the model takes advantage of the fact that, given an estimate of the rate at which a stock price will change, future stock prices can be estimated within a reasonable band of certainty using mathematical distribution tables.

The result is a model which produces options prices that closely mirror market prices. Furthermore, because the binomial model splits its calculations into tiny time portions, it can easily cope with the effect of dividends on stock prices and, hence, option values. This is an important factor with which the more widely used black-scholes option pricing model copes less capably.

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.

Balance sheet

The financial statement of what a company owns and what it owes at a particular date, known as the statement of financial position in the United States. Traditionally, the left-hand side of the balance sheet is a schedule of the company's assets (land, buildings, plant and equipment, cash and inventories); the right-hand side is a statement of the liabilities, either real or potential. Real liabilities comprise the debts the company must pay - that is, creditors - plus its loans. Potential liabilities are the allowances that are likely to be paid: deferred taxes and, increasingly, post-retirement benefits for employees. The remaining item on the right-hand side is the shareholders' interest in the business. This is technically not a liability at all, but a statement of the risk capital subscribed to the business adjusted by the aggregate of retained earnings and (possibly) revaluation of some assets.

Alpha

A term borrowed from statistics which is used to show how much of the investment performance of a stock or portfolio of stocks is independent of the stock market in which they trade.

• Within a simplified pricing model used to identify those portfolios of investments that deliver the best combination of risk and return, alpha is used to describe the expected return from a security or a portfolio assuming that the return from the market is zero. Thus in this model the expected return for, say, an ordinary share would be its alpha plus the market return leveraged by the share's sensitivity to market returns (its beta). Here both alpha and beta are estimated based on comparison of the historical returns of the share and the market (see also single index model).

• In measuring portfolio performance, alpha is used to define to what extent a portfolio has done better or worse than it should have done, given the amount of risk it held. If it is accepted that a portfolio's performance will (simply speaking) depend on market returns times the portfolio's sensitivity to the market, then alpha quantifies the extent to which the portfolio's return varies from its expected return. Thus it measures the extent to which the manager adds or erodes value.

Arbitrage pricing theory (APT)

A theory which aims to estimate returns and, by implication, the correct prices of investments. Intellectually, it is an extension of the capital asset pricing model. It says that the CAP-M is inadequate because it assumes that only one factor - the market - determines the price of an investment, whereas common sense tells us that several factors will have a major impact on its price in the long term. Put those factors into a model and you are making progress.

Thus arbitrage pricing theory (APT) defines expected returns on, say, an ordinary share as the risk-free rate of return plus the sum of the share's sensitivity to various independent factors. (Here sensitivity, as with the CAP-M, is defined by the share's BETA.) The problem is to identify which factors to choose. This difficulty is compounded by academic studies which have come up with varying conclusions about the number and identity of the key factors, although benchmarks for interest rates, inflation, industrial activity and exchange rates loom large in tests.

In practice, the aim of using APT would be simultaneously to buy and sell a range of shares whose sensitivity to the chosen factors was such that a profit could be made while all exposure to the effect of the key variables and all capital outlay were canceled out. To the extent that APT assumes that markets always seek equilibrium, it says that the market would rapidly price away such arbitrage profits.

Alternatively, a portfolio could be chosen which could be expected to outperform the market if there were unexpected changes in one or more key factors used in the model, say industrial activity and interest rates. As such, however, that would be doing little more than betting on changes in industrial production and interest rates and would not have much to do with minimizing risk for a given return. Resolving problems such as these means that APT gives greater cause for thought to academics than to investors.

Arbitrage

To arbitrage is to make a profit without risk and, therefore, with no net exposure of capital. In practice, it requires an arbitrager simultaneously to buy and sell the same asset - or, more likely, two bundles of assets that amount to the same - and pocket the difference. Before financial markets were truly global, arbitraging was most readily identified with selling a currency in one financial center and buying it more cheaply in another. The game has now moved on a little, but, for example, there would be the potential to make risk-free profits if dollar interest rates were sufficiently high to allow traders to swap their euros for dollars and be left with extra income after they had covered the cost of their currency insurance by selling dollars forward in the futures market. Similarly, arbitrage opportunities can be exploited by replicating the features of a portfolio of shares through a combination of equity futures and bonds then simultaneously selling the actual stocks in the market. (See risk arbitrage)