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.

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)