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.

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.