## The Black-Scholes model

##### Published: Oct 2004

The Black-Scholes model is the best known of all the techniques for valuing options. It is used in the pricing of both options and futures and forms the basis for formulas pricing exotic options such as barriers, compounds and Asian options. It also enabled the derivation of the ‘Greeks’ of option pricing. It is more difficult to adapt to value interest rate options because one of the model’s underlying assumptions is a constant interest rate.

The model takes its name from two economists, Fischer Black and Myrton Scholes, who published a research paper, “The Pricing of Options and Corporate Liabilities”, in the Journal of Political Economy in 1973.

The value of the call option (c) is a function of the present values of both purchasing the stock outright and paying the exercise price on the day the option expires. This is shown by the two parts of the model in the following equation:

$$C = A \:- \: B$$
Where component A calculates the expected benefit of purchasing the stock outright. This is effectively the current stock price multiplied by the chance that the option will be exercised. The expected benefit is given by:

$$A = P \:\times\: Nd_1$$

Component B is the present value of exercising the option on the expiration day. This is determined by the exercise price of the option (k) discounted to a present value $$e^{\:-\:rT}$$ multiplied by the likelihood of the option being ‘in the money’ on expiration day. It is given by:

$$B = K \:\times\: e^{-\:rT} \:\times\: Nd_2$$
such that

$$C = p \:\times\: Nd_1 \: – \: K \:\times\: e^{\:-\:rT} \:\times\: Nd_2$$

This is intuitively correct. It says that the value of the call option is the difference between the present value of the stock on expiration day and the present value of exercising the option on expiration day.

It can be seen most clearly by looking at the value on expiration day itself.

• If the option is exercised (it is ‘in the money’), the value will be the difference between the cost of buying the stock on the open market and the cost of exercising the option. In the equation A is greater than B.
• If the option is ‘out of the money’ the exercise price will be higher than the stock price. As a result the option will not be exercised and the option value will be zero.

Where

• P = current stock price
• N = cumulative standard normal distribution
• K = option strike price
• r = risk-free interest rate
• T = time to option expiry (in years)

And where

• In = the natural logarithm
• $$\sigma\:$$ = the standard deviation of stock returns ( $$\sigma^2\:$$is the variance from the mean)