The Black-Scholes model for calculating the premium of an option was introduced in 1973 in a paper entitled, "The Pricing of Options and Corporate Liabilities" published in the Journal of Political Economy. The formula, developed by three economists – Fischer Black, Myron Scholes and Robert Merton – is perhaps the world's most well-known options pricing model. Black passed away two years before Scholes and Merton were awarded the 1997 Nobel Prize in Economics for their work in finding a new method to determine the value of derivatives (the Nobel Prize is not given posthumously; however, the Nobel committee acknowledged Black's role in the Black-Scholes model). The Black-Scholes model is used to calculate the theoretical price of European put and call options, ignoring any dividends paid during the option's lifetime. While the original Black-Scholes model did not take into consideration the effects of dividends paid during the life of the option, the model can be adapted to account for dividends by determining the ex-dividend date value of the underlying stock. Many empirical tests have shown that the Black–Scholes price is "reasonably close" to the observed prices, although there are exceptions such as the "option smile" which are widely known in the investment field.

Some points to reiterate - Black-Scholes option pricing model (also called Black-Scholes-Merton Model) values a European-style call or put option based on the current price of the underlying (asset), the option’s exercise price, the underlying’s volatility, the option’s time to expiration and the annual risk-free rate of return.

The Black-Scholes model values a call option by weighting the current price of the underlying asset with the probability that the stock price will be higher than the exercise price and subtracting the probability-weighted present value of the exercise price.

The value of a call option at expiration equals the spot price of the underlying asset minus its exercise price (also called the strike price) i.e. at which the option entitles you to purchase the underlying asset. This can be expressed mathematically as follows:

C = X - S

Where C is the value of call option (all called call premium), S is the spot price of the underlying and X is the exercise price. At any time before the expiration, the value of the call option equals the current stock price minus the present value of the strike price, this can be express as follows:

C = S - X/(1 + r) ^t Where r is the risk-free rate of interest and t is the time to expiration.

The above expression gives us the value of the call in a static scenario i.e. in a scenario in which we know we will or won’t exercise the option. If we want to know the value of a call option based on our expectation, we can write the following crude expression of probability weighted cash inflows and out flows:, which leads to -

C = S * N(Nd1) - Xe^-n * N(d2)

Where N(d1) and N(d2) represent the standardized normal distribution probability that a random variable will be less than d1 and d2 respectively when d1 and d2 are given by the following equation:

d1 = (ln(s/x) + (r + s^2) * t)/ SQROOT(s^2 * t) where s = standard deviation of return of the asset

d2 = (ln(s/x) + (r - s^2) * t)/ SQROOT(s^2 * t)

Worked Example

A 6-month call option with an exercise price of £50 on a stock that is trading at £52 costs £4.50. Determine whether you should buy the option if the annual risk-free rate is 5% and the annual standard deviation of the stock returns is 12%.

We need to determine the value of the call option using Black-Scholes option pricing model and then compare it with the current price of the option and purchase the option if it is fairly priced.

Plugging in these values into the above equations we get d1 = 0.7993 and d2 = 0.7144

Now we can find the standardized normal distribution probability using NORMSDIST in zoho sheets or excel d1 = 0.7879 & d2 = 0.7625

We can now plug these values into Black Scholes C = £52 * 0.7879 - £50 * e^-0.05 * 0.5 * 0.7625 = £3.788

The option value as calculated is lower than the current traded call option.

Black Scholes D1 and D2 calculations using zoho sheets

Vodafone Option Chain. Use this as Aide Memoire in Option calculations

BP Option Chain. Use this as Aide Memoire in Option calculations