# Black-Scholes Model

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:

$$ \text{C}=\text{S}-\text{X} $$

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:

$$ \text{C}=\text{S}-\frac{\text{X}}{(\text{1}+\text{r})^\text{t}} $$

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:

$$ \text{C}=\text{S}\times \text{p}\ -\frac{\text{X}}{(\text{1}+\text{r})^\text{t}}\times \text{p} $$

Where p is the probability.

## Formula

The Black-Scholes formula is a refined form of the expression above. Given a stock price S, exercise price X, annual risk-free rate r, time to maturity t and annual standard deviation of return of the underlying asset σ, we can determine the value of call option using the following formula:

$$ \text{C}=\text{S}\times \text{N}(\text{d} _ \text{1})\ -\text{Xe}^{-\text{rt}}\times \text{N}(\text{d} _ \text{2}) $$

Where N(d_{1}) and N(d_{2}) represent the standardized normal distribution probability that a random variable will be less than d_{1} and d_{2} respectively when d_{1} and d_{2} are given by the following equation:

$$ \text{d} _ \text{1}=\frac{\ln{\frac{\text{S}}{\text{X}}+(\text{r}+}\frac{\sigma^\text{2}}{\text{2}})\times \text{t}}{\sqrt{\sigma^\text{2}\times \text{t}}} $$

$$ \text{d} _ \text{2}=\frac{\ln{\frac{\text{S}}{\text{X}}+(\text{r}-}\frac{\sigma^\text{2}}{\text{2}})\times \text{t}}{\sqrt{\sigma^\text{2}\times \text{t}}} $$

N(d_{1}) and N(d_{2}) roughly represent that probability that the exercise price of the option will be higher than the current stock price and hence the option will be in-the-money and hence valuable.

The Black-Scholes option formula can also be used to estimated implied volatility based on the current call premiums.

## Example

A 6-month call option with an exercise price of $50 on a stock that is trading at $52 costs $4.5. 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.

We first need to find d_{1} and d_{2}:

$$ \text{d} _ \text{1}=\frac{\ln{\frac{\text{60}}{\text{50}}+(\text{5%}+}\frac{{\text{12%}}^\text{2}}{\text{2}})\times\text{0.5}}{\sqrt{{\text{12%}}^\text{2}\times\text{0.5}}}=\ \text{0.7993} $$

$$ \text{d} _ \text{2}=\frac{\ln{\frac{\text{60}}{\text{50}}+(\text{5%}-}\frac{{\text{12%}}^\text{2}}{\text{2}})\times\text{0.5}}{\sqrt{{\text{12%}}^\text{2}\times\text{0.5}}}\ =\ \text{0.7144}\ $$

Next, we can find the standardized normal distribution probability using Microsoft Excel NORMSDIST function. N(d_{1}) and N(d_{2}) equal 0.7879 and 0.7625 respectively.

Once we have N(d_{1}) and N(d_{2}), we can plug-in the relevant numbers in the Black-Scholes formula:

C = 52×0.7879 − 50×e^{-0.05×0.5}×0.7625

C = $3.788

The option value as per the model is lower than the premium on the call options currently traded. It might be because the option is overvalued or because our estimate of the volatility is lower.

If we have current value of call premium C, stock price S, exercise price X, time to maturity t and risk-free rate r, we can work back to find out the **implied volatility**. In the above example, the market estimate of annual standard deviation of return based on call premium of $4.5 is 18.06%. It can be worked out using Excel Goal Seek tool.

by Obaidullah Jan, ACA, CFA and last modified on