## Explaining Black-Scholes

##### Published: Dec 2004

In October 2004 we explained the Black-Scholes Model for valuing options as the difference between the present values of the stock on expiration day (the first part of the equation) and of exercising the option on expiration day (the second part of the equation).

The value of a call option (c) can be written as:

$$c = p * Nd_1 – k*e^{ \:- \:rT }* Nd_2$$

In November 2004, we outlined the main assumptions underpinning the model. In this article, we analyse Black-Scholes in more detail.

On expiration day, the value of the option is equal to the current stock price minus the option strike price. Symbolically:

$$c = p \: – k$$
This shows that if the stock price is above the strike price on expiration day, the option will be exercised and the value to the holder is the difference between the two prices. On the other hand, if the stock price on expiration day is equal to or below the strike price, the option will not be exercised and the value of the option is zero. This is the intrinsic value of the option.

Because prices vary over time, we also have to consider the time value of the option. In our basic equation, the current stock price (p) varies over time, meaning it does not have to be adjusted in the equation. However, because the option strike price (K) is fixed, we need to adjust this to take account of time.

This is done by multiplying the strike price by$$e^{\:-rT}$$. Black-Scholes assumes a constant risk-free interest rate, r, where no dividend is payable, which is continuously compounded over the time to expiration (T).$$\:k*e^{\:-rT}$$is the present value of the strike price. This represents the amount the writer of the option will have to invest today (at the risk-free rate) to meet its obligation on expiration day, if the option is exercised. Symbolically:

$$c = p \: – \: k * e^{\:-rT}$$
Finally, the value of the option will be greater, the more volatile the underlying instrument (in this case the stock price). Future volatility cannot be calculated, so this is based on historic data and is represented by components d1 and d2.

#### Key

• 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:

• $$d_1 = \frac{In \frac{p}{k} \:+ \:r \:+\: \frac{\sigma^2}{2} *T}{\sigma \: * \: \sqrt{T}}$$
• $$d_2 = d_1 \: – \: \sigma \: * \: \sqrt{T}$$

Where:

• In = the natural logarithm
• σ = the standard deviation of stock returns ( σ2 is the variance from the mean)