## Volatility in Black-Scholes

##### Published: Jan 2005

In December 2004’s calculator corner, we described how the Black-Scholes Model for valuing options incorporates:

• the intrinsic value of the option (PK),
• the time value of the option (achieved by multiplying the strike price by e rT) and
• volatility (defined as d1and d 2).

In this article, we look at the role of volatility in the model.

According to the model, the value of a call option (C) can be written as:

$$C = P*Nd_1 – K*e^{-rt} * Nd_2$$

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:

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

Where:

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

The value of an option depends on the likelihood of the option being exercised. In other words, it depends on the likelihood that the stock price will be higher than the strike price on expiration day. This depends on the volatility of the price and is shown in components
d1 and d2 above.

There are four determinants of d1 , which gives a measure of the volatility of the stock price:

1. The ratio of the current stock price to the strike price $$(\frac{p}{k})$$. The higher the value of $$\frac{p}{k}$$, the more the chance the option will be exercised. As a relative value, this provides a good indication of the likelihood of the option being exercised.
2. The risk-free interest rate (r). One of the weaknesses of the Black-Scholes model is the assumption that there is a risk-free rate, which can be calculated. In practice this assumption is unrealistic as interest rates change and there is no single risk-free rate. But, a risk-free rate must be assumed so the model can compute the time value of money.
3. The historic volatility of the stock price (σ2). The more volatile the price of the underlying instrument, the higher the value (and the cost) of the option will be. This is because the harder it is to predict the future value of the stock, the more volatile the underlying stock price. Again, the danger here is that it is impossible to predict future stock prices and thus volatility. This means the volatility is based on historic observation.
4. The time to expiration (T). The longer until expiration, the more uncertain the stock price on expiration day. In other words, the closer it is to expiration day, the easier it will be to predict the stock price on that day. In addition, the closer the option is to expiration, the less important the time value of money is in the calculation.

The second component d2 is a function of the same determinants as d1 and gives a measure of the volatility of the time value of the strike price.