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Fraction Factors
The fractions used in the Black-Scholes equation are determined by five factors:
- current market value of the underlying stock
- exercise price of the option
- risk-free interest rate
- time to maturity of the option
- volatility of the underlying stock
Let’s examine each of the 5 factors…
Current market rate (S)
The current market price of the underlying stock is a readily accessible factor. This factor can change daily, so there may be corresponding changes that affect the values each time the analysis is run.
Q: What if there is no market?
A: Then the only figure that can be used is an estimate of the company’s value.
Exercise price (L)
For employee stock options, the exercise price is the strike price.
Risk-free interest rate (r)
The risk-preferences question stumped the people that tried to develop models in the area of derivatives.
Then the argument was made that risk preferences don’t affect the option price. This way, anyone can assume that all investors have a zero-risk tolerance.
In other words…
Two investors with very different risk preferences can agree on the value of an option.
The risk-free interest rate is a theoretical rate at which an investment may earn interest without incurring risk. In practice, the risk-free rate is assumed to be the short-term treasury rate.
Time to maturity (T)
Time to maturity is calculated using the stock option expiration date.
Volatility of the underlying stock (s)
Volatility of the underlying stock is the only unobservable factor. Volatility is the degree of random variability in the price of the stock.
| High volatility | Low volatility |
|---|---|
| If the price of the stock varies widely over time, the stock is considered highly volatile. | If the price of the stock has low variability, the stock has low volatility. |
In finance, volatility is measured using the standard deviation in price change of a stock’s price against its price at any given time.
There are two ways to estimate volatility:
- Implied volatility. Volatility is inferred from the movement of the price of the stock option over time. Implied volatility reflects the market’s perception of the variable’s volatility. It’s a time-sensitive measure — reflecting the market's perceptions today. Implied volatilities can be biased, especially if they are based on options that are thinly traded AND can only be estimated for a variable if options are traded on that variable.
- Historical volatility. The recent historical movement of the stock price is examined. Historical volatility is a retrospective measure of volatility. It reflects how volatile the variable has been in the recent past. Historical volatility is a highly objective measure. Furthermore, historical volatility can be calculated for any variable for which historical data is tracked.
Memory Jogger
Which of the five factors is NOT easily observable?