Looking to Benchmark Pay?
Theory
What’s the theory that supports this formula?
If you have a call option on a stock and it expires:
- the value is either zero
- OR
- the option is underwater
- OR
- the value is the difference between the current market value and the exercise price of the option
Here’s an example…
Edward buys a call option on Ajax Ltd. with an exercise price of $50. The expiration date is in one month. What happens at the end of that month?
- If the price of the stock is less than $50, the option is worthless.
- BUT
- If the market value is $60, then the option is worth $10.
The payoff to any call option is the positive difference between the current market value and the exercise price.
According to Black-Scholes, an investor can precisely replicate the payoff to a call option by buying the underlying stock and financing part of the stock purchase by borrowing.
For example…
Instead of purchasing the call option, Edward purchased a share of Ajax Ltd. and borrowed the $50 exercise price. Upon the option’s expiration date, Edward sold the stock for $60 and paid back the loan plus interest.
This produces an equivalent result to the option.
The same thing can be done to value the call option before it expires, creating a replicating portfolio.
| Replicating portfolio | A way to match the future payoff of a call option. You buy a fraction of a share of the stock and borrow a fraction of the exercise price. You can use the Black-Scholes formula to determine those fractions. |
|---|
The Black-Scholes formula states that:
The call option (C) is equal to a fraction (N(d1)) of the stock’s current market value (S) minus a fraction (N(d2)) of the exercise price (X).
If the current market value is way above the exercise price, these fractions approach a value of 1. So, the call option is approximately the difference between the current market value and the present discounted value of the exercise price. The opposite is true if the current market value falls below the exercise price and the call has little value. We’ll look at these factors in the next section.
Memory Jogger
Black and Scholes figured out that the payoff on a call option could be replicated by: