Understanding Option Valuation
A useful concept
By Kaye A. Thomas
Posted June 11, 2009
Simplified example provides insight.
The value of a stock option is usually determined according to the Black-Scholes formula. With its statistical and logarithmic functions, this formula is too complicated for mere mortals to fully understand. If we strip away some of the complexity, though, we can gain an appreciation for the thought process behind the formula, and the significance of some of the factors that go into the calculation.
Simplified situation
Suppose we have to determine the value of an option to buy a single share of stock when we're given the following facts. On the date we're determining the value, the stock is trading at $100, and this is the exercise price of the option. In addition, we somehow know that on the expiration date, when we'll have to decide whether to exercise the option, the stock will be trading at either $90 or $110, although we don't know which. That's an unrealistic assumption, of course, but analyzing this idealized situation will provide insight into the value of options in the real world.
Option payoffs
We begin by determining the possible payoffs for the option. If the stock goes to $110, we exercise the option, paying $100 for a share of stock, and sell it immediately for a $10 profit. If the stock goes to $90 we allow the option to expire for a payoff of $0. This tells us the value of the option must be equal to the value of the opportunity to obtain these payoffs:
$10 when the stock goes to $110
$0 when the stock goes to $90
Another way to get these payoffs
Our strategy for finding the value of the option is to look for another way to get the same payoffs from something else that has a value we can more easily determine. Here's how to do that. First, buy half a share of stock. We know the stock is currently trading at $100, so the cost of a half share is a known quantity: $50. At the same time sell a debt obligation that will pay $45 on the date the option expires. The amount we'll receive for this debt obligation is easy to determine if we know the interest rate and the amount of time until the expiration date. For the sake of argument, let's say the debt obligation sells for $42. In other words, we've borrowed $42 and agreed to repay that amount with $3 interest on the option expiration date.
Now let's look at the payoff. If the stock goes to $110, we sell our half share for $55 and use $45 to satisfy the debt obligation, so the payoff is $10. If the stock goes to $90, we sell our half share for $45, the same amount needed to satisfy the debt obligation, so our payoff is $0. We've replicated the payoff of the option:
$10 when the stock goes to $110
$0 when the stock goes to $90
We know exactly how much it cost to get these payoffs: the half share of stock cost $50 and we sold the debt obligation for $42, with a net cost of $8. As a result, the option has to be worth $8. No one would pay more than $8 for the option because they could get the same result as the option for $8 as described above. And no one would sell the option for less than $8 because it's possible to do the reverse of the transaction just described: sell a half share of stock and lend $42. The value of the option has to be $8 because that's the cost of replicating the option's economics.
Implications
This analysis doesn't get us anywhere near the complexity of the Black-Scholes formula, but it's based on the same thought process used to develop that formula and reveals some aspects of how option values work. Here's what we learn.
- Interest rates. One factor in the value of the option is the interest rate. We borrowed $42 and paid $3 of interest. If interest rates had been higher, we would have been forced to borrow a smaller amount, because the loan payoff has to be $45. For example, we might have had to borrow $41.50 and pay $3.50 of interest. In that case the option value would have been $8.50. This shows us that a higher interest rate leads to a higher option value.
- Volatility. Another factor in the value of the option is the volatility of the stock price. We specified that the stock would go up or down by $10. If the stock went up or down by a larger number (in other words, it had greater volatility), we would have been forced to use a smaller debt obligation to create an equivalent position. For example, if the stock will end up at either $120 or $80 (up or down $20), the debt obligation would have to pay off at $40 instead of $45. The half share of stock would still cost $50, so the net cost of the equivalent position would be greater. This shows us that greater volatility leads to a higher option value.
- Probability. The value of the option is determined without regard to the probability of the two possible outcomes. We know the stock will go to $110 or $90, but we don't know which one is more likely, and when it comes to determining option value, we don't care. The option value comes from its equivalence with other items that are priced in the marketplace, and isn't affected by our view of whether the stock is likely to go up or down.
This last point may seem surprising. Our intuition tells us the value of the option should be greater when we're optimistic about the company's prospects. The analysis shows that our assessment of the likely performance of the stock must be disregarded. If we take it into account, the option will be priced incorrectly.
Option values don't represent opinions about how the stock is going to perform. They're derived from current stock values, interest rates, and expected volatility.
If that seems paradoxical, think of it this way. If the option price seems wrong based on your estimate of the stock price probabilities, then your estimate must be inconsistent with one of the inputs to the formula. You may be implicitly assuming a higher or lower interest rate, or greater or smaller volatility. Otherwise, you're implicitly assuming the market has undervalued or overvalued the stock.
