Robustness
Suppose you believe a stock has a historical annualized 100-day volatility of 60%, that the stock is selling now for $100, a September 100 call on the stock is selling for $4.50, and you compute the value of the call to see if it is overpriced or underpriced. Using an interest rate of 2%, you will get a value of $7.64. The call seems to be underpriced.
You might try an interest rate of 4% just to make sure. If you do, you will get a value of $7.73, about the same. An interest rate of 0% will give you a value of $7.55. It looks like the undervalued nature of the call is robust relative to the interest rate, meaning that even allowing for reasonable errors in estimating the interest rate, the call remains undervalued.
What about the volatility? You have a value for the recent historical volatility of the stock, but that value may not be applicable to the future, for various reasons. Suppose you think the future volatility might be poorly computed or estimated, or perhaps that it is correctly computed from the past data but that the future volatility will be different. Is the underpricing of the option robust relative to the volatility?
The table at right shows the values for this option for volatilities 10%, 20%, 30%, 40% and 50%:
| Annualized Volatility | Value |
|---|---|
| 10% | $1.36 |
| 20% | $2.62 |
| 30% | $3.87 |
| 40% | $5.13 |
| 50% | $6.38 |
As you can see clearly, if you do not have good reason to believe that your volatility estimate of 60% is correct, and if in fact the actual future volatility turns out to be much lower, this option may not be underpriced at all. We will be speaking more about this problem in the next few commentaries.