We recently met an individual who is working towards a PhD in mathematical finance. He had a masters degree in something technical, too, and if we recall correctly, has been in the PhD program for four years.
He wanted us to read his paper regarding the volatility of volatility. We didn’t have the time, but a quick skim revealed that it was suitably loaded with math and graphs—quite technical. 1
When individuals discuss volatility, they usually speak of one of two kinds: (1) historical or realized vol or (2) implied vol. Historical or realized vol is the measured randomness in a past sequence of observations of a particular variable.
Implied vol is an estimate of future randomness, and it is called “implied” because it is found by solving a model, which will have any number of assumptions and restrictions. For example, imagine a pricing model (a function) of, say, input five variables. 2 Now, imagine that one can observe the actual price and four of the five input variables. Then, under suitable conditions, one can solve for the fifth, possibly unknown, variable that is implied by the function or model.
If one is using the Black-Scholes option pricing model or one of its variants, then (usually) the price of the option is known, and four of the five input variables are known or can be independently estimated: (1) the exercise value of the underlying variable, (2) the current value of the underlying variable, (3) the time until expiration, and (4) the risk-free rate. Using the appropriate algorithm, one can then find the implied vol that sets the function equal to the observed option price.
All of that just to say that when folks are concerned about the vol-of-vol, it almost always involves options, and when people are concerned about options or instruments with embedded options, they almost always use Black-Scholes or some variant. So, Black-Scholes and vol-of-vol are kind of like bread and peanut butter. A lot things can be eaten with bread (Black-Scholes), but peanut butter (vol-of-vol) is used almost exclusively with bread (B-S). Of course, we know about peanut butter pie and crackers and celery, but it is a decent analogy, and we are always willing to hear of better ones.
So, what does all of this have to do with our new acquaintance? We asked him to describe the Black-Scholes model, which is like the bread of any vol sandwich. His reply: when he had left his country of birth—about six years ago—they did not have equity options; so, he didn’t really know it well enough to explain it to others. Not the answer that we sought. Morals: (1) make certain that you have the right person asking the right questions, and (2) thought before calculation is the preferred order for the two.
Footnotes:
- Now for those of you who are unaware or unconcerned about the volatility of volatility, it basically involves the randomness of the randomness of an underlying (random) variable. For some market variables, we may have periods of great variability interspersed with other periods of near certainty—little change from day-to-day. Our short-term measures of uncertainty—like, say, the standard deviation over thirty days—would then vary, too. ↩
- These five variables will determine the price or value. ↩
