Spero Consulting

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.

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