# Inherent volatility of selling longterm options and buying short term options

A two-month option has an implied vol of 60%, the corresponding 2-year option has an implied vol of 34%. You buy the short terms and sell the long terms. What is the inherent volatility of the total position (show a calculation).

a. higher than 60%

b. between 60% and 34%

c below 34%

I dont even know how to attack this question. Not even sure how this position is balanced, but lets say 1 long term versus 1 short term.

I do understand this position but not the question. If you're in this spread you want a high realized volatility (especially first two monhts) and a decreasing implied volatility. Your short leg has more gamma than vega, and your long leg vice versa.

• I think you have mixed up your question - if you are long the front end options, you want a high volatility for the short term and then for it to collapse into low vol – will Mar 23 at 20:29
• True, my bad. edited – Cindy88 Mar 24 at 5:03
• Given the possible answers, the answer they want is obviously b. But as everyone has pointed out, it's a bit of a weird question. – will Mar 24 at 10:00

## 1 Answer

It is hard to know what "inherent volatility" refers to, as this term is somewhat non-standard. I will interpret it as the long term equilibrium level of volatility $$\bar{\sigma}$$ to which all volatilities are expected to revert.

Clearly a short term vol of 60 and a long term vol of 34 is a highly unusual situation. The market expect volatility to be very high for 2 months (perhaps as a result of a recent stock market crash and/or uncertainty about near term events) and then return to a lower value.

Using the fact that variances are additive over successive intervals of time we can compute the volatility over the next 24 months as the weighted average of the volatility over the next two months and the volatility over the next 22 months:

$$0.34^2 = \frac{2}{24} 0.60^2 +\frac{22}{24} \bar{\sigma}^2$$

Solving his we get $$\bar{\sigma}=$$ 0.3055

So in this simple model, we expect, after the next two months are over for volatility to equal 30.55% a year at all maturities.

• Thanks. I found the formulation weird as well. But I think it has to be related to the position itself. So shorting the longer terms and buying the shortterms. – Cindy88 Mar 23 at 2:55
• I see. All I can say is it is a very skewed position. You profit from an (unlikely) further rise in short term vol, so that is kind of like an insurance (crash protection). But you lose in most cases when the sh term vol mean reverts back to more usual values; those losses are like the (expensive) premium you pay on the insurance. But how to calculate this from the few numbers given, I don't know. You would need extensive data on past vol dynamics... – Alex C Mar 23 at 13:26
• FYI Alex - the crude oil market in December had the exact situation described here. These structures are not uncommon in commodity markets – will Mar 23 at 20:31