Inferring Forward Volatility from Short- and Long-Term Implied Volatility
Summary
The document interprets a steep implied-volatility term structure by treating “inherent volatility” as the volatility expected after the near-term period. It uses the principle that variances accumulate over time: the total variance implied for a two-year horizon is decomposed into the variance for the first two months and the variance for the remaining twenty-two months. Solving this weighted variance relationship gives an implied volatility for the later period of about 30.55% per year, below both quoted maturities’ implied volatilities.
The answer presents this as a simple equilibrium model: after the unusually volatile initial period, volatility is assumed to settle at one level across maturities. The question’s phrase “inherent volatility” is not standard, so the result depends on that interpretation. It is an inference from implied volatilities, not a forecast proven by realized outcomes, and it does not specify option quantities or neutralize Greeks for an actual spread.
Key ideas
- Variances, rather than volatilities, add across successive time intervals.
- A longer-term implied variance can be decomposed into near-term and later-period variance contributions.
- Under the stated constant later-volatility assumption, the implied volatility after the initial period is about 30.55% annually.
- The calculation interprets a nonstandard term and does not by itself define a balanced options position.
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Full text
# Inherent volatility of selling longterm options and buying short term options
# 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.
## Answer by Alex C (score 4)
https://quant.stackexchange.com/a/44733
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.