Why Swaption Implied Volatility Varies by Expiration and Tenor
Summary
The document discusses the two-dimensional term structure of lognormal implied volatility for European swaptions, varying by option expiration and underlying swap tenor. It describes a typical pattern in which volatility may rise over the shortest expirations, peak, and then decline for long-dated expirations. A proposed explanation is that bounded interest rates make annualized volatility fall approximately with the square root of expiration at long horizons.
Across swap tenors, the answer reports that rates in the intermediate 2-to-5-year area have often been more volatile than very long rates, possibly because new information has a smaller effect on distant forward rates. It also notes a model limitation: when rates are very low, forward-rate behavior can be more volatile than a lognormal model implies, producing high short-expiration, short-tenor implied volatilities. These are described as empirical tendencies and hypotheses, not universal laws; the document gives no dataset or detailed model comparison.
Key ideas
- Swaption implied volatility varies across both option expiration and swap tenor.
- Short-expiration volatility may rise before reaching a peak and declining at long expirations.
- Bounded interest rates are offered as a possible reason for lower long-horizon annualized volatility.
- Intermediate swap tenors are described as more volatile than the long end in observed markets.
- Low rates can make lognormal implied volatility a poor representation of forward-rate behavior.
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Full text
# European Swaptions: does implied volatility of swap rates decreases both with start and tenor?
# European Swaptions: does implied volatility of swap rates decreases both with start and tenor?
Does implied volatility of swap rates decreases both with start and tenor?
Given a Swaption price and a discount curve I calculate the swap_rate from the curve, then I define implied volatility as the volatility $V_{impl}$ such that the price returned by BS formula with spot = strike = swap_rate and volatility = $V_{impl}$ exactly matches the given swaption price.
I've seen such implied volatility decreasing both with tenor and with start. This surprises me.
I wonder if there is a good reason for that.
## Answer by dm63 (score 7)
https://quant.stackexchange.com/a/21871
You are asking about the term structure of lognormal implied volatilities for European swaptions, which is a two dimensional function (expiration and tenor).
First expiration: typically (but not always), implied volatilities are increasing in the 0 to 6 month sector, because the immediate future is often more predictable than the medium term. At some point, volatilities max out and they always start decreasing for very long dated swaptions. One hypothesis for this effect is that interest rates are fundamentally bounded (they very rarely go below 0 or above 15%, say), so the annual implied vol of long dated swaptions has to decrease approximately at 1/sqrt(expiration).
Tenor: empirically, one finds that the most volatile rates over time are those in the 2yr-5yr sector of the curve. The long end (30yrs and above) tends to be less volatile. Presumably that's because new information affects long dated forward rates less than shorter dated forward rates, but that's a hypothesis.
There is also a model effect in lognormal volatilities. When rates get very low (as is now the case at the front end of many yield curves), the actual behavior of forward rates tends to be more volatile than the lognormal model would predict. Hence implied lognormal volatilities are quite high for very short expitation, short tenor optionsShown 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.