# 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.

• If you are given a price for a particular swaption why are you later setting strike to something else? – Brian B Oct 22 '13 at 14:18
• @BrianB The price I'm given is the price of the option: the option on the swap. The strike I'm setting is a rate, I exercise if the rate at maturity (of the option) is above the strike (rate). – jimifiki Oct 22 '13 at 14:46

## 1 Answer

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 options