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Say you have two identical payer swaptions, exception for their terms and tenors. In other words, suppose you have two payer swaptions: $1y10y$ and $10y1y$.

All other things being equal, according to the Black model, am I right in thinking $10y1y$ is more expensive, and if so, by approximately how much? Is it approximately X$\sqrt{10}$ more expensive?

It's well know that with all other beings equal, larger $T$ (tenor) implies a higher price, for the intuitive reason that there's a high probability of landing in-the-money. But how does $t$ (the swap term/length) impact the price too?

I know you discount the Black model by an annuity factor (the only term where t is found in the formula).

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  • $\begingroup$ Given the same yield volatility a bond that matures in 10 years has a higher price volatility than a bond that matures in 1 year due to higher duration. The same applies to swaps. $\endgroup$ – RRL Jul 1 '18 at 9:15
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Currently the USD 10Y swaprate is $2.93 \%$ and the ATMF 1Yx10Y implied volatility (relative) is $22.5 \%$ which corresponds to the Black model (absolute) volatility of about $4.15$ bp/day. The 1Y swaprate is $2.60 \%$ and the ATMF 10Yx1Y implied volatility is $25.0 \%$ which corresponds to the Black model volatility of about $4.10$ bp/day.

With the current flat term structure (both for rates and volatility) the 10Y swaption price would be greater than the price of a 1Y swaption with the same tail by the $\sqrt{10}$ factor. However, the annuity factor is roughly equivalent to duration which is about $9$ times larger for a 10Y tail.

Hence, the 1Yx10Y ATMF swaption price should be about 3 times larger than the 10Yx1Y ATMF swaption price.

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