# 1y10y vs. 10y1y Swaption

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).

• 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.
– RRL
Jul 1, 2018 at 9:15

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.

This is an old but nice question and already has a good answer. A quick way to think about it is using the old "trader's formula" for a (ATMF) swaption: $$c\equiv 0.4 \sigma\sqrt{t}\cdot dv01$$ where $$t=$$ swaption maturity, $$\sigma=$$ atm normal vol, $$dv01 =$$ duration of the underlying swap. Assume both swaptions have the same vols and the durations of the 1Y and 10Y swaps are 1 and 10, respectively. Then $$c_{1Y10Y}=\sqrt{10}\cdot c_{10Y1Y}$$. Or, more generally, if the 10Y1Y and 1Y10Y vols are $$\sigma_{10Y1Y}$$ and $$\sigma_{1Y10Y}$$, respectively, then $$c_{1Y10Y}=\sqrt{10}\frac{\sigma_{1Y10Y}}{\sigma_{10Y1Y}}\cdot c_{10Y1Y}.$$

Edit:

Request in comment for derivation of "trader's formula". We use a vanilla caplet under Bachelier model to illustrate this. A strike $$K$$ caplet with expiry $$t$$, accrual factor $$\delta$$, vol $$\sigma$$, forward $$F$$, discount factor $$df$$ and nominal $$N$$ is: $$c=\left[N(d)(F-K) +n(d)\sigma\sqrt{t}\right]\cdot \delta\cdot df\cdot N \tag{1}$$ with $$d=\frac{F-K}{\sigma\sqrt{t}}$$ where $$n(x)=\frac{e^\frac{-x^2}{2}}{\sqrt{2\pi}}\tag{2}$$ is the pdf of a standard normal distribution and $$N(x)$$ the corresponding cdf. See this paper as a reference for (1). With $$F=K$$, $$d=0$$ so by (2) $$n(0)=1/\sqrt{2\pi} \approx 0.4$$ and (1) simplifies to $$c=0.4\sigma\sqrt{t}\cdot A\cdot N$$ where $$A=\delta \cdot df$$ is the annuity factor of the caplet. For swaptions $$A=\sum_i \delta_i\cdot df_i$$ for the duration of the underlying swap.

• Minor comment, I think it would be more correct to speak about $\text{PV}01$ rather than $\text{DV}01$ in this case, see this answer. The former corresponds to an annuity in the case of a swap. Sep 19, 2023 at 17:51
• Thank you. Is there a source for this "trader's formula"? Thanks Oct 27, 2023 at 11:21
• sure, updated my answer to add a derivation for this Oct 28, 2023 at 6:04