1y10y vs. 10y1y Swaption

Question

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).

Answer 1

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.

Answer 2

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.