# Forward swap rate calculation from the market

Following my question

Swaption valuation across time using vcub

where I wanted to know how to value a swaption across time using bloomberg's vcub, I remark that I have to calculate myself the forward swap rate $$s_t$$, even if $$s_0$$ is quoted on bloom.

What techniques are available to do this ? If the formula from discount curve and forward curve valid / precise enough for this ?

it requires a model to do it correctly but often i might just do a simple forward math calculation especially if it's not very far forward. So for 1yr fwd 2yr i'd do ((1+yield(3yr))^3 /(1+yield(1yr)^1)^(1/2)-1. It's better to do this with zero coupon bonds but often those yields aren't that different these days anyway.

Let's say $$I$$ is the Libor index for our underlying swap and $$D$$ is our discount curve.

If at time $$t$$ we have a forecast of all the relevant future Libor fixings $$L_{I}(t, T_{i}, T_{i}+\tau)$$ for our swap, where $$\tau$$ is the accrual factor for our Libor index and $$T_{i}$$ is the time of the $$i$$th fixing for our swap, we just need to solve for the fixed swap rate $$s$$ that equates the NPV of the fixed leg of our swap with the NPV of the floating leg.

The fixed leg NPV is given by

$$V_{fixed} = \sum_{i}{s \tau D(t, T_{i})} = s \sum_{i}^{n}{\tau D(t, T_{i})} = s * PV01$$

where $$D(t, T_{i})$$ is the time $$t$$ discount factor on our discount curve for time $$T_{i}$$.

Similarly, the floating leg NPV is given by

$$V_{float} = \sum_{j}{ L_{I}(t, T_{j}, T_{j} + \tau) \tau D(t, T_{j}) }$$

For a par swap, we know that $$V_{fixed} + V_{float} = 0$$, therefore we can substitute in for $$V_{fixed}$$ and divide by the fixed leg PV01 (sometimes called the level or annuity of the swap) to obtain

$$s = \frac{-V_{float}}{PV01}$$

In reality, each individual period's $$\tau$$ may be slightly different due to day count conventions, but it's fairly clear that the swap rate $$s$$ is just a weighted average of the forward Libor rates $$L_{I}$$ on the floating leg of the swap

• How did can you calculate the PV01 of the swaption ? – Gogo78 Sep 6 '19 at 7:06
• The PV01 is for the underlying swap - we're trying to determine the at-the-money forward rate for the swaption, which is just the current market rate for its underlying swap. – thetableed Sep 9 '19 at 2:59