# Does an Interest Rate Swap has a Vega component?

I am a bit confused on how you calculate vega for Interest Rate Swap.

One argument is that IR Swap is a combination of fixed rate bond and floating rate bond. Since a bond has no vega component, IR Swap has no vega component.

Another argument is that IR Swap can be synthetically reproduced using a cap and floor. For pay fixed and receive floating side, it is a long a cap and short a floor. Since both cap and floors are options on IR, there is a vega component.

Since these two arguments contradict themselves, which is right and why is the other wrong?

Need some guidance on this.

## 2 Answers

There is no contradiction.

If the strike of the floor and cap are both equal to the swap rate, and all accrual/payment frequencies, etc. are the same, then put-call partiy implies $$C_{t}-F_{t}=S_{t},$$ where $C_{t},F_{t},S_{t}$ are the values of the cap, floor and swap instruments at time $t$.

Since the (theoretical Black-Scholes) volatility is independent of the option-type, the Vegas on the LHS cancel so that the swap Vega is $0$.

An interest rate swap (IRS) can have a vega component if it is not a standard IRS.

If you are familiar with the convexity adjustment for FRAs (and single period IRSs) compared with their respective short term interest rate (STIR) future, you will be aware that it is the different gamma components of these products that result in profit-and-loss (PnL) over their lifetimes. One of the main over-arching reasons for this is the timing of payments of any PnL cashflows. The effect, and the discrepancy in pricing adjustments, is largest for higher assumed volatility.

For interest rate swaps that customise the payment date, i.e. they bring forward or lag the payment compared with the vanilla product then this introduces the same concern. Investment bank mark-to-market (MTM) valuations of IRSs factor this into their portfolio assessments, although generally this is a small effect - a few hundredths of a basis point depending on the size of payment lag.

This question expands on futures convexity, and this reference includes an example of precisely the effect I mention in the section on gamma.