Why Standard Interest Rate Swaps Usually Have Zero Vega
Summary
The document explains why a standard interest rate swap can be represented both as fixed and floating rate bonds and, under matching terms, as a cap minus a floor. These descriptions do not conflict: when the cap and floor share the swap rate as their strike and have matching accrual and payment conventions, put-call parity equates their value difference with the swap value. Under the same theoretical volatility assumption, the options’ vegas offset, leaving the standard swap with zero vega in this representation.
The answer also describes an exception. Changing payment dates, such as by bringing a payment forward or delaying it, can create convexity effects because the timing of cash flows differs from the vanilla instrument. Those effects depend on volatility and may be reflected in mark-to-market valuations, though the source characterizes them as generally small. The conclusion relies on matched contract terms and the stated option-pricing setup; customized swaps need separate analysis.
Key ideas
- A standard swap can be valued as a fixed rate bond position combined with a floating rate bond position.
- With matched terms and a common strike at the swap rate, cap-floor parity reproduces the swap value.
- Under the same theoretical volatility, the cap and floor vegas offset in that parity relationship.
- Customized payment dates can introduce convexity effects and a nonzero volatility sensitivity.
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Full text
# Does an Interest Rate Swap has a Vega component?
# 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.
## Answer by Sargera (score 7, accepted)
https://quant.stackexchange.com/a/16744
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 $0$.
## Answer by Attack68 (score 4)
https://quant.stackexchange.com/a/35896
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.