Measuring Interest Rate Swap Volatility in Basis Points

Summary

The document explains that interest rate swap volatility is commonly expressed as an absolute change in the swap rate, measured in basis points, rather than as a percentage change in the rate. It describes a normal-volatility convention in which a daily basis-point volatility is scaled by the square root of the number of business days to obtain an annualized measure. Under the stated normal-distribution assumption, the resulting volatility describes a range around the forward swap rate over a given horizon.

For realized volatility analysis, the response recommends measuring day-to-day absolute changes in swap or forward rates, matching the units used for implied swaption volatility quotes. The interpretation depends on the assumed normal distribution and the chosen horizon and day-count convention. This is a rate-volatility convention; it differs from calculating percentage returns on a traded instrument's price.

Key ideas

• Interest rate swap volatility is commonly quoted as an absolute move in basis points per day.
• The daily basis-point volatility can be annualized by scaling with the square root of business days per year.
• Under a normal-rate assumption, volatility describes a range around the forward swap rate over the chosen horizon.
• Historical realized volatility can be estimated from daily absolute changes in swap or forward rates.
• Using percentage changes in the rate would not match the stated market quoting convention.

Tags

# Calculating the volatility of an interest rate swap


# Calculating the volatility of an interest rate swap












At its most basic the volatility of an instrument is the standard deviation of its return series over time calculated as percentage change of the price series.

How would this work for interest rate swaps where the "price" is a rate and is expressed as a percentage? I would be inclined to proceed using the percentage change in the swap rate to calculate the volatility. Is this the correct approach?

## Answer by user35980 (score 2, accepted)

https://quant.stackexchange.com/a/75772

Volatility is a measure of the deviation from expected value over a given time horizon. The expected value of an IRS is the forward rate of the IRS to that time horizon. Let's assume this forward rate is normally distributed. Standard volatility quote convention for vols in IRS vol markets is basis points/day $\sigma_d$ (this is an absolute change in bps of the swap rate) and is related to the annualized bp volatility $\sigma$ via $\sigma=\sigma_d\sqrt{252}$ (for 252 business days in a year). So if the $n$-year vol of an $N$-year swap is $\sigma_d$ bps/day and the $n$-year forward rate is $F$, then this means that there is a 68% probability that the $N$-year swap rate in $n$-years time will be in the range $F\pm\sigma\sqrt{n}$.

In case you're doing some time series analysis of realized IRS vols, then it would make sense to compute the historical "bps/day" vol as this is how the implied vols are quoted in the market i.e. look at historical day/day absolute changes in IRS/forward rates.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.