Converting Bond Volatility into Yield Standard Deviation

Summary

The note explains how to express annualized bond yield volatility in basis points. Normal volatility is already an absolute yield change, while Black volatility is relative and can be approximated in yield terms by multiplying it by the forward yield. With the stated inputs, the two conventions produce similar annualized standard deviations.

To convert annualized volatility into the standard deviation over an option’s life, multiply by the square root of the time to expiry, measured in years. The example applies this scaling to a six-month option. The calculation assumes a normally distributed underlying; the approximation between Black and normal volatility also depends on the yield level and should not be treated as an exact equivalence in all cases.

Key ideas

• Normal volatility directly gives annualized yield volatility in basis points.
• Black volatility can be converted approximately to yield terms by multiplying it by the forward yield.
• Scale annualized standard deviation by the square root of expiry in years to get the option-period value.
• The time scaling assumes a normal distribution for the underlying.

Tags

# Convert volatility to yield


# Convert volatility to yield












I have an elementary bond volatility math question - if 5y black vol is 38 and normal vol is 68, and the yield is 1.85%, how do I calculate 1 stdev in yield terms? As in how many basis points (or percentage) would 1 standard deviation be.

## Answer by dm63 (score 1)

https://quant.stackexchange.com/a/46797

The annualized basis point standard deviation is equal to (a) the normal vol and (b) the lognormal (Black) vol times the forward yield of the instrument. In the above question, (a)=68bp and (b)= 0.38*185bp = 70bp so these are roughly in agreement.

The actual standard deviation is equal to the sqrt(expiration) times the annualized s.d , so for example, a 6 month option, sqrt(1/2) times 68bp = about 48bp. (This assumes the actual distribution of the underlying is a normal distribution).

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.