How Normal Volatility Scales with the Forward Price

Summary

The document addresses confusion about converting lognormal volatility to normal volatility for a bond option. A calculation using a 5.9% lognormal volatility, a forward of 134.5, a strike of 132.5, and a short maturity produces a normal volatility near 7.87 in price units, which the questioner interprets as 787%. The response explains that this large figure is not inherently an error or evidence of a problem with a named formula.

The key distinction is that normal volatility is expressed in absolute price units and scales with the underlying forward, while lognormal volatility is a relative percentage measure. Near at-the-money, the response gives the approximation that normal volatility is lognormal volatility multiplied by the forward price. This explains why the conversion can yield a value around 784 price units when the forward is around 134.5. The explanation is approximate and focuses on the units and scale of the two volatility conventions rather than a full derivation of the conversion formula.

Key ideas

• Normal volatility is measured in absolute price units, while lognormal volatility is relative to the underlying price.
• Near the money, normal volatility is approximately lognormal volatility multiplied by the forward price.
• A large normal-volatility figure can therefore be consistent with a much smaller percentage lognormal volatility.
• The response explains scale and interpretation rather than deriving the complete conversion formula.

Tags

# Hagan formula for normal volatility


# Hagan formula for normal volatility












I am not sure I understand how Hagan normal volatility formula works. Basically I have:

Lognormal volatility of 0.059 (5.9%) Forward price of the bond 134.5 Bond Strike price 132.5 Option maturity 0.16 years

Normal volatility = 0.059 * (134.5-132.5) / ln ( 134.5/132.5) * ( 1 - 0.059 * 0.059 * 0.16/24) = 7.87

This means that with a 5.9% lognormal volatility I get 787% normal volatility!! I am doing anything wrong?

For the context these data are taken from bond option

## Answer by jherek (score 1)

https://quant.stackexchange.com/a/80717

This has nothing to do with Hagan formula but only with what lognormal and normal volatilities mean.

Normal volatilities are highly dependent on the underlying price, where as lognormal volatilities are not.

If your underlying forward is $F=134.5$, at-the-money, the normal vol is approximately $\sigma_N = \sigma_{LN} F$.

Hence the value of 784% is normal for a normal vol, when $\sigma_{LN} \approx 6\%$ and $F=134.5$.

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.