Normal vs Log normal implied volatility

Question

I am referring to an earlier discussion at How do we know if the volatility which is quoted in market is Normal (Bachelier model) or log normal (Black 76)?

For the short rate case, is there any approximate relation between these 2 types of volatilities, given that we have quote for log-normal.

Answer 1

Pat Hagan describes this well in the famous SABR paper Managing smile risk. An approximate relation given in equation (B.64) reads $$\sigma_N \approx \sigma_B \frac{f-K}{\ln f/K}\left(1-\frac{\sigma_B^2 T}{24}\right),$$ where $\sigma_N$ is the normal (or Bachelier) vol, $\sigma_B$ is the Black-Scholes volatility, $f$ is the forward price, $T$ the option time to maturity, and $K$ the option strike. In particular, at-the-money, we have $\sigma_N \approx \sigma_B f$.

There exists very fast algorithms which allow to convert a Black vol to a normal (or b.p. vol) vol with near machine epsilon accuracy. They start from an option price, you would just use the Black-Scholes formula with $\sigma_B$ to obtain it.

Answer 2

Choi et al (2022) have a slightly better approximation for the volatility conversion:

Eq. (17): $$\sigma_N(K) \approx \sigma_B F_0 \sqrt{k}\left(1+\frac{\log^2 k}{24}\right) \Big/ \left(1 + \frac{\sigma_B^2}{24} T \right) \;\;\text{for}\;\; k=\frac{K}{F_0}. $$ is better than Eq. (16) which from Grunspan (2011): $$\sigma_N(K) \approx \sigma_B F_0 \frac{k-1}{\log k} \left(1 - \log \left(\frac{k-1}{\sqrt{k}\log k}\right) \frac{\sigma_B^2 T}{\log^2 k}\right) \;\;\text{for}\;\; k=\frac{K}{F_0}.$$

References:

• Choi J, Kwak M, Tee CW, Wang Y (2022) A Black–Scholes user’s guide to the Bachelier model. Journal of Futures Markets 42:959–980. https://doi.org/10.1002/fut.22315. [Arxiv Download]
• Grunspan C (2011) A Note on the Equivalence between the Normal and the Lognormal Implied Volatility : A Model Free Approach. arXiv:11121782 [q-fin]

Answer 3

This classical article by Patrick Hagan might help you: http://janroman.dhis.org/finance/Norm%20-%20LogNorm/Hagan%20Normvol.pdf