Converting Between Black Lognormal and Bachelier Normal Volatility

# Normal vs Log normal implied volatility


# Normal vs Log normal implied volatility












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 by jherek (score 12, accepted)

https://quant.stackexchange.com/a/58570

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 expiry, 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 by jaehyukchoi49 (score 5)

https://quant.stackexchange.com/a/71149

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 by Frido (score 2)

https://quant.stackexchange.com/a/85511

Remind me again why you would even want an approximation if you can apply a root finder to solve $$ C(K) = C^{BS}(K,\Sigma^{LN}) = C^{Bachelier}(K,\Sigma^N) $$

But if you must use an approximation, I rather like an elegant first order approximation due to Fei Zhou: $$ \Sigma^{LN} \approx \frac{\Sigma^N}{\sqrt{FK}} $$ where $F$ is the forward price.

## Answer by Jesper Tidblom (score 1)

https://quant.stackexchange.com/a/58569

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