How Black-Scholes and Bachelier Implied Volatility Differ

# Does it matter that Bachelier IV differs from BS IV for a given option price?


# Does it matter that Bachelier IV differs from BS IV for a given option price?












In one sense, it’s just an accounting convention, so it doesn't matter. In another sense, the implied volatility can be interpreted as the minimum realised volatility which implies that your option price was ≤ fair value (realized via dynamic hedging/gamma scalping, see Gamma Pnl vs Vega Pnl from BS.)

So does it matter, or does it not? Which is the correct hurdle realized volatilty?

## Answer by Hasek (score 6)

https://quant.stackexchange.com/a/70653

Neither of them is correct or incorrect, these are just two different numerical inputs that one should plug-in into two different formulas to get the market price of an option given all other information.

The Black-76 formula (i.e. Black-Scholes in terms of forward price rather than a spot price) for pricing a call option is

$$C_{BS}(K) = F_0 N(d_1) - K N(d_2)$$ where $d_{1,2} = \frac{\log(F_0/K)}{\sigma_{BS}\sqrt{T}}\pm\frac{\sigma_{BS}\sqrt{T}}{2}$.

The Bachelier formula for a call is $$C_N(K) = (F_0-K) N(d_N) + \sigma_N\sqrt{T}n(d_N)$$ where $d_N = \frac{F_0-K}{\sigma_N\sqrt{T}}$.

It's clear that giving the same $K$, $F_0$, $T$ you have to use two different implied volatilites $\sigma_{BS}$ and $\sigma_N$ in order to match the market price of an option with two different formulas.

The more qualitative explanation is that volatility has different meanings in each model. The Black-Scholes model assumes a lognormal distribution of the underlying. The Bachelier model assumes a normal distribution. The Black-Scholes volatility $\sigma_{BS}$ measures the relative change in $F_t$, while the Bachelier volatility $\sigma_N$ measures the absolute change in $F_t$. In other words, the probability of the forward rate going from $1\%$ to $2\%$ is the same as the probability of it going from $2\%$ to $4\%$ in Black-Scholes and from $2\%$ to $3\%$ in Bachelier.

Thus these are just two different pricing conventions and one can more or less freely go back-and-forth between them. More details on conversion between Black-Scholes and Bachelier volatilities can be found in such articles as A Black-Scholes user's guide to the Bachelier model or Volatility conversion calculators if needed. Hope it helps.