Bachelier vs Black Volatility Mapping

Worked Solution

How to Think About It: The Black model assumes the underlying follows geometric Brownian motion (lognormal returns), while the Bachelier model assumes arithmetic Brownian motion (normal returns). Both produce perfectly good option prices -- the difference is in how they parameterize uncertainty. Black quotes vol as a percentage of the underlying, Bachelier quotes it in absolute dollar terms. At ATM, the two models agree to first order because percentage and absolute moves are nearly the same when the move is small relative to the level. The mapping between the two vols is essentially a unit conversion -- like Fahrenheit to Celsius, but with a nonlinear correction away from ATM.

Quick Sanity Checks:

• At ATM, the normal vol $\sigma_N$ should be roughly $S_0 \cdot \sigma_B$ -- percentage vol times the level gives you dollar vol. This is the most important formula to know.
• Away from ATM, the mapping should reduce to the ATM formula when $K = S_0$.
• For negative rates or spreads near zero, the lognormal model breaks down (you cannot take the log of a negative number), so the normal model must be preferred there.

Derivation:

Part 1: ATM mapping.

Under the Black model, the ATM ($K = S_0$) European call price is:

$C_{\text{Black}} = S_0 \left[\Phi\left(\frac{\sigma_B \sqrt{T}}{2}\right) - \Phi\left(-\frac{\sigma_B \sqrt{T}}{2}\right)\right]$

Using the identity $\Phi(x) - \Phi(-x) = 2\Phi(x) - 1$ and the small-$x$ expansion $\Phi(x) \approx \frac{1}{2} + \frac{x}{\sqrt{2\pi}}$, for small $\sigma_B \sqrt{T}$:

$C_{\text{Black}} \approx S_0 \cdot \frac{\sigma_B \sqrt{T}}{\sqrt{2\pi}}$

Under the Bachelier model, the ATM call price is:

$C_{\text{Bach}} = \sigma_N \sqrt{T} \cdot \phi(0) = \frac{\sigma_N \sqrt{T}}{\sqrt{2\pi}}$

where $\phi(0) = 1/\sqrt{2\pi}$ is the standard normal PDF at zero.

Setting $C_{\text{Black}} = C_{\text{Bach}}$:

$S_0 \cdot \frac{\sigma_B \sqrt{T}}{\sqrt{2\pi}} = \frac{\sigma_N \sqrt{T}}{\sqrt{2\pi}}$

The $\sqrt{T}$ and $\sqrt{2\pi}$ cancel, giving the first-order ATM mapping:

$\boxed{\sigma_N \approx S_0 \, \sigma_B}$

This is the key formula: normal vol equals spot times lognormal vol.

Part 2: General strike via vega matching.

For $K \neq S_0$, we match the vegas (price sensitivities to their respective vol parameters) of the two models.

The Black vega is:

$\mathcal{V}_{\text{Black}} = \frac{\partial C_{\text{Black}}}{\partial \sigma_B} = S_0 \sqrt{T} \, \phi(d_1)$

where $d_1 = \frac{\ln(S_0/K)}{\sigma_B \sqrt{T}} + \frac{\sigma_B \sqrt{T}}{2}$.

The Bachelier vega is:

$\mathcal{V}_{\text{Bach}} = \frac{\partial C_{\text{Bach}}}{\partial \sigma_N} = \sqrt{T} \, \phi(d_N)$

where $d_N = \frac{S_0 - K}{\sigma_N \sqrt{T}}$.

Setting the price changes equal for a small perturbation ($\mathcal{V}_{\text{Black}} \, d\sigma_B = \mathcal{V}_{\text{Bach}} \, d\sigma_N$) and taking the ratio:

$\frac{d\sigma_N}{d\sigma_B} = \frac{S_0 \, \phi(d_1)}{\phi(d_N)}$

To first order in $\sigma_B \sqrt{T}$ (small-$T$ regime), $d_1 \approx \frac{\ln(S_0/K)}{\sigma_B \sqrt{T}}$ and $d_N \approx \frac{S_0 - K}{\sigma_N \sqrt{T}}$. Using $S_0 - K \approx S_0 \ln(S_0/K)$ for $K$ near $S_0$ and the ATM relation $\sigma_N \approx S_0 \sigma_B$, the arguments match to leading order and the PDF ratio is approximately 1.

The refined small-$T$ approximation (valid for moderate moneyness) is:

$\boxed{\sigma_N \approx \sigma_B \cdot \frac{S_0 - K}{\ln(S_0/K)}}$

Note that $\frac{S_0 - K}{\ln(S_0/K)}$ is the logarithmic mean of $S_0$ and $K$. At ATM ($K = S_0$), L'Hopital's rule gives $\frac{S_0 - K}{\ln(S_0/K)} \to S_0$, recovering $\sigma_N \approx S_0 \sigma_B$.

Part 3: When to prefer the Bachelier model.

• Negative or near-zero underlyings: The Black model requires $S > 0$ since it models $\ln S$. For interest rates (which can go negative), credit spreads near zero, or calendar spreads, the Bachelier model is the natural choice.
• Additive payoffs: When the payoff depends on absolute moves rather than percentage moves (e.g., basis point changes in rates), normal vol is the natural unit.
• Skew stability: In rates markets, the normal vol smile is often more stable than the lognormal smile across different rate levels. When rates double from 1% to 2%, lognormal vol halves mechanically, but normal vol stays roughly constant -- this makes normal vol a better quoting convention.
• Near-zero forwards: As the forward approaches zero, Black implied vol explodes (since $\sigma_B \approx \sigma_N / F$), making it useless as a quoting convention.

Practical Interpretation: On a rates desk, you live in normal vol. On an equity desk, you live in lognormal vol. The mapping $\sigma_N \approx S_0 \sigma_B$ (or its log-mean generalization) is what you use to translate between the two worlds. When someone quotes you "80 bps normal vol" on a 10-year swap rate at 4%, that is roughly $0.0080 / 0.04 = 20\%$ lognormal vol. This back-of-envelope conversion is used constantly.

Answer: The ATM mapping is $\sigma_N \approx S_0 \, \sigma_B$. The general small-$T$ approximation via vega matching is $\sigma_N \approx \sigma_B \cdot \frac{S_0 - K}{\ln(S_0/K)}$, where the fraction is the logarithmic mean of $S_0$ and $K$. The Bachelier model is preferred when the underlying can be negative, near zero, or when absolute (rather than percentage) moves are the natural unit -- most commonly in interest rate and credit markets.