Comparing Bachelier and Black–Scholes Volatility Units in Call Pricing
Summary
The document explains why using the same numerical volatility input in the Bachelier and Black–Scholes models can produce very different call prices. Although both models describe uncertainty in an underlying price, their volatility parameters use different units: Black–Scholes volatility is proportional to the asset price, while Bachelier volatility is measured in absolute price units. Equal numeric inputs therefore do not represent equal uncertainty.
For the at-the-money example shown, the response converts the Black–Scholes percentage volatility into an equivalent Bachelier price-unit volatility by multiplying by the initial underlying price. The document reports the resulting conversion and the two model prices, illustrating the mismatch. This is a unit-comparison explanation rather than a claim that the models are generally equivalent; the conversion is tied to the stated price level and setup, and the models have different distributional assumptions.
Key ideas
- Black–Scholes volatility is expressed proportionally to the asset price, while Bachelier volatility is in price units.
- Using the same numeric volatility in both models does not give the same scale of price uncertainty.
- The example converts percentage volatility to Bachelier units by multiplying by the initial underlying price.
- The conversion illustrates the example’s mismatch but does not make the two pricing models generally identical.
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Full text
# Bachelier model VS Black Scholes in call option pricing. Why are they so different?
# Bachelier model VS Black Scholes in call option pricing. Why are they so different?
I have been working with Bachelier model for some days but when I experimented with the model I saw some unwanted result with huge differences from the Black Scholes model. Bachelier model is described in detail here: Bachelier model call option pricing formula
Here is an numerical experiment: No interest rate; $\sigma=0.15$ for both models.
At time 0 I want to price a ATM European Call with $T=1$ and strike $K=55$ when $S_0=55$
The BS result: $C=3.29$
The Bachelier result: $C=0.06$
Why is there such a huge gap? I have tried to make sense of it by simply looking at the models but it is complicated with the CDFs and PDFs. Bachelier model is normally distributed and the BS model is log-normally distributed. Can we use that for explaining the big difference by claiming that the two processes is way different with the same volatility constant $\sigma$
The no Arbitrage pricing function for the Bachelier model with zero rate can be looked up a lot of places: $$ C_t = (S_t-K)*\Phi((S_t-K)/u)+v_t*\phi((S_t-K)/u) $$ where $u = \sigma \sqrt{T-t}$
## Answer by Bob Jansen (score 5)
https://quant.stackexchange.com/a/39581
Combining the comments:
You used $\sigma=0.15$ for both models. That is not right, the two sigma's are defined differently. $\sigma$ for BS is in percentage terms, $\sigma$ for Bachelier is in dollar terms.
So the equivalent volatility in Bachelier is $0.15 \times 55 = 8.25$ where 55 is the original stock price.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.