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 $u = \sigma \sqrt{T-t}$

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    $\begingroup$ You used "σ=0.15 for both models". I am not sure that is right, the two sigmas are defined differently. $\sigma$ for BS is in percentage terms, $\sigma$ for Bachelier is in dollar (or french franc? ;) ) terms. $\endgroup$ – Alex C Mar 26 '18 at 2:13
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    $\begingroup$ To elaborate on @Alexc comment , if $\sigma=0.15$ in the BS model , then the equivalent vol in the Bachelirr is 0.15*55= 8.25. $\endgroup$ – dm63 Mar 26 '18 at 3:54
  • $\begingroup$ Why mulitply with 55? $\endgroup$ – Lisa Mar 26 '18 at 10:49
  • $\begingroup$ The random variable which is added or subtracted to the stock price under Bachelier, is equal to 15% of the original stock price $S_0$, i.e. 0.15*55 $\endgroup$ – Alex C Mar 26 '18 at 13:32

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.


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