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I am currently writing a small paper comparing the Black-Scholes formula to the Bachelier model. However I am wondering how exactly I should compare the two models?

Obviously I am comparing the prices given by the two models, but the whole point of implementing the Bachelier model (in a context of energy option specifically) is that it allows for negative prices on the underlying asset. The Black-Scholes model does not allow for negative prices (or strikes) as input and hence I cannot directly compare the prices of the options when the price of the underlying turns negative.

How can I meaningfully compare the two models? So far I am trying to model the squared difference between the two prices but I would like to do more.

Also, if anybody has a link to analytical formulas for the greeks of the Bachelier model I would greatly appreciate it.

Thanks a lot!

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You can compute the implied volatility in terms of the Bachelier model. That is, compute the volatility parameter in the Bachelier that corresponds to the BS price.

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    $\begingroup$ Thanks a lot! This is a very good idea. I was wondering about implementing a comparison of the IV so thank you a lot for this! $\endgroup$ – Vetlekos Feb 28 at 10:44
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Not sure if I am allowed to post them (or if you expect them to - because it may affect your own research) but; there have been some research done already on the Bachelier vs. BSM vs. other models.

With regards to the negative price range, my opinion is that you simply should leave it out from the comparison.

In addition to the RMS of the difference between BSM and Bachelier, you could also compare the differences of the Greeks. There are some papers that discuss the same.

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  • $\begingroup$ Thank you very much for your input. I agree about the negative prices, if I cannot compare them they should just be left out. I am also in the process of comparing the greeks. Again, thanks! $\endgroup$ – Vetlekos Feb 28 at 10:45
  • $\begingroup$ Hello again, do you perhaps have a reference to the papers discussing the greeks in the Bachelier model? Thank you! $\endgroup$ – Vetlekos Mar 3 at 7:41

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