Comparing Black–Scholes and Bachelier Option Pricing Models
Summary
The document asks how to compare Black–Scholes and Bachelier option prices in an energy-market setting, where the underlying can become negative. Since Black–Scholes does not accept negative underlying prices or strikes, a direct price comparison cannot cover the full range of market conditions. The questioner has considered measuring squared price differences and asks about Bachelier Greeks.
The responses suggest comparing model-implied volatilities by finding the Bachelier volatility that reproduces a Black–Scholes price, and comparing Greeks as well as prices. One response advises excluding negative-price observations from the comparison. These are brief suggestions rather than a developed evaluation method: the document provides no data, calculations, or empirical results, and does not explain how to handle the models' different assumptions or calibrations. Excluding negative prices also leaves out the region that motivates using Bachelier, so conclusions from such a comparison may not address that use case.
Key ideas
- Black–Scholes cannot price options with negative underlying values or strikes, which limits direct comparisons in markets where negative prices are possible.
- A Bachelier implied volatility can be calculated to match a price generated by Black–Scholes.
- Comparisons could include model Greeks in addition to price differences.
- Excluding negative-price observations avoids an invalid Black–Scholes calculation but omits a key use case for Bachelier.
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Full text
# Method of comparing two option pricing models? # Method of comparing two option pricing models? 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! ## Answer by tcpedersen (score 1) https://quant.stackexchange.com/a/61409 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. ## Answer by Kulendra 'KJ' Janaka (score 0) https://quant.stackexchange.com/a/61406 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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