Bachelier model

The Bachelier model is a model of an asset price under Brownian motion presented by Louis Bachelier on his PhD thesis The Theory of Speculation (Théorie de la spéculation, published 1900). It is also called "Normal Model" equivalently (as opposed to "Log-Normal Model" or "Black-Scholes Model"). One early criticism of the Bachelier model is that the probability distribution which he chose to use to describe stock prices allowed for negative prices. (His doctoral dissertation was graded down because of that feature.) The (much) later Black-Scholes-(Merton) Model addresses that issue by positing stock prices as following a log-normal distribution which does not allow negative values. This in turn, implies that returns follow a normal distribution.

On April 8, 2020, the CME Group posted the note CME Clearing Plan to Address the Potential of a Negative Underlying in Certain Energy Options Contracts,^[1] saying that after a threshold on price, it would change its standard energy options model from one based on Geometric Brownian Motion and the Black–Scholes model to the Bachelier model. On April 20, 2020, oil futures reached negative values for the first time in history,^[2] where Bachelier model took an important role in option pricing and risk management.

The European analytic formula for this model based on a risk neutral argument is derived in Analytic Formula for the European Normal Black Scholes Formula (Kazuhiro Iwasawa, New York University, December 2, 2001).^[3]

The implied volatility under the Bachelier model can be obtained by an accurate numerical approximation.^[4]

For an extensive review of the Bachelier model, see the review paper, A Black-Scholes User's Guide to the Bachelier Model ^[5], which summarizes the results on volatility conversion, risk management, stochastic volatility, and barrier options pricing to facilitate the model transition. The paper also connects the Black-Scholes and Bachelier models by using the displaced Black-Scholes model as a model family.

References

1. "CME Clearing Plan to Address the Potential of a Negative Underlying in Certain Energy Options Contracts". www.cmegroup.com. Retrieved 2020-04-21.
2. "An oil futures contract expiring Tuesday went negative in bizarre move showing a demand collapse". CNBC. 15 December 2003. Retrieved 21 April 2020.
3. "Analytic Formula for the European Normal Black Scholes Formula". New York University. 2 December 2001.
4. Choi, Jaehyuk; Kim, Kwangmoon; Kwak, Minsuk (2009). "Numerical Approximation of the Implied Volatility Under Arithmetic Brownian Motion". Applied Mathematical Finance. 16 (3): 261–268. doi:10.1080/13504860802583436. S2CID 120908084. SSRN 990747.
5. Choi, Jaehyuk; Kwak, Minsuk; Tee, Chyng Wen; Wang, Yumeng (2022). "A Black-Scholes User's Guide to the Bachelier Model". Journal of Futures Markets. 42 (5): 959–980. arXiv:2104.08686. doi:10.1002/fut.22315. S2CID 246867256. SSRN 3828310.