Bachelier model

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The Bachelier model is a mathematical pricing model considered to be particularly useful in pricing options when the value of the underlying becomes or may become negative. It is an alternative to the Black-Merton-Scholes and other option pricing models and is attractive because it does not rely on logarithms which cannot represent negative values.[1] It is named after the French mathematician Louis Bachelier, who was active in France in the first half of the 20th century.

Use at CME Group

Against a market backdrop of low and falling crude oil and related energy prices in April 2020, CME Group announced its plans for its internal pricing of futures options in the event of negative traded prices of underlying futures contracts on August 8.[2][3] The plan called for a shift from the Whaley options pricing model to the Bachelier model, and on April 21 CME Group said that it was implementing Bachelier pricing effective with the next day's margin calculations for energy contracts.[4]

Bachelier in finance

Bachelier defended his doctoral thesis, Théorie de la Spéculation, at the Sorbonne in Paris, France, in 1900.

Paul Samuelson, the first American to win the Nobel Prize in Economics, introduced Bachelier to modern financial economists in his 1972 article "Mathematics of Speculative Price," which appeared in Mathematical Topics in Economic Theory and Computation. Samuelson said that Bachelier, ". . . seems to have had something of a one-track mind. But what a track!" Samuelson attributed to Bachelier the discovery of "Brownian motion" five years prior to Einstein's famous and much lauded similar discovery of the same phenomenon. Samuelson's encomium, most of which appears in a footnote in the article, reveals that Bachelier's work contributed to physics and mathematics before it was picked up by economists.[5]


  1. How Close Are the Option Pricing Formulas of Bachelier and Black-Merton-Scholes?. arXiv (Cornell University).
  2. Advisroy Notice:CME Clearing Plan to Address the Potential of a Negative Underlying in Certain Energy Options Contracts. CME Group.
  3. Episode 993: Negative Oil.
  4. Advisory Notice: Switch to Bachelier Options Pricing Model. CME Group.
  5. Robert C. Merton, Paul Samuelson and Financial Economics. JSTOR.