In his 1900 thesis “Théorie de la spéculation”, mathematician Louis Bachelier established the axiomatic foundations of long-term product pricing, emphasizing two key ideas:

• Trades can occur only when, at a given moment, buyers and sellers share the same “view” (probability law)regarding future price fluctuations. The principle of intertemporal price coherence is assumed by market participants.
• From this, he deduced that the distribution used by market participants to model price variation could only be that of a “Brownian motion.”
  The existence of a “pricing” probability distinct from statistical probability is central to Bachelier’s thinking.

These original insights were applied to financial markets only after being connected to the idea of a hedging portfolio, proposed by Black, Scholes, and Merton in the 1970s. Nevertheless, Bachelier’s work paved the way for major mathematical developments, notably the emergence of stochastic calculus, whose applications now span many fields. These new tools ultimately made the effective use of hedging portfolios possible.

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