The article "Théorie de la Spéculation" by L. Bachelier, published in the *Annales scientifiques de l'É.N.S.* in 1900, explores the mathematical theory of speculation in financial markets. Bachelier discusses the numerous factors influencing stock price movements, including past, present, and future events, as well as artificial causes such as the self-referential nature of the market. He argues that while it is impossible to predict market movements mathematically, it is possible to study the static state of the market at a given instant, determining the probability distribution of price changes. Bachelier introduces two types of operations in the stock market: forward operations and options. Forward operations are similar to spot operations but are settled at a fixed date, while options allow for more flexibility in buying or selling. He also discusses the concept of premiums, where buyers pay a premium to lock in a higher price for future delivery, and the response of premiums, which occur before the settlement date. The article delves into the probabilities associated with these operations, distinguishing between mathematical probabilities (a priori) and future-dependent probabilities. Bachelier uses the concept of mathematical expectation to analyze the expected outcomes of different operations, showing that the expected mathematical expectation of a speculator is zero. He derives the Gaussian distribution as the probability distribution governing price changes over time, and discusses the mean and variance of these changes. Bachelier's work provides a foundational framework for understanding the statistical behavior of financial markets, emphasizing the importance of considering both historical and future probabilities in speculative activities.The article "Théorie de la Spéculation" by L. Bachelier, published in the *Annales scientifiques de l'É.N.S.* in 1900, explores the mathematical theory of speculation in financial markets. Bachelier discusses the numerous factors influencing stock price movements, including past, present, and future events, as well as artificial causes such as the self-referential nature of the market. He argues that while it is impossible to predict market movements mathematically, it is possible to study the static state of the market at a given instant, determining the probability distribution of price changes. Bachelier introduces two types of operations in the stock market: forward operations and options. Forward operations are similar to spot operations but are settled at a fixed date, while options allow for more flexibility in buying or selling. He also discusses the concept of premiums, where buyers pay a premium to lock in a higher price for future delivery, and the response of premiums, which occur before the settlement date. The article delves into the probabilities associated with these operations, distinguishing between mathematical probabilities (a priori) and future-dependent probabilities. Bachelier uses the concept of mathematical expectation to analyze the expected outcomes of different operations, showing that the expected mathematical expectation of a speculator is zero. He derives the Gaussian distribution as the probability distribution governing price changes over time, and discusses the mean and variance of these changes. Bachelier's work provides a foundational framework for understanding the statistical behavior of financial markets, emphasizing the importance of considering both historical and future probabilities in speculative activities.