**Summary:** This paper by L. Bachelier presents a mathematical analysis of the stock market, focusing on the theory of speculation. Bachelier examines the behavior of stock prices and the factors influencing them, noting that while the market is influenced by numerous unpredictable events, it can be analyzed statistically. He introduces the concept of "true course" (cours vrai), which represents the expected price at a given time, and argues that the market does not predict future movements but rather assigns probabilities to them. Bachelier discusses various financial operations, including "firm" (ferme) and "prime" (prime) trades, and their implications on market behavior. He introduces the idea of "equivalent courses" (cours équivalents), where different prices are considered equivalent based on time and interest rates. He also explores the concept of "options" (options), which are intermediate between firm and prime trades. The paper then delves into the mathematical modeling of market probabilities. Bachelier proposes a probability distribution function that describes the likelihood of stock price movements over time. He derives a formula for the probability density function, which is based on the assumption that stock price changes follow a normal distribution. This function is used to calculate the expected value of a trade and to determine the probability of a stock price reaching a certain level at a given time. Bachelier also discusses the concept of "expected value" (espérance mathématique) in trading, noting that the expected value of a trade is the product of the potential profit and the probability of that profit occurring. He argues that the expected value of a trade is zero in a fair market, meaning that neither the buyer nor the seller has an advantage. The paper concludes with a detailed analysis of the probability distribution of stock price movements, showing that the probability of a stock price reaching a certain level decreases over time. Bachelier's work laid the foundation for modern financial mathematics and is considered a pioneering contribution to the field of quantitative finance.**Summary:** This paper by L. Bachelier presents a mathematical analysis of the stock market, focusing on the theory of speculation. Bachelier examines the behavior of stock prices and the factors influencing them, noting that while the market is influenced by numerous unpredictable events, it can be analyzed statistically. He introduces the concept of "true course" (cours vrai), which represents the expected price at a given time, and argues that the market does not predict future movements but rather assigns probabilities to them. Bachelier discusses various financial operations, including "firm" (ferme) and "prime" (prime) trades, and their implications on market behavior. He introduces the idea of "equivalent courses" (cours équivalents), where different prices are considered equivalent based on time and interest rates. He also explores the concept of "options" (options), which are intermediate between firm and prime trades. The paper then delves into the mathematical modeling of market probabilities. Bachelier proposes a probability distribution function that describes the likelihood of stock price movements over time. He derives a formula for the probability density function, which is based on the assumption that stock price changes follow a normal distribution. This function is used to calculate the expected value of a trade and to determine the probability of a stock price reaching a certain level at a given time. Bachelier also discusses the concept of "expected value" (espérance mathématique) in trading, noting that the expected value of a trade is the product of the potential profit and the probability of that profit occurring. He argues that the expected value of a trade is zero in a fair market, meaning that neither the buyer nor the seller has an advantage. The paper concludes with a detailed analysis of the probability distribution of stock price movements, showing that the probability of a stock price reaching a certain level decreases over time. Bachelier's work laid the foundation for modern financial mathematics and is considered a pioneering contribution to the field of quantitative finance.