Bachelier's Theory of Speculation Revisited: A Reproducible Reconstruction of the Origins of Quantitative Finance

Louis Bachelier’s 1900 doctoral thesis, Théorie de la Spéculation, is one of the earliest mathematical foundations of modern quantitative finance. Long before the standardization of stochastic calculus in financial economics, Bachelier introduced a probabilistic framework for market price fluctuations and derived option-pricing relations based on what is now recognized as arithmetic Brownian motion. This paper provides a modern, reproducible reconstruction of Bachelier’s theory. We reformulate the core model in contemporary notation, derive its martingale and variance-scaling properties, recover the Bachelier European call pricing formula, and validate the formula numerically through Monte Carlo simulation. We further compare the Bachelier model with Black–Scholes in a low-relative-volatility regime, emphasizing both the historical continuity and the structural differences between arithmetic and geometric price dynamics. The contribution of this paper is not to introduce a new option-pricing theory, but to provide an open, transparent, and computationally reproducible bridge between Bachelier’s original work and modern quantitative finance. The paper concludes by discussing the limitations of the arithmetic model, especially the possibility of negative prices, and by proposing a reproducibility framework for historical financial mathematics.