The famous volatility smile in options pricing is actually just a fundamental geometric property of information theory called Fisher curvature.
April 26, 2026
Original Paper
A Geometric Generalization of Black-Scholes-Merton: The Volatility Smile as Fisher Curvature
SSRN · 6630259
The Takeaway
This discovery maps a messy, real-world financial phenomenon to a clean law of geometry. The smile has long been treated as an empirical quirk that traders had to account for using messy math. This paper shows it is actually a natural curve on a Gaussian Fisher manifold. Identifying this connection could replace existing heuristic models with a more stable, geometric foundation for pricing. It bridges the gap between abstract physics and the chaos of the derivatives market.
From the abstract
We place the Black-Scholes-Merton (BSM) formula inside the information geometry of the Gaussian Fisher manifold and identify the volatility smile with the curvature of a leverage-corrected Poincaré extension of that manifold, made visible to the pricing problem precisely when stochastic volatility pulls the state off the one-dimensional flat slice on which BSM is exact. Three results organize the paper. First, BSM is exact on the one-dimensional flat sub-manifold ℒ_σ₀ = {σ = σ₀} of the Gaussian