Finance-Informed Neural Network: Learning the Geometry of Option Pricing

We propose a Finance-Informed Neural Network (FINN) for option pricing and hedging that integrates financial theory directly into machine learning. Instead of training on observed option prices, FINN is learned through a self-supervised replication objective based on dynamic hedging, ensuring economic consistency by construction. We show theoretically that minimizing replication error recovers the arbitrage-free pricing operator and yields economically meaningful sensitivities. Empirically, FINN accurately recovers classical Black–Scholes prices and performs robustly in stochastic volatility environments, including the Heston model, while remaining stable in settings where analytical solutions are unavailable or unreliable. Fundamental pricing relationships such as put–call parity emerge endogenously. When applied to implied-volatility surface reconstruction, FINN produces surfaces that are consistently closer to observed market-implied volatilities than those obtained from Heston calibrations, indicating superior out-of-sample adaptability and reduced structural bias. Importantly, FINN extends beyond liquid option markets: it can be trained directly on historical spot prices to construct coherent option prices and Greeks for assets with no listed options. More broadly, FINN defines a new paradigm for financial pricing, in which prices are learned from replication and risk-control principles rather than inferred from parametric assumptions or direct supervision on option prices. By reframing option pricing as the learning of a pricing operator rather than the fitting of prices, FINN offers practitioners a practical and scalable tool for pricing, hedging, and risk management across both established and emerging financial markets.

💡 Research Summary

The paper introduces the Finance‑Informed Neural Network (FINN), a novel framework that learns option pricing functions directly from the economic principle of dynamic replication rather than from observed option prices. FINN parameterizes the pricing map (g_\theta(t,S)) with a deep neural network and constructs a self‑financing hedging portfolio (\Pi_t = -g_\theta(t,S_t) + \alpha_t S_t + \eta_t H_t + \beta_t B_t), where (S_t) is the underlying, (H_t) an optional traded derivative used for gamma hedging, and (B_t) the risk‑free asset. The training objective is the squared replication error (|d\Pi_t - r\Pi_t dt|^2), i.e., the deviation of the portfolio’s drift from the risk‑free rate. By minimizing this loss, the network is forced to produce prices and Greeks that make the portfolio locally risk‑neutral, thereby embedding no‑arbitrage constraints directly into learning.

The authors provide rigorous theoretical results. First, they show that when the replication error converges to zero, the learned function coincides with the unique arbitrage‑free price given by the risk‑neutral expectation (E_Q