Author(s)

Hao Li^*, Martin Keller

Affiliation(s)

School of Finance, University of St. Gallen, German-Speaking, Switzerland.

Corresponding Author

Hao Li

ABSTRACT

The construction of arbitrage-free volatility surfaces represents a fundamental challenge in quantitative finance, particularly in incomplete markets where hedging portfolios cannot perfectly replicate option payoffs. Traditional parametric models often fail to capture the complex dynamics of market-observed implied volatility structures, including the characteristic smile and skew patterns, while simultaneously satisfying no-arbitrage constraints. This paper introduces a novel framework that leverages Physics-Informed Neural Networks (PINNs) to construct arbitrage-free volatility surfaces in incomplete market settings. The proposed methodology integrates partial differential equation constraints derived from arbitrage-free conditions directly into the neural network training process through automatic differentiation and soft constraint penalties. By incorporating Dupire's local volatility equation and calendar-butterfly arbitrage constraints into a multi-objective loss function, our approach generates smooth, arbitrage-free implied volatility surfaces that accurately fit market data across different strikes and maturities. Numerical experiments using both synthetic data and real market observations from S&P 500 and VIX options demonstrate that the PINN-based framework substantially reduces calibration errors while maintaining theoretical consistency. The method exhibits particular strength in handling incomplete market scenarios where traditional parametric approaches produce inconsistent surfaces or violate no-arbitrage conditions.

KEYWORDS

Physics-informed neural networks; Arbitrage-free constraints; Volatility surface; Incomplete markets; Volatility smile; Heston model

CITE THIS PAPER

Hao Li, Martin Keller. Physics-informed neural networks for arbitrage-free volatility surface construction in incomplete markets. AI and Data Science Journal. 2025, 2(2): 50-60. DOI: https://doi.org/10.61784/adsj3026.

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