Constrained deep learning for pricing and hedging european options in incomplete markets
Researchers present a deep learning framework for pricing and hedging European options in incomplete markets, where standard no-arbitrage methods fail. By embedding the terminal payoff into neural networks and using constrained architectures, they achieve higher accuracy, especially for exotic options with complex payoffs. A single network outputs both price and hedging strategy, trained to minimize profit-and-loss dispersion. The direct P&L loss method outperforms classical approaches, though it may not always ensure coherent prices under severe model errors.
What it examines
This paper presents a deep learning framework for pricing and hedging European options in incomplete markets, where perfect hedging is impossible. It uses a single neural network to learn both the option price and hedging strategy, focusing on minimizing profit and loss errors, especially for non-smooth or complex payoffs.
What it concludes
The study shows that embedding the terminal payoff condition in neural networks greatly improves hedging performance, even for exotic or non-smooth options. This approach can be applied to real-world financial markets for better risk management and pricing, especially when markets are incomplete or models are misspecified.
Evidence objects
Researchers unveil a deep learning framework that embeds terminal payoff conditions into neural networks, dramatically boosting pricing accuracy and hedging for European options, even with non-smooth or discontinuous payoffs in incomplete markets.
key_findings bullet 1 · key_findings · validation V0
A single neural network outputs both option price and its gradient (hedging strategy), trained with a loss function enforcing self-financing and minimizing profit-and-loss (P&L) dispersion, outperforming classical methods in out-of-sample hedging.
key_findings bullet 2 · key_findings · validation V0
Constrained and zero-target neural architectures excel at handling exotic payoffs like the Equinox option, but the P&L loss alone may not always ensure coherent prices, especially under severe model misspecification.
key_findings bullet 3 · key_findings · validation V0
This paper presents a novel constrained deep learning framework for pricing and hedging European options in incomplete markets, jointly outputting prices and hedges while embedding terminal payoff conditions inspired by PDEs. Its explicit treatment of non-smooth payoffs and robustness to market jumps make it an original, compelling advancement in quantitative finance.
key_findings bullet 4 · key_findings · validation V0
Raw abstract and provenance
Abstract: In incomplete financial markets, pricing and hedging European options lack a unique no-arbitrage solution due to unhedgeable risks. This paper introduces a constrained deep learning approach to determine option prices and hedging strategies that minimize the Profit and Loss (P&L) distribution around zero. We employ a single neural network to represent the option price function, with its gradient s… ▽ More In incomplete financial markets, pricing and hedging European options lack a unique no-arbitrage solution due to unhedgeable risks. This paper introduces a constrained deep learning approach to determine option prices and hedging strategies that minimize the Profit and Loss (P&L) distribution around zero. We employ a single neural network to represent the option price function, with its gradient serving as the hedging strategy, optimized via a loss function enforcing the self-financing portfolio condition. A key challenge arises from the non-smooth nature of option payoffs (e.g., vanilla calls are non-differentiable at-the-money, while digital options are discontinuous), which conflicts with the inherent smoothness of standard neural networks. To address this, we compare unconstrained networks against constrained architectures that explicitly embed the terminal payoff condition, drawing inspiration from PDE-solving techniques. Our framework assumes two tradable assets: the underlying and a liquid call option capturing volatility dynamics. Numerical experiments evaluate the method on simple options with varying non-smoothness, the exotic Equinox option, and scenarios with market jumps for robustness. Results demonstrate superior P&L distributions, highlighting the efficacy of constrained networks in handling realistic payoffs. This work advances machine learning applications in quantitative finance by integrating boundary constraints, offering a practical tool for pricing and hedging in incomplete markets. △ Less
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