Optimal calculations for the space-time fractional derivative option pricing models

What is it about?

This paper introduces novel space-time fractional derivative models (CLH and CFLH) for European option pricing, integrating stochastic liquidity risk, stochastic volatility, and Caputo/Caputo-Fabrizio fractional derivatives to capture financial market nonlinearity, memory effects, and non-stationarity. Traditional models like Black-Scholes and Heston overlook liquidity risk or assume constant volatility, limiting their realism. The proposed hybrid models address these gaps by incorporating a liquidity discount factor and extending the Heston framework with fractional calculus for multi-scale dynamics. A combination neural network (NN) architecture—combining exponential and polynomial terms in test solutions—is designed to solve high-dimensional fractional PDEs, overcoming challenges faced by traditional numerical methods. The NN integrates physical prior knowledge (PDEs, boundary conditions) and market data via composite loss minimization. Empirical validation using CSI 300 and SSE 50 ETF options demonstrates that the CLH and CFLH models achieve prediction errors below 0.01, outperforming the classical Heston model and integer-order liquidity models. The CFLH model, utilizing Caputo-Fabrizio derivatives, shows faster training than CLH while maintaining accuracy. Key innovations include unifying liquidity risk, volatility, and fractional derivatives, and leveraging neural networks for efficient PDE solutions. Results highlight the models’ adaptability to real market data, offering a robust framework for derivatives pricing under realistic conditions. This work bridges fractional calculus in finance with deep learning, providing a flexible tool for financial decision-making.

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Photo by Олег Мороз on Unsplash

Photo by Олег Мороз on Unsplash

Why is it important?

This paper is important for several key reasons: 1. Addressing Critical Gaps in Financial Models: Traditional models like Black-Scholes and Heston oversimplify market dynamics by ignoring liquidity risk or assuming constant volatility. This work integrates stochastic liquidity risk, stochastic volatility, and fractional calculus into a unified framework, capturing real-world complexities such as memory effects, multi-scale dynamics, and non-stationarity in asset prices. 2. Innovative Methodology: The authors design a combination neural network to solve high-dimensional fractional PDEs, overcoming limitations of traditional numerical methods (e.g., finite difference). The NN architecture leverages physical prior knowledge (PDEs, boundary conditions) and market data, achieving prediction errors below 0.01 while handling mixed fractional derivatives and multi-dimensional boundaries efficiently. 3. Empirical Validation: The models (CLH and CFLH) are rigorously tested on real market data (CSI 300 and SSE 50 ETF options). Results demonstrate superior accuracy compared to classical Heston and integer-order models, proving their practical utility for derivatives pricing in imperfectly liquid markets. 4. Bridging Disciplines: By merging fractional calculus with deep learning, the work advances computational finance, offering a flexible framework for pricing under realistic conditions. The Caputo-Fabrizio-based CFLH model further shows faster training than its Caputo counterpart, enhancing computational efficiency. 5. Broader Implications: The paper provides tools for better risk management, trading strategies, and financial decision-making in markets with liquidity constraints and volatility clustering. Its methodology opens avenues for future research at the intersection of machine learning, fractional calculus, and quantitative finance. Overall, this research advances both theoretical modeling and practical applications, addressing longstanding challenges in financial engineering with innovative, data-driven solutions.

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