Mathematical Modeling of Option Pricing with an Extended Black-Scholes Framework
This study enhances option pricing by extending Black-Scholes with stochastic volatility and interest rates, comparing PDE and LSTM models.
What it examines
The paper extends the Black-Scholes model by incorporating stochastic volatility and interest rate variations. It compares a finite difference method for solving the resulting PDE with an LSTM machine learning model to improve option pricing accuracy under realistic market conditions.
What it concludes
The study finds that while the extended Black-Scholes model is computationally efficient, the LSTM model offers superior predictive accuracy. Their combined use can enhance real-time option pricing, risk management, and financial decision-making, with further research needed to refine parameters and apply the models in diverse markets.
Evidence objects
Researchers extend the classic Black-Scholes model by integrating stochastic volatility and variable interest rates into a complex partial differential equation solved via finite difference methods, significantly enhancing option pricing realism.
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Surprisingly, while the extended model offered rapid computations and robust performance under varied market conditions, an LSTM machine learning model achieved lower prediction errors, greatly outperforming standard finite difference approaches.
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The study combines traditional analytical models with machine learning, using historical data and backtesting, while exposing high parameter sensitivity and substantial LSTM data requirements for robust, practical, comprehensive financial modeling.
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This paper extends the classical $$\mathrm{Black\text{-}Scholes}$$ framework by incorporating stochastic volatility and interest rate variation, and compares it with an LSTM model. It integrates numerical methods with machine learning, providing a fresh yet familiar hybrid approach. Although not revolutionary, its innovative adaptation presents significant implications for advanced financial modeling applications.
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Raw abstract and provenance
Abstract: This study investigates enhancing option pricing by extending the Black-Scholes model to include stochastic volatility and interest rate variability within the Partial Differential Equation (PDE). The PDE is solved using the finite difference method. The extended Black-Scholes model and a machine learning-based LSTM model are developed and evaluated for pricing Google stock options. Both models we… ▽ More This study investigates enhancing option pricing by extending the Black-Scholes model to include stochastic volatility and interest rate variability within the Partial Differential Equation (PDE). The PDE is solved using the finite difference method. The extended Black-Scholes model and a machine learning-based LSTM model are developed and evaluated for pricing Google stock options. Both models were backtested using historical market data. While the LSTM model exhibited higher predictive accuracy, the finite difference method demonstrated superior computational efficiency. This work provides insights into model performance under varying market conditions and emphasizes the potential of hybrid approaches for robust financial modeling. △ Less
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