Mathematical Modeling of Option Pricing with an Extended Black-Scholes Framework
Nikhil Shivakumar Nayak
TL;DR
The paper tackles option pricing under realistic market dynamics by extending the Black-Scholes PDE to include stochastic volatility $\sigma(t)$ and stochastic interest rate $r(t)$, and by benchmarking a data-driven LSTM predictor against this extended model on Google options. The extended PDE is solved on a finite-difference grid, incorporating $\sigma$ and $r$ dynamics such as $d\sigma^2 = \kappa(\theta - \sigma^2)dt + \xi\sqrt{\sigma^2} dW_\sigma$ and $dr = a(b - r)dt + s dW_r$, while the LSTM model uses historical price, volatility, and rate inputs to forecast the option price $V$. Empirical results show the LSTM achieves lower RMSE than the extended PDE but requires more computation, whereas the finite-difference PDE offers faster inference albeit with higher sensitivity to parameter estimates and less flexibility in capturing non-lognormal features. The findings highlight complementary strengths of physics-based and machine-learned approaches, advocating hybrid strategies for robust option pricing across varying market conditions.
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 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.