GARCH-PDE models for option pricing under stochastic volatility and their finite difference solvers

This paper presents numerical solvers for generative and hybrid option pricing models that unify econometric and diffusion-based approaches. These models are formulated as systems of continuous partial differential equations (PDEs), with stochastic volatility updated at discrete reset dates according to generalized autoregressive conditional heteroskedasticity (GARCH) dynamics. In contrast to approaches that estimate volatility from option prices using the Black–Scholes model or Monte Carlo simulations, this method simplifies option pricing under stochastic volatility by exogenously supplying and updating latent intraday volatility using discrete return data. We then develop and analyze numerical techniques for solving the resulting system of parabolic partial differential equations, which feature time-varying diffusion coefficients governed by the stochastic volatility paths inferred from the return data. Convergence and stability analyses of the numerical schemes attest to the option pricing accuracy of the proposed framework using available discrete implied volatility samples without compromising its computational accuracy. Several sets of numerical tests using SPX data are presented to illustrate our approach and demonstrate its superiority over other empirically well-tested pricing methods.