Analytical Formula for Pricing European Options with Stochastic Volatility under the GARCH-PDE Approximation

This article introduces a class of generative models based on the (G)ARCH-like continuous-time framework to unify econometric and diffusion-based methods for pricing European options. The authors formulate a partial differential equation (PDE) for the option price when the volatility of the underlying asset is described by a broad class of discrete-time GARCH models. The GARCH-PDE framework combines discrete data from the physical market with a latent stochastic volatility process, and provides flexibility in the pricing process to accommodate any underlying realized volatility dynamics. The authors reduce solving a two-dimensional degenerate PDE to finding the solutions of a system of one-dimensional PDEs, and then obtain closed-form analytical formulas for option pricing under stochastic volatility, which is typically achieved through Monte Carlo simulations. Convergence and error analysis of the analytical formula 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 are presented to illustrate the authors' approach and demonstrate its superiority over other empirically well-tested pricing methods.