Equilibrium Option Pricing in Stochastic Differential Games

This paper develops a rigorous game-theoretic framework for robust option pricing under model uncertainty, formulating the hedging problem as a zero-sum stochastic differential game between a hedger and an adversarial representation of market uncertainty. The resulting value function is shown to satisfy a Hamilton–Jacobi–Bellman–Isaacs (HJBI) equation, derived transparently via dynamic programming rather than imposed axiomatically. The analysis emphasises viscosity solutions as the natural solution concept, providing verification results, uniqueness via comparison principles, and a fully worked analytical example illustrating equilibrium structure and optimal controls. Numerical finite-difference schemes and Monte Carlo simulations consistent with viscosity theory are developed to visualise state-dependent regimes and worst-case behaviour. Rather than proposing a new pricing PDE, the contribution lies in unifying financial interpretation, rigorous PDE theory, and computational implementation within a coherent robust-hedging framework, clarifying when and why HJBI formulations arise in practice.