Lie symmetry analysis of the Black–Scholes equation with endogenous stochastic arbitrage

Abstract Existing research on relaxing the no-arbitrage assumption within the Black–Scholes framework remains relatively limited. To bridge the gap between this idealized assumption and the presence of arbitrage opportunities in real markets, we derive an option pricing model with endogenous stochastic arbitrage. This provides an analytical framework for characterizing localized fluctuations in option prices. Using empirical data on Nasdaq-100 E-mini futures options, we identify and characterize a general form of the arbitrage perturbation. We then apply the Lie symmetry method to the modified Black–Scholes equation and derive its Lie symmetry group and infinitesimal generators. Based on these infinitesimal generators, we derive invariants by solving the associated characteristic equations, from which the exact solutions are obtained. Finally, the numerical results show that, in the presence of localized price fluctuations, the proposed endogenous stochastic arbitrage model reduces pricing errors more effectively than the classic Black–Scholes model. This study not only extends option pricing analysis under non-equilibrium market conditions from the perspective of localized fluctuations, but also enriches the study of symmetry analysis and explicit solutions for modified Black–Scholes-type equations.