The Black–Scholes equation in the presence of arbitrage
运用几何套利理论，证明了在特定多维伊藤过程下，无免费午餐与消失风险条件与期望效用最大化等价，并将布莱克-舒尔斯方程推广到允许套利的市场。
We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which do not need stochastic differential geometry in their formulation. First, for generic market dynamics given by a subclass of multidimensional Itô processes we specify and prove the equivalence between No-Free-Lunch-with-Vanishing-Risk (NFLVR) and expected utility maximization. As a by-product, we provide a geometric characterization of the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition given by the zero curvature (ZC) condition for this subclass of Itô processes. Finally, we extend the Black–Scholes partial differential equation to markets allowing arbitrage.