Nobel-Prize-winning Black-Scholes equations are actively used to estimate the price of options and other financial instruments. In practice, they provide a good estimate for the price, but the problem is that their original derivation is based on many simplifying statistical assumptions which are, in general, not valid for financial time series. The fact that these equations are effective way beyond their usual assumptions leads to a natural conclusion that there must be an alternative derivation for these equations, a derivation that does not use the usual too-strong assumptions. In this paper, we provide such a derivation in which the only substantial assumption is a natural symmetry: namely, scale-invariance of the corresponding processes. Scale-invariance also allows us to describe possible generalizations of Black-Scholes equations, generalizations that we hope will lead to even more accurate estimates for the corresponding prices.

中文翻译：

---

为什么 Black-Scholes 方程的有效性超出了通常的假设：基于对称性的解释

获得诺贝尔奖的 Black-Scholes 方程被积极用于估计期权和其他金融工具的价格。在实践中，它们为价格提供了一个很好的估计，但问题是它们的原始推导是基于许多简化的统计假设，这些假设通常对金融时间序列无效。这些方程的有效方式超出了它们通常的假设这一事实导致了一个自然的结论，即这些方程必须有一个替代推导，这种推导不使用通常的太强假设。在本文中，我们提供了这样一个推导，其中唯一的实质性假设是自然对称性：即相应过程的尺度不变性。尺度不变性还允许我们描述 Black-Scholes 方程的可能推广，