Why Black-Scholes Equations Are Effective Beyond Their Usual Assumptions: Symmetry-Based Explanation

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.