A Few Uncomfortable Truths About the Black and Scholes (B-S) Equation

Abstract

We examine the performance of the Black-Scholes (B-S) formulas around (i.e., before, during and after) two periods of market stress: the subprime crisis (October, 2018) and the onset of the COVID-19 pandemic (March, 2020). We find, in agreement with previous studies under different circumstances, that the accuracy of these formulas is very poor. We also demonstrate that the usual derivation of the B-S formulas is unnecessarily complicated as one can arrive at the same expressions invoking less restrictive assumptions, dispensing with stochastic calculus altogether, and using only undergraduate-level statistics. Moreover, we challenge: (i) the relative merits of assuming that stock prices are better described by means of log-normal distributions, as opposed to normal distributions (both assumptions seem equally inadequate); and (ii) the idea of estimating the price of European options by focusing on modeling the asset price-process between t = 0 and expiration (T). We show that what is relevant is estimating the asset price at T; not the price trajectory between 0 and T. In short, in reference to the asset price, what really matters is the destination, not the journey. This represents an important shift in terms of how we should think about European-options pricing schemes.