macrocephalopod

A short thread about the Black derivation of Black-Scholes ________________ I first learned the Merton derivation of BS, pretty much the standard one now. You hedge the option with delta shares, find delta to make the portfolio riskless, then it must earn the riskless return 1/n

… That pde, using stochastic calculus, gives you the BS pde. In my youth I liked that precise argument. And when I learned that derivation I didn’t even know what CAPM was, and when I learned it I was unimpressed. BUT … 2/n

… When I look at it now, I’m kind of impressed by the CAPM/Black derivation of BS. Because what it’s saying is that the Sharpe ration of the option must = the Sharpe ratio of the stock. (When you apply stochastic calculus to the Sharpe of the option, you get the BS pde.) 3/n

Equating their Sharpe ratios is a statistical statement, because Sharpe ratios are measured statistically. So this is a coarser statement than the precise one of Merton’s. And courses is perhaps better when a model isn’t really describing the world accurately. 4/n …

… And furthermore the equality of Sharpe ratios is saying that “the expected return per unit of risk” for both the stock and the option should be the same. True, with stochastic calculus, that translates to the BS PDE. BUT … 5/n

"Equal expected return per unit of risk”(EERPUOFR) is a more general statement than pinning it down with stochastic calculus which works only for geometric Brownian motion, which doesn’t actually hold. … 6/n