Black-Scholes or Bust

There are two ways to look at the derivation of Black-Scholes:

(i) the way Black and Scholes derived it originally, which was based on the hope that the market for a stock and its options would be in equilibrium when their Sharpe ratios were equal; and

(ii) via dynamic riskless replication of a portfolio that has the same payoff of the option.

Everyone nowadays learns the second method, Merton’s elegant discovery.

The two methods are equivalent as long as you assume geometric Brownian motion (GBM) for the stock, and lead to the same PDE.

But over the years, after I learned method (i) rather than method (ii), I’ve come to see it as more realistic and capable of extension, in other words, robust.

For GBM, the Sharpe ratio or risk premium is the excess return per unit of risk, and method (i) says that the stock and option will have the same price when they provide the same risk bang for the buck. But even when you don’t have GBM, the idea that the prices will equilibrate when both securities provide the same excess return per unit of risk is a good idea and seems like a more general truth.

Irrespective of subtleties, the risk of the stock and the risk of the option, whatever nature they take, are clearly related, so that equating their risk premiums gives a sensible constraint on their relative prices. That makes the Black-Scholes model robust and a rarity among financial models. In contrast, CAPM, from which it stems, is also based on GBM, but it works much less well for stock valuation, because stock prices suffer risks more diverse and wild than those associated with diffusion. There are many other huge risks – that prices can jump, that volatility can spike, that liquidity can dry up, that counterparties can all fail together in a crisis – that GBM ignores and that can therefore invalidate many of its results. But the idea behind BS can survive the invalidity of GBM