# Raiders of the Lost Black Scholes Derivation

I love the tingling sensation when some new idea or revelation becomes clear. Perhaps not 100% clear, but you get close enough that you can almost taste it.

That happened to me when I read about the Pythagorean interpretation of special relativity.

Nice to understand 20th century physics first in terms of Euclidean space and *then* make a small jump into Minkowski space.

Let's attempt something similar with jumping from a world of well known statistics into Black, Scholes & Merton.

Everyone understands Sharpe ratios. Risk adjusted performance. Expected return divided by volatility, measuring bang for each dollar of risk incurred.

But you rarely hear Sharpe ratios and Black Scholes mentioned in the same breath.

Which is odd. In 1973 CAPM; covariances and betas were referenced a lot in their first paper.

In any case Black Scholes give an alternate derivation of their equation using the CAPM model. Actually, they originally thought about the problem in that context (although I cannot find a reference to back this up!).

The key idea is that the Sharpe ratios for both the option and the stock should be the same; they then find the beta for the option and stock; and the rest is history.

Starting with replicating a call option with a risk free bond and a number (delta) of stocks; the Sharpe of both the stock and option are equal.

See the original paper for the full derivation.

There are two really important takeaways from this approach.

Firstly, statistics are mostly read as 'on average such and such happens'. Black Scholes steps out of this frequentist philosophy and is telling us that this relationship holds steadfastly, and makes the first step into financial engineering.

Secondly. We don't need to assume that stocks move in a Brownian motion. Our chief insight in this formulation is that the risk adjusted performance of both stock and option are the same. It's a more general and perhaps more powerful concept.

The Brownian motion assumption is prescriptive. While precise in one way, it's simply wrong because stocks don't move that way.

The Sharpe assumption however, is perhaps more robust because the source of risk for both stock and option is the same. It is descriptive not prescriptive.