Thursday, September 23, 2010

How the Black-Scholes equation was derived

Invoking CAPM, [Fischer] Black used the betas to obtain equations for the expected return on the option and the stock. Expanding the term for the return on the option, and working out some simple algebra, he arrived at the crucial differential equation...

After "many, many days" trying unsuccessfully to solve the equation, Black put the problem aside. ... In Black's mind, just as in [Robert] Merton's, the warrant valuation problem was simply not that important. It was a kind of ivory tower curiosity, important to some academics but not to the wider world. Who really cared about warrant valuation except for speculators...? ...

One day, in summer or early fall of 1969, at one of their regular meetings on the Wells Fargo case at Fischer's office in Belmont, Myron Scholes brought up the topic of options pricing. ...

Indeed, success cannot have appeared very likely, since Scholes seemed no better equipped to solve the equation than Black.

Nevertheless, working together, they did manage to solve it, and in a most unlikely way. Bringing a number cruncher's practical empiricist approach to the problem, Scholes was naturally attracted to the analysis of Case Sprenkle, a graduate student at Yale who had come up with an incomplete formula for the option price containing parameters that he estimated from the data. Having this proto-formula in mind, Black and Scholes achieved the key breakthrough by thinking not about what had to be in the formula but rather about what had to be absent from it. ...

Sprenkle had provided a formula for the option price that required the user to provide two inputs: the expected return on the stock, and the discount rate for valuing the payoffs of the option. But Black and Scholes knew from the differential equation that the expected return on the stock could not enter the correct formula. They concluded that, without loss of generality, they could assume that the option was written on a zero-beta stock, which meant they could set the expected return on the stock equal to the interest rate. Further, since the option on a zero-beta stock would have a zero beta as well, they could use the same interest rate as the appropriate discount rate. Plugging these rates into Sprenkle's formula, they got a formula for the value of an option on a zero-beta stock. But the formula also satisfied the differential equation, which meant that it was also the formula for the value of an option on a non-zero-beta stock. The problem was solved.
--Perry Mehrling, Fischer Black and the Revolutionary Idea of Finance, on the stumble towards discovery

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