252 Frequently Asked Questions In Quantitative Finance

T

he ten different ways of deriving the Black–Scholes

equation or formulæ that follow use different types

of mathematics, with different amounts of complexity

and mathematical baggage. Some derivations are useful

in that they can be generalized, and some are very

specific to this one problem. Naturally we will spend

more time on those derivations that are most useful or

give the most insight. The first eight ways of deriving

the Black–Scholes equation/formulæ are taken from the

excellent paper by Jesper Andreason, Bjarke Jensen and

Rolf Poulsen (1998).

In most cases we work within a framework in which

the stock path is continuous, the returns are normally

distributed, there aren’t any dividends, or transac-

tion costs, etc. To get the closed-form formulæ (the

Black–Scholesformulæ) we need to assume that volatil-

ity is constant, or perhaps time dependent, but for

the derivations of the equations relating the greeks

(the Black–Scholesequation) the assumptions can be

weaker, if we don’t mind not finding a closed-form

solution.

In many cases, some assumptions can be dropped.

The final derivation, Black–Scholes for accountants,

uses perhaps the least amount of formal mathematics

and is easy to generalize. It also has the advantage

that it highlights one of the main reasons why the

Black–Scholes model is less than perfect in real life. I

will spend more time on that derivation than most of

the others.

I am curious to know which derivation(s) readers prefer.

Please email your comments [email protected]

you know of other derivations please let me know.