268 Frequently Asked Questions In Quantitative Finance
The steps along the way to finding the Black–Scholes
formulæ are as follows. We work within a single-period
framework, so that the concept of continuous hedging,
or indeed anything continuous at all, is not needed.
We assume that the stock price at the terminal time
(which will shortly also be an option’s expiration) and
the consumption are both lognormally distributed with
some correlation. We choose a utility function that is
a power of the consumption. A valuation expression
results. For the market to be in equilibrium requires
a relationship between the stock’s and consumption’s
expected growths and volatilities, the above-mentioned
correlation and the degree of risk aversion in the utility
function. Finally, we use the valuation expression for an
option, with the expiration being the terminal date. This
valuation expression can be interpreted as an expecta-
tion, with the usual and oft-repeated interpretation.
A Diffusion Equation
The penultimate derivation of the Black–Scholes partial
differential equation is rather unusual in that it uses just
pure thought about the nature of Brownian motion and
a couple of trivial observations. It also has a very neat
punchline that makes the derivation helpful in other
modelling situations.
It goes like this.
Stock prices can be modelled as Brownian motion, the
stock price plays the role of the position of the ‘pollen
particle’ and time is time. In mathematical terms Brow-
nian motion is just an example of a diffusion equation.
So let’s write down a diffusion equation for the value
of an option as a function of space and time, i.e. stock
price and time, that’sV(S,t). What’s the general linear
