Chapter 4: Ten Different Ways to Derive Black–Scholes 269
diffusion equation? It is
∂V
∂t
+a
∂^2 V
∂S^2
+b
∂V
∂S
+cV= 0.
Note the coefficientsa,bandc. At the moment these
could be anything.
Now for the two trivial observations.
First, cash in the bank must be a solution of this
equation. Financial contracts don’t come any simpler
than this. So plugV=ertinto this diffusion equation
to get
rert+ 0 + 0 +cert= 0.
Soc=−r.
Second, surely the stock price itself must also be a
solution? After all, you could think of it as being a call
option with zero strike. So plugV=Sinto the general
diffusion equation. We find
0 + 0 +b+cS= 0.
Sob=−cS=rS.
Puttingbandcback into the general diffusion equation
we find
∂V
∂t
+a
∂^2 V
∂S^2
+rS
∂V
∂S
−rV= 0.
This is the risk-neutral Black–Scholes equation. Two
of the coefficients (those ofVand∂V/∂S) have been
pinned down exactly without any modelling at all. Ok,
so it doesn’t tell us what the coefficient of the second
derivative term is, but even that has a nice interpreta-
tion. It means at least a couple of interesting things.
First, if we do start to move outside the Black–Scholes
world then chances are it will be the diffusion coefficient
