Chapter 4: Ten Different Ways to Derive Black–Scholes 259
unit of currency. But if we rewrite the Black–Scholes
equation in terms ofV′using
∂V
∂t
=S
∂V′
∂t
,
∂V
∂S
=S
∂V′
∂S
+V′,
and
∂^2 V
∂S^2
=S
∂^2 V′
∂S^2
+ 2 S
∂V′
∂S
,
then we have
∂V′
∂t
+^12 σ^2 S^2
∂^2 V′
∂S^2
+(r+σ^2 )S
∂V′
∂S
= 0.
The functionV′can now be interpreted, using the
same comparison with the Fokker–Planck equation,
as an expectation, but this time with respect to the
random walk
dS=(r+σ^2 )Sdt+σSdW ̃t′.
And there is no present valuing to be done. Since at
expiration we have for the call option
max(ST−K,0)
ST
we can write the option value as
StE
Q′
t
[
(ST−K)H(S−K)
ST
]
.
where
dSt=(r+σ^2 )Sdt+σSdW ̃t′.
Change of numeraire is no more than a change of depen-
dent variable.
Local Time
The most obscure of the derivations is the one involving
the concept from stochastic calculus known as ‘local
time.’ Local time is a very technical idea involving the
time a random walk spends in the vicinity of a point.
