258 Frequently Asked Questions In Quantitative Finance
measures. The end result is the Black–Scholes formula
for a call option.
This method is most useful for simplifying valuation
problems, perhaps even finding closed-form solutions,
by using the most suitable traded contract to use for
the numeraire.
The relationship between the change of numeraire result
and the partial differential equation approach is very
simple, and informative.
First let us make the comparison between the risk-
neutral expectation and the Black–Scholes equation
as transparent as possible. When we write
e−r(T−t)E
Q
t[max(ST−K,0)]
we are saying that the option value is the present
value of the expected payoff under the risk-neutral ran-
dom walk
dS=rS dt+σSdW ̃t.
The partial differential equation
∂V
∂t
+^12 σ^2 S^2
∂^2 V
∂S^2
+rS
∂V
∂S
−rV= 0
means exactly the same because of the relationship
between it and the Fokker–Planck equation. In this
equation the diffusion coefficient is always just one half
of the square of the randomness indS. The coefficient
of∂V/∂Sis always the risk-neutral driftrSand the coef-
ficient ofVis always minus the interest rate,−r,and
represents the present valuing from expiration to now.
If we write the option value asV=SV′then we can
think ofV′as the number of shares the option is equiv-
alent to, in value terms. It is like using the stock as the
