Chapter 4: Ten Different Ways to Derive Black–Scholes 257
A simplification of this using the cumulative distribution
function for the standardized normal distribution results
in the well-known call option formula.
Change of Numeraire
The following is a derivation of the Black–Scholes call
(or put) formula, not the equation, and is really just a
trick for simplifying some of the integration.
It starts from the result that the option value is
e−r(T−t)EQt[max(ST−K,0)].
This can also be written as
e−r(T−t)EQt[(ST−K)H(S−K)],
whereH(S−K) is the Heaviside function, which is zero
forS<Kand 1 forS>K.
Now define another equivalent martingale measureQ′
such that
W ̃t′=Wt+ηt−σt.
The option value can then be written as
StE
Q′
t
[
(ST−K)H(S−K)
ST
]
.
where
dSt=(r+σ^2 )Sdt+σSdW ̃t′.
It can also be written as acombinationof the two
expressions,
StE
Q′
t
[
STH(S−K)
ST
]
−Ke−r(T−t)E
Q
t[H(S−K)].
Notice that the same calculation is to be performed,
an expectation ofH(S−K), but under two different
