The Mathematics of Arbitrage

68 4 Bachelier and Black-Scholes

Why was the calculation ofEQ[f 2 ] so easy? Simply because the factor
Ke−rTappearing inf 2 is a constant (expressed in terms of the present num ́e-
raire, namely the bond); hence the calculation of the expectation was reduced
to the calculation of the probability of an event, namely the probability that
the option will be exercised, with respect to the probability measureQ.

To proceed similarly with the calculation ofEQ[f 1 ] we make a change of

num ́eraire, now choosing the risky assetŜin the Black-Scholes model (4.12)
as num ́eraire. Under this num ́eraire the model reads

B̂t

Ŝt

=

1

St

=S 0 −^1 e−σWt+

(

r−μ+σ 22

)

t

Ŝt

Ŝt

≡ 1

whereW is a standard Brownian motion underP. The reader has certainly
noticed the symmetry with (4.13). But what is the probability measureQˇ

under which the process
B̂t
Ŝt=

1

St becomes a martingale? Using Girsanov we

can explicitly calculate the densityd
Qˇ
dP; but, in fact, we don’t really need this
information. All we need is to observe that we may write

1

St

=S− 01 e−σ

Wˇt−σ 22 t

,

whereWˇ is a standard Brownian motion underQˇ(the reader worried by the
minus sign in front ofσWˇtmay note that−Wˇ is also a standard Brownian
motion underQˇ). We now apply the change of num ́eraire theorem (in the
form of Theorem 2.5.4) to calculateEQ[f 1 ]. In fact, we have only proved this
theorem for the case of finite Ω, but we rely on the reader’s faith that it also
applies to the present case (for a thorough investigation for the validity of
this theorem for general locally bounded semi-martingale models we refer to
Chap. 11 below). Applying this theorem we obtain

EQ[f 1 ]=EQ

[

STχ{ST≥e−rTK}

]

=S 0 EQˇ

[

ST

ST

χ{ 1

ST≤erTK−^1

}

]

=S 0 EQˇ

[

χ{

S− 01 e−σWˇT−σ

2

2 T≤erTK−^1

}

]

=S 0 Qˇ

[

S 0 eσ

WˇT+σ 22 T

≥e−rTK

]

.

Noting that
W√ˇT
T isN(0,1)-distributed under
Qˇ, this expression is completely
analogous to the one appearing in (4.21), with the exception of the plus in

front of the termσ

2

2 T. Hence we get