4.4 The Black-Scholes Model 61The parameterrmodels the “riskless rate of interest”, while the parameter
μmodels the average increase of the stock price. Indeed using Itˆo’s formula
one may describe the model equivalently by the differential equations:
dB̂t
B̂t=rdt,dŜt
Ŝt=μdt+σdWt.The num ́eraire in this model is just the relevant currency (saye). In order
to remain consistent with the above theory, we shall rather follow our usual
procedure of taking a traded asset as num ́eraire, namely the bond, to use
discounted terms. We then have
Bt=B̂t
B̂t= 1 (4.13)
St=Ŝt
B̂t=S 0 e
σWt+(
μ−r−σ 22)
t
.We shall writeνforμ−rwhich is called the “excess return”. The only
thing we have to keep in mind when passing to the bond as num ́eraire is that
now quantities have to be expressed in terms of the bond: in particular, ifK
denotes the strike price of an option at timeT(expressed ineat timeT), we
have to express it asKe−rTunits of the bond.
Contrary to Bachelier’s setting, the process
St=S 0 e
σWt+(
ν−σ 22)
t
, 0 ≤t≤T,isnota martingale underP(unlessν= 0, which typically is not the case).
The unique martingale measureQforS(which is absolutelyP-continuous)
is given by Girsanov’s theorem (see [RY 91] or any introductory text to
stochastic calculus)
dQ
dP=exp(
−
ν
σWT−
ν^2
2 σ^2T
)
. (4.14)
Let us price and hedge the contingent claimf(ω)=(
ST(ω)−Ke−rT)
+,
which is the pay-off function of the European call option with exercise time
Tand a strike price ofKEuros (expressed in terms of the bond num ́eraire).
Noting that (Wt+νt)∞t=0is a standard Brownian motion underQand
applying Theorem 4.2.1 to theQ-martingaleS, we may calculate, similarly
as in (4.6) above.
C(S 0 ,T)=EQ[f]=EQ[(
S 0 eσ(WT+νT)−σ^2
2 T−Ke−rT)
+]
(4.15)
=S 0 EQ
[
eσ√
TZ−σ^22 Tχ
{ST≥K}]
−Ke−rTQ[ST≥K],whereZ=WT√+TνTis aN(0,1)-distributed random variable underQ.