
190 Chapter 5
Remark 4.2 Using relation (4.31), we get relation (3.15) for t=0 or
T
=
.
The interpretation is of course already given in section 3.
4.4.2 The Risk Neutral Measure And The Martingale Property
As for the CRR model, it is possible to construct another probability measure Q
on
()
,,( )
t
Ωℑ ℑ , called the risk neutral measure, such that the value of the call
given by formula (4.38) is simply the expectation value of the present value of
the “gain” at maturity time T.
Using a change of probability measure for going from P to Q, with the famous
Girsanov theorem (see for example Gikhman and Skorokhod,vol.III
(1975),p.250), it can be shown that the new measure Q, which moreover is
unique, can be defined by replacing in the stochastic differential equation (4.16)
the trend
by r.
Doing so, the explicit form of S(t) given by relation (4.26) becomes:
2
2
'( )
0
()
rt
t
St Se e
σ
σ
⎛⎞
−
⎜⎟
⎜⎟
⎝⎠
= (4.39)
where the process
'
still an adapted standard Brownian motion and the value of
C can be computed as the present value of the expectation of the final “gain” of
the call at time T:
()
(,) sup () ,0 .
rT t
Q
CSt e E ST K
−−
=− (4.40)
The risk neutral measure gives another important property for the process of
present values of the asset values on [0,T]:
]
.,0),( TttSe
rt
∈
−
(4.41)
Indeed, under Q, this process is a martingale, so that (see Chapter 1, section 8)
for all s and t such that :
t<
()
() ( ) ( ).
rt
s
eSt st Ss
−
ℑ<= (4.42)
This means that at every time s, the best statistical estimation of S(t) is given by
the observed value at time s, a result consistent with the assumption of an
efficient market.
From relation (4.42), we get in particular:
0
() .
rt
eSt S
−
=
(4.43)
So, on average, the present value of the asset at any time t equals its value at time
0.
In conclusion, we see that the knowledge of the risk neutral measure avoids the
resolution of the partial derivative equation and replaces it by the computation of
an expectation, which is in general easier, as it only uses the marginal
distribution of S(T).