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Black & Scholes extensions 187
The fact of having the log-normality confirms the CRR process at the limit as,
indeed, a lot of empirical studies show that, for an efficient market, stock prices
are well adjusted with such a distribution.
From properties of the log-normal distribution (see Chapter 1, section 5.4), we
obtain:
2
0
2
0
()
,
()
var ( 1).
t
tt
St
Ee
S
St
ee
S
μ
μσ
⎛⎞
=
⎜⎟
⎝⎠
⎛⎞
−
⎜⎟
⎝⎠
(4.21)
So, we see that the mean value of the asset at time t is given as if the initial
amount S
0
was invested at the non-risky instantaneous interest rate
and that its
value is above or below S
0
following the “hazard” variations modelled with the
Brownian motion.
We also see that the variance of S(t) increases with time in conformity with the
fact that, for long time periods, variations of the asset are very difficult to predict.
The explicit relation (4.20) can also be written in the form:
2
2
()
0
()
.
t
Bt
St
e
Se
σ
μ
σ
⎛⎞
−
⎜⎟
⎜⎟
⎝⎠
= (4.22)
This allows us to distinguish three cases:
(i)
2
.
2
μ
= (4.23)
If so, the evolution of the asset is that of a pure Brownian exponential.
(ii)
2
.
2
μ
> (4.24)
Here, S(t) will vary faster than the pure Brownian exponential and so, we may
expect at certain times large gains but also large losses parallel with the time
evolution of the pure brownian exponential
(iii)
2
.
2
μ
< (4.25)
Here, the situation is similar but the evolution is opposed to that of the pure
Brownian exponential.
From the second result of (4.21), it is also clear that the expectations of large
gains - and losses! - are better for large values of
; that is why
is called the
volatility of the considered asset.
It follows that a market with high volatility will attract risk lover investors and
not risk averse investors
From the explicit form, it is not difficult to simulate trajectories of the S-process.
The next figure shows a typical form.