Unformatted text preview: EC3070 FINANCIAL DERIVATIVES
CONTINUOUSTIME STOCHASTIC PROCESSES
DiscreteTime Random Walk . A standardised random walk deﬁned
over the set of integers {t = 0, ±1, ±2, . . .}, is a sequence by w(t) =
{wt ; t = 0, ±1, ±2, . . .} in which
wt+1 = wt + εt+1 , (1) where ε(t) = {εt ; t = 0, ±1, ±2, . . .} is a sequence of independently
distributed of standard normal random elements with E (εt ) = 0 and
V (εt ) = 1, for all t, described as a whitenoise process.
By a process of backsubstitution, the following expression can be derived:
wt = w0 + εt + εt−1 + · · · + ε1 . (2) This is the sum of an initial value w0 and of an accumulation of stochastic
increments. If w0 has a ﬁxed ﬁnite value, then the mean and the variance
of wt , conditional of this value, are be given by
E (wt w0 ) = w0 and V (wt w0 ) = t.
1 (3) EC3070 FINANCIAL DERIVATIVES 25
20
15
10
5
0
−5
−10
0 100 200 300 400 500 Figure 1. The graph of 500 observations on simulated randomwalk process generated by the equation yt = yt−1 + εt . 2 EC3070 FINANCIAL DERIVATIVES To reduce the random walk to a stationary process, we take its ﬁrst differences:
(4)
wt+1 − wt = εt+1 .
We can take longer steps through time without fundamentally altering the
nature of the process. Let h be any integral number of time periods. Then
√ h wt+h − wt = εt+j = ζt+h h, (5) j =1 where ζt+h is an element of a sequence {ζt+jh ; j = 0, ±1, ±2, . . .} of independently and identically distributed standard normal variates.
√
The factor h is present because we are now taking steps through time
of length h instead of steps of unit length. 3 EC3070 FINANCIAL DERIVATIVES A ﬁrstorder random walk over a surface is know as Brownian motion. One
can imagine small particles, such a pollen grains, ﬂoating on the surface
of a viscous liquid. The viscosity is expected to bring the particles to a
halt quickly. However, if they are very light, then they will dart hither
and thither on the surface of the liquid under the impact of its molecules,
which are in constant motion.
A Wiener process is the consequence of allowing the intervals of a
discretetime random walk to tend to zero. The dates at which the process is deﬁned become a continuum—and the process becomes continuous
almost everywhere, but nowhere diﬀerentiable.
When sampled at regular intervals, a Wiener process has the same mathematical description as the discretetime process. However, whereas the
random walk is deﬁned only on the set of integers, the Wiener process is
deﬁned for all points on a real line that represents continuous time. 4 EC3070 FINANCIAL DERIVATIVES To generalise of equation (5), replace the integer h by an inﬁnitesimally
small increment dt. Then, the equation becomes
√
(6)
dw(t) = w(t + dt) − w(t) = ζ (t + dt) dt
This equation describes a standard Wiener process. The process fulﬁls the
following conditions:
(a) w(0) is ﬁnite,
(b) E {w(t)} = 0, for all t,
(c) w(t) is normally distributed,
(d) dw(s), dw(t) for all t = s are independent stationary increments,
(e) V {w(t + h) − w(t)} = h for h > 0. 5 EC3070 FINANCIAL DERIVATIVES
Arithmetic Brownian Motion The standard Wiener process is inappropriate to much of ﬁnancial modelling.
A generalisation is the socalled random walk with drift. In discrete time,
this can be represented by the equation
x(t + 1) = x(t) + µ + σε(t + 1). (7) The continuoustime analogue of this process is described by
dx(t) = µdt + σdw(t). (8) A generalisation of the latter is the Ito process, where the drift parameter
µ and the variance or volatility parameter σ 2 become timedependent
functions of the level of the process:
dx(t) = µ(x, t)dt + σ (x, t)dw(t). 6 (9) EC3070 FINANCIAL DERIVATIVES
Geometric Brownian Motion The domain of a normally distributed
random variable extends from −∞ to +∞. In ﬁnance, nominal interest
rates must be nonnegative. Also, asset values cannot become negative.
The diﬃculty can be overcome by taking logarithms of the variables. The
logarithmic transformation maps from [0, ∞) to (−∞, ∞).
The logarithmic version of the random walk is described by
ln x(t + 1) = ln x(t) + σε(t + 1). (10) The corresponding continuoustime version can be written as
d{ln x(t)} = σdw(t). (11) Given that 1 dx
dx
d
ln x =
or
d{ln x(t)} =
,
dt
x dt
x
it follows that equation (11) can also be written as
dx = σxdw(t). 7 (12) EC3070 FINANCIAL DERIVATIVES Observe that the process has an absorbing barrier at zero. That is to say,
if x = 0 at any time, then it will remain at that value thereafter.
A more general equation incorporates a drift term:
dx = xµdt + σxdw(t). (13) On the strength of the preceding reasoning, it might be imagined that this
is synonymous with the process described by the equation d{ln y (t)} =
µdt + σdw(t). It is interesting to discover that this is not the case.
To ﬁnd the logarithm of the process described by (13), we must use Ito’s
Lemma. This indicates that, for any process described by equation (9),
and for any continuous diﬀerentiable function f (x), there is
df (x, t) = 1 ∂2f
∂f
∂f
2
µ(x, t)
+
+ σ (x, t)
∂x
∂t
2 ∂x2 8 dt + σ (x, t) ∂f
dw.
∂x (14) ...
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 Spring '12
 D.S.G.Pollock
 Normal Distribution, Stochastic process, µ, Ito, interestrates

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