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3. The Wiener Process and Rare Events in

Financial Markets

The Wiener process works as a mathematical

model for continuous time stock price.

To make the process even more realistic oc-casional rare events causing jumps into the

sample paths are added by modeling with

some point processes.

Consider first the modeling the ”ordinary”events with Wiener process (Brownian mo-

tion) (see here for a heuristic construction of

the Brownian motion).

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Definition 3.1: A Wiener process, W t, rela-

tive to a family of information sets {I t} (fil-

tration), is a stochastic process such that

1. W 0 = 0 (with probability one).

2. W t is continuous.

3. W t is adapted to the filtration {I t}.

4. For s ≤ t, W t − W s is independent of I s,with E[W t − W s] = 0 and Var[W t − W s] =

E(W t − W s)2 = t − s.

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Equivalently we can define:

Definition 3.2: A random process Bt, t ∈[0, T ] is a (standard) Brownian motion if 1. B0 = 0.2. Bt is continuous.3. Increments of Bt are independent. I.e., if

0 ≤ t0 < t1 < · · · < tn, then Bt1 −Bt0, . . . , Btn −Btn−1 are independent.4. If 0 ≤ s ≤ t, Bt − Bs ∼ N (0, t − s).

-20

-10

0

10

20

30

80 82 84 86 88 90 92 94 96 98 00

Time

W ( t )

+std

-std

Figure 3.1: Three sample paths of a standard

Brownian motion.3

Properties:

(a) W t is an I t-martingale.

(b) W t has independent increments.

(c) E[W t] = 0 for all t.

(d) Var[W t] = t.

(e) The law of iterated logarithms

lim supt→∞

W t√ 2t log log t

= 1 (with probability one)

(f) W t/t → 0 w.p.1 as t → ∞.

(g) W 2t − t is an I t-martingale.

(h) exp{σW t − (σ2/2)t} is an I t-martingale.

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Proof : (a) Let s ≤

t, then E[W t−

W s|I

s] =

E[W t − W s] = 0. So E[W t|I s] = W s.

(b) Let 0 ≤ s < t < u then W t − W s is I t-

measurable (W s, W t ∈ I t ⊂ I u) and W u −W t is independent of I t, which implies that

W u − W t is independent of W t − W s.

(c) 0 = E[W t − W 0] = E[W t].(d) Var[W t] = Var[W t − W 0] = t − 0 = t.

(e) Will not be proven, but means that if b >

1, for sufficiently large t, W t < b√

2t log log t.

(f) Follows from the law of iterated loga-

rithms.

(g) and (f) Left as exercises.

5

-40

-30

-20

-10

0

10

20

30

40

80 82 84 86 88 90 92 94 96 98 00

W_

t

Time

std

log-log

Figure 3.2: The law of iterated logarithms.

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Example 3.1:

S t = S 0e(µ−12

σ2)t+σW t ,where S 0 > 0, and µ and σ > 0 are constants. This is an exampleof generalized Brownian motion, called also a Geometric Brow-nian motion, and serves as a basic model for stock prices. Thedistribution of log S t is normal with mean

log S 0 +

µ − 1

2σ2

t

and variance σ2

t. The continuously compounded return to thestock, per unit of time (e.g. if annual, then annualized cumula-tive returns), over time interval [t, t + u], u > 0, is

Ru = 1

u (log S t+u − log S t) = µ − 1

2σ2 +

1

uσ (W t+u − W t).

Because for fixed t, W t+u − W t is a standard Brownian motionwith time index u, property (f) above implies that

Ru → µ −1

2σ2

as u → ∞.

Note. E [S t] = S 0eµt.

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0

400

800

1200

1600

2000

80 82 84 86 88 90 92 94 96 98 00

S ( t )

Time

E[S(t)]

Figure 3.3: Sample paths of monthly stock price processes S t =

S 0e(µ−12

σ2)t+σW t with S 0 = 100, and annual rate of return µ = 10%

and volatility σ = 25%.

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-30

-20

-10

0

10

20

30

80 82 84 86 88 90 92 94 96 98 00

M o n t h l y r e t u r n ( % )

Month

Figure 3.4: Monthly log-returns Rt = 100(log S t − log S t−s) for

the sample period 1980:1 to 2001:12.

-60

-40

-20

0

20

40

60

80 82 84 86 88 90 92 94 96 98 00

A n n u a l i z e d c u m u l a t i v e r e t u r n ( % )

Month

Figure 3.5: Annualized cumulative log-returns RAt = 100 12

t (log S t−

log S 0) for the period 1980:1 to 2001:12 (monthly observations).

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Remark 3.1: In the above example the geometric Brow-

nian motion is usually given in the differential form

dS t

S t= µ dt + σ dW t,

(”model of the return”) or

dS t = µ S t dt + σ S t dW t.

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Rare Events and Poisson Process

In stock markets we occasionally observe sud-

den jumps in the prices due to some impor-

tant or extreme event that affects the price.

To model these a reasonable assumption couldbe that the jumps—being consequences of

extreme shocks—are independent of the usual

information driving innovations dW t.

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Following Merton∗ we can introduce a model

where the asset price has jumps superim-

posed upon a geometric Brownian motion

such that

dS t

S t= (µ − λ ν )dt + σ dW t + dJ t,(1)

where µ the asset’s expected return, λ isthe jump intensity (e.g. average number of

jumps per year), ν is the average size of the

jump as a percentage of the asset price, σ

is the return volatility, and J t is related to

the (independent) Poisson process generat-

ing the jumps.

Once the jump occurs we can superimpose it

into the process by assuming that it is gen-

erated from a normal distribution with mean

ν and standard deviation equaling the jump

volatility.

∗Merton, R.C. (1976). Option pricing when underlying stock

returns are discontinuous. Journal of Financial Economics , No.

3, 125–144.

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Example 3.2: Consider a situation where the underly-

ing process has a jump frequency one per year (λ = 1/2)(i.e., on average once every two year). The averagepercentage jump size ν is assumed for simplicity to beequal to zero and the volatility of the jump is 20%.

Suppose the stock price now is 100, the drift is 8%,σ = 15%. Below are four sample baths from the dif-fusion process

dS t

S t= µ dt + σ dW t + dJ t.

We model dJ t as dJ t = 0.20 Z U , where Z is a standardnormal variable and U is a random variable such thatP (U = 1) = λdt and P (U = 0) = 1 − λdt. In additionZ and U are independent.

A discrete approximation of the above difference equa-tion is

S t = S t−h

1 + 0.08h + 0.15

√ h Z t + dJ t

,

13

0

100

200

300

400

500

600

700

90 92 94 96 98 00 02 04 06

S1 S2 S3 S4

Sample paths of a jump process:S(t) = S(t-dt)*[1 + 0.08dt + 0.15 sqrt(dt) Z(t) + dJ(t)],dJ(t) = 0.2 Z U(t), with P(U(t) = 1) = 0.5 dt

Month

S(t)

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We can rewrite the diffusion process in this example

asdS t

S t= (µ − λν )dt + σ dW t

if the Poisson event does not occur, and

dS t

S t= (µ − λν )dt + σ dW t + 0.20Z

if the Poisson event occurs.

It can be shown that the process then is

S t = S 0 exp

µ − 1

2σ2 − λν

t + σW t

J (N t),

where

J (N t) = exp N t

i=1

J i

with J i = 0.20Z i, and Z i are independent N (0, 1) ran-

dom variables. N t has the Poisson distribution, dis-

cussed more closely below.

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In order to model the jump intensity, let N tdenote the total number of extreme (unordi-

nary) shocks until time t (counting process).

Then once the event occurs, N t changes by

one unit and remains otherwise unchanged.

Conceptually we can model this within an in-

finitesimal interval dt by

dN t =

1 with probability λ dt0 with probability 1 − λ dt

(2)

This causes a discrete jump (once it occurs),because the size does not depend on dt (the

size can be a random variable as in the pre-

vious example).

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Poisson process has the following properties:

(1) During a small interval h, at most one

event occurs with probability ≈ 1.

(2) The information up to time t does not

help to predict the occurrence of the event

in the next instant.

(3) Events occur at a constant rate λ.

Remark 3.2: Within any time interval [t, t + h], ∆N t+h =N t+h − N t has the Poisson distribution

P (∆N t+h = k) = (λh)k

k! eλh, k = 0, 1, . . .(3)

E[∆N t+h] = λh = Var[∆N t+h].(4)However, if h is small

P (∆N t+h = 1) = (λh)e−λh ≈ λh(5)

as stated above. We denote ∆N t+h ∼ Po(λh). Thus

because N 0 = 0, we have in the previous example

N t ∼ Po(λt).

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