Behaviour of Stock Prices

8/6/2019 Behaviour of Stock Prices

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10.1

Options, Futures, and Other Derivatives, 4th edition 1999 by John C. Hull

Model of theBehavior

of Stock Prices

Chapter 10


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10.2


Categorization of Stochastic

Processes

Discrete time; discrete variable

Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable


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10.3


Modeling Stock Prices We can use any of the four types of

stochastic processes to model stockprices

The continuous time, continuousvariable process proves to be the most

useful for the purposes of valuingderivative securities


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10.4


Markov Processes (See pages 218-9)

In a Markov process futuremovements in a variable depend onlyon where we are, not the history ofhow we got where we are

We will assume that stock pricesfollow Markov processes


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10.5


Weak-Form Market

Efficiency The assertion is that it is impossible to

produce consistently superior returnswith a trading rule based on the past

history of stock prices. In other wordstechnical analysis does not work.

A Markov process for stock prices isclearly consistent with weak-form marketefficiency


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10.6


Example of a Discrete Time

Continuous Variable Model

A stock price is currently at $40At the end of 1 year it is

considered that it will have aprobability distribution of

J(40,10) where J(Q,W) is anormal distribution with mean Qand standard deviation W


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10.7


Questions

What is the probability distribution ofthe stock price at the end of 2years?

years? years? (tyears?Taking limits we have defined acontinuous variable, continuoustime process


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10.8


Variances & Standard

Deviations

In Markov processes changes insuccessive periods of time areindependent

This means that variances are additive

Standard deviations are not additive


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10.9


Variances & Standard

Deviations (continued)

In our example it is correct to saythat the variance is 100 per year.

It is strictly speaking not correct tosay that the standard deviation is 10

per year.


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10.10


AWiener Process (See pages 220-1)

We consider a variable z whose valuechanges continuously

The change in a small interval of time (t is(z

The variable follows a Wiener process if

1.2. The values of(z for any 2 different (non-overlapping) periods of time are independent

( (z t! I I J(0,1)where is a random drawing from


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10.11


Properties of a Wiener

Process Mean of [z (T ) z (0)] is 0

Variance of [z (T ) z (0)] is T

Standard deviation of [z (T ) z (0)]is T


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10.12


Taking Limits ...

What does an expression involving dz and dtmean?

It should be interpreted as meaning that thecorresponding expression involving (z and (tis

true in the limit as (t tends to zero

In this respect, stochastic calculus is analogous toordinary calculus


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10.13


GeneralizedWiener Processes(See page 221-4)

A Wiener process has a drift rate(ie average change per unit time) of

0 and a variance rate of 1

In a generalized Wiener processthe drift rate & the variance rate

can be set equal to any chosenconstants


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10.14


GeneralizedWiener Processes(continued)

The variable x follows a generalized

Wiener process with a drift rate ofa& a variance rate ofb2 if

dx=adt+bdz


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10.15


GeneralizedWiener Processes(continued)

Mean change in x in time T isaT

Variance of change in x in time Tis b2T

Standard deviation of change inx in time T is

( ( (x a t b t ! I

b T


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10.16


The Example Revisited

A stock price starts at 40 & has a probabilitydistribution ofJ(40,10) at the end of the year

If we assume the stochastic process is Markov

with no drift then the process isdS = 10dz

If the stock price were expected to grow by $8on average during the year, so that the year-

end distribution is J(48,10), the process isdS = 8dt + 10dz


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10.17


Ito Process (See pages 224-5)

In an Ito process the drift rate and thevariance rate are functions of time

dx=a(x,t)dt+b(x,t)dz

The discrete time equivalent

is only true in the limit as (ttends to

zero

( ( ( x a x t t b x t t! ( , ) ( , )I


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10.18


Why a GeneralizedWiener Process

is not Appropriate for Stocks

For a stock price we can conjecture that itsexpected proportional change in a short period

of time remains constant not its expectedabsolute change in a short period of time

We can also conjecture that our uncertainty asto the size of future stock price movements isproportional to the level of the stock price


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10.19


An Ito Process for Stock Prices(See pages 225-6)

where Q is the expected return Wis the volatility.

The discrete time equivalent is

dS Sdt Sdz! Q W

( ( (S S t S t ! Q W I


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10.20


Monte Carlo Simulation We can sample random paths for the

stock price by sampling values forI

Suppose Q= 0.14, W= 0.20, and (t=0.01, then

(S S S! 0 0014 0 02. . I


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10.21


Monte Carlo Simulation One Path(continued. See Table 10.1)

Perit ck Price tt rt f Peri

RleforI

Change in t ckPrice, (

. .5 . 36

. 36 . .6

. 7 - . 6 - .3 9

3 .5 . 6 .6

. 6 - .69 - . 6


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10.22


Itos Lemma (See pages 229-231) If we know the stochastic process

followed by x, Itos lemma tells us the

stochastic process followed by somefunction G (x, t)

Since a derivative security is a function ofthe price of the underlying & time, Itos

lemma plays an important part in theanalysis of derivative securities


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10.23


Taylor Series ExpansionA Taylors series expansion of

G (x , t) gives

( ( ( (

( ( (

GG

xx

G

tt

G

xx

G

x tx t

G

tt

!

x

x

x

x

x

x

x

x x

x

x

2

2

2

2 2

2

2-


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10.24


Ignoring Terms of Higher Order

Than (t

In r inar calc l s eget

stic calc l s eget

ecause has a c nent hich is f r er

In st cha

( ( (

( ( ( (

( (

G

G

xx

G

t t

G

G

x x

G

t t

G

x x

x t

!

!

x

x

x

x

x

x

x

x

x

x

2

2

2


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10.25


Substituting for (x

Suppose

( , ) ( , )

so that= +

Then ignoring terms of higher order than

dx a x t dt b x t dz

x a t b t

t

GG

xx

G

tt

G

xb t

!

!

( ( (

(

( ( ( (

I

x

x

x

x

x

xI

2

2

2 2


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10.26


The I2(tTerm

Since

It follows that

The variance of is proportional to and can

be ignored. Hence

2

I J II I

I

I

x

x

x

x

x

x

} ! !

!

!

!

( , ) ( )

( ) [ ( )]

( )

( )

0 1 0

1

1

1

2

2 2

2

2

2

2

2

E

E E

E

E t t

t t

GG

xx

G

tt

G

xb t

( (

( (

( ( ( (


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10.27


Taking LimitsTaking limits

Substituting

We obtain

This is Ito's Lemma

dGG

xdx

G

tdt

G

xb dt

dx a dt b d z

dGG

xa

G

t

G

xb dt

G

xb dz

!

!

!

x

x

x

x

x

x

x

x

x

x

x

x

x

x

2

2

2

2

2

2


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10.28


Application ofItos Lemma

to a Stock Price ProcessThe stock price process is

For a function of &

d S S dt S d z

G S t

dGG

SS

G

t

G

SS dt

G

SS dz

!

!

Q W

xx

Qxx

x

xW

xx

W2

2

2 2


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10.29


Examples

1. The forward price of a stock for a contractmaturing at time

e

2.

T

G S

dG rG dt G dz

G S

dG dt dz

r T t!

!

!

!

( )

( )

ln

Q W

Q W W2

2

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Behaviour of Stock Prices