FACULTY SCIENCE AND TECHOLOGY

DEPARTMENT OF MATHEMATICS

SEMESTER II 2011/2012

MKE 4500 STOCHASTIC DIFFERENTIAL EQUATION

ASSIGNMENT 2

BROWNIAN MOTION

NAME MATRIX NUMBER

LEE MUN YEE UK20119

CHAI SHUK YNG UK20160

LEONG LEE SHIN UK20323

LEE WAN YEE UK20355

PROGRAMME: FINANCIAL MATHEMATICS

LECTURER:

INTRODUCTION OF BROWNIAN MOTION

Definition Of Brownian Motion

A real-valued stochastic process W(·) is called a Brownian Motion or Wiener process

if

(i) W(0) = 0 a.s.,

(ii) W(t) −W(s) is N (0, t − s) for all t ≥ s ≥ 0,

(iii) for all times 0 < t1 < t2 < · · · < tn , the random variables W(t1),W(t2) –

W(t1), . . . , W(tn) −W(tn-1) are independent (“independent increments”).

Notice in particular that

E (W (t)) = 0 and E (W2 (t)) = t for each time t ≥ 0.

The Central Limit Theorem provides some further motivation for our definition of

Brownian motion, since we can expect that any suitably scaled sum of independent,

random disturbances affecting the position of a moving particle will result in a

Gaussian distribution.

Motivation of Brownian Motion

In 1900 L. Bachelier attempted to describe fluctuations in stock prices mathematicallyand essentially discovered first certain results later rederived and extended by A. Einstein in 1905. Einstein studied the Brownian phenomena this way. Let us consider a long, thin tube filled with clear water, into which we inject at time t= 0 a unit amount of ink, at the locationx=0. Now let f (x ,t) denote the density of ink particles at position x∈ R and time t ≥ 0. Initially we have

f ( x , 0 )=δ0, the unit mass at 0.Next, suppose that the probability density of the event that an ink particle moves from x to x+ y in (small) time τ is ρ(τ , y ). Then

f ( x , t+τ )=∫−∞

∞

f ( x− y , t ) ρ (τ , y ) dy

(1)

¿∫−∞

∞

( f −f x y+12

f xx y2+…)dy ρ (τ , y ) dy

But since ρ is a probability density,∫−∞

∞

ρ dy=1 ; whereas ρ (T ,− y )=ρ(T , y) by

symmetry. Consequently ∫−∞

∞

ρ dy =0. We further assume that ∫−∞

∞

y2 ρ dy, the variance ρ

, is linear in T :

∫−∞

∞

y2 ρdy=DT , D>0.

We insert these identities into (1), thereby to obtain

f ( x ,t +T )−f (x , t)T

=D f xx(x , t)

2{+higher order terms }.Sending now τ → 0 , we

discover

f t=D2

f xx

This is the diffusion equation, also known as the heat equation. This partial

differential equation, with the initial conditionf ( x , 0 )=δ0, has the solution

f ( x , t )= 1

(2πDt )12

e−x2

2 DtThis says the probability density at time t isN (0 , Dt ), for

some constant D .

In fact, Einstein computed:

D=RT

N A f, where { R=gas constant

T=absolute temperaturef =friction coefficient

N A=Avogadr o' s number .

This equation and the observed properties of Brownian motion allowed J.Perrin to

compute N A(≈ 6 ×1023= number of molecules in a mole) and help to confirm the

atomic theory of matter.

Expectation

Following note from:

http://www.doc.ic.ac.uk/~nd/surprise_95/journal/vol1/skh1/article1.html

Geometric Brownian Motion Assumptions

GBM describe the probability distribution of the future price of a stock. The basic

assumption of the model is as follow:

The return on a stock price between now and some very short time later (T-t)

is normally distributed

The standard deviation of tis distribution can be estimated from historical data

ST volatility is good predictor of the LT volatility

The stock price models employed in option pricing are not predictive but

probabilistic: they assume a distribution of the future prices derived from historical

data and current market conditions.

Geometric Brownian Motion– Mean and Standard Deviation

The mean of the distribution is μ times the amount of time μ(T−t ) (The expected rate

of return changes in proportion to time)

(μ−σ2

2)(T−t)

Where: μ= instantaneous expected return; σ = instantaneous standard deviation

It says that short term returns alone are not a good predictor of long term returns.

Volatility tends to depress the expected returns below what the short term returns

suggest (the average amount the stochastic component depresses returns in a single

move is σ2

2)

The standard deviation of return increases in proportion to the square root of the

amount of time (Bachelier, 1900)

σ √T−t

APPLICATIONS OF BROWNIAN MOTION TO MARKET ANALYSIS

Brownian Motion in the Stock Market

In the middle of this century, work done by M.F.M Osborne showed that the

logarithms of common-stock prices, and the value of money, can be regarded as an

ensemble of decisions in statistical equilibrium, and that this ensemble of logarithms

of prices, each varying with time, has a close analogy with the ensemble of

coordinates of a large number of molecules. Using a probability distribution function

and the prices of the same random stock choice at random times, he was able to derive

a steady state distribution function, which is precisely the probability distribution for a

particle in Brownian motion. A similar distribution holds for the value of money,

measured approximately by stock market indices. Sufficient, but not necessary

conditions to derive this distribution quantitatively are given by the conditions of

trading, and the Weber-Fechner law. (The Weber-Fechner law states that equal ratios

of physical stimulus, for example, sound frequency in vibrations/sec, correspond to

equal intervals of subjective sensation, such as pitch. The value of a subjective

sensation, like absolute position in physical space, is not measurable, but changes or

differences in sensation are, since by experiment they can be equated, and reproduced,

thus fulfilling the criteria of measurability).

A consequence of the distribution function is that the expectation values for

price itself increases, with increasing time intervals 't', at a rate of 3 to 5 percent per

year, with increasing fluctuation, or dispersion, of Price. This secular increase has

nothing to do with long-term inflation, or the growth of assets in a capitalistic

economy, since the expected reciprocal of price, or number of shares purchasable in

the future, per dollar, increases with time in an identical fashion.

Thus, it was shown in his paper that prices in the market did vary in a similar

fashion to molecules in Brownian motion. In another paper presented around the same

period, it was also found that there is definite evidence of periodic in time structure

(of the prices in Brownian motion) corresponding to intervals of a day, week, quarter

and year : these being simply the cycles of human attention span.

With compounding evidence and widespread acceptance that Brownian

motion exists in market structures, many researches and studies have since taken

place, revolving and evolving around this theory.

For example, a statistical analysis of the New York Stock Exchange composite

index to show that Levy Processes do exist in it was carried out by R.N.Mantegna,

and he showed that the daily variations of the of the price index are distributed on a

'Levy' stable probability distribution, and that the spectral density of the price index is

close to one expected for a Brownian motion.

Investment, Uncertainty, and Price Stabilization Schemes

Another application of Einstein's theory is seen in the paper done by William Smith,

who uses the method of regulated Brownian motion to analyse the effects of price

stabilization schemes on investment when demand is uncertain. He investigates the

behaviour of investment when price is random, but subject to an exogenous ceiling,

and with the aid of the mathematics of regulated Brownian motion, demonstrated that

price controls mitigate the response of investment to changes in price, even when

controls are not binding. The conclusions developed would be applicable to any

economic situation involving smooth costs of adjustments of stocks when prices are

uncertain but subject to government control (ie. rent controls, hiring/firing decisions

in the presence of a minimum wage).

A Brownian Motion Model for Decision Making

The Brownian model was also made use of by L.Romanow to develop a model for a

decision making process in which action is taken when a threshold criterion level is

reached. The model was developed with reference to career mobility, and it provides

an explanation of an important feature of promotion processes in internal labour

markets. The model assumes continuous observation of behavior (of employees) and

that the only route for leaving a job is by promotion. This suggests that the important

mechanisms in the process are the basic evaluation procedure -- rating which includes

a random component (Brownian motion theory), and the decision rule -- promote

when an estimated average reaches a criterion level. The model was able to provide

substantive qualitative results and hence is of good use to the 'real' world in decision

making policies.

Stock Prices and Its Relation with Brownian Motion

Prices changes can be decomposed into two components:

Deterministic component (the guaranteed 10 percent compounding each year)

Stochastic or random component (the plus or minus 5 percent “jump”

experienced each year on top of the guaranteed 10%)

The stochastic component is normally distributed with an expected value of zero (is

symmetric about zero, just as the random “jump” was symmetric about zero)

The standard deviation of the stochastic component controls “how much” volatility

there is on top of the deterministic component.

The long term returns on a stock are proportional to

μ−σ 2

2(¿not ¿μ)

Because X2 itself represent the result of two price moves, the average amount the

stochastic component depresses returns in a single move is σ 2/2.

If a random variable representing the stochastic component of Brownian Motion, then

we have

VAR ( X )=E ( X 2)−E ( X )2=E (X ¿¿2)=σ2¿

For example:

μ = 10% per annum

σ = 35% per annum

The model predicts that the five-year returns are normally distributed,

with mean (0.10−0.202/2 ) 5=19.73 % and standard deviation 35 % √5=78.2%.

If we observed the average one-day returns and the standard deviation of one-day

returns, respectively, we would find that they are approximately

10 %× (1/365 ) and 35 %× 1/√365

Geometric Brownian Motion and The Real World

Geometric brownian motion (GBM) states that the mean and standard deviation of a

stock are constant. Clearly this is not the case with the rate of return on a stock (in

fact, the rate of return cannot observe directly).

Fortunately,only the instantaneous standard deviation is important for option pricing

and standard deviation is less difficult to predict if we compare the change in the

stock price over small period of time, infinitesimally close.

μ is a drift term or growth parameter that increases at a factor of time steps δ t .

σ is the volatility parameter, growing at a rate of the square root of time, and ε is a

simulated variable, usually following a normal distribution with a mean of zero and a

variance of one.

Now, we calibrate the model by computing its parameters over very short time

intervals, and then using the conclusions to infer information about the long-term

returns and volatility.

Geometric Brownian Motion As A Tool for Studying The Stock Markets

Stock Price Returns Frequency Distribution

Geometric Brownian Motion- Empirical Evidence

Geometric brownian motion states stock returns are normally distributed, and should

be proportional to elapsed time and standard deviation should be proportional to the

square root of elapsed time.

What does the data say?

Large movements in stock prices are more likely than GBM predicts (the

likelihood of returns near the mean and of large returns is greater than GBM

predicts, while other returns tend to be less likely).

Downward jumps three standard deviations from the mean is three times more

likely than a normal distribution would predict (the theory underestimates the

likelihood of large downward jumps).

Monthly and quarterly volatilities are higher than annual volatility and daily

volatilities are lower than annual volatilities.

The Distribution of Stock Prices

GBM model concludes that stock returns are normally distributed and the

stock prices are lognormally distributed

The annualized return is given by 1T−t0

ln (ST

S t0

)

The above equation is equal to 1T−t0

ln ST−1

T−t 0

ln St 0

Let’s write x for this random variable X= 1T−t 0

ln ST−1

T−t 0

ln S t0

Rearranging a bit, we have a new random

variable: X ( the return, a random

variable, normally distributed) plus a

constant (not random)

X+ 1T−t 0

ln St 0= 1

T−t0

ln ST

Therefore, the natural logarithm of the

future stock price ( ln ST) is normally

distributed

X (T−t 0 )+ ln St 0=ln ST