introduction of brownian motion
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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
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