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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
Lecture
Financial Econometrics and Statistical Arbitrage
Master of Science Program in Mathematical FinanceNew York University
Lecture 1: Basics on Time Series Analysis
Fall 2005
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureAdministrative Details
• Instructors: Farshid Maghami ASL and Lee Maclin
• Email: fma1@nyu.edu
• Course Web sites:– Blackboard
– http://homepages.nyu.edu/~fma1
• Teaching Assistant: Junyoep Park
• Email: junyoep@gmail.com
• Office Hours: Mondays 5-7 pm
• Office Location: WWH 606
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureAdministrative Details (cont.)
Time: Mondays, 7:10 – 9 pm First Class: September 12, 2005, Last Class: December 12, 2005. There will be no class on Columbus Day (October 10, 2005). We will make it up on Wednesday 11/23/05.
Homework and Exam:
• There will be six homework sets which will be assigned every other week. Students must write up and turn in their solutions individually within one week.
• Computer assignments can be solved by C/C++/C#, MATLAB, R. For other tools, please coordinate with the TA or the instructor.
• There will be one final exam (no mid-term).
• Final grade will be evaluated based on homework solutions (30%) and the final exam (70%).
• Lecture Notes and Homework will be posted on the course website as they become available
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureAdministrative Details (cont.)
Textbooks: Lectures are drawn from many sources including the following books:
1. Alexander, C. “Market Models,” John Wiley and Sons, 2001 2. Brockwell, P.J., Davis, R.A., “Introduction to Time Series and
Forecasting,” Springer3. Javaheri, A. “Inside Volatility Arbitrage : The Secrets of Skewness”
Wiley4. Tsay, R. S., “Analysis of Financial Time Series,” Wiley, 20025. Wilmott, P. “Derivatives: The Theory and Practice of Financial
Engineering,” Wiley Frontiers in Finance Series6. Pandit S.M., Wu S.M., “Time Series and System Analysis with
Applications.” Krieger Publishing, Malabar, FL, 20017. Hamilton J. D. “Time Series Analysis.” Princeton University Press, 1994
A number of research articles will be posted on the course webpage.
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureExpected Background
• Prior knowledge of Linear Algebra, Probability and Statistics is required
• I assume you have taken the following courses:– Derivative Securities– Continuous Time Finance– Scientific Computing / Computing for Finance
• Programming in C/C++ or MATLAB/R is required
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
Lecture
Market Microstructur Theory(Transaction costs and Optimal Control, Algorithmic Trading,…)
Risk Management(Practical Risk Measurement and Management Technics)
Financial Econometrics(Time Series Review and Volatility modeling)
Strategies and Implementation Process(Cointegration based pairs trading, Volatility trading, …)
What is Statistical Arbitrage?
• Statistical Arbitrage covers any trading strategy which uses statistical tools and time series analysis to identify approximate arbitrage opportunities while evaluating the risks inherent in the trades considering the transaction costs and other practical aspects.
• Arbitrage is a riskless profit. “Arbitrage Strategy” is a trading strategy that locks in a riskless profit.
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureCourse Outline
• Financial Econometrics (8 weeks)– Time Series Models Review and Analysis– Volatility and Correlation Models in Financial Systems– Calibration and Estimation Methods
• Cointegration and Market Microstructure in Practice (3 weeks)– Cointegration and Pairs Trading– Transaction Costs, and Market Friction– Trade Execution Strategies
• Practical Simulation and Risk Management (1 week)
• More on Trading Strategies (1-2 week)
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTypical Behavior of Financial Assets
• The unpredictability inherent in asset prices is the main feature of financial modeling.
• Because there is so much randomness, any mathematical model of a financial asset must acknowledge the randomness and have a probabilistic foundation.
0 1000 2000 3000 4000 5000 6000 7000 8000 90000
2000
4000
6000
8000
10000
12000
Time in days from 1/1/1975 to 07/30/2005
Dow
Jon
es In
dex
5
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureIntroduction to Financial Modeling
• There are three general types of analysis used in finance and trading
1. Fundamental Analysis2. Technical Analysis3. Quantitative Analysis
• Return in financial assetsBy return we mean the percentage growth in the value of an asset, together with accumulated dividends, over some period:
Change in value of the asset + accumulated cashflowsOriginal value of the asset
Return =
• Denoting the asset value on the i-th day by Si, the return from day ito day i+1 is given by
i
iii S
SSR −= +1
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureIntroduction to Financial Modeling
• Supposing that we believe that the empirical returns are close enough to Normal for this to be a good approximation.
• For start, we write the returns as a random variable drawn from a Normal distribution with a known, constant, non-zero mean and a known, constant, non-zero standard deviation:
0 1000 2000 3000 4000 5000 6000 7000 8000 9000-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Time in days from 1/1/1975 to 07/30/2005
Diff
eren
ce o
f the
Log
Tra
nsfo
rm o
f Dow
Jon
es In
dex
i
iii S
SSR −= +1 = mean + standard deviation x φ
2
21
21)1,0(
φ
π
−= eN
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureIntroduction to Financial Modeling
i
iii S
SSR −= +1 = mean + standard deviation x φ
φσδµδ ×+=−
= + 2/11 ttS
SSRi
iii
• Time scale δt
tµδ
Mean return over period δt is
2/1tσδ
Standard deviation over period δt is
δXAnd in the limit δt 0
tt
t dXdtS
dSR σµ +==
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
Lecture
020
4060
80100
1
2
3
4
50.9
1
1.1
1.2
1.3
x 104
Basic Review
STOCHASTIC PROCESS:
A stochastic process is a collection of random variables defined on a probability space . ( )PF ,,Ω
τω ∈tX t ),(
State 2State 1
State 3State 4
State 5
Ω
Time
Time
ω
ωFor a fixed , a realization of stochastic process is a function of time (t).
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureSimulation of a Stochastic Process
020
4060
80100
0
50
100
150
2000.6
0.8
1
1.2
1.4
1.6
x 104
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureDefinition
Time Series:
A time series is a stochastic process where τ is a set of discrete points in time. In other words, it is a discrete time, continuous state process.
In this course we consider τ = all integers
X1 X2 X3 …
Xk
k0 5 10 15 20 25 30 35 40
-4
-3
-2
-1
0
1
2
3
8
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
Lecture
We want to forecast distributions
Goals of Studying Time Series
1- Forecasting
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
2000
4000
6000
8000
10000
12000
Time in days from 1/1/1975 to 07/30/2005
Dow
Jone
s Ind
ex
2- Understanding the statistical characteristics and buildingtrading strategies based on them
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBasic Review
An Example of a Time Series:
0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0- 1 0
- 8
- 6
- 4
- 2
0
2
4
6
8
1 0
Xk
k
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-8
-6
-4
-2
0
2
4
6
8
10
Xk
Xk-1-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
Xk
Xk-2-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
Xk
Xk-3-10 -8 -6 -4 -2 0 2 4 6 8 10
-10
-8
-6
-4
-2
0
2
4
6
8
10
Xk
Xk-10
Xk= ϕ 1 Xk-1+ ϕ 2 Xk-2+…+ekAuto-Regression as a Dynamic System?We will get back to this
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureDefinition
Autocovariance Function: Let Xt be a time series. The autocovariance function of process Xt for all
integers r and s is:),cov(),( srX XXsr =γ
))]())(([(),( ssrrX XEXXEXEsr −−=γ
)]()()()([),( srrssrsrX XEXEXEXXEXXXEsr +−−=γ
)()()()()()()(),( srrssrsrX XEXEXEXEXEXEXXEsr +−−=γ
)()()(),( srsrX XEXEXXEsr −=γ
0)var()()(),( 22 ≥=−= rrrX XXEXErrγNote that
= 0
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureAutocovariance Function
0 2 4 6 8 10 12 14 16 18 20
Lag
Sample Autocovariance Function
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureDefinition
Stationary Process: A time series Xt is stationary (weakly) if:
),(),(.3)(.2)(.1 2
tstrsrXEXE
XX
t
t
++==
∞<
γγSome constant m for all t
i.e. Cov(Xr,Xs) only depends on r and s and not on t.
)()0,(),(),( srsrsssrsr XXXX −=−=−−= γγγγ
Note: If Xt is stationary, then
),cov()()( httXX XXhsr +==− γγDefine h=r-s
Does not depend on t
A strict (strong) stationary time seriesXt , t=1,2,…,n
is defined by the condition that realizations(X1, X2, …, Xn) and (X1+h, X2+h, …, Xn+h)have the same joint distributions for all integers h and n>0.
Note:
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureDefinition
Note:
Strict Stationary(Strong)
Weak Stationary(Covariance Stationary)
Not generally true except for the Gaussian processes
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureStationary Process
Stationary Process and Mean Reversion• We are interested in stationary time series because many models and
tools are developed for stationary processes.
• A stationary process can never drift too far from its mean because of
the finite variance. The speed of mean-reversion is determined by the
autocovariance function: Mean-reversion is quick when autocovariances
are small and slow when autocovariances are large.
• Trends and periodic components make a time series non-stationary.
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureStationary Process
0 50 100 150 200 250 300-20
0
20
40
60
80
0 50 100 150 200 250 300-6
-4
-2
0
2
4
Stationary Process
Non-Stationary Process
Mean-Reversion
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureGeneral Approach to Time Series
Time Series Analysis
1. Plot time series and check for trends or sharp changes in behavior
(most of the time non-stationary)
2. Transform into a stationary time series
3. Fit a model
4. Perform diagnostic tests (residual analysis,…)
5. Generate forecasts (find predictive distributions) and invert the
transformations performed in 2.
Note for option pricing:
6. Find a risk neutral version of the model
7. Obtain predictive distributions under the risk neutral model
If bad
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBuilding Blocks of Financial Models
White Noise Process
=
=otherwise
srsrX 0),(
2σγ
0 1000 2000 3000 4000 5000 6000 7000
-5
0
5
WN
If Xt is a sequence of random variables with , and 0)( =tXE
)( 2 ∞<σ
2σXt is called White Noise and it is written as WN(0, )
22 )( σ=tXE
Note that E[Xt Xs]=0 for t=s Uncorrelated r.v.’s2σIf Xt and Xs independent for t=s IID(0, )
-5 0 5-5
0
5Xk
Xk-1
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBuilding Blocks of Financial Models
0)( =tXE
)( 2 ∞<σ
=
=otherwise
srsrX 0),(
2σγ
White Noise Process (Is it Stationary?)
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBuilding Blocks of Financial Models
Random Walk Process
0 1000 2000 3000 4000 5000 6000 7000-100
0
100
Ran
dom
Wal
k
If Xt be a sequence of random variables , a sequence St with S0=0 and
Is called a Random Walk.
2σIID(0, )
∑ ==
t
j jt XS1
(Integrated Process)
-10 -5 0 5 10-10
-5
0
5
10Sk
Sk-1
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBuilding Blocks of Financial Models
Random Walk Process (Is it Stationary?)
2σIID(0, )∑ ==
t
j jt XS1
Xt
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBuilding Blocks of Financial Models
Moving Average ProcessLet Xt be WN(0, ), and consider the process
Where θ could be any constant. This time series model is called a first-order moving average process, denoted MA(1).
The term “Moving Average” comes from the fact that Yt is constructed from a weighted sum of the two most recent values of Xt.
1−+= ttt XXY θ
2σ
Yk
-4 -2 0 2 4-4
-2
0
2
4
Yk-10 1000 2000 3000 4000 5000 6000
θ =0.5
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBuilding Blocks of Financial Models
Moving Average Process (Is it Stationary?)Xt is WN(0, )
1−+= ttt XXY θ2σ
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBuilding Blocks of Financial Models
Autoregressive ProcessLet Zt be WN(0, ), and consider the process
Where |φ |<1 and Zt is uncorrelated with Xs for each s<t. This time series model is called a first-order Autoregressive process, denoted AR(1).
2σ
ttt ZXX += − 1φ
It is easy to show that E(Xt)=0
0 100 200 300 400 500 600 700
-5
0
5
Ran
dom
Wal
k φ=0.7
-10 -5 0 5 10-10
-5
0
5
10Xk
Xk-1
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBuilding Blocks of Financial Models
Autoregressive Process (Is it Stationary?)Zt is WN(0, ), and
Where |φ |<1 and Zt is uncorrelated with Xs for each s<t.
2σ ttt ZXX += − 1φ
We will see this later
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureBuilding Blocks of Financial Models
ttt ZXX += − 1φ
0 50 100 150 200 250 300-10
0
10
20
30
-10 0 10 20 30-10
0
10
20
30
φ = 1Random Walk
0 50 100 150 200 250 300-10
-5
0
5
10
-10 -5 0 5 10-10
-5
0
5
10
φ = 0.9AR(1)
0 50 100 150 200 250 300-4
-2
0
2
4
-4 -2 0 2 4-4
-2
0
2
4
φ = 0.1AR(1)
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTransforming a Non-Stationary Process to a Stationary Process
Classical Decomposition
tttt YSmX ++=
Original Time series
(Nonstationary)Trend
Seasonalcomponent
StationaryTime series(zero-mean)
∑=
=d
jjS
1
0
Seasonal component St satisfies
St+d=St where d= period of seasonality
Also for mathematical convenience assume
Most observed time series are non-stationary but they can be transformed to stationary processes.
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTransforming a Non-Stationary Process to a Stationary Process
Classical Decompositiontttt YSmX ++=
tttt SmXX^^
* +−=
Idea of transformation is to estimate mt and St by mt and St, then work with the stationary process:
Assume there is no seasonal component (St=0)
ttt YmX +=
2210
^tataamt ++=
Consider a parametric form for mt e.g.
2
1
^)(∑
=
−n
ttt mX
Using observed data X1, X2, … Xn, choose α0, α1, α2 to minimize
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTransforming a Non-Stationary Process to a Stationary Process
0 1000 2000 3000 4000 5000 6000 7000 8000 90000
2000
4000
6000
8000
10000
12000
Time in days from 1/1/1975 to 07/30/2005
Dow
Jone
s Ind
ex
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTransforming a Non-Stationary Process to a Stationary Process
0 1000 2000 3000 4000 5000 6000 7000 8000 90005
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Time in days from 1/1/1975 to 07/30/2005
Log
Tran
sfor
m o
f Dow
Jone
s Ind
ex
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTransforming a Non-Stationary Process to a Stationary Process
0 1000 2000 3000 4000 5000 6000 7000 8000 90005
5.5
6
6.5
7
7.5
8
8.5
9
9.5
10
Time in days from 1/1/1975 to 07/30/2005
Log
Tran
sfor
m o
f Dow
Jone
s Ind
ex
tmt 0004.01513.6 +=
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTransforming a Non-Stationary Process to a Stationary Process
0 1000 2000 3000 4000 5000 6000 7000 8000 9000-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Time in days from 1/1/1975 to 07/30/2005
Diff
eren
ce o
f the
Log
Tra
nsfo
rm o
f Dow
Jone
s Ind
ex
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
Lecture
Forecast
Transforming a Non-Stationary Process to a Stationary Process
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
Time in days from 1/1/1975 to 07/30/2005
Fore
cast
of t
he m
odel
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
Lecture
Forecast
Transforming a Non-Stationary Process to a Stationary Process
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100006
6.5
7
7.5
8
8.5
9
9.5
10
Time in days from 1/1/1975 to 07/30/2005
Con
vert
back
the
diff
eren
ce in
the
Fore
cast
of t
he m
odel
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
Lecture
Forecast
Transforming a Non-Stationary Process to a Stationary Process
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
5000
10000
15000
Time in days from 1/1/1975 to 07/30/2005
conv
ert b
ack
the
Log
of th
e Fo
reca
st o
f the
mod
el
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTransforming a Non-Stationary Process to a Stationary Process
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
4
Time in days from 1/1/1975 to 07/30/2005
Mon
te C
arlo
Sim
ulat
ion
of th
e Fo
reca
st
22
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTransforming a Non-Stationary Process to a Stationary Process
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2x 10
4
Time in days from 1/1/1975 to 07/30/2005
Mon
te C
arlo
Sim
ulat
ion
of th
e Fo
reca
st
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G63.2707 - Financial Econometrics and Statistical ArbitrageFarshid Magami Asl
LectureTransforming a Non-Stationary Process to a Stationary Process
0 10 20 30 40 50 60 70 80 90 1000.8
0.9
1
1.1
1.2
1.3x 10
4
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
x 104
0
20
40
60