Financial Econometrics and Statistical Arbitrage

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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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LectureAdministrative Details

• Instructors: Farshid Maghami ASL and Lee Maclin

• Email: [email protected]

• Course Web sites:– Blackboard

– http://homepages.nyu.edu/~fma1

• Teaching Assistant: Junyoep Park

• Email: [email protected]

• Office Hours: Mondays 5-7 pm

• Office Location: WWH 606

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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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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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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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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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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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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.

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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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• 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:

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i

iii S

SSR −= +1 = mean + standard deviation x φ

2

21

21)1,0(

φ

π

−= eN

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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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Lecture

020

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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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LectureSimulation of a Stochastic Process

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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

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Lecture

We want to forecast distributions

Goals of Studying Time Series

1- Forecasting

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2- Understanding the statistical characteristics and buildingtrading strategies based on them

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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

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1 0

Xk

k

-10 -8 -6 -4 -2 0 2 4 6 8 10-10

-8

-6

-4

-2

0

2

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6

8

10

Xk

Xk-1-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

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10

Xk

Xk-2-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

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6

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10

Xk

Xk-3-10 -8 -6 -4 -2 0 2 4 6 8 10

-10

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-2

0

2

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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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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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LectureAutocovariance Function

0 2 4 6 8 10 12 14 16 18 20

Lag

Sample Autocovariance Function

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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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LectureDefinition

Note:

Strict Stationary(Strong)

Weak Stationary(Covariance Stationary)

Not generally true except for the Gaussian processes

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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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LectureStationary Process

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Stationary Process

Non-Stationary Process

Mean-Reversion

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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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LectureBuilding Blocks of Financial Models

White Noise Process

=

=otherwise

srsrX 0),(

2σγ

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-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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0)( =tXE

)( 2 ∞<σ

=

=otherwise

srsrX 0),(

2σγ

White Noise Process (Is it Stationary?)

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Random Walk Process

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100

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dom

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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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Random Walk Process (Is it Stationary?)

2σIID(0, )∑ ==

t

j jt XS1

Xt

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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

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θ =0.5

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Moving Average Process (Is it Stationary?)Xt is WN(0, )

1−+= ttt XXY θ2σ

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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

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k φ=0.7

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10Xk

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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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ttt ZXX += − 1φ

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φ = 1Random Walk

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φ = 0.9AR(1)

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φ = 0.1AR(1)

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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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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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tmt 0004.01513.6 +=

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Lecture

Forecast

Transforming a Non-Stationary Process to a Stationary Process

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Lecture

Forecast


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Lecture

Forecast


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conv

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Financial Econometrics and Statistical Arbitrage

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