Econometrics Forecasting Financial Markets-Chaos Theory

Copyright © 1999 -2006 Investment Analytics Forecasting Financial Markets – Chaos Theory Slide: 1

Forecasting Financial Markets

Chaos Theory


OverviewFractals

Self-similarityFractal Market HypothesisLong Term Memory Processes

Rescale Range Analysis Biased Random walkHurst Exponent

CyclesPhase SpaceChaos & the Capital Markets


The Chaos GamePlot point P at randomRoll diceProceed halfway from P to point labeled with rolled number & plot new pointRepeat 10,000 times

P

A (1, 2)

B (3, 4)C (5, 6)


The Sierpinksi Triangle


ChaosLocal randomness, global determinism

Apparently random process may contain deterministic pattern

Stable, self-similar structureSierpinksi Triangle

Plot order impossible to predictBut odds of plotting each point are not equal• Empty spaces in each triangle have zero probability• Local randomness does not equate to independence


Characteristics of FractalsSelf-similarity

The part is similar to the whole• Precise similarity in case of Sierpinski triangle

Scale InvarianceSub-parts not to same scale as parent

DimensionEuclidean space features integer dimensionsFractals occupy fractional dimension• E.g dimension of Sierpinski triangle is more than a

line but less than a plane (1 < d < 2)


Fractal Time Series

Dimension measures how “jagged” series isStraight line has fractal dimension of 1Random time series has fractal dimension of 1.5• 50% chance of rising or falling

A line can have fractal dimension between 1 and 2At values ≠ 1.5 series is less or more jagged than a random series• Non-Gaussian


Non-Gaussian Properties of Financial Markets

Distribution of ReturnsHigher peak at mean than NormalFatter tails• Uniformly fatter

– As many observations at 4σ away from mean as at 2σ

Markets tend to stay still or make major moves more often than theory predicts• Reflected in option volatility smiles

Term Structure of VolatilityScales at faster rate than T1/2


Example: Returns on the DJIADow Jones Industrials Returns

0.00

0.10

0.20

0.30

0.40

0.50

0.60

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Standard Deviations

Normal

30 Day Returns


Volatility Term StructureDJIA Volatility Term structure 1888-1900

1000 Days

-2.5

-2.0

-1.5

-1.0

-0.5

0.00.0 1.0 2.0 3.0 4.0 5.0

Log(Days)

Log(

SD)

Actual

Theoretical


Regression Analysis of Term Structure

Chart indicates clear breakdown in volatility term structure after n = 1,000 daysRegression analysis confirms this:

Days < 1,000

Days > 1,000

Intercept -1.95 -1.38Slope 0.53 0.31R2 99.3% 47.2%F 1823.0 5.4p 0.00% 5.99%


Conclusion Re term Structure

Risk-ReturnRiskier to invest for period < 4 yearsIncreasingly less risk incurred beyond 4 yearsTied to business cycle?

Sharpe RatioGets larger for longer time horizons


Sharpe Ratio & Time Horizon

DJIA Sharpe Ratio and Time Horizon

y = 9E-08x2 - 0.0001x + 1.1433R2 = 0.7465

0.00.51.01.52.02.53.03.54.04.55.0

0 1000 2000 3000 4000 5000 6000 7000

Days

Sha

rpe

Ratio


Fractal Theory of MarketsStable markets

Investors with many investment horizons• Ensures liquidity

Information set depends on investment horizonShort term: market sentiment & technical factorsLong term: fundamental analysis

Unstable marketsOccur when LT traders exit market or trade ST

Prices set by combination of ST & LT valuationST trends are noise. LT trends tied to economic cycles


Rescaled Range AnalysisDeveloped by H.E. Hurst 1950’sBrownian Motion

Distance traveled R ∝ T0.5

Hurst Exponent(R/S)T = cTH

• H is the Hurst Exponent• c is a constant• T is # observations• (R/S)T is the rescaled range, a standardized measure

of distance traveled• Note for random time series H = 0.5


Hurst Exponent & Market Behavior

H measures persistenceCorrelation C = 2(2H-1) - 1White Noise: H = 0.5, C = 0Black Noise: 0.5 < H < 1 , 0 < C < 1

Persistent, trend reinforcing series“Long memory”

Pink Noise: 0 < H < 0.5, C < 0Antipersistent, mean-revertingChoppier, more volatile than random series


White Noise ProcessFractal Random Walk

-140

-120

-100

-80

-60

-40

-20

0

20

H = 0.5


Black Noise ProcessFractal Random Walk

-600

-500

-400

-300

-200

-100

0

100

H = 0.9• Smoother series• Trend


Fractal Random Walk

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

Pink Noise Process

H = 0.1• More volatile• Antipersistent

– Mean reverting


Simulating a Fractal Random WalkFeder (1988):

• Ei is a strict white noise process, No(0, 1)• M is the number of periods for which long memory

is generated• n is set to 5• t is set to 1• H is Hurst exponent

[ ]⎭⎬⎫

⎩⎨⎧

−++×⎟⎟⎠

⎞⎜⎜⎝

⎛+Γ

=∆ ∑ ∑=

−

=−+−+

−−−++

−− nt

i

Mn

iitMn

HHiMn

HH

H EiinEiHnty

1

)1(

1))1(1(

)5.0()5.0())1(1(

)5.0( )()5.0(

)(


Calculating (R/S)Form series of returns

rt = Ln(Pt / Pt -1) for t = 1, 2, . . . , TDivide into A contiguous sub-periods

Length n, such that An = TCompute average for each sub-periodForm cumulative series

Define range Ra = Max(Xk,a) - Min(Xk,a)

∑=

=n

kaka rr

1

( )∑=

−=k

iaiaka rrX

1


Calculating (R/S)Compute standard deviation for each subperiod

Calculate average R/S for each n

Use OLS Regression to Estimate HLn(R/S)n = Ln(c) + H Ln(n)

( )2/12

1

1⎥⎥⎦

⎤

⎢⎢⎣

⎡−= ∑

=

n

kakaa rr

nS

∑=

=A

aaan SR

ASR

1)/(1)/(


Example: RS WorksheetGenerate random sequence using RAND() fn.

Periods of length n = 10, 20, . . ., 100Calculate mean and SD for each sub-periodForm cumulative series Calculate R, R/S and Ln(R/S) for each sub-periodRepeat 10 timesPlot Ln(n) against average Ln(R/S)

Fit linear trend OLS slope estimate = H


Example: RS AnalysisRescaled Range Analysis

y = 0.507x + 0.0572R2 = 0.9619

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

-1.0 0.0 1.0 2.0 3.0 4.0 5.0

Ln(N)

Ln(R

/S)


Hurst Exponent for Random Series1,000 Simulations

Simulated Values of the Hurst Exponenent

0

50

100

150

200

250

300

350

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0


Testing (R/S): E(R/Sn)

Anis & Lloyd (1976)

For large n (>350) Use Stirling Function

∑−

=

−Γ

−Γ=

1

1)5.0()]1(5.0[)/(

n

in i

inn

nSREπ

∑−

=

− −⎟⎠⎞

⎜⎝⎛⎟⎠⎞

⎜⎝⎛ −

=1

1

5.0

25.0)/(

n

in i

innn

nSRE π


Testing (R/S)

E(H) = Ln(E[R/Sn]) / Ln(n)Var(H) = 1 / T

If underlying process is random Gaussian, H will be Normally distributed

V-StatisticRecall (R / Sn) = cnH

V-Statistic• Divide by √n• V(n) = (R / Sn) / √n = cn(H-0.5)


V-Statistic

V-Statistic V(n) = (R / Sn) / √n = cn(H-0.5)

For Persistent Series H > 0.5V(n) is increasing fn. of n

For Random Process H = 0.5V(n) is constant

For Antipersistent process H < 0.5V(n) is declining fn. of n


V Statistic for E(R/Sn)V Statistic for E(R/Sn)

0.8

0.9

0.9

1.0

1.0

1.1

1.1

1.2

1.2

1.3

1.3

2.0 3.0 4.0 5.0 6.0 7.0 8.0Log(n)

V n =

E(R

/Sn)

/ n0.

5


V-Statistic for Random SeriesV-Statistic

0.0

0.5

1.0

1.5

2.0

2.0 3.0 4.0 5.0


Example: Long memory Process in DJIA (20-day Returns)

R/S Analysis DJIA (20 Day Returns)

y = 0.6119x - 0.148R2 = 0.9898

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.0 0.5 1.0 1.5 2.0 2.5 3.0Log(n)

Log(

R/S

)

DowE(R/S)Linear (Dow)


Example: Long memory Process in DJIA (20-day Returns)

V-Statistic

n=52

0.70

0.80

0.90

1.00

1.10

1.20

1.30

1.40

0 100 200 300 400 500 600 700n

V

DowE(R/S)


Dow R/S Regression AnalysisEstimated Hurst Exponent

H = 0.62 • OLS estimate of regression slop coefficient

Indicates fractal persistent memory process10 < n < 52

H = 0.71, R2 = 99.9%• Highly persistent Hurst process

52 < n < 650H = 0.49, R2 = 99.8%• White noise process


Dow: Conclusions

R/S analysis indicates long memory processAverage cycle length approx 4 years

Tied to economic cycleEvents occurring today affected by events up to 4 years ago

Long memory effects dissipated after 4 years


R/S Analysis of Stocks

Hurst CycleExponent (months)

IBM 0.72 18Xerox 0.73 18Apple 0.75 18Coca-Cola 0.70 42McDonald’s 0.65 42Con Edison 0.68 90


R/S Analysis: Conclusions About Stock

Innovative, high growth firmsHave high H and short cycles

Stable, low growth firmsHave low H and long cycles

Implications for riskHigh H firms are less risky

• Less noise in series• Contradicts standard theory

What about diversification?• Dow index has one of the highest H exponents


Lab: R/S Analysis of S&P 500 IndexMonthly returns April 75 to Feb 99

Detrended to remove short term memory effectsCalculate

RS, E(R/S), v-statistic (actual and expected)Plot

Ln(n) vs. Ln(R/S)N vs. V-statistic

EstimateCycle lengthHurst exponents pre and post cycle


Lab: R/S Analysis of S&P 500 IndexMonthly S&P500 Index Returns Apr 75 - Feb 99

-25.00%

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

Apr-

75

Apr-

77

Apr-

79

Apr-

81

Apr-

83

Apr-

85

Apr-

87

Apr-

89

Apr-

91

Apr-

93

Apr-

95

Apr-

97


Solution: R/S Analysis of S&P 500 Index - Ln(R/S) Plot

R/S Analysis S&P500 Monthly Returns

y = 0.5153x - 0.0637R2 = 0.9692

1.01.21.41.61.82.02.22.42.62.8

2.0 3.0 4.0 5.0Ln(n)

Ln(R

/S)

SP500

E(R/S)

Linear (SP500)


Solution: R/S Analysis of S&P 500 Index - V-Statistic

V-Statistic

0.65

0.75

0.85

0.95

1.05

1.15

1.25

0 20 40 60 80 100 120 140n

V

SP500E(R/S)


SP500 Regression Analysis


Economic Indicators

Industrial Production IndexH = 0.79• Based on monthly data, Jan 1946 - Jan 1999• Strongly persistent

Cycle 42 months• Shorter than 4-year cycle accepted by economists

Ties in with S&P 500 Index


R/S Analysis: CurrenciesTrue Hurst process: no cycle length

R/S continues scaling at rate H indefinitely with nInfinite memory processNot tied to economic cycles No “fundamental” valuation of currencyLess persistent, more volatile than stocks

H Exponents (based on daily data) • Yen: H = 0.64• GBP: H = 0.63• DM: H = 0.62


R/S Analysis: Other Financial MarketsTreasury Bonds

H = 0.68• Based on daily yields, Jan 1950-Dec 1989

Cycle length 5 yearsGold

Some evidence of 4-year cycleH = 0.58, but not significant

VolatilityA true pink noise antipersistent processH = 0.31 for S&P 500 Index vol. (realized)


Chaos Theory & Financial Markets -Summary

R/S Analysis confirms aperiodic cycles in many series

Stocks, stock indices, bonds, economic indicatorsCycle length related to economic cycle

Currencies are true Hurst processScale indefinitely

Volatility is only known antipersistent financial time series (apart from Wheat futures!)

Mean-reverting?

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Econometrics Forecasting Financial Markets-Chaos Theory