risk management > volatility

1

Volatility Modeling

Copyright © 2000 – 2006Investment Analytics

Copyright 2001-2006 Investment Analytics Volatility Slide: 2

Asset Return Characteristics

The Standard Gaussian ModelThick TailsNon-Normal DistributionVolatility ClusteringLeverageVolatility & Correlation


Standard Gaussian ModelAsset returns follow random walk

Return this period independent of past return

Asset returns are normally distributedThese assumptions underlie all major financial theories

CAPMBlack-Scholes model


Thick Tails, Non-Normal DistributionMandelbrot (1963), Fama (1963, 1965)

Skewness = -0.6

Kurtosis = 5.7


Tests for NormalityError Distribution Moments

Skewness: should be ~ 0Kurtosis: should be ~ 3

Statistical TestsLilliefors Kolmagorov-Smirnov nonparametric testShapiro-Wilk test

More powerful

Jarque-Bera Testn[Skewness / 6 + (Kurtosis – 3)2 / 24] ~ χ2(2)


Volatility is StochasticVolatility - DOW Stocks

0%

20%

40%

60%

80%

100%

120%

140%

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

DJIA IBM INTC IP JNJ


Volatility ClusteringHigh vol. followed by high vol.Decay back to normal levels

SP500 Index Volatility

0%

20%

40%

60%

80%

100%

120%

140%

160%

Jan-

50

Jan-

52

Jan-

54

Jan-

56

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60

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62

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64

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66

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72

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86

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88

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90

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92

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94

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96

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98

Jan-

00


The Volatility ConeV

olat

ility

(%)

Maximum

Minimum

Average

Days0 600 3


Leverage Effect

Black (1976)Stock price changes negatively correlated with volatilityFixed costs provide a partial explanation

Firm with equity and debt becomes more leveraged as stock fallsRaises equity returns risk/volatility

Correlation too large to be explained by leverage alone

Christe (1982), Schwert (1989)


Volatility Seasonality

DOW Volatility Seasonality

40%

60%

80%

100%

120%

140%

160%

JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC


Volatility Correlation

Volatilities tend to change togetherStocks: Black (1976)FX: Diebold & Nerlove (1989)

Also across marketsStock & bond volatilities move together (Schwert, 1989)


Volatility CorrelationCorrelation: IBM vs JNJ Volatility

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

May-85 May-88 May-91 May-94 May-97 May-00


Asset Characteristics –Conclusions

The Bad Newsiid Gaussian model inappropriate

The Good NewsCorrelation suggests few common factors may explain variationVolatility models (GARCH, etc.)


Volatility MetricsConsider statistic f of log asset price siH,(i+1)H

If f is homogeneous in some power γ of volatility,then

and

Where s* is standardized diffusion with unit volatility

)()( *1)(, 1)(, HiiH

sfsf iHHiiH +=+

γσ

)( 1)(,*

)1(, lnln)(ln HiiHiHHiiH sfsf ++ += σγ


Standard Volatility MetricsSquared or absolute returns

γ only scales proxy, does not affect distributionVery noisyNon-Gaussian

Skew –1.09, kurtosis 5.0Problems with bias in Gaussian QMLE

Andersen & Sorensen (1997)

**1)(1)(, lnln)(ln iHHiiHHiiH sssf −+= ++ γσγ


Log Absolute ReturnsLog Absolute Returns SP500 Index Jan 1983- Jul 2002

X <= -3.502295.0%

X <= -7.44175.0%

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

-12 -10 -8 -6 -4 -2 0


Realized Volatility

Anderson, Bollerslev, Diebold (2000)Dow 30 stock volatilityUses high frequency dataIdea: diffusion coefficients can be estimated arbitrarily well

Given fine enough samplingMerton (1980), Nelson (1992)


Realized VolatilityMultivariate process

Ω is NxN positive definite diffusion matrix

Distribution of continuously compounded returns is:

Convergence:

tttt dWdtdp Ω+= µ

[ ] ⎟⎠⎞⎜

⎝⎛ ΩΩ ∫ ∫ ++=+++

h h

tth

tttht ddNr0 00, ,~, ττµµσ τττττ

∫∑ →Ω−′•h

drr 0τ+∆∆+∆∆+ tjtj

jt 0,, τ


Modeling with Realized Volatility

Distribution propertiesRealized volatility lognormally distributedReturns standardized by realized volatility are approximately Gaussian

Andersen & Bollerslev (1998)Foreign exchange rate volatilityR2 increases with sampling frequency

Daily ~ 7%, 5 min ~ 48%


Example: DD Volatility Histogram: LogStDev

K-S d=.03328, p> .20; Lilliefors p> .20Shapiro-Wilk W=.99517, p=.13844

-3 -2.8 -2.6 -2.4 -2.2 -2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0X <= Category Boundary

0

20

40

60

80

100

120

No.

of o

bs.


Realized Volatility: Conclusion

Significant gains to forecast accuracy with high frequency estimationDaily returns well described by continuous normal-lognormal mixture

See Mixture of Distributions Hypothesis (Clark 1973)


The Log Range

Difference in log of highest and lowest log prices

⎥⎦⎤

⎢⎣⎡ −=

+<<+<<+ tHtiHt

HtiHHiiH sssf

1)(11)(1)1(, infsupln)(ln

⎥⎦⎤

⎢⎣⎡ −+=

+<<+<<t

HtiHt

HtiHiH ss *

1)(1

*

1)(1infsuplnlnσ


Range in Volatility EstimationIntuition

Days with large intraday movesClose happens to be close to openRange reflect true, higher intraday volatility

Historical ApplicationsParkinson (1980)Garman & Klass (1980)Rogers & Satchell (1991)


Other Range-Based Metrics

Parkinson (5x efficiency)

Garman & Klass (7 x efficiency)

)/()2(2

1ii LHLn

LnN ∑=σ

σ =−

−

∑

∑ −

A B SN

Ln H L

NLn Ln C C

i i

i i

[ [ ( / )]

( ( ) )[ ( / )] ]

1 12

1 2 2 1

2

12


Properties of Log RangeDistribution

Very close to NormalDt ~N[0.43 + lnht, 0.292]

Where ht = σ / 2521/2

Typical skewness 0.28, kurtosis 3.2

EfficiencyStdev approx ¼ of log abs return

RobustnessNot affected by bid/ask bounce to same degree as realized volatility


Log Range for SP500 IndexLog Range SP500 Index Jan 1983- Jul 2002

X <= -3.673395.0%

X <= -5.50265.0%

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

-7 -6 -5 -4 -3 -2 -1


Example: GE Log RangeHistogram: GE

K-S d=.05215, p> .20; Lilliefors p> .20Shapiro-Wilk W=.99131, p=.81256

-3.5 -3 -2.5 -2 -1.5 -1

X <= Category Boundary

0

5

10

15

20

25

30

35

40

45

50

No.

of o

bs.


Robustness of Log Range vsRealized Volatility

Alizadeh, Brandt, Diebold (2001)Simulated performance of log range vsrealized volatility with bid/ask spreads

Actual daily vol was set at 1.87%

Volatility EstimatesInterval Realized Range

5-min 9.35 2.1140-min 3.72 1.8712 hour 1.79 0.68


Volatility Models

Key volatility characteristicsLong memoryVolatility of volatility

Univariate modelsMultivariate models


Time Series Models

Autoregressive AR(1):yt = a0 + a1yt-1 + εt

εt = sequence of independent random variablesIndependentZero meanConstant variance σ2

Differenced series (yt – (a0 + a1yt-1)) = εtWhite noise


White NoiseMean is constant (zero)

E(εt) = µ (0)Variance is constant

Var(εt) = E(εt2) = σ2

UncorrelatedCov(εt , εt-j) = 0 for j < > 0 and t

Gaussian White NoiseIf εt is also normally distributed

Strict White Noiseεt are independent


StationarityWeak (covariance) stationarity

Population moments are time-independent:E(yt) = µVar(yt) = σ2

Cov(yt, yt-j) = γj

Example: white noise εt

Strong stationarityIn addition, yt is normally distributed


Stationary Series


Stationarity of AR(1) ProcessAR(1) Process: yt = a0 + a1yt-1 + εt

Expected value E(yt) is time-dependent:

If |a1| < 1, then as t →∞, process is stationaryLim E(yt) = a0 / (1 - a1)

Hence mean of yt is finite and time independentAlso Var(yt) = σ2/[1 - (a1)2]

And Cov(yt, ys) = σ2 (a1)s /[1 - (a1)2]

01

1

010)( yaaayE t

t

i

it += ∑

−

=


Random Walk ProcessRandom Walk with drift

yt = a0 + a1yt -1 + εtWith a1 = 1A non-stationary process


Random Walk Process

Random Walk without driftyt = a0 + a1yt -1 + εt

With a1 = 1, a0 = 0

A non-stationary processVariance of yt gets larger over time

Hence not independent of time.

2

1

2 2)( σεεε nEyVarn

ststtt =⎥⎦

⎤⎢⎣

⎡+= ∑ ∑

≠


Random Walk Integration

First difference of RW is stationaryyt - yt -1 = εt

Changes in random walk are random white noise

Integrated process, order 1Denoted I(1)


Near-Random Walk Process

AR process with coefficient < 1Very difficult to distinguish from random walkBut difference is huge

AR(1) stationary, RW is not

Dickey-Fuller testBest available, but not powerful


Long Memory

Idea: shocks persist over long time periodLong Memory autocorrelation function

Hyperbolic decay

Short Memory autocorrelation function

ARMA models only have short memory

210,)(~)( 12 <<∞→− dtasttLt dρ

10,0|)(| || <<>≤ rCsomeforCrt tρ


Volatility Long Memory

Volatility is highly persistentEvents have sustained influence on future volatilityIn principle, process is very forecastable


Volatility Autocorrelations

Volatility Autocorrelations

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

1 4 7 10 13 16 19 22

Months

DJIABADDGEHWPIBM


Evidence for Volatility Long Memory

Bollerslev & Mikkelsen (1996)High persistence & fractional integration in SP500 index volatility

Baillie, Bollerslev, Mikkelsen (1996)FX processes well modeled by FIGARCH

Grau-Carles (2000)Long memory effects confirmed in volatility processes for all major stock markets

Brunetti & Gilbert (2000)Volatility in crude oil markets has long memory and is fractionally cointegrated


Theories of Long Memory in Volatility

Andersen & Bollerslev (1997)Results from aggregation of a news arrival process with different persistence levels

Zin & Bachus (1993)Spread from other variables, e.g. inflation, which themselves have long memory

Lamoureux & Lastrapes (1990)Caused by regime switching


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 Exponentc is a constantT is # observations(R/S)T is the rescaled range, a standardized measure of distance traveledNote 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 = 0.9Smoother seriesTrend


Pink Noise ProcessFractal Random Walk

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

H = 0.1More volatileAntipersistent

Mean reverting


Simulating A Fractal Random Walk

Feder (1988):

Ei is a strict white noise process, No(0, 1)M is the number of periods for which long memory is generatedn is set to 5t is set to 1H 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

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)/(


Volatility R/S AnalysisGE - Rescaled Range Analysis

y = 0.843x - 0.825R2 = 99%

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Ln(Months)

Ln

(R/

S)


DOW Stock Volatility –Hurst Exponents

Hurst Exponents - DOW Stocks

0.75

0.80

0.85

0.90

0.95

1.00

DJI

AA

AA

XP

BA C

CA

TD

DD

IS EK GE

GM HD

HO

NH

WP

IBM

INTC IP JN

JJP

M KO

MC

DM

MM

MO

MR

KM

SFT

PG

SBC T

UTX

WM

TX

OM


Other Methods for Estimating Fractional Integration

Lo (1991)Modified R/S statistic

Peng et al (1994)Detrended fluctuation analysis

Geweke & Porter-Hudak (1983)Spectral regression

Sowell (1992)Spectral analysis


Lo’s Modified R/S

Lack of robustness in R/SIn presence of short memory effects

Lo’s statistic replaces standard deviationUses consistent estimator of standard deviation of partial sum of x

2/1

1 11

2 ))((1

2/)()(⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−−

++−= ∑ ∑∑

= +=−

=

q

j

T

jijii

T

iiT xxxx

qj

TTxxqs


Comments on Lo’s Method

Lo shows modified R/S is robust to short-range dependenceTeverlosky et al (1999)

Lo test tends to reject long range dependenceChoice of truncation lag q is critical


Peng’s DFA Analysis

Distinguishes between long memory and non-stationaritiesMethod

Obtain integrated seriesDivide into non-overlapping intervals

Each containing m data points

Fit regression line to each interval

∑′

=

=′t

Ttxty

1)()(


Peng’s DFA Analysis

Calculate fluctuation around regression line ym(t)

For series with long memory F(m) ∝ ma

a > 1/2

[ ]∑=′

′−′=T

tm tyty

TmF

1

2)]()(1)(


Geweke & Porter-HudakSpectral density regression

I(ωλ) is the periodogram at frequencies ωλ = 2πλ/Tλ =1, . . .,(T-1),T is #observationsThe slope of the OLS regression provides estimate of fractional differencing parameter d

λλ

λ ηωω +⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛+=

2sin4ln)(ln 2baI


Sowell MethodCalculates autocovariance in terms of spectral density function f(w)

Estimates ARFIMA model using maximum likelihoodIncludes fractional differencing parameter

∫=π

πγ

2

0

)(21)( dwewfk iwk


ARFIMA ModelsGeneralized ARIMA models

ARFIMA(p,d,q)Fractional differencing parameter d = H - 0.5φ and θ are polynomials order p and q

Models fractal Brownian motionShort memory effectsLong memory effects

ttd LyLL εθφ )()1)(( =−

∑∞

= +Γ−Γ−Γ

=−0 )1()(

)()1(j

jd

jdLdjL


ARFIMA(1, d, 0)Process: (1 - αLd) yt = εt

Combines long and short term memory processes

Correlation function

F(a,b;C,z) is the Hypergeometric function

);1;1,1()1()!1()1()!( 12

2 αααρ

ddFk

dd d

k −+×

−−+−

=−


AR(1) vs ARFIMA(1,d,0)ARCH Error Process εt

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

ARMA vs ARFIMA Process

-3.0-2.5-2.0-1.5-1.0-0.50.00.51.01.52.02.5

1 3 5 7 9 11 13 15 17 19

ARMA ARFIMA

Parametersd = 0.4a1 = 0.5

ARFIMA shocks are more persistent


AR(1) vs. ARFIMA(1, d, 0)

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25Lag

Cor

rela

tion

AR(1)

ARFIMA(1,d,0)

Example: AR(1) vs. ARFIMA(1,d, 0)AR(1): a = 0.711ARFIMA(1, d, 0): d = 0.2, a = 0.5


GARCH Models

AR(1) process: yt+1 = a0 + a1yt + εt+1

Conditional ForecastEt(yt+1) = a0 + a1yt

Forecast Error VarianceEt[yt+1 - Et(yt+1)]2 = Et[yt+1 - (a0 + a1yt)]2 = Et(εt+1 )2 = σ2


Unconditional ForecastUnconditional Expectation is Constant

E(yt+1) = a0 /(1 - a1) i.e. the long run mean

Unconditional Variance is ConstantE[yt+1 - E(yt+1)]2 = E[yt+1 - a0 / (1 - a1)]2 = σ2 / (1 - a1)2

Unconditional forecast has greater varianceSince 1 / (1 - a1) > 1

Conditional Variance is ConstantEt(εt+1

2) = Et(yt+1 - a0 + a1yt )2 = σ2


ARCH ProcessSuppose conditional variance is not constantModel conditional variance as an AR(p) process

εt2 = α0 + α1 (εt-1)2 + α2 (εt-2)2 + . . . + αq (εt-q)2 + vt

vt is white noise

Multiplicative ARCH model (Engle):εt

2 = [α0 + α1 (εt-1)2] vt2

is white noise with σ2v = 1

εt are independent of each other

α0 > 0 and 0 < α1 < 1


Key Points about ARCH

Errors MomentsZero mean, covariance, unconditional variance

Error variance fluctuatesFor large εt , variance of εt will be largePeriods of tranquility & volatility in y

Errors are not independent Related through second moment

Parameter valuesRestricted to ensure variance > 0 and series is stable

α0 > 0 and 0 < α1 < 1


ARCH ExampleA R C H E rr o r P ro c e ss ε t

- 2 .5

- 2 .0

- 1 .5

- 1 .0

- 0 .5

0 .0

0 .5

1 .0

1 .5

2 .0

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0

ARCH Process yt = a1yt-1 + ε t

-5.0

-4.0

-3.0

-2.0

-1.0

0.0

1.0

2.0

1 3 5 7 9 11 13 15 17 19

yt

y't

Parametersα0 = 0.3, α1 = 0.9a1 = 0.25 & 0.9

Effects & InteractionsLarger α1, morepersistent are shocks in εtLarger a1, more persistent is change in yt


GARCH ModelsGARCH(p, q)

Error Process εt = vt√ σt

vt is white noise No(0,1)

Error Process εtConditional mean and variance are zeroConditional variance is σt

2

∑ ∑= =

−− ++=q

i

p

iitiitit

1 1

220

2 σβεαασ


Properties of GARCHDisturbances of series yt follow ARMA process

ARMA(p, q) process in series εt2

Estimating a GARCH ModelFit ARMA model to series yt Evaluate sample autocorrelations of squared residuals

Should suggest an ARMA(p, q) process in series εt2

∑ ∑= =

−−− ++==q

i

p

iitiititttE

1 1

220

221 σβεαασε


ARFIMA-GARCH

Returns follow ARFIMA processVolatility follows GARCH processExample

ARFIMA(1,d,1)-GARCH(1,1)

21

2110

2

)1()1)(1(

−− ++=

−=−−

ttt

ttd LyLL

βσεαασ

εθφ


Fractionally Integrated GARCHBaillie, Bollerslev, Mikkelsen (1996)

φ and β are polynomials order p and qd is fractional differencing parameter

FI(p,d,q) is strictly stationary

ttd vLLL )]([1))(1( 2 βωεφ −+=−22tttv σε −=


GARCH vs FIGARCH

GARCHShocks to variance process die away at fast exponential rate

FIGARCHShocks die away much more slowly (hypergeometeric)Has “long memory”


FIGARCH Research: Stock indices

Grau-Carles (2000)FIGARCH models for major indices

Volatility processes:Absolute and squared returns

Estimated fractional differencing parameterHurst exponentDetrended fluctuation analysis (Peng)Sowell’s spectral density method


FIGARCH for Stock IndicesIndex Estimated d*DOW 0.27 – 0.31SP500 0.32 – 0.37FTSE 0.11 – 0.17NIKKEI 0.29 - 0.42

* based on absolute returns

Conclusion:Compelling evidence of long range autocorrelations in stock index volatility


Volatility Direction Prediction

ARFIMA-GARCH modelsAccount for 50% of variation in conditional volatility

Sign predictionVaries, but 70% is typicalHighly statistically significant

Pesaran-Timmerman sign test


Pesaran-Timmerman Test

Test of market timing abilityBased on correct sign predictions

Test statistic ~ No(0,1)

zt+n = 1 if (yt+n ft,n )>0; 0 otherwiseP* = pr(zt+n = 1) = pr (yt+n ft,n )> 0 = PyPf + (1-Py)(1-pf)Py = pr(yt+n > 0) ; pf = pr(ft,n > 0 )

2/12ˆ

2ˆ

*

ˆˆ

ˆ

*PPn

PPSσσ −−

= ∑=

+−=

T

tntz

TTP

11

1ˆ


Volatility Direction Prediction

Volatility Direction Forecast Accuracy

40%

50%

60%

70%

80%

90%

100%

90 91 92 93 94 95 96 97 98 99 00 01

BMY CCE GEIBM JNJ SP500


Economic Value of Volatility Forecasting

Fleming, Kirby & Ostdiek, 2000Addresses issue of whether volatility forecasting is economically worthwhileStocks, bonds, gold and cash

Volatility timing strategiesRe-estimate conditional covariances every periodConsistently outperform static strategies

in 84% - 92% of trials

Sharpe ratio ~ 0.85


Stochastic Volatility Models

Asset process S with instantaneous drift µ and volatility σBoth drift and volatility depend on latent state variable v which also evolves as a diffusion

Stttttt dWSSdS ),(),( νσνµ +=

tttttt dWSSd ννβναν ),(),( +=


Streamlined Model

Log volatility is the state variableEvolves as a mean-reverting Ornstein-Uhlenbeck process

Sttt

t dWdtS

dS σµ +=

ttt dWdtd νβσσασ +−= )ln(lnln


Euler Discretized Model

tss stiHttt ∆+= ∆− εσHviiHHHi βεσσρσσ +−+=+ )ln(lnlnln )1(

Where iH < t <= (i+1)Hεst and εvt are independent N[0,1]

innovations


Multifactor Models

HiHiHi )1(,2)1(,1)1( lnlnlnln +++ ++= σσσσ

HiiHHHi H )1(,11,1,1)1(,1 lnln ++ += νβσρσ

HiiHHHi H )1(,22,2,2)1(,2 lnln ++ += νβσρσ

Volatility component innovations v1 and v2 are independent N[0,1] variates


Applying Multifactor ModelsAlizadeh, Brandt, Diebold (2001)

Apply single and multifactor modelsUsing log rangeGBP, CAN$, DM, YEN, SFr

Single factor modelsPoor fitLong term autocorrelations in residualsUnable to account for long memory


Multifactor Models ResultsCURR lnσbar ρ1 β1 ρ2 β2

GBP -2.5 .98 .94 .19 5.14CAD -3.34 .98 1.2 .16 4.26DM -2.47 .97 1.23 .05 4.64YEN -2.53 .97 1.43 .15 5.68SFr -2.32 .97 1.05 .03 4.50


Multifactor EGARCH models

Brandt & Jones (2002)Multifactor log-range REGARCH modelsAllow for volatility asymmetryApplied to SP500 index Outperform single factor models and

multifactor models based on log returns


Conclusions on Multifactor ModelsVolatility model must explain two factors

Persistent volatility (autocorrelation)Transient Volatility (volatility of volatility)

Single factor models mis-specifySignificant gains to using log range

NormalityGreater efficiencyBetter at modeling the Vvol


Fokker-Planck ModelsAssume stochastic volatility model

Then (to leading order)

From regression, we can estimate

dWdtd )()( σβσασ +=

tδφσβδσ 222 )()( =[ ]( ) ( ) )ln()ln()(ln2)(ln 2 σδσβδσ batE +=+=

γνσσβ =)(


Estimating the Volatility DriftFokker-Planck equation

P(σ,t) is the pdf of σSteady state distribution p∞(σ,t)

Hence

)()(21 2

2

2

pptp α

σβ

σ ∂∂

−∂∂

=∂∂

)()(210 2

2

2

∞∞ ∂∂

−∂∂

= pp ασ

βσ

)(2

1)( 2∞

∞

= pdd

pβ

σσα


The Steady State Distribution

p∞ is approximately lognormal

Hence drift

22 ))/)(ln(2/1(

21 σσ

σπae

ap −

∞ =

⎟⎠⎞

⎜⎝⎛ −−= − )/ln(

21

21)( 2

122 σσγσνσα γ

a


Fokker-Planck SimulationFokker-Planck Volatility Model

0%

5%

10%

15%

20%

25%

30%


Multivariate Volatility Models

Relationships between volatility processesCointegration and fractional cointegration


Bi-Variate GARCH & FIGARCH

Bi-variate GARCH(1,1) Bollerslev (1990)Bi-variate FIGARCH, Brunetti & Gilbert (1998)

E.g. bi-variate FIGARCH (1,d,1)

(1)1)( 2

,2

,jj

jtjjjtjj L

βω

ελσ−

+=

1/22,

2,

2, ],[ tjjtiitij σσρσ =

)](]/[1)))(1([(1 LLL jjd

jjjjj βφλ −−−=


Multivariate FIGARCH

General Form

Where ∆ has diagonal elements (1-L)dj

tt LL νΒ∆Φ ))(()( 2 −Ι+= ωε


CointegrationGranger (1986) and Engle (1987)General idea:

Processes that “move together”Individually non-stationarySome (linear) function of them is stationary

ExampleSpot & futures pricesIndividually non-stationaryDifference (basis) is stationary


Cointegration –Formal Definition

Components of vector yt are said to be cointegrated of order (d, b) ifAll components of are integrated of order d > 0

Ldyt is stationary

There exists vector β = (β1, β2, . . . βν) such thatβ1y1t + β2y2t + . . . + βnynt is I(d-b)

b > 0

Vector β is called cointegrating vector


Cointegration ExamplesForward rates

Expectations theory Et[st+1] = ft

Error process εt+1 = st+1 – ftεt+1 must be a stationary process

Otherwise arbitrage

Even though st and ft are nonstationary I(1) processes

Currencies: Purchasing Power Parity Difference in real exchange rates must be stationary

Econometric models in generale.g. Money demand as linear function of prices, real income and interest rate


Example: CI(1,1,) System

Two random walk processesyt = µt + εyt

zt = µt + εzt

µt is random walk representing trend

Processes yt and zt are I(1) Cointegrated C(1,1) process because:

(yt - zt) = εit is stationary error process I(0) Cointegrating vector (1,-1)


Example: CI(1,1) SystemCI(1,1) Process

yt = µ t + ε yt

zt = µ t + εyt

µ t = µ t-1 + εt

-6.0

-4.0

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

0 5 10 15 20


Error Process is StationaryError Process yt - zt

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

1 6 11 16


Scatter Plot of System Variables

Scatter Plot of System Variables

y = 0.4179x - 0.524R2 = 0.4355

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0

Y(t)

Z(t)


Fractional Cointegration

Robinson & Marinucci (1989)Chueng & Lai (1993)Baillie & Bollerslev (1994)

Parent series may be fractionally integratedSub-process may also be fractionally cointegrated


Implications for Investment

Volatility processes fractionally cointegratedDivergences in volatilities less persistent than the volatilities themselves

Implication:Opportunities for statistical arbitrage between cointegrated volatility markets

Entails relatively low degree of risk


Investment Strategy

Volatility ModelsIdentify key factors underlying volatilityIdentify key stock volatility processes

Within a defined group, e.g. DOW 30

Stock SelectionIdentify stock baskets with cointegratedvolatility processes


Modeling Procedure

Estimate fractional order of vol processesUsing univariate FIGARCH models

Test hypothesis that fractional integration parameters are equalEstimate linear cointegrating vectorTest for fractional cointegration


Example: NYMEX - IPE


Brunetti & Gilbert (2000)Modeled variance as:

Absolute returnsSquared returns

Estimate ARFIMA models for two volatility processes

Find common fractional integration ~ 0.2Model difference in volatility processes

i.e cointegrating vector is (1,-1)Find it is cointegrated I(0)

IPE volatility reacts to shocks in NYMEX volatility more strongly than NYMEX reacts to IPE


Identifying Cointegrated Volatility Processes

Exploratory Multivariate AnalysisCluster AnalysisFactor AnalysisRegression Analysis

Fractional Cointegration AnalysisFit FIGARCH models to volatility processesTest for cointegrationEstimate cointegrating vector


Cluster AnalysisTree Diagram for Variables

Single LinkageEuclidean distances

13 14 15 16 17 18 19 20 21 22

Linkage Distance

INTCMSFT

HDHWP

JNJCT

SBCJPMAXPIBM

WMTGMBA

CATEK

MOMRK

AAMCDDIS

HONPGKO

UTXXOM

DDMMM

IPGE

DJIA

Primary grouping:Capital goods/


Factor ModelsPlot of Eigenvalues

Number of Eigenvalues0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Valu

e

“Raw Materials & Cap Goods”:XOM, DD, IP, AA, MMM, GE, CAT

“New Technology”:MSFT, INTC

“Finance & Technology:C, AXP, JPM, HWP, T

“Drugs & Consumer Goods”:MRK, JNJ, PG, MO, KO, MCD


Factor ModelsFactor Loadings, Factor 1 vs. Factor 2

Rotation: Varimax rawExtraction: Principal components

DJIA

AAAXP

BA

C CATDD

DIS

EK

GE

GMHD

HON

HWP

IBM

INTC

IP

JNJ

JPM

KO

MCD

MMM

MOMRK

MSFTPG

SBC

T

UTX

WMT

XOM

-0.2 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

Factor 1

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

Fact

or 2


Regression-Cointegration Models

Predicted vs. Observed ValuesDependent variable: DJIA

-2 -1 0 1 2 3 4 5 6

Predicted Values

-3

-2

-1

0

1

2

3

4

5

6

Obs

erve

d Va

lues

95% confidence

DJIADD GE IP MMMMRK UTX XOM

R2 = 78%


Volatility Portfolio ConstructionVolatility modeling & forecasting

FIGARCH models For cointegrated volatility processes

Portfolio optimizationRisk adjusted returnMarket neutrality & other constraints

HedgingPlatinum hedge


Summary

Key theoretical conceptsVolatility measuresLong MemoryFIGARCH & Multifactor modelsVolatility cointegration


ReferencesAndersen, Bollerslev, Diebold, Labys (2001)

Modeling and forecasting realized volatility, 2001Realized volatility & correlation, 1999

Lien & Yiu Kuen Tse (1999)Forecasting the Nikkei Spot Index with fractional cointegration, Journal of Economterics (1999)

Bollerslev, Mikkelsen (1996)Modeling & pricing long memory in stock market volatility, Journal of Economterics

Lamoureaux & Lastrapes (1990)Persistence in variance, structural change and the GARCH modelJournal of Business and Economic Statistics 8 pp225-235


ReferencesBrunetti & Gilbert (2000)

Bivariate FIGARCH and fractional cointegration, Journal of Empirical Finance 7 pp509-530

Grau-Carles (2000)Empirical evidence of long-range correlations in stock returns, Physica A 287 pp396-404

Andersen & Bollerslev (1997)Heterogeneous information arrivals and return volatility dynamics, Journal of Finance 52

risk management > volatility

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