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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
Copyright 2001-2006 Investment Analytics Volatility Slide: 3
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 4
Thick Tails, Non-Normal DistributionMandelbrot (1963), Fama (1963, 1965)
Skewness = -0.6
Kurtosis = 5.7
Copyright 2001-2006 Investment Analytics Volatility Slide: 5
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)
Copyright 2001-2006 Investment Analytics Volatility Slide: 6
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 7
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
Jan-
58
Jan-
60
Jan-
62
Jan-
64
Jan-
66
Jan-
68
Jan-
70
Jan-
72
Jan-
74
Jan-
76
Jan-
78
Jan-
80
Jan-
82
Jan-
84
Jan-
86
Jan-
88
Jan-
90
Jan-
92
Jan-
94
Jan-
96
Jan-
98
Jan-
00
Copyright 2001-2006 Investment Analytics Volatility Slide: 8
The Volatility ConeV
olat
ility
(%)
Maximum
Minimum
Average
Days0 600 3
Copyright 2001-2006 Investment Analytics Volatility Slide: 9
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)
Copyright 2001-2006 Investment Analytics Volatility Slide: 10
Volatility Seasonality
DOW Volatility Seasonality
40%
60%
80%
100%
120%
140%
160%
JAN FEB MAR APR MAY JUN JUL AUG SEP OCT NOV DEC
Copyright 2001-2006 Investment Analytics Volatility Slide: 11
Volatility Correlation
Volatilities tend to change togetherStocks: Black (1976)FX: Diebold & Nerlove (1989)
Also across marketsStock & bond volatilities move together (Schwert, 1989)
Copyright 2001-2006 Investment Analytics Volatility Slide: 12
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 13
Asset Characteristics –Conclusions
The Bad Newsiid Gaussian model inappropriate
The Good NewsCorrelation suggests few common factors may explain variationVolatility models (GARCH, etc.)
Copyright 2001-2006 Investment Analytics Volatility Slide: 14
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 15
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 16
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 17
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)
Copyright 2001-2006 Investment Analytics Volatility Slide: 18
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 19
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%
Copyright 2001-2006 Investment Analytics Volatility Slide: 20
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.
Copyright 2001-2006 Investment Analytics Volatility Slide: 21
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)
Copyright 2001-2006 Investment Analytics Volatility Slide: 22
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 23
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)
Copyright 2001-2006 Investment Analytics Volatility Slide: 24
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 25
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 26
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 27
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.
Copyright 2001-2006 Investment Analytics Volatility Slide: 28
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 29
Volatility Models
Key volatility characteristicsLong memoryVolatility of volatility
Univariate modelsMultivariate models
Copyright 2001-2006 Investment Analytics Volatility Slide: 30
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 31
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 32
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 33
Stationary Series
Copyright 2001-2006 Investment Analytics Volatility Slide: 34
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 += ∑
−
=
Copyright 2001-2006 Investment Analytics Volatility Slide: 35
Random Walk ProcessRandom Walk with drift
yt = a0 + a1yt -1 + εtWith a1 = 1A non-stationary process
Copyright 2001-2006 Investment Analytics Volatility Slide: 36
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 =⎥⎦
⎤⎢⎣
⎡+= ∑ ∑
≠
Copyright 2001-2006 Investment Analytics Volatility Slide: 37
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)
Copyright 2001-2006 Investment Analytics Volatility Slide: 38
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 39
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 40
Volatility Long Memory
Volatility is highly persistentEvents have sustained influence on future volatilityIn principle, process is very forecastable
Copyright 2001-2006 Investment Analytics Volatility Slide: 41
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 42
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 43
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 44
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 45
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 46
White Noise ProcessFractal Random Walk
-140
-120
-100
-80
-60
-40
-20
0
20
H = 0.5
Copyright 2001-2006 Investment Analytics Volatility Slide: 47
Black Noise ProcessFractal Random Walk
-600
-500
-400
-300
-200
-100
0
100
H = 0.9Smoother seriesTrend
Copyright 2001-2006 Investment Analytics Volatility Slide: 48
Pink Noise ProcessFractal Random Walk
-35
-30
-25
-20
-15
-10
-5
0
5
10
15
H = 0.1More volatileAntipersistent
Mean reverting
Copyright 2001-2006 Investment Analytics Volatility Slide: 49
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(
)(
Copyright 2001-2006 Investment Analytics Volatility Slide: 50
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 51
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)/(
Copyright 2001-2006 Investment Analytics Volatility Slide: 52
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)
Copyright 2001-2006 Investment Analytics Volatility Slide: 53
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 54
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 55
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 56
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 57
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)()(
Copyright 2001-2006 Investment Analytics Volatility Slide: 58
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)(
Copyright 2001-2006 Investment Analytics Volatility Slide: 59
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 60
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 61
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 62
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 −+×
−−+−
=−
Copyright 2001-2006 Investment Analytics Volatility Slide: 63
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 64
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 65
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 66
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 67
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 68
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 69
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 70
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 71
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 72
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 73
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 −=
Copyright 2001-2006 Investment Analytics Volatility Slide: 74
GARCH vs FIGARCH
GARCHShocks to variance process die away at fast exponential rate
FIGARCHShocks die away much more slowly (hypergeometeric)Has “long memory”
Copyright 2001-2006 Investment Analytics Volatility Slide: 75
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
Copyright 2001-2006 Investment Analytics Volatility Slide: 76
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
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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
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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
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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
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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
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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 ),(),( +=
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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
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Euler Discretized Model
tss stiHttt ∆+= ∆− εσHviiHHHi βεσσρσσ +−+=+ )ln(lnlnln )1(
Where iH < t <= (i+1)Hεst and εvt are independent N[0,1]
innovations
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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
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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
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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
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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
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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
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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 =)(
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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
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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
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Fokker-Planck SimulationFokker-Planck Volatility Model
0%
5%
10%
15%
20%
25%
30%
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Multivariate Volatility Models
Relationships between volatility processesCointegration and fractional cointegration
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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 −−−=
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Multivariate FIGARCH
General Form
Where ∆ has diagonal elements (1-L)dj
tt LL νΒ∆Φ ))(()( 2 −Ι+= ωε
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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
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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
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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
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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)
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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
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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
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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)
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Fractional Cointegration
Robinson & Marinucci (1989)Chueng & Lai (1993)Baillie & Bollerslev (1994)
Parent series may be fractionally integratedSub-process may also be fractionally cointegrated
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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
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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
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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
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Example: NYMEX - IPE
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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
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Identifying Cointegrated Volatility Processes
Exploratory Multivariate AnalysisCluster AnalysisFactor AnalysisRegression Analysis
Fractional Cointegration AnalysisFit FIGARCH models to volatility processesTest for cointegrationEstimate cointegrating vector
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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/
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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
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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
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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%
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Volatility Portfolio ConstructionVolatility modeling & forecasting
FIGARCH models For cointegrated volatility processes
Portfolio optimizationRisk adjusted returnMarket neutrality & other constraints
HedgingPlatinum hedge
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Summary
Key theoretical conceptsVolatility measuresLong MemoryFIGARCH & Multifactor modelsVolatility cointegration
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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
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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