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Volatility Modeling

LS 2011

Lucia Jarešová

Econometrics and Operational Research

Charles UniversityFaculty of Mathematics and PhysicsPrague, Czech Republic

28th March 2011

VolatilityModeling

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Market DataDataHistoricalVolatilityImplied VolatilityGARCH

EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility

LeverageEffectExtensions ofGARCH

Literature

Outline

1 Market DataDataHistorical VolatilityImplied VolatilityGARCH

2 EWMA EstimatorsEWMAHistorical EstimatorsStochastic Volatility ModelsForecasting Volatility

3 Leverage EffectExtensions of GARCH

4 Literature2/68 ,

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Market Data

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Market Data

Market Volatility

. . . is probably nowadays one of the most investigatedphenomenon in finance.

Reasons:

1 Still increasing amount and types of traded derivatives.

2 Growing interest in improving risk management.

3 Attempt to measure current risk and to predict future risk.

Consequences:

+/− Enormous collection of literature.

− Hard to find out new results.

+ More quoted prices, new types of data, new types oftraded derivatives.

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Market Data Data

Basic Definitions - Returns

St price process

pt logarithmic price process, pt := log(St)

continuously compounded return over [s, t], 0 ≤ s ≤ t

r(s, t) := pt − ps = log(St/Ss).

discrete time return (log-returns), h > 0 is a given time lag

r (h)n := r ((n − 1)h, nh) , n ∈ N.

When only one time lag is used, index (h) will be omitted.

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Market Data Data

Empirical Properties of Log-returns

Leptokurtic DistributionKurtosis is mostly much greater than 3.⇒ Nonnormality.

Typical Shape of the DistributionHighly peaked and fat-tailed distributions.⇒ Characteristics of mixtures distributions with different

variances.

Volatility Clustering

⇒ Volatility (or variance) is autocorrelated.

Leverage Effect

⇒ Negative returns have a higher effect on volatility thanpositive returns (volatility increases more after bad news).

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Market Data Data

Figure: Market Indices - St7/68 ,

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Market Data Data

Figure: Market Indices - Log-Values - pt8/68 ,

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Market Data Data

Figure: Market Indices - Log-Returns - r (1/260)n

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Market Data Data

−0.2 −0.1 0.0 0.1

United States

82 yearskurt = 23.4

−0.15 −0.05 0.00 0.05 0.10

Germany

50 yearskurt = 11.9

−0.15 −0.05 0.05

39 yearskurt = 14.5

−0.15 −0.05 0.05 0.10

Czech Republic

15 yearskurt = 16.5

Figure: Market Indices - Density of Log-Returns10/68 ,

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Market Data Data

Figure: Market Indices - QQ Plots of Log-Returns11/68 ,

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Market Data Data

Volatility

The word volatility means rapid or unexpected changes.

In finance:

Volatility = measure of the dispersion of returns

. . . refers to the amount of uncertainty or risk about the size ofchanges in a security’s value

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Market Data Historical Volatility

We observe the price process St and gather following daily datafor i = 0, . . . ,N.

Ci closing price

Oi opening price

Li daily low price

Hi daily high price

σi volatility (usually implied)

The daily log-returns r1, . . . , rN are computed from the closingprices as

ri = r 1/260i = log(Ci/Ci−1).

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Historical Close-to-Close Volatility

. . . is equal to standard deviation scaled to one year

r̄ =1N

N∑i=1

σcc =

√√√√ 260N − 1

N∑i=1

(ri − r̄)2

Sometimes (when drift is small) is used

σcc =

√√√√ 260N − 1

N∑i=1

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Historical High-Low Volatility:Parkinson

√√√√ 2604N log(2)

N∑i=1

• 5 times more efficient than the close-to-close estimate

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Historical Open-High-Low-Close Volatility:Garman and Klass

σgk =

√√√√260N

N∑i=1

− (2 log 2− 1)

• assumes Brownian motion with zero drift• assumes no opening jumps (opening = previous close)• 7.4 times more efficient than the close-to-close estimate

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Historical Open-High-Low-Close Volatility:Garman and Klass (Yang Zhang extension)

σgkyz =

√√√√√260N

N∑i=1

(log OiCi−1

(log HiLi

−(2 log 2− 1)(

log CiOi

• currently the preferred version of OHLC volatility estimator• assumes Brownian motion with zero drift• allows for opening jumps• 8 times more efficient than the close-to-close estimate

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Historical Open-High-Low-Close Volatility:Rogers Satchell

σrs =

√√√√260N

N∑i=1

logHiOi

+ logLiCi

logLiOi

• allows for nonzero drift• assumes no opening jumps

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Market Data Implied Volatility

Geometric Brownian Motion

GBM is a basic (very simple) model for asset prices S .

dSt = µStdt + σStdWt ,

where drift µ is the expected return on the asset, volatility σmeasures the variability around µ and W is the standardBrownian motion.

Properties:

• The logarithm of the ending price is distributed as

log(ST ) = log(St) + (µ− σ2/2)τ + σ√τε,

i.e. ST given St is lognormally distributed.• Log-returns are normally distributed.

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Implied Volatility

Assumptions of BS Model:• S follows a GBM, Perfect financial markets.• Asset yield: known and constant.• Volatility: known and constant.

price(European Option) = function(S , K , σ, T − t)

observed market pricesof options⇓

Implied volatility⇓

Volatility surface20/68 ,

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Volatility (Fear) Indices

Utilizes a wide variety of strike prices of options to obtain oneindicator of market volatility.

USA: Index VIX

• CBOE Chicago Board Options Exchange

• Launched in 2003

• Data begins 1990

Germany: Index VDAX

• Deutsche Börse AG

• Launched in 2005.

• Data begins 1992.21/68 ,

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Volatility in USA vs Volatility in Germany

USA − VIX index

1992 1994 1996 1998 2000 2002 2004 2006 2008

Germany − VDAX index

1992 1994 1996 1998 2000 2002 2004 2006 2008

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Market Data GARCH

Generalised AutoRegressive ConditionalHeteroscedasticity

ARCH models

• Robert F. Engle (1982)• 2003 shared a Nobel price with Clive Granger

GARCH models

• Tim Bollerslev (1986), doctoral student of R. Engle

and later NGARCH, NAGARCH, IGARCH, EGARCH, GARCH-M,QGARCH, GJR-GARCH, TGARCH, APARCH, FIGARCH, FIEGARCH,FIAPARCH, FCKARCH, HYGARCH, fGARCH,. . .look into Bollerslev Glossary of ARCH models (more then 100)

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Market Data GARCH

Definition 1Proces {rn}, n ∈ Z is (strong) GARCH(p, q), if

i. E[rn|Fn−1] = 0 conditional mean ⇒ unpredictable

ii. Var[rn|Fn−1] = σ2t conditional variance

iii. εn = rn/σt is i.i.d.

σ2n = ω +

q∑i=1

αi r 2n−i +

p∑j=1

βjσ2n−j .

Unconditional varianceσ2 = ω/(1−

∑αi −

∑βj)

persistence =∑αi +

∑βj (< 1 if σ2 exists)

Financial data: usually high persistence (near 1)24/68 ,

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Market Data GARCH

Kurtosis of GARCH(1, 1)

σ2n = ω + αr 2

n−1 + βσ2n−1; εn = rn/σn ∼ N(0, 1)

Kurt(rn) =E[r 4n ]

(E[r 2n ])2 = 3 +

1− β2 − 2αβ − 3α2 ≥ 3

⇒ leptokurtic distribution

0.40.6

0.81.0

0.60.8

Kurtosis

0.0 0.2 0.4 0.6 0.8 1.0

15 20 25

finite variancefinite kurtosis

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Market Data GARCH

Estimation of GARCH(1, 1)

Estimation was done in R using 5 years history till 23.4.2009.Index ω̂ α̂ β̂ Pers. Vol. p.a. Kurt(rt) Kurt(Zt)SPX 0.000 0.076 0.916 0.992 18% 15.129 4.633

(USA) *** *** ***DAX 0.000 0.102 0.884 0.987 22% 13.675 4.219

(Germany) ** *** ***

1992 1994 1996 1998 2000 2002 2004 2006 2008

SPX Index and volatility VIX Index

1992 1994 1996 1998 2000 2002 2004 2006 2008

DAX Index and volatility VDAX Index

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Market Data GARCH

Closer to Normality: res. = rt/σt

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Market Data GARCH

Performance of GARCH(1,1) in ForecastingVolatility (2009)

GARCH(1,1) is estimated using 3Y history or returns. Modelswere estimated for t equal to dates from 3.5.2002 to 24.4.2009.

In the table are correlation squared (R2) of the modelestimators with the actual values of volatility index.

Indexprevious

indexvalue

GARCHGARCHforecast

3M hist.volatility

SPX 97.47% 88.43% 89.21% 84.24%DAX 98.23% 86.71% 87.44% 83.36%

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EWMA Estimators

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EWMA Estimators EWMA

Exponentially Weighted Moving Average

GARCH(1,1):σ2n = ω + αr 2

n−1 + βσ2n−1

σ2n = (1− λ)

∞∑j=1

λj−1(rn−j − r̄)2 = (1− λ)(rn−1 − r̄)2 + λσ2n−1

σ2n = (1− λ)

∞∑j=1

λj−1r 2n−j = (1− λ)r 2

n−1 + λσ2n−1

where λ is a decay factor. Typical values of the decay factor areclose to one (0.93 - 0.98).

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EWMA Estimators EWMA

EWMA estimator

σE ,n =

√√√√260(1− λ)

1− λNN∑i=1

λi−1(rn−i − r̄)2

σE ,n =

√√√√260(1− λ)

1− λNN∑i=1

λi−1r 2n−i

n is a time indexN is the number of used observations

The EWMA estimator is approaching the historical volatilityestimator for λ→ 1.

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EWMA Estimators Historical Estimators

Explanatory Variables in HistoricalEstimators

A = {log(Ci/Ci−1)2} . . . close-to-close change squared

B = {log(Hi/Li )2} . . . extreme daily change squared

C = {log(Oi/Ci−1)2} . . . opening jump squared

D = {log(Ci/Oi )2} . . . trading daily change squared

E = {log(Hi/Ci ) log(Hi/Oi )} . . . RS part 1

F = {log(Li/Ci ) log(Li/Oi )} . . . RS part 2

Explained VariableY = {σ2

i /260} . . . actual one day variance

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Correlation Matrix

SPXvolatility VIX Y A B C D E FY 1.00 0.53 0.71 0.13 0.60 0.52 0.45A 0.53 1.00 0.81 0.04 0.67 0.19 0.20B 0.71 0.81 1.00 0.11 0.70 0.49 0.66C 0.13 0.04 0.11 1.00 0.08 0.01 0.05D 0.60 0.67 0.70 0.08 1.00 0.47 0.24E 0.52 0.19 0.49 0.01 0.47 1.00 0.19F 0.45 0.20 0.66 0.05 0.24 0.19 1.00

DAXvolatility VDAX Y A B C D E FY 1.00 0.48 0.68 0.23 0.55 0.51 0.45A 0.48 1.00 0.74 0.17 0.66 0.22 0.24B 0.68 0.74 1.00 0.26 0.66 0.61 0.66C 0.23 0.17 0.26 1.00 0.31 0.24 0.19D 0.55 0.66 0.66 0.31 1.00 0.40 0.28E 0.51 0.22 0.61 0.24 0.40 1.00 0.23F 0.45 0.24 0.66 0.19 0.28 0.23 1.00

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Cross-Correlation FunctionsDashed line is the approximation of cross-correlation by fuctionf (h) = kλh, where h is the time shift. Estimated functions are verysimilar, estimated parameter λ is alwas equal to 0.99 (after roundingto 2 decimal places).

−250 −200 −150 −100 −50 0

time shift (days)

ABCDEFY

−250 −200 −150 −100 −50 0

time shift (days)

ABCDEFY

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EWMA Estimators Stochastic Volatility Models

Stochastic Volatility Models

Definition 2Process {rn, n ∈ N}, is a discrete time stochastic volatility (SV)model, if

rn = σnεn

where {σn, n ∈ Z} is a non-negative volatility process and{εn, n ∈ Z} is a noise sequence of i.i.d. random variablesindependent of the process {σn, n ∈ Z}.

• SV model defines the distribution of returns rn indirectly,via the structure of the model

• GARCH model specifies the distribution directly via thepast values of returns

• ⇒ the GARCH model does not belong to stochasticvolatility models

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Lemma 1

• ε, X and Y are random variables• ε is independent of X and Y

ThenCov(X , εY ) = E[ε]Cov(X ,Y ).

Proof:E[εXY ] = E[ε]E[XY ]E[εY ] = E[ε]E[Y ]

Cov(X , εY ) = E[X εY ]− E[X ]E[εY ] == E[ε]E[XY ]− E[X ]E[ε]E[Y ] == E[ε] (E[XY ]− E[X ]E[Y ]) == E[ε]Cov(X ,Y )

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Auto- and Cross- Correlation Functions

Modelr 2t = Ytε2

t εt ∼ N(0, 1) Yt = σ2

where Yt is independent of {εt} and stationary• mean E[Yt ] = M and variance Var[Yt ] = V• ρ(i) := Cor(Yt ,Yt−i ), i = 0, 1, 2, . . .

Then1 Cov(Yt , r 2

t−i ) = Cov(Yt ,Yt−iε2t−i ) =

= E[ε2t−i ]Cov(Yt ,Yt−i ) = V ρ(i)

2 E[r 4t ] = E[Y 2

t ε4t ] = E[ε4

t ]E[Y 2t ] = 3(V + M2)

3 E[r 2t ] = E[Ytε2

t ] = E[ε2t ]E[Yt ] = M

4 Var(r 2t ) = E[r 4

t ]− (E[r 2t ])2 = 3V + 2M2

and the cross correlation between {Yt} and {r 2t } is

Cor(Yt , r 2t−i ) =

V ρ(i)3V + 2M2 ∝ ρ(i)

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Stochastic Volatility Models

SV models:

• SV model are close to continuous time models, where it ismore convenient to model the volatility or variance of assetprices as separate process independent of the past returns.

• Unfortunately the likelihood function of these models is ingeneral not directly available and consequently theestimation of these models is much more difficult.

• I follow the notation of [3]. According to this notation, theGARCH model does not belong to stochastic volatilitymodels.

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EWMA Estimators Forecasting Volatility

2010 article in AUC

http://auc.karlin.mff.cuni.cz/

L. JarešováEWMA Historical Volatility EstimatorsAUC, Vol.51-2, pages 17–282010

currently in press39/68 ,

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Declining Amount of Information

−250 −200 −150 −100 −50 0

time shift (days)

YABCDEFY

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New EWMA-Style Estimators

Estimator of the form

σ2 =260N

N∑i=1

(. . .t−i )2

is changed to an EWMA-style estimator by weights wi

σ2EWMA = 260

N∑i=1

wi (. . .t−i )2

(1−λ)λi−1

1−λN λ ∈ (0, 1)

1N λ = 1

It is clear that∑Ni=1 wi = 1 and that EWMA style estimator

converges to classical estimator for λ→ 1.41/68 ,

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EWMA weights

−120 −80 −60 −40 −20 0

time lag (days)

equal weightslambda = 0.93lambda = 0.94lambda = 0.95lambda = 0.96lambda = 0.97lambda = 0.98

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Empirical Study

• The historical estimators introduces before are changed toEWMA style estimators.

• The used decay factor is λ = 0.96.

Description of estimators (λ is a decay factor):GARCH . . . GARCH forecast from the last year empirical study(was the best performing)EWMA(λ) . . . EWMA estimatorPARK(λ) . . . EWMA-style Parkinson estimatorGK(λ) . . . EWMA-style Garman Klass estimatorGKYZ(λ) . . . EWMA-style Garman Klass(Yang Zhang)estimatorλRS(λ) . . . EWMA-style Rogers Satchell estimator

Note: estimator with decay factor 1 is equal to classical historical estimator.43/68 ,

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Definition of Performance Measures

Models were estimated for t equal to dates from 12.4.2002 to2.4.2010.Estimator with λ = 0.96 are based on 2Y history andestimators with λ = 1 are based on 3M history.

y = (y1, . . . , yn) . . . volatility indexx = (y1, . . . , yn) . . . estimator value

R2 = Cor(x, y)2

meanA = mean(x− y)sdA =

√Var(x− y)

MSEA =√

mean((x− y)2)

meanR = mean(x/y)sdR =

√Var(x/y − 1)

MSER =√

mean((x/y − 1)2)

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Performance - SPX Index

R2 meanA sdA MSEA meanR sdR MSERGARCH 87.8% -3.0% 4.3% 5.2% -16.5% 13.7% 21.4%

EWMA(1) 83.6% -2.8% 5.0% 5.7% -15.8% 17.5% 23.6%EWMA(0.96) 91.0% -2.8% 3.8% 4.7% -16.6% 14.7% 22.2%

PARK(1) 83.3% -5.6% 4.4% 7.1% -27.4 % 13.4% 30.5%PARK(0.96) 91.4% -5.6% 3.2% 6.5% -28.0% 10.7% 29.9%

GK(1) 83.4% 0.4% 5.7% 5.7% 0.1% 18.8% 18.8%GK(0.96) 91.4% 0.4% 4.5% 4.5% -0.8% 15.1% 15.1%GKYZ(1) 83.5% 0.7% 5.8% 5.8% 1.0% 19.0% 19.1%

GKYZ(0.96) 91.5% 0.6% 4.6% 4.7% 0.2% 15.3% 15.3%RS(1) 82.5% -7.0% 4.7% 8.4% -33.2% 12.1% 35.3%

RS(0.96) 90.7% -7.0% 3.7% 7.9% -33.7% 9.8% 35.1%index yest. 97.4% 0.0% 1.8% 1.8% 0.2% 5.9% 5.9%

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Performance - DAX Index

R2 meanA sdA MSEA meanR sdR MSERGARCH 86.8% -0.7% 5.1% 5.1% -5.3% 16.0% 16.8%

EWMA(1) 83.4% -0.5% 5.3% 5.3% -4.1% 18.2% 18.7%EWMA(0.96) 91.0% -0.5% 4.2% 4.2% -5.0% 14.7% 15.5%

PARK(1) 84.4% -3.7% 4.5% 5.8% -18.2% 15.8% 24.1%PARK(0.96) 91.4% -3.8% 3.4% 5.0% -18.9% 13.2% 23.0%

GK(1) 84.1% 3.4% 7.1% 7.8% 10.4% 21.5% 23.9%GK(0.96) 91.2% 3.3% 6.2% 7.0% 9.5% 18.1% 20.4%GKYZ(1) 84.0% 4.4% 7.4% 8.6% 15.1% 21.7% 26.4%

GKYZ(0.96) 91.3% 4.4% 6.5% 7.8% 14.2% 17.8% 22.7%RS(1) 84.9% -4.4% 4.3% 6.2% -21.0% 15.2% 25.9%

RS(0.96) 91.2% -4.5% 3.3% 5.5% -21.6% 13.0% 25.2%index yest. 98.2% 0.0% 1.5% 1.5% 0.1% 4.7% 4.7%

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Volatility Forecasting

0 500 1000 1500 2000

SPX − Volatility index

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0 500 1000 1500 2000

yabs_S50 −0.0298abs_SM −0.0304abs_SD 0.0427abs_MSE 0.0524

rel_S50 −0.1803rel_SM −0.1648rel_SD 0.1369rel_MSE 0.2142

R 0.9368R^2 0.8775

volest

SPX − Forecast of GARCH volatility tomorrow

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0 500 1000 1500 2000

R 0.9537R^2 0.9095

volest

SPX − EWMA volatility (0.96)

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0 500 1000 1500 2000

R 0.9562R^2 0.9143

volest

SPX − EWMA Park volatility (0.96)

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0 500 1000 1500 2000

yabs_S50 −0.0054abs_SM 0.004abs_SD 0.045abs_MSE 0.0452

R 0.956R^2 0.9139

volest

SPX − EWMA GK volatility (0.96)

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0 500 1000 1500 2000

yabs_S50 −0.0042abs_SM 0.0064abs_SD 0.0461abs_MSE 0.0465

rel_S50 −0.0263rel_SM 0.002rel_SD 0.1527rel_MSE 0.1526

R 0.9563R^2 0.9145

volest

SPX − EWMA GKYZ volatility (0.96)

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0 500 1000 1500 2000

R 0.9523R^2 0.9068

volest

SPX − EWMA RS volatility (0.96)

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0 500 1000 1500 2000

yabs_S50 4e−04abs_SM 0abs_SD 0.0175abs_MSE 0.0175

rel_S50 0.0027rel_SM 0.0018rel_SD 0.0592rel_MSE 0.0592

R 0.9868R^2 0.9737

volest

SPX − Volatility index yesterday

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Leverage Effect

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Leverage Effect Extensions of GARCH

Leverage Effect

Standard GARCH models assume that positive and negativeerror terms have a symmmetric effect on volatility.

According to Black(1976)”a drop in the value of the firm will cause a negative return onits stock, and will ussually increase the leverage of thestock.[. . . ]That rise in the debt-equity ratio will surely mean a rise in thevolatility of the stock.”

Empirical fingings: volatility reacts asymmetrically to the signof the shocks

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Exponential GARCH Model

Nelson(1991) suggested following EGARCH(p,q) model forconditional variance σ2

n, rn = σnεn

log(σ2n) = α0 +

q∑j=1

αjg(εn−j) +

p∑i=1

βi log(σ2n−i ),

where αj , βi are dereministic coefficients, εn ∼ (0, σ2) and

g(X ) = θX︸︷︷︸sign effect

+ γ(|X | − E|X |)︸︷︷︸size effect

, typicallyθ < 0, γ > 0.

It holds that E[g(X )] = 0 (for X ∼ (0, σ2)).

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Properties of E-GARCH

• Conditional variance σ2t is an explicit multiplicative

function of lagged innovations• Volatility can react assymetrically to the good and the bad

news.• No parameter restriction.

As in ARMA processes we can express EGARCH in the form

log(σ2n) = c0 +

∞∑k=1

ckg(εn−k).

If c0 = 0 and∑∞k=1 c2

k <∞, then σ2t is strictly starionary and

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Threshold GARCH Models

Idea: different multiplicative coefficient for positive andnegative innovations.

TGARCH(p,q) model for conditional variance σ2n, rn = σnεn

σ2n = ω

p∑i=1

βiσ2n−i +

q∑j=1

αj r 2n−j +

q∑j=1

α−j r 2n−j I[rn−j<0],

with the indicator function I(·) and αj , α−j , βi are dereministiccoefficients.

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Estimation Results for the DAX returns

The example is taken from the book Statistics of FinancialMarkets (J.Franke, W.Härdle, C.Hafner), page 256

Basic modelrt = µt + σtεt

Mean value is either modelled by AR(1)

µt = ν + φrt−1

or by ARCH-M (ARCH in Mean)

µt = ν + λσt

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Estimation Results for the DAX returns 2

Volatility σt is modelled as• GARCH

(ω, α(r), β(σ)

)• TGARCH

(ω, α(r), α−(r), β(σ)

)• EGARCH

(ω, β(g(ε)), γ(size), θ(sign)

)Estimation results of models applied to DAX index returns1974-1996 are on the following slide. Parenthesis show thet-statistic based on the QML asymptotic standard error.

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Estimation Results for the DAX returns 3

rt = µt + σtεtAR µt = ν + φrt−1, ARCH-M µt = ν + λσt

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Comments and Future

It the previous example is the leverage effect on the border ofsignifficance (absolute value of t-statistic is with one exceptionin all cases less than 2).

My future direction:leverage effect of slightly different type that is highlysignifficant and intuitively used in practice, even if thereasoning is sometimes questionable . . .

See next year :-)

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T.G.Andersen, T.G.Bollerslev and F.X.DieboldParametric and nonparametric volatility measurement, inHandbook of Financial Econometrics: Volume 1 - Toolsand TechniquesNorth-Holland2010

A.M.LindnerContinuous time approximation to GARCH and stochasticvolatility models, in Handbook of Financial Time SeriesSpringer2009

N.Shephard and T.G.AndersenStochastic volatility: Origins and overview, in Handbook ofFinancial Time SeriesSpringer2009

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J. GatheralThe Volatility SurfaceJohn Wiley and Sons2006

P. WilmottPaul Wilmott on Quantitative FinanceJohn Wiley and Sons2000

P. JorionFinancial Risk Manager HandbookJohn Wiley and Sons2007

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R. EngleNew Frontiers for ARCH ModelsJ. Appl. Econ. 17: 425-4462002

T. BollerslevGlossary to ARCH (GARCH)CREATES Research Paper2008

R. S. TsayAnalysis of Financial Time SeriesJohn Wiley and Sons2002

J. Franke, W. K. Härdle, CH. M. HafnerStatistics of Financial MarketsSpringer2007

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R. RebonatoVolatility and CorrelationJohn Wiley and Sons2004

F. BlackStudies in stock price volatility changesProceedings of the 1976 Meeting of the Business andEconomic Statistics Section, American StatisticalAssociation1976

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