Upload
others
View
3
Download
3
Embed Size (px)
Citation preview
Volatility Modeling
LS 2011
Lucia Jarešová
Econometrics and Operational Research
Charles UniversityFaculty of Mathematics and PhysicsPrague, Czech Republic
28th March 2011
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Outline
Outline
1 Market DataDataHistorical VolatilityImplied VolatilityGARCH
2 EWMA EstimatorsEWMAHistorical EstimatorsStochastic Volatility ModelsForecasting Volatility
3 Leverage EffectExtensions of GARCH
4 Literature2/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data
Market Data
3/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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.
4/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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.
5/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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).
6/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Data
Figure: Market Indices - St7/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Data
Figure: Market Indices - Log-Values - pt8/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Data
Figure: Market Indices - Log-Returns - r (1/260)n
9/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Data
−0.2 −0.1 0.0 0.1
010
3050
70
United States
SPX
82 yearskurt = 23.4
−0.15 −0.05 0.00 0.05 0.10
010
2030
4050
Germany
DAX
50 yearskurt = 11.9
−0.15 −0.05 0.05
010
2030
4050
60
Japan
NKY
39 yearskurt = 14.5
−0.15 −0.05 0.05 0.10
010
2030
4050
Czech Republic
PX
15 yearskurt = 16.5
Figure: Market Indices - Density of Log-Returns10/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Data
Figure: Market Indices - QQ Plots of Log-Returns11/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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
12/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Historical Volatility
Data
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).
13/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Historical Volatility
Historical Close-to-Close Volatility
. . . is equal to standard deviation scaled to one year
r̄ =1N
N∑i=1
ri
σcc =
√√√√ 260N − 1
N∑i=1
(ri − r̄)2
Sometimes (when drift is small) is used
σcc =
√√√√ 260N − 1
N∑i=1
r 2i
14/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Historical Volatility
Historical High-Low Volatility:Parkinson
σp =
√√√√ 2604N log(2)
N∑i=1
(log
HiLi
)2
• 5 times more efficient than the close-to-close estimate
15/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Historical Volatility
Historical Open-High-Low-Close Volatility:Garman and Klass
σgk =
√√√√260N
N∑i=1
[12
(log
HiLi
)2
− (2 log 2− 1)
(log
CiOi
)2]
• assumes Brownian motion with zero drift• assumes no opening jumps (opening = previous close)• 7.4 times more efficient than the close-to-close estimate
16/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Historical Volatility
Historical Open-High-Low-Close Volatility:Garman and Klass (Yang Zhang extension)
σgkyz =
√√√√√260N
N∑i=1
(log OiCi−1
)2+ 1
2
(log HiLi
)2−
−(2 log 2− 1)(
log CiOi
)2
• 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
17/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Historical Volatility
Historical Open-High-Low-Close Volatility:Rogers Satchell
σrs =
√√√√260N
N∑i=1
[log
HiCi
logHiOi
+ logLiCi
logLiOi
]
• allows for nonzero drift• assumes no opening jumps
18/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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.
19/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Implied Volatility
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 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Implied Volatility
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 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data Implied Volatility
Volatility in USA vs Volatility in Germany
USA − VIX index
Time
1992 1994 1996 1998 2000 2002 2004 2006 2008
020
4060
80
Germany − VDAX index
Time
1992 1994 1996 1998 2000 2002 2004 2006 2008
020
4060
80
22/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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)
23/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data GARCH
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.
where
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 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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 +
6α2
1− β2 − 2αβ − 3α2 ≥ 3
⇒ leptokurtic distribution
alpha
0.0
0.2
0.40.6
0.81.0
beta
0.0
0.2
0.4
0.60.8
1.0
Kurtosis
0
2
4
6
8
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
alpha
beta
5 10
15 20 25
30 30
35 40
45
45 50
55
finite variancefinite kurtosis
25/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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) ** *** ***
−0.
10−
0.05
0.00
0.05
0.10
SP
X r
etur
ns
1020
3040
5060
7080
VIX
inde
x
2040
6080
1992 1994 1996 1998 2000 2002 2004 2006 2008
GA
RC
H v
olat
ility
Time
SPX Index and volatility VIX Index
−0.
050.
000.
050.
10
DA
X r
etur
ns
1020
3040
5060
70
VD
AX
inde
x
1020
3040
5060
7080
1992 1994 1996 1998 2000 2002 2004 2006 2008
GA
RC
H v
olat
ility
Time
DAX Index and volatility VDAX Index
26/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Market Data GARCH
Closer to Normality: res. = rt/σt
27/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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%
28/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators
EWMA Estimators
29/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators EWMA
Exponentially Weighted Moving Average
GARCH(1,1):σ2n = ω + αr 2
n−1 + βσ2n−1
EWMA
σ2n = (1− λ)
∞∑j=1
λj−1(rn−j − r̄)2 = (1− λ)(rn−1 − r̄)2 + λσ2n−1
or
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).
30/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators EWMA
EWMA estimator
σE ,n =
√√√√260(1− λ)
1− λNN∑i=1
λi−1(rn−i − r̄)2
or
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.
31/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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
32/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Historical Estimators
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
33/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Historical Estimators
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
0.0
0.2
0.4
0.6
0.8
1.0
SPX
time shift (days)
corr
elat
ion
with
Y
ABCDEFY
−250 −200 −150 −100 −50 0
0.0
0.2
0.4
0.6
0.8
1.0
DAX
time shift (days)
corr
elat
ion
with
Y
ABCDEFY
34/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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
35/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Stochastic Volatility Models
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 )
36/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Stochastic Volatility Models
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)
37/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Stochastic Volatility Models
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.
38/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
Declining Amount of Information
−250 −200 −150 −100 −50 0
0.0
0.2
0.4
0.6
0.8
1.0
SPX
time shift (days)
corr
elat
ion
with
YABCDEFY
40/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
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
where
wi =
(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 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
EWMA weights
−120 −80 −60 −40 −20 0
0.00
0.02
0.04
0.06
time lag (days)
wei
ght
equal weightslambda = 0.93lambda = 0.94lambda = 0.95lambda = 0.96lambda = 0.97lambda = 0.98
equal weightslambda = 0.93lambda = 0.94lambda = 0.95lambda = 0.96lambda = 0.97lambda = 0.98
equal weightslambda = 0.93lambda = 0.94lambda = 0.95lambda = 0.96lambda = 0.97lambda = 0.98
equal weightslambda = 0.93lambda = 0.94lambda = 0.95lambda = 0.96lambda = 0.97lambda = 0.98
equal weightslambda = 0.93lambda = 0.94lambda = 0.95lambda = 0.96lambda = 0.97lambda = 0.98
equal weightslambda = 0.93lambda = 0.94lambda = 0.95lambda = 0.96lambda = 0.97lambda = 0.98
42/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
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 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
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)
44/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
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%
45/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
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%
46/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
Volatility Forecasting
0 500 1000 1500 2000
0.0
0.2
0.4
0.6
0.8
1.0
SPX − Volatility index
Index
Qvo
l
47/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
Volatility Forecasting
0 500 1000 1500 2000
0.0
0.2
0.4
0.6
0.8
1.0
Index
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
48/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
Volatility Forecasting
0 500 1000 1500 2000
0.0
0.2
0.4
0.6
0.8
1.0
Index
yabs_S50 −0.0319abs_SM −0.0283abs_SD 0.0378abs_MSE 0.0473
rel_S50 −0.1949rel_SM −0.1656rel_SD 0.1471rel_MSE 0.2215
R 0.9537R^2 0.9095
volest
SPX − EWMA volatility (0.96)
49/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
Volatility Forecasting
0 500 1000 1500 2000
0.0
0.2
0.4
0.6
0.8
1.0
Index
yabs_S50 −0.0509abs_SM −0.0562abs_SD 0.0321abs_MSE 0.0647
rel_S50 −0.2977rel_SM −0.2796rel_SD 0.1068rel_MSE 0.2993
R 0.9562R^2 0.9143
volest
SPX − EWMA Park volatility (0.96)
50/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
Volatility Forecasting
0 500 1000 1500 2000
0.0
0.2
0.4
0.6
0.8
1.0
Index
yabs_S50 −0.0054abs_SM 0.004abs_SD 0.045abs_MSE 0.0452
rel_S50 −0.0351rel_SM −0.0078rel_SD 0.1506rel_MSE 0.1507
R 0.956R^2 0.9139
volest
SPX − EWMA GK volatility (0.96)
51/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
Volatility Forecasting
0 500 1000 1500 2000
0.0
0.2
0.4
0.6
0.8
1.0
Index
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)
52/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
Volatility Forecasting
0 500 1000 1500 2000
0.0
0.2
0.4
0.6
0.8
1.0
Index
yabs_S50 −0.0613abs_SM −0.07abs_SD 0.0368abs_MSE 0.0791
rel_S50 −0.3474rel_SM −0.3374rel_SD 0.098rel_MSE 0.3513
R 0.9523R^2 0.9068
volest
SPX − EWMA RS volatility (0.96)
53/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
EWMA Estimators Forecasting Volatility
Volatility Forecasting
0 500 1000 1500 2000
0.0
0.2
0.4
0.6
0.8
1.0
Index
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
54/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Leverage Effect
Leverage Effect
55/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
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
56/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Leverage Effect Extensions of GARCH
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)).
57/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Leverage Effect Extensions of GARCH
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
ergodic.58/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Leverage Effect Extensions of GARCH
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.
59/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Leverage Effect Extensions of GARCH
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
60/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Leverage Effect Extensions of GARCH
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.
61/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Leverage Effect Extensions of GARCH
Estimation Results for the DAX returns 3
rt = µt + σtεtAR µt = ν + φrt−1, ARCH-M µt = ν + λσt
62/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Leverage Effect Extensions of GARCH
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 :-)
63/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Literature
Literature
64/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Literature
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
65/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Literature
J. GatheralThe Volatility SurfaceJohn Wiley and Sons2006
P. WilmottPaul Wilmott on Quantitative FinanceJohn Wiley and Sons2000
P. JorionFinancial Risk Manager HandbookJohn Wiley and Sons2007
66/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Literature
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
67/68 ,
VolatilityModeling
Outline
Market DataDataHistoricalVolatilityImplied VolatilityGARCH
EWMAEstimatorsEWMAHistoricalEstimatorsStochasticVolatility ModelsForecastingVolatility
LeverageEffectExtensions ofGARCH
Literature
Literature
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
68/68 ,