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Quantitative Finance

Conditional Heteroskedastic Models

Miloslav S. VosvrdaDept of Econometrics

ÚTIA AV ČR

Robert EngleMV1

Professor of FinanceMichael Armellino Professorship in the

Management of Financial Services

Joined Stern 2000Phone: (212) 998-0710Fax : (212) 995-4220

Email: rengle@stern.nyu.eduOffice: KMEC 9-6244 West Fourth Street

Suite 9-62 New York, NY 10012-1126

Snímek 2

MV1 Prof Engle je nositelem Nobelovy ceny za ekonomii pro rok 2003 Miloslav Vosvrda; 23.3.2004

Recent Awards

The Bank of Sweden Prize in Economic Sciencesin

Memory of Alfred Nobel 2003for

methods of analyzing economic time series with time-varying volatility (ARCH)

The Pivotal Interest in almost all

financial applications

The Predictabilityof

Price Changes

The Pivotal Requirement in almost all

financial applications

Any Volatility ModelMust Be Capable to

Forecast the Volatility

Financial Time SeriesThe volatile behavior in financial markets is usuallyreferred to as the “volatility”. Volatility has become a very important concept in different areas in financial theory and practice. Volatility is usually measured by variance, or standard deviation. The financial marketsare sometimes more volatile, and sometimes less active. Therefore a conditional heteroskedasticity and stylized facts of the volatility behavior observed in financial time series are tuypical features for each financial market.

ARCH, GARCH

This one is a class of (generalized) autoregressive conditional heteroskedasticitymodels which are capable of modeling of time varying volatility and capturing many of the stylized facts of the volatility behavior observed in financial time series.

Stylized Facts1) Non-Gaussian, heavy-tailed, and skewed

distributions. The empirical density function of returns has a higher peak around its mean, butfatter tails than that of the corresponding normaldistribution. The empirical density function istailer, skinnier, but with a wider support than thecorresponding normal density.Figure

2) Volatility clustering (ARCH-effects).Figure3) Temporal dependence of the tail behavior.

6 4 2 0 2 4 60

dnormx 0, 0.7,( )

dt x 3,( )

x Back

Series and Conditional SD

Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q11984 1985 1986 1987 1988 1989 1990 1991 1992

Conditional SD

Original Series

4) Short- and long-range dependence.

5) Asymmetry-Leverage effects. There is evidencethat the distribution of stock returns is slightlynegatively skewed. Agents react more strongly to negative information than to positive information.Figure

0 200 400 600 800 100020

60EGARCH(1,1)

54.403re

10.126−

1 103×1 n

Capital Market Price Models• The Random Walk Model (RWM)• Martingale (MGL)• Conditional Heteroscedastic Model (CHM)

– Models of the first category• ARCH• GARCH

– IGARCH– EGARCH– CHARMA

• ARCH-M• GARCH-M

– Models of the second category• Stochastic volatility model Next

The Random Walk Model(RWM)

the price observed at the beginning of time tS t

( ) ( ) 2an error term with =0 and Var values of the independent of each other

t t tE εε ε ε σ=

1t t tS S ε−= + 1t t tS S ε−− =t

t jS1jε

= ∑RWM was first hypothesis about how financial prices move. Fama (1965) compiled EMH.

A Prediction by the RWM

( ) ( )1 11 , , ...h h h hˆ

hS E S S S S+ −= =

( ) ( )( ) ( )

ˆ 2 , ,...ˆ, ,... 1

h h h h

h h h h h h

S E S S S

E S S S S Sε

+ + −

= + = =

( )ˆh hS l S=

The 1-step ahead forecast at the forecast origin h is

The 2-step ahead forecast at the forecast origin h is

For any l > 0 forecast horizon we have

Therefore, the RWM is not mean-reverting

Forecast Error

( ) 1h h l he l ε ε+ += + +

( ) 2hVar e l l εσ=⎡ ⎤⎣ ⎦

l→∞

The l-step ahead forecast error is

So thatwhich diverges to infinity as

The RWM is not predictable.

RWM with drift

( )1t tE S Sµ = − −

1t t tS Sµ ε−= + +RWM with drift for the price S

Var t 2

iεε σ⎛ ⎞

=∑⎜ ⎟⎝ ⎠

Then S S t01

µ ε=

= + +∑and

Therefore the conditional standard deviationgrows slower than the conditional expectation

2t εσ

Le Roy’s critique of RWMThe critique of RWM by Le Roy (1989) led to some serious questions about RWM as the theoretical model of financial markets. The assumption that price changes are independent was found to be too restrictive. Therefore was, after broad discussion, suggested the following model

1t t tt

S S DrS

+ − +=

wherer is a returnD is a dividend

We assume that ( ) is a constantt tE r rℑ =Taking expectations at time t of the both sides

1t t tt

S S DrS

+ − +=

1t t t

E S DS

r+ + ℑ

( )1 1 1t t t t th S h S D+ + +⋅ = ⋅ +

and we get

We assume that reinvesting of dividends isand t t tx h S= ⋅

( ) ( ) ( )( ) ( )1 1t t t

1 1 1 1t t t t t t t t+ + + +E x E h S h S D

r h S r x

ℑ = ⋅ ℑ = ⋅ + =

= + ⋅ = +i.e. that is a martingalex . For , is a submartingale, because

t0r > x

For , is a supermartingale, because tx

( )E x x1t t t+ ℑ ≥

( )E x x1t t t+ ℑ ≤

Very Important DistinctionA stochastic process following a RWM is morerestrictive than the stochastic process thatfollows a martingale.A financial series is known to go throughprotracted both quiet periods and periods of turbulence. This type of behaviorcould be modeled by a process with conditionalvariances. Such a specification is consistent with a martingale, but not with RWM.

Martingale processes lead to non-linear stochastic processes that are capable of modeling higher conditional moments. Such models are called

Conditional Heteroskedastic Models.These ones are very important for a modeling of the volatility. The volatility is an very important factor in options trading. The volatility means the conditional variance of underlying asset return

( ) ( )( ) ( )221 1 1,where t t t t t t t t tVar r E r E rσ µ µ− − −= ℑ = − ℑ = ℑ

Conditional HeteroskedasticModel (CHM)

( ) ( )( ) ( )22 21 1 1t t t t t t t tVar r E r Eσ µ ε− − −= ℑ = − ℑ = ℑ

( )E r 1t t t

t t tr

t i t i i t ii i

µ µ φ θ ε

− −= =

= + ⋅ + ⋅∑ ∑

Let rt is a stationary ARMA(p,q) model, i.e.,

We distinguish two categories of CHM• The first category: An exact function to a

governing of the evolution of the st2 is used

• The second category: A stochastic equation to a describing of the st

2 is usedThe models coming under the first category are models of type ARCH or GARCH. Thesemodels may catch three out of the five stylizedfeatures, namely 1), 2), and 4).

The models coming under the second category are models of type Stochastic Volatility Model.

ARCH modelThis type of model was introduced by Engle in the paper "Autoregressive Conditional Heteroskedasticity with Estimates of the Varianceof U.K. Inflation," Econometrica 50 (1982),pp. 987-1008, for modeling the predictive variance for U.K. inflationrates.

ARCH modelThe basic idea t t tr µ ε= +

t tr tε µ= −

2 2 20 1 1

t tσ η= ⋅ innovations

t t m t mσ α α ε α ε− −= + + +

( ) ( ), 0 , 1tt t

0 10 , 0 , , 0m

E V a rη η ηα α α

> ≥ ≥

( ) ( )0,1 o rt tN tη η ⋅∼ ∼

ARCH(1)

0, 0ε σ ηα α= ⋅

2 20 1 1t t

σ α α ε −= +

( )0 1/ 1 .α α−

so that

This process is stationary if, and only if, Unconditional variance is equal

1 1.α <

The fourth moment is finite if 213 1.α <

( )0,1t Nη ∼

The kurtosis is so given by ( ) ( )2 21 13 1 / 1 3 .α α− −

0 200 400 600 800 10005

10ARCH(1)

rns r n

µn β0 β1 n⋅+:=rn µn εn α0 α1 εn 1−( )2⋅+⎡

⎣⎤⎦⋅+:=

α0 0.1:= α1 0.1:= β0 0.02:= β1 0.007:=

0 100 200 300 400 50010

5ARCH(1)

5.327−

rn µn−

5001 n

0 200 400 600 800 10000

1.5ARCH(1) Volatility

tility

α0 α1 εn 1−( )2⋅+

1 103×1 n

0 1 2 3 4 5 60.5

r-mi(r-mi)^2+Stadard deviation-Stadard deviation

Autocorrelation Function1

0.05−

σε−

0 1 2 3 4 5 60.5

r-mi(r-mi)^2+Stadard deviation-Stadard deviation

Partial Autocorrelation Function1

0.069−

σε−

Simulated ARCH(1) errorse(

0 100 200 300 400 500

1.0Simulated ARCH(1) volatility

0 100 200 300 400 500

Sample Quantiles: min 1Q median 3Q max

-0.9662 -0.1044 0.005786 0.1096 1.225

Sample Moments: mean std skewness kurtosis

0.002704 0.2315 0.1384 7.016

Number of Observations: 500

moment rr 0, 4,( ) 6.214=

moment rrr 0, 4,( ) 2.402 10 3×=

moment rr 0, 3,( ) 1.85=

moment rrr 0, 3,( ) 108.796=

The fourth

The third

Daily Stock Returns of FORD

Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1

1984 1985 1986 1987 1988 1989 1990 1991 1992

Series : ford.s

0 2 4 6 8 10 12

Series : ford.s^2

The kurtosis exceeds 3, so that the unconditional distribution of ε is fatter tailed than the normal.

Testing for ARCH effects.This test is constructed on a simple LagrangeMultiplier (LM) test. The null hyphothesis is that:There are no ARCH effects,i.e.,

The test statistic is

1 0mα α= = =

2 2~ ( )LM N R pχ= ⋅

Test for ARCH Effects: LM Test

Null Hypothesis: no ARCH effects

Test Statistics:FORD

Test Stat 112.6884p.value 0.0000

Dist. under Null: chi-square with 12 degrees of freedom

Total Observ.: 2000

A practical problem with this model is that, withm increasing, the estimation of coefficients oftenleads to the violation of the non-negativity of thea’s coefficients that are needed to ensure that conditional variance st is always positive. A natural way to achieve positiveness of the conditional variance is to rewrite an ARCH (m) model as

( )20 , 1 , 1 , 1 1, , , TT

t m t m t m t t t mσ α ε ε− − − − −= + Ε ΩΕ Ε = …

ForecastingForecasts of the ARCH model can be obtained recursively. Consider an ARCH(m) model. At forecastorigin h, the 1-step ahead forecast of is

( )2 2 20 1 11h h m h mσ α α ε α ε + −= + + +

( ) ( )2 2 2 20 1 2 22 1h h h m h mσ α α σ α ε α ε + −= + + + +

21hσ +

( ) ( )2 20

h i hi

l l iσ α α σ=

= + −∑

( )2 2 if 0h h l il i l iσ ε + −− = − ≤where

The 2-step ahead forecast of is22hσ +

The l-step ahead forecast of is2h lσ +

0 200 400 600 800 1000 12005

15ARCH(1) Forecasting

Weakness of ARCH Models• The model assumes that positive and negative shocks have

the same effects on volatility because it depends on the square of the previous shocks.

• The ARCH model is a little bit restrictive because for a finite fourth moment is necessary to be This constraint becomes complicated for higher order of the ARCH.

• The ARCH model provides only way to describe the behavior of the conditional variance. It gives no any new insight for understanding the source of variations of a financial time series.

• ARCH models are likely to overpredict the volatility because they respond slowly to large isolated shocks .

[ )2 0,1/ 3 .α ∈1

Although the ARCH model is simple, it often requires many parameters to adequately describethe volatility process. To obtain more flexibility, the ARCH model was generalised to GARCH model. This model works with the conditional variance function.

GARCH model

2 2 20

,t t t t t tm s

t i t i j t ji j

r µ ε ε σ η

σ α α ε β σ− −= =

= + = ⋅

= + +∑ ∑

( ) ( )

( ) 1, and 0 ,i i i

, 0, 10, 0, 0,

E Varη η ηα α β

> ≥ ≥

for i m

for j s

α β α+ < = >=∑

A Connection to the ARMA processFor better underestanding the GARCH model is usefulto use the following transform:

2 2 2 2 2 2 so that , and t t t t t t t i t i t iξ ε σ σ ε ξ σ ε ξ− − −= − = − = −

( )( )max ,

t i i t i t j t j

Substitute these expressins into GARCH equation andwe get as follows

1 1i j

ε α α β ε ξ β ξ− −= + + + −∑ ∑= =

( )0, cov , 0, for 1.t t t jE jξ ξ ξ −= = ≥

The process is a martingale difference series, i.e.,

The process

GARCH(max(m,s),s)

is an application of the ARMA idea to the squared series . Thus2

( )( )max ,m s s

1 1t i i t i t j t j

ε α α β ε ξ β ξ− −= =

= + + + −∑ ∑

( )( )max ,

/ 1m s

t i ii

Eε α α β=

⎛ ⎞= − +⎜ ⎟⎜ ⎟

⎝ ⎠∑

GARCH(1,1)

( )221 1 11 2 0tε α β−− − + >

21tε −

•A large gives rise to a large . A large tends to be followed by large generating the well-known behavior of volatility clustering in financial time series.•If then kurtosis

•The model provides a simple parametric function for describing the volatility evolution.

2tσ2 2

1 1t torε σ− −2tε

( )( ) ( )2 221 1 1 1 13 1 /1 2 3tα β ε α β−− + − − + >

2 2 20 1 1 1 1t t tσ α α ε β σ− −= + +

( )0 1 1 1 10 , 1 0 , 1 0 , 1α α β α β> > ≥ > ≥ + <

0 200 400 600 800 10005

10GARCH(1,1)

1.229−

1 103×1 n

GARCH(1,1) Volatility3

0 200 400 600 800 10000

tility

1 103×1 ntime

Forecasting by GARCH(1,1)

2 21 1

α ε β σ

We assume that the forecast origin is h. For 1-step ahead forecast For 2-step ahead forecast

For l-step ahead forecast

( )( ) ( )

α β σ

( )( )( )

( ) ( )1

0 1 1 12 21 1

α α βσ α β σ

−− +

= + +− −

Therefore ( )2 0

1 11h ll ασα β→∞⎯⎯⎯→

− −

The multistep ahead volatility forecasts of GARCH(1,1) model converge to the unconditional

infinity provided that Var(εt) exists.variance of ε as the forecast horizon increases to

0 200 400 600 800 1000 12005

original realizationforecasting+standard deviation-standard deviation

GARCH(1,1) Forecasting

2.887−

rrr nn

1.1 103×1 n nn,

Mean Equation: ford.s ~ 1

Conditional Variance Equation: ~ garch(1, 1)

Coefficients:

C 7.708e-004A 6.534e-006

ARCH(1) 7.454e-002GARCH(1) 9.102e-001

Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q1 Q2 Q3 Q4 Q11984 1985 1986 1987 1988 1989 1990 1991 1992

Conditional SD

Original Series

esSeries and Conditional SD

Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q4 Q2 Q41984 1985 1986 1987 1988 1989 1990 1991 1992

ford.mod11

esSeries with 2 Conditional SD Superimposed

0 10 20 30

ACF of Observations

0 10 20 30

ACF of Squared Observations

0 20 40 60

Cross Correlation

residuals

lsGARCH Model Residuals

volatility

DGARCH Volatility

residuals

GARCH Standardized Residuals

0 10 20 30

ACF of Std. Residuals

0 10 20 30

ACF of Squared Std. Residuals

-2 0 2

QQ-Plot

10/13/1989

12/23/1991

10/19/1987

Quantiles of gaussian distribution

QQ-Plot of Standardized Residuals

garch(formula.mean = ford.s ~ 1, formula.var = ~ garch(1,1))Mean Equation: ford.s ~ 1Conditional Variance Equation: ~ garch(1, 1)Conditional Distribution: gaussian--------------------------------------------------------------

Estimated Coefficients:--------------------------------------------------------------

Value Std.Error t value Pr(>|t|) C 7.708e-004 3.763e-004 2.049 0.02031225A 6.534e-006 1.745e-006 3.744 0.00009313

ARCH(1) 7.454e-002 5.362e-003 13.902 0.00GARCH(1) 9.102e-001 8.762e-003 103.883 0.00

AIC(4) = -10503.79BIC(4) = -10481.39

Normality Test:--------------------------------------------------------------

Jarque-Bera P-value 364.2 0

Shapiro-Wilk P-value 0.9915 0.9777

Ljung-Box test for standardized residuals:--------------------------------------------------------------

Statistic P-value Chi^2-d.f. 14.82 0.2516 12

Ljung-Box test for squared standardized residuals:--------------------------------------------------------------

Statistic P-value Chi^2-d.f. 14.04 0.2984 12

Lagrange multiplier test:--------------------------------------------------------------Lag 1 Lag 2 Lag 3 Lag 4 Lag 5 Lag 6 Lag 72.135 -1.085 -2.149 -0.1347 -0.9144 -0.2228 0.708

Lag 8 Lag 9 Lag 10 Lag 11 Lag 12 C

-0.2314 -0.6905 -1.131 -0.3081 -0.1018 0.9825

TR^2 P-value F-stat P-value14.77 0.2545 1.352 0.2989

Test for Residual Autocorrelation

Test for Autocorrelation: Ljung-Box

Null Hypothesis: no autocorrelation

Test Statistics: Test Stat 14.8161p.value 0.2516

Test for Residual^2 Autocorrelation

Test for Autocorrelation: Ljung-Box

Null Hypothesis: no autocorrelation

Test Statistics: Test Stat 14.0361p.value 0.2984

0 200 400 600 800 10000.6

0.4GARCH(1,1)

0.538−

1 103×1 n

0 100 200 300 400 5004

4GARCH(1,1) Volatility

tility

3.114−

5001 nk

0 200 400 600 800 1000 12000.6

original realizationforecasting+standard deviation-standard deviation

GARCH(1,1) Forecasting

0.538−

rrr nn

1.1 10 3×1 n nn,

Weakness of GARCH Models

This model encounters the same weakness as theARCH model. In addition, recent empirical studies ofhigh-frequency financial time series indicate that the tail behavior of GARCH models remains too short even with standardised Student t-innovations.

ARCH-M modelThe return of a security may depend on its volatility.If we take conditional deviation as a measure for risk,it is possible to use risk as a regressor in returnsmodeling. To model such a phenomenon, one may consider the ARCH-M model, where M meansin mean. This type model was introduced by Engle in the paper "Estimation of Time Varying Risk Premia inthe Term Structure:the ARCH-M Model," (with David Lilien and Russell Robins), Econometrica 55 (1987): 391-407.

A simple ARCH-M(1) model

2 2 20 1 1

t t t t

t t m t m

r cµ σ η

σ α α ε α ε− −

= + ⋅

= + + +The parameter c is called the risk premium parameter. A positive c indicates that the return is positivelyrelated to its past volatility.

0 200 400 600 800 10002

8ARCH-M(1)

0.08−

1 103×1 n

GARCH-M ModelA simple GARCH(1,1)-M model

2 2 20 1 1 1 1

t t t t

ε σ ηr cµ σ ε= + ⋅ +

σ α α ε β σ− −

The parameter c is called the risk premium parameter.A positive c indicates that return is positively related to its past volatility. By formulation we can see thatthere are serial correlations in the return series rt.

0 200 400 600 800 10005

10GARCH-M(1,1)

1.181−

1 103×1 n

IGARCHIGARCH models are unit-root (intergrated) GARCHmodels. This type model was introduced by Engle and Bollerslev in the paper “Modelling the Persistence of ConditionalVariances," Econometric Reviews,5, (1986), pp. 1-50.A key feature of IGARCH models is that the impactof past squared shocks for onis persistent.

2 2t i t i t iζ ε σ− − −= − 0i > 2

IGARCH(1,1)An IGARCH(1,1) model can be written as

where is defined as before and

2 2 20 1 1 1 1

, ,t t t t t t

r µ ε ε σ η

σ α β σ α ε− −

tε 11 0.β> >

0 200 400 600 800 100010

15IGARCH(1,1)

10.025

6.151−

1 103×1 n

Forecasting by IGARCH(1,1)When repeated substitutions in

givewhere h is the forecast origin. The effect of on futurevolatilities is also persistent, and the volatility forecasts form astraight line with slope . The case of is ofparticular interest in studying the IGARCH(1,1) model. Thevolatility forecasts are simply for all forecast horizons.

( ) ( ) ( )2 20 1 1 1h hl lσ α α β σ= + + −

1 1 1α β+ =

( ) ( ) ( )2 201 1 , 1,h hσ σ α= + − ≥

( )2 1hσ

0α0 0α =

( )2 1hσ

0 200 400 600 800 1000 120010

IGARCH(1,1) Forecasting40

31.071

6.151−

1.1 103×1 n nn,

FARIMAGeneral IGARCH(p,q) process is called FARIMA(p,d,q):

, ,t t t t t tr µ ε ε σ η= + =2 2 2 2 2

0 1 1 1 1 1 1 1( ) (1 ) ( ) 2t t t tσ tα β σ ε α ε α β ε− − −= + − + + + + −

EGARCH ModelThis model allows to consider asymmetric effectsbetween positive and negative asset returns throughthe weighted innovation

( ) ( )t t t tg Eη θη γ η η⎡ ⎤= + −⎣ ⎦where θ and γ are real constant.

Leverage Effects

Negative shocks tend to have a larger impact on volatility than positive shocks. Negative shocks tend to drive down the shock price, thus increasing the leverage ( i.e. the debt-equity ratio) of the shock and causing the shock to be more volatile. The asymmetricnews impact is usually refferred to as the leverage effect.

( )t tEη η− ( ) 0.tE g η =⎡ ⎤⎣ ⎦

,rt t t t t tµ ε ε= + σ η= ⋅

Because is iid sequence then

So EGARCH(m,s) can be written as

( )2 10 1

L L gL L

β βσ α ηα α −

⎛ ⎞+ + += +⎜ ⎟− − −⎝ ⎠

EGARCH(1,0)

( )20 1

t t t t t t

ε ε σ η= ⋅r µ= +

σ α ηα −

⎛ ⎞= +⎜ ⎟−⎝ ⎠

( )( ) ( )( ) ( )

if 0t t t

tt t t

θ γ η γ η ηη

θ γ η γ η η

⎧ + − ≥⎪= ⎨− − <⎪⎩

( )tg ηThe asymmetry of can easily be seen as

( )( )

1t−12

12 21 0 1

exp if 0

exp 1 2 /

exp if 0

εθ γ εσ

σ σ α α γ πεθ γ εσ

−−

⎧ ⎡ ⎤⎪ ⎢ ⎥+ ≥⎪ ⎢ ⎥⎪ ⎣ ⎦⎡ ⎤= − − ⎨⎣ ⎦ ⎡ ⎤⎪

⎢ ⎥− <⎪⎢ ⎥⎪ ⎣ ⎦⎩

0 200 400 600 800 100020

60EGARCH(1,1)

54.403

10.126−

1 103×1 n

garch(formula.mean = hp.s ~ 1, formula.var = ~ egarch(1,1), leverage = T, trace = F)

Mean Equation: hp.s ~ 1Conditional Variance Equation: ~ egarch(1, 1)

Coefficients:

C 0.000313A -1.037907

GARCH(1) 0.886652LEV(1) -0.133998

ARCH(1) 0.227878

CHARMAThis model uses random coefficients to producea conditional heteroscedasticity. The CHARMA model has the following form

( )20, ,t ηη σ∼ ( ) 1 ,..., Tt t mtδ δ δ=

1 1 2 2

t t t t t mt t m t

r µ εε δ ε δ ε δ ε η− − −

= += + + + +

whereis a sequence of iid random vectors with mean zeroand non-negative definite covariance matrix Ώ and

tδ is independent of tη

CHARMA(2)

1 1 2 2,t t t t t t t t tr µ ε ε δ ε δ ε η− −= + = + +

The CHARMA model can easily be generalized sothat the volatility of may depend on someexplanatory variables. Let be m explanatoryvariables available at time t. Consider the model

t t t t it i t ti

r xµ ε ε δ η−=

= + = +∑where and are sequences of random vectors and random variables .

tη ( ) 1 ,..., Tt t mtδ δ δ=

0 200 400 600 800 10005

15CHARMA(2)

3.262−

1 103×2 n

Then the conditional variance of istε

( ) ( )2 21, 1 , 1 1, 1 , 1,..., ,...,

Tt t m t t m tx x x xησ σ − − − −= + Ω

is a diagonal matrixΩwhere

Stochastic Volatility Model

2 20 1 1

, ,t t t t t t

r gµ ε ε= + = σ η

σ α α σ υ−= + +

( )0,1iidtη ∼

( )20,t iid υυ σ∼Back

t tη υ⊥where

The Routes Ahead

• High frequency volatility• High dimension correlation• Derivative pricing• Modeling non-negative processes• Analysing conditional simulations by

Least Squares Monte Carlo

Thank you very muchfor your attention

quantitative finance - vosvrdaweb

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