Introduction ARCH Model Garch Models Forecasting Volatility using Garch Models Results Dataset, Software and References Time Series Analysis of GARCH Models for Volatality Sumit Sourabh Ravi Ranjan Singh Sheetanshu Gupta Sahil Singhal November 12, 2010 Sumit Sourabh Ravi Ranjan Singh Sheetanshu Gupta Sahil Singhal Time Series Analysis of GARCH Models for Volatality
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IntroductionARCH ModelGarch Models
Forecasting Volatility using Garch ModelsResults
Dataset, Software and References
Time Series Analysis of GARCH Models forVolatality
Sumit SourabhRavi Ranjan SinghSheetanshu Gupta
Sahil Singhal
November 12, 2010
Sumit Sourabh Ravi Ranjan Singh Sheetanshu Gupta Sahil SinghalTime Series Analysis of GARCH Models for Volatality
IntroductionARCH ModelGarch Models
Forecasting Volatility using Garch ModelsResults
Dataset, Software and References
introduction
Introduction
Financial Time Series such as exchange rates, inflation ratesand stock prices exhibit volatility which varies over time.
Statistically speaking, it means that conditional variance forthe given past or in other words volatility may beheteroskedastic.
σt =√
Var(Xt |Xt−1, . . . ,X1)
Engel (1982) modeled this heteroskedasticity by relatingconditional variance of the disturbance term at time t to sizeof squared disturbance terms in the past.
Volatality estimation is needed for a lot of applications namelyOption Pricing, Asset Pricing etc.
Sumit Sourabh Ravi Ranjan Singh Sheetanshu Gupta Sahil SinghalTime Series Analysis of GARCH Models for Volatality
IntroductionARCH ModelGarch Models
Forecasting Volatility using Garch ModelsResults
Dataset, Software and References
Model Descriptionl
ARCH Model
We define the variable ui as the continuously compoundedreturn during the day which are assumed to be normallydistributed
ui = lnSi
Si−1
We want an estimate of the σn2 , the volatility of the market
variable on the nth day.
In order to have a current estimate of volatility we considerthe past m compounded square returns and a long runaverage variance rate.
Sumit Sourabh Ravi Ranjan Singh Sheetanshu Gupta Sahil SinghalTime Series Analysis of GARCH Models for Volatality
IntroductionARCH ModelGarch Models
Forecasting Volatility using Garch ModelsResults
Dataset, Software and References
Model Descriptionl
ARCH Model
According to the simple Autoregressive ConditionallyHeteroskedastic ARCH(m) model the variance is given by
σn2 = γVL +
m∑i=1
αiu2n−i
where γ +m∑i=1
αi = 1, VL is the long term variance rate , γ
and α′i s are the respective weights assigned.We can use Generalised Least Squares or maximum likelihoodestimation to estimate the ARCH models.Defining ω = γVL the above equation can be rewritten as
σn2 = ω +
m∑i=1
αiu2n−i
Sumit Sourabh Ravi Ranjan Singh Sheetanshu Gupta Sahil SinghalTime Series Analysis of GARCH Models for Volatality
IntroductionARCH ModelGarch Models
Forecasting Volatility using Garch ModelsResults
Dataset, Software and References
The Garch ModelEstimation of Garch Models
The Garch Model
The Garch (1,1) model was proposed by Bollerslev in 1986. Incase of Garch we also include the past variance rate σ2n−1when estimating the current vairance σ2n.
Formally put the equation for a Garch(1,1) model is
σn2 = γVL + αu2
n−1 + βσ2n−1
where the weights add up to 1 or γ + α + β = 1
The general Garch(p,q) thus is given by
σn2 = γVL +
p∑i=1
αiu2n−i +
q∑j=1
βjσ2n−j
Sumit Sourabh Ravi Ranjan Singh Sheetanshu Gupta Sahil SinghalTime Series Analysis of GARCH Models for Volatality
IntroductionARCH ModelGarch Models
Forecasting Volatility using Garch ModelsResults
Dataset, Software and References
The Garch ModelEstimation of Garch Models
Estimation of Garch Models
We consider how Maximum Likelihood method can be used forestimating the Garch parameters. We assume that probabilitydistribution of ui conditional on the variance is normal
The Log Likelihood function is given by
m∏i=1
1√2πσi
exp
(−u2
i
2σi
)This is same as maximaizing
m∑i=1
[−ln(σi )−
u2i
σi
]We use the equation for the garch model and search iterativelyto find the parameters which maximizes the above equation.
Sumit Sourabh Ravi Ranjan Singh Sheetanshu Gupta Sahil SinghalTime Series Analysis of GARCH Models for Volatality
IntroductionARCH ModelGarch Models
Forecasting Volatility using Garch ModelsResults
Dataset, Software and References
Forecasting Volatility using Garch Models
The volatility estimated used a Garch (1,1) model is
σn2 = (1− α− β)VL + αu2
n−1 + βσ2n−1
For estimating the volatality after on (n + t)th day we use thefollowing