Time Series Project Presentation

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




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.





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.





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





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





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.





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

E [σ2n+t − VL] = αE [un+t−1 − VL] + βE [σ2n+t−1 − VL]= (α + β)t(E [σ2n+t−1 − VL]

Using the above equation repeatedly gives us

E [σ2n+t − VL] = (α + β)t(σ2n − VL)

orσ2n+t = VL + (α + β)t(σ2n − VL)





Plot of returns and log returnsPlot of ACF and PACFThe Estimated ModelPlot of Estimated Volatalities

Plot of Returns

We used the end of the day BSE index for the last couple of years






Plot of log Returns






Plot of ACF






Plot of PACF






The Estimated Model

The estimated GARCH (1,1) model for our case is

σ2n = 4.9687e − 006 + 0.087243u2n + 0.8705σ2n−1

Using estimated model we can predict the future volatalities.We have the plot of the volatalities on the next slide using theforecasting method.






Plot of Estimated Volatalities





Dataset and SoftwareReferences

Dataset and Software

The end of day data for the stocks listed in National Stockexchange is freely available at http:\\finance.yahoo.com.

For estimating the ARCH and GARCH model parameters weuse the MATLAB Garch toolbox which uses iterations tomaximize the maximum likelihood fucntion.






References

Cambell,John Y., Lo,Andrew W. & Craig, A., TheEconometrics of Financial Markets, Princeton UniversityPress, 1997.

Engle, R. ,Autorregressive Conditional Heteroskedasticitywith Estimates of United Kingdom Inflation Econometrica, 50,987-1008, 1982.

Engel, R. GARCH 101: The Use of ARCH/GARCH Modelsin Applied Econometrics, Journal of Economic PerspectivesVolume 15, Number 4, 2001.

Bollerslev, T.Generalized Autorregressive ConditionalHeteroskedasticity, Journal of Econometrics, 31, 307-327,1986






Thank You


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