Modelling volatility - ARCH and GARCH volatility - ARCH and GARCH models BetaStehlkov Timeseriesanalysis Modellingvolatility-ARCHandGARCHmodels –p.1/33

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  • Modelling volatility - ARCH and GARCHmodels

    Beta Stehlkov

    Time series analysis

    Modelling volatility - ARCH and GARCH models p.1/33

  • Stock prices

    Weekly stock prices (library quantmod)

    Continuous returns:

    At the beginning of the term we analyzed theirautocorrelations in a HW

    Modelling volatility - ARCH and GARCH models p.2/33

  • Returns

    Time evolution:

    Modelling volatility - ARCH and GARCH models p.3/33

  • Returns

    Based on ACF, they look like a white noise:

    Modelling volatility - ARCH and GARCH models p.4/33

  • Returns

    We model them as a white noise:

    residuals are just - up to a contant - the returns

    If the absolute value of a residual is small, usuallyfollows a residual with a small absolute value

    Similarly, after a residual with a large absolute value,there is often another residual with a large absolutevalue - it can be positive or negative, so it cannot beseen on the ACF

    Second powers will likely be correlated (but this doesnot hold for a white noise)

    Modelling volatility - ARCH and GARCH models p.5/33

  • Returns

    ACF of squared residuals:

    significant autocorrelation

    QUESTION:Which model can capture this property?

    Modelling volatility - ARCH and GARCH models p.6/33

  • Returns

    Possible explanation: nonconstant variance

    Modelling volatility - ARCH and GARCH models p.7/33

  • ARCH and GARCH models

    u is not a white noise, butut =

    2t t,

    where is a white noise with unit variance, i.e.,

    ut N(0, 2

    t )

    ARCH model (autoregressive conditionalheteroskedasticity) - equation for variance 2t :

    2t = + 1u2

    t1 + . . . qu2

    tq

    Constraints on parameters: variance has to be positive:

    > 0, 1, . . . , q1 0, q > 0

    stationarity:1 + . . . + q < 1

    Modelling volatility - ARCH and GARCH models p.8/33

  • ARCH and GARCH models

    Disadvantages of ARCH models: a small number of terms u2ti is often not sufficient

    - squares of residuals are still often correlated for a larger number of terms, these are often not

    significant or the constraints on paramters are notsatisfied

    Generalization: GARCH models - solve theseproblems

    Modelling volatility - ARCH and GARCH models p.9/33

  • ARCH and GARCH models

    GARCH(p,q) model (generalized autoregressiveconditional heteroskedasticity) - equation for variance2t :

    2t = + 1u2

    t1 + . . .+ qu2

    tq

    +12

    t1 + . . . + p2

    tp

    Constraints on parameters: variance has to be positive:

    > 0, 1, . . . , q1 0, q > 0

    1, . . . , p1 0, p > 0

    stationarity:(1 + . . .+ q) + (1 + . . . p) < 1

    A popular model is GARCH(1,1).Modelling volatility - ARCH and GARCH models p.10/33

  • GARCH models in R

    Modelling YHOO returns - continued

    In R: library fGarch function garchFit, model is writen for example like

    arma(1,1)+garch(1,1) parameter trace=FALSE - we do not want the

    details about optimization process

    We have a model constant + noise; we try to model thenoise by ARCH/GARCH models

    Modelling volatility - ARCH and GARCH models p.11/33

  • ARCH(1)

    Estimation of ARCH(1) model:

    We check1. ACF of standardized residuals2. ACF of squared standardized residuals3. summary with tests about standardized residuals

    and their squares

    Modelling volatility - ARCH and GARCH models p.12/33

  • GARCH models in v R

    Useful values: @fitted - fitted values @residuals - residuals @h.t - estimated variance @sigma.t - estimated standard deviation

    Standardized residuals - residuals divided by theirstandard deviation rezdu vydelen ich tadardnou -should be a white noise

    Also their squares should be a white noise

    Modelling volatility - ARCH and GARCH models p.13/33

  • ARCH(1)

    Residuals:

    Modelling volatility - ARCH and GARCH models p.14/33

  • ARCH(1)

    Squares:

    Modelling volatility - ARCH and GARCH models p.15/33

  • Tests about residuals

    Tests:

    We have: normality test, Ljung-Box for standardizedresiduals and their sqaures

    What is new: testing homoskedasticity for the residuals

    Modelling volatility - ARCH and GARCH models p.16/33

  • ARCH(2)

    We try ARCH(2) - results of the tests:

    Modelling volatility - ARCH and GARCH models p.17/33

  • ARCH(3)

    ARCH(3) - results of the tests:

    Modelling volatility - ARCH and GARCH models p.18/33

  • ARCH(4)

    ARCH(4) - results of the tests:

    No autocorrelation in residuals and their squares.

    Modelling volatility - ARCH and GARCH models p.19/33

  • ARCH(4)

    ACF of squared residuals:

    without significant correlation

    Modelling volatility - ARCH and GARCH models p.20/33

  • ARCH(4)

    But ARCH coefficients i are not significant:

    Modelling volatility - ARCH and GARCH models p.21/33

  • GARCH(1,1)

    We try GARCH(1,1)

    Tests:

    Modelling volatility - ARCH and GARCH models p.22/33

  • GARCH(1,1)

    Estimates:

    Modelling volatility - ARCH and GARCH models p.23/33

  • Estimated standard deviation

    We obtain it using @sigma.t :

    Modelling volatility - ARCH and GARCH models p.24/33

  • Estimated standard deviation

    Another access to the graphs - plot(model11):

    Modelling volatility - ARCH and GARCH models p.25/33

  • Predictions

    We use the function predict with parameter n.ahead(number of observations)

    Modelling volatility - ARCH and GARCH models p.26/33

  • Predictions

    Parameter nx - we can change the number ofobservations from the data which are shown in the plot(here nx=100:

    Modelling volatility - ARCH and GARCH models p.27/33

  • Predictions

    Predicted standard deviation: plot(ts(predictions[3]))

    Modelling volatility - ARCH and GARCH models p.28/33

  • Predictions

    For a longer time (exercise: compute its limit):

    Modelling volatility - ARCH and GARCH models p.29/33

  • Application: Value at risk (VaR)

    Value at risk (VaR) is basicly a quantile

    Let X be a portfolio value, then

    P(X V aR) = ,

    for example for = 0.05

    A standard GARCH assumes normal distribution - wecan compute quantiles

    Shortcomings: normality assumptions there are also better risk measures than VaR

    Modelling volatility - ARCH and GARCH models p.30/33

  • Apication: Value at risk (VaR)

    WHAT WE WILL DO: Start with N observations of returns Estimate the GARCH model. Make a prediction for standard deviation and using

    the prediction we construct VaR for returns for thefollowing day

    Every day move the window with data (we have anew observation), estimate GARCH again andcompute thes new VaR

    Modelling volatility - ARCH and GARCH models p.31/33

  • Not required - for those interested

    https://systematicinvestor.wordpress.com/2012/01/06/trading-using-garch-volatility-forecast/

    "... Now, lets create a strategy that switches betweenmean-reversion and trend-following strategies basedon GARCH(1,1) volatility forecast." + R code

    From the website:

    Modelling volatility - ARCH and GARCH models p.32/33

  • Other models for volatility

    Threshold GARCH: ut > 0 - "good news", ut < 0 - "bad news" TARCH can model their different effect on

    volatility leverage effect: bad news have a higher impact

    We do not model variance (as in ARCH/GARCHmodels), but its logarithm exponential GARCH any power of standard deviation power GARCH

    and others...

    Modelling volatility - ARCH and GARCH models p.33/33

    Stock pricesReturnsReturnsReturnsReturnsReturnsARCH and GARCH modelsARCH and GARCH modelsARCH and GARCH modelsGARCH models in RARCH(1)

    GARCH models in v RARCH(1)

    ARCH(1)

    Tests about residualsARCH(2)

    ARCH(3)

    ARCH(4)

    ARCH(4)

    ARCH(4)

    GARCH(1,1)

    GARCH(1,1)

    Estimated standard deviationEstimated standard deviationPredictionsPredictionsPredictionsPredictionsApplication: Value at risk (VaR)Apication: Value at risk (VaR)Not required - for those interestedOther models for volatility

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