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1、Modelling Financial Volatilities:ARCH,GARCH,CARR and Other Models,Ray Y.Chou 周雨田Academia Sinica,&National Chiao-Tung UniversityPresented at南開大學經濟學院4/11-12/2007,2,Plan of the talks,1.Overview of ARCH Modeling2.Frontiers of ARCH Modeling 3.CARR and ACARR Models4.CARR DCC and Covariance Forecasting5.CA
2、RR in Dynamic Hedge Ratio,3,2003 Nobel Prize WinnerRobert Engle,4,2003 Nobel Laureate,Engles Nobel citation was explicitly for methods of analyzing economic time series with time-varying volatility(ARCH).,5,25 years of ARCH modeling,ARCH survey papers:Engle(1982)EconometricaBollerslev,Chou and Krone
3、r(1992)Journal of Econometrics,Bollerslev,Engle and Nelson(1994)Handbooks of EconometricsEngle(2002)Journal of Applied EconometricsEngle(2004)(Nobel lecture)American Economic Review,6,The First ARCH Model,Rolling Volatility or“Historical”Volatility EstimatorWeights are equal for jNWhat is N?,7,1982
4、ARCH Paper,Weights can be estimatedARCH(p),8,WHAT IS ARCH?,Autoregressive Conditional HeteroskedasticityPredictive(conditional)Uncertainty(heteroskedasticity)That fluctuates over time(autoregressive),9,THE SIMPLEST PROBLEM WHAT IS VOLATILITY NOW?,One answer is the standard deviation over the last 5
5、yearsBut this will include lots of old information that may not be relevant for short term forecastingAnother answer is the standard deviation over the last 5 daysBut this will be highly variable because there is so little information,10,THE ARCH ANSWER,Use a weighted average of the volatility over
6、a long period with higher weights on the recent past and small but non-zero weights on the distant past.Choose these weights by looking at the past data;what forecasting model would have been best historically?This is a statistical estimation problem.,11,ROLLING WINDOW VOLATILITIESNUMBER OF DAYS=5,2
7、60,1300,12,ARCH/GARCH VOLATILITIES,13,CONFIDENCE INTERVALS,14,VALUE AT RISK,Future losses are uncertain.Find a LOSS that you are 99%sure is worse than whatever will occur.This is the Value at Risk.One day in advanceMany days in advanceThis single number(a quantile)is used to represent a full distrib
8、ution.It can be misleading.,15,CALCULATING VaR,Forecast the one day standard deviation GARCH style models are widely used.Then:Assuming normality,multiply by 2.33Without assuming normality,multiply by the quantile of the standardized residuals.For the example,multiplier=2.65,16,MULTI-DAY HORIZONS,If
9、 volatility were constant,then the multi-day volatility would simply require multiplying by the square root of the days.Because volatility is dynamic and asymmetric,the lower tail is more extreme and the VaR should be greater.,17,TWO PERIOD RETURNS,Two period return is the sum of two one period cont
10、inuously compounded returnsLook at binomial tree versionAsymmetry gives negative skewness,High variance,Low variance,18,MULTIPLIER FOR 10 DAYS,For a 10 day 99%value at risk,conventional practice multiplies the daily standard deviation by 7.36For the same multiplier with asymmetric GARCH it is simula
11、ted from the example to be 7.88Bootstrapping from the residuals the multiplier becomes 8.52,19,OPTIONS,Traded options always have multiple days to expiration.Hence the distribution of future price levels is negatively skewed.Thus the Black Scholes implied volatility should depend on strike if option
12、s are priced by GARCH.A skew in implied volatility will result from Asymmetric GARCH,at least for short maturities.,20,IMPLIED VOLATILITY SKEW FOR 10 DAY OPTION,From simulated(risk neutral)final values,find average put option payoff for each strike.Calculate Black Scholes implied volatilities and pl
13、ot against strike.Notice the clear downward slope.This would be zero for constant volatility.,21,WHAT ABOUT HETEROSKEDASTICITY?,22,EXPONENTIAL SMOOTHER,Another Simple ModelWeights are decliningNo finite cutoffWhat is lambda?(Riskmetrics=.06),23,The GARCH Model,The variance of rt is a weighted averag
14、e of three componentsa constant or unconditional varianceyesterdays forecastyesterdays news,24,25,FORECASTING WITH GARCH,GARCH(1,1)can be written as ARMA(1,1)The autoregressive coefficient is The moving average coefficient is,26,GARCH(1,1)Forecasts,27,FORECASTING AVERAGE VOLATILITY,Annualized Vol=sq
15、uare root of 252 times the average daily standard deviationAssume that returns are uncorrelated.,28,Variance Targeting,Rewriting the GARCH modelwhere is easily seen to be the unconditional or long run variancethis parameter can be constrained to be equal to some number such as the sample variance.ML
16、E only estimates the dynamics,29,The Component Model,Engle and Lee(1999)q is long run component and(h-q)is transitoryvolatility mean reverts to a slowly moving long run component,30,TAYLOR-SCHWERT,Standard deviation model,31,Asymmetric Models-The Leverage Effect,Engle and Ng(1993)following Nelson(19
17、89)News Impact Curve relates todays returns to tomorrows volatilityDefine d as a dummy variable which is 1 for down days,32,NEWS IMPACT CURVE,33,Other Asymmetric Models,34,NEW ARCH MODELS,GJR-GARCHTARCHSTARCHAARCHNARCHMARCHSWARCHSNPARCHAPARCHTAYLOR-SCHWERT,FIGARCHFIEGARCHComponent Asymmetric Compone
18、ntSQGARCHCESGARCHStudent tGEDSPARCH,35,FINANCIAL ECONOMETRICS,THIS MAY ALSO BE THE BIRTH OF FINANCIAL ECONOMETRICSSTATISTICAL MODELS DEVELOPED SPECIFICALLY FOR FINANCIAL APPLICATIONSTODAY THIS IS A VERY POPULAR AND ACTIVE RESEARCH AREA WITH MANY APPLICATIONS,36,EXOGENOUS VARIABLES IN A GARCH MODEL,I
19、nclude predetermined variables into the variance equationEasy to estimate and forecast one stepMulti-step forecasting is difficult,37,EXAMPLES,Non-linear effectsDeterministic EffectsNews from other marketsHeat waves vs.Meteor ShowersOther assetsImplied VolatilitiesIndex volatilityMacroVariables or E
20、vents,38,STOCHASTIC VOLATILITY MODELS,Easy to simulate modelsEasy to calculate realized volatilityDifficult to summarize past information setHow to define innovation,39,SV MODELS,Taylor(1982),40,41,WHAT HAVE BEEN ACCOMPLISHED IN 25 YEARS?,Stochastic Properties of ARCH Models the Alphabet soup of ARC
21、H type modelsCulminating in Figlewskis YAARCHComparison with latent or stochastic volatilityMacro ApplicationsInflation,PolicySimple Options trading strategiesEfficiency of Options MarketsModeling the Risk Return Trade-offAsset Pricing,CAPMMeasuring Risk-VaR,42,NEW FRONTIERS,High Frequency Volatilit
22、yClock timeTick timeUse high frequency data to improve daily volatility estimates,43,ANOTHER FRONTIER,Multivariate ARCHHow general should a Multivariate GARCH model be?The Dynamic Conditional Correlation Model Engle(2002),Engle and Sheppard(2002)Highly restricted parameterizationSeparates the volati
23、lity and correlation parametersHow to measure correlations with high frequency data Epps(1972),Zebedee(2001)?,44,THE MULTIVARIATE PROBLEM,Asset Allocation and Risk Management problems require large covariance matricesCredit Risk now also requires big correlation matrices to accurately model loss or
24、default correlationsMultivariate GARCH has never been widely used it is too difficult to specify and estimate,45,Dynamic Conditional Correlation,DCC is a new type of multivariate GARCH model that is particularly convenient for big systems.See Engle(2002)or Engle(2004).,46,DCC,Estimate volatilities f
25、or each asset and compute the standardized residuals or volatility adjusted returns.Estimate the time varying covariances between these using a maximum likelihood criterion and one of several models for the correlations.Form the correlation matrix and covariance matrix.They are guaranteed to be posi
26、tive definite.,47,HOW IT WORKS,When two assets move in the same direction,the correlation is increased slightly.When they move in the opposite direction it is decreased.This effect may be stronger in down markets.The correlations often are assumed to only temporarily deviate from a long run mean,48,
27、STILL MORE FRONTIERS,OPTIONS Pricing and HedgingSimulation Methods Engle Mustafa(1992)Tree Methods Trevor and Richken(1999)Risk Neutralization by local quadratic approximation Duan(1995)Empirical Pricing Kernel Rosenberg and Engle(2001)Path dependent options,49,TWO MORE FRONTIERS:,Modeling Non-Negat
28、ive ProcessesUsing ARCH/GARCH models for a wider range of time series problemsSimulation Methods for Model Analysis(See Engles paper for details),50,MODELING NON-NEGATIVE PROCESSES,Suppose is a non-negative process which has non-zero probability of being zero at any time A model such asmight commonl
29、y be employed where the conditional mean and variance depend on predetermined and weakly exogenous variables.However D cannot have a constant rangeD is unlikely to have constant varianceLeast squares is consistent but inefficient,51,Multiplicative Error Model,Instead,consider the modelNow the error
30、distribution can be i.i.d.without violating the assumptions of the model.Of course it still might not be.Forecasts of x do not depend upon the error density,52,ESTIMATION OF MEM,Specify the mean.For example assume it to be linear in lagged dependent variables,lagged means and other predetermined and
31、 weakly exogenous variables zSpecify the error density and any heteroskedasticity it may have.For example,assume it is unit exponential,53,LOG LIKELIHOOD,In this case:now if this is simply GARCH(p,q)with Gaussian likelihood function and exogenous variables z,54,ESTIMATION WITH GARCH,This likelihood
32、can be maximized with a Gaussian GARCH program such as Eviews.Simply consider square root of x as the dependent variable,mean of zero,and error is GARCHEstimated variance is the mean of xMultistep forecasts are computed conventionally,55,ANOTHER EXAMPLE-DURATION,The ACD model of Engle and Russell(19
33、98)The“duration”or waiting time between two events is a non-negative time series x.,56,ANOTHER EXAMPLE-VOLUME,Manganelli(2001)models the volume of shares traded in a transaction assuming a multiplicative error modelThe expected volume depends upon lagged durations,volatility and volume and their exp
34、ectations,57,ANOTHER EXAMPLE-REALIZED VOLATILITY,Andersen,Bollerslev and Diebold in a series of papers compute daily realized volatility from intra-daily returns.Engle and Gallo(2001)model this realized volatility as a function of lagged realized volatility,daily ranges,and their expected values.,58,ANOTHER EXAMPLE Price Range of stocks,Chou(2005)models the high-low range of S&P500 stock index with an MEM called a Conditional Auto-Regressive Range(CARR)model.The CARR model significantly outperforms the regular GARCH model in volatility forecasts.,59,
