留学生金融学课程作业需求:Models of time varying v

Empirical Finance
 Models of time varying volatility in empirical finance:
ARCH and GARCH models.
Module Leader: Dr Stuart Fraser
[email protected]
留学生dissertation网Room D1.18 (Social Studies)Warwick Business School 2
Introduction
Generalized-Autoregressive-Conditional-Heteroscedastic(GARCH)processes:
•Motivation for GARCH models.
•Examples of different types of GARCH model.
•Some applications of GARCH in finance.
•Identifying, estimating and testing GARCH models.
⇒Seminar 5: Modelling time-varying volatility in the FTSE All-Share Index excess returns.Warwick Business School 3
Motivation for GARCH processes
Two empirical features of financial return series are relevant in this context.
Firstly, there is the observation that over short horizons/holding periods of up to one month:
–Return volatility is time varying(there are periods of tranquilityand turbulence).
–Volatility clustering: large (small) price changes tend to be followed by further large (small) changes.
⇒volatility (risk) is positively correlated.
⇒Non-linear dependence in returns.
Secondly, the unconditional distributions of short-horizon returns have fat tails (leptokurtosis).Warwick Business School 4
1. Volatility clustering (see handout for Seminars 1 and 2)-.08-.06-.04-.02.00.02.04.06949596979899000102030405SP500_RETURNSWarwick Business School 5
⇒Non –linear dependence in returns
Correlogramof squaredSP500 returns
AutocorrelaPartial CorrelationAC PAC Q-Stat Prob |** |** 10.2070.207130.550 |** |* 20.20.164252.340 |** |* 30.2280.171410.190 |* |* 40.180.093508.640 |** |* 50.2050.115636.520 |* | 60.1550.044709.210 |* |* 70.1740.072801.50 |* | 80.1520.039871.820 |* | 90.1560.051946.10 |* | 100.160.051024.40Warwick Business School 6
2. Leptokurtic unconditional return distributions0100200300400500600700800-0.075-0.050-0.0250.0000.0250.050Series: SP500_RETURNSSample 1/03/1994 1/11/2006Observations 3033Mean 0.000412Median 0.000645Maximum 0.054248Minimum -0.070376Std. Dev. 0.010419Skewness -0.138163Kurtosis 6.818984Jarque-Bera 1852.783Probability 0.000000Warwick Business School 7
Volatility clusteringVolatility clustering implies volatility, h(r), is predictable from past information:A common measure of ex-antevolatility is the conditional variance: The ex-post (realized) volatility is: ()1−Ω=ttfhΩincludes past volatility and other relevant information.h is the conditional volatility.This measures ex-antevolatility.()()12−Ω−=tttrEhμThis is a one-step forecast of the varianceconditional on 1−Ωt()22ttrr≅−μμ≅0 for short horizon returnsestimator Range lnlnreturnsPower ,0 ,returns Absolute lowtthighttPPrr−>θθAlternative volatility/risk measures include:Warwick Business School 8#p#分页标题#e#
General framework for modellingexpected returns and time varying volatility
The following general framework sets out a conditional mean equation(to predict expected returns) and a conditional variance equation(to predict risk):()()1,0~122NIDvEvrtttttttttt−Ω==+=εσσεεμ()()()2111,0~0ttttttttNvEEσεσε−−−Ω⇒=Ω=ΩThe conditional variance depends on information from previous periods (⇒volatility clustering).Mean equation describing equilibrium returns (μcould include the conditional variance to model a time-varying risk premium–see GARCH-M below).The assumption that the standardized residuals (v) are Gaussian ⇒the residuals (ε) are conditionallynormally distributed.This assumption is important for estimation of the model (see below).Warwick Business School 9
ARCH(q) model (Autoregressive Conditional Heteroscedasticity)
Different assumptions about Ωgenerate different models of time varying risk.Suppose thenThis is an ARCH(q) model. A sufficient condition for a positiveconditional variance (variances cannot be negative) is that: Engle shared the Nobel Prize in Economics (2003): “for methods of analyzing economic time series with time-varying volatility (ARCH)”{}2211,...,qttt−−−=Ωεε221102...qtqtt−−+++=εαεαασqii,...,1 ,0 ,00=≥>ααThe ARCH model was firstproposed by Engle (1982)in the context of modelinginflation uncertainty.Warwick Business School 10
ARCH(q) modelThe ARCH(q) model can be written as an AR(q) model in the squared residualsThe process is stationary if all the roots of the characteristicequation lie outside of the unit circle:The long run/unconditionalvariance can be found from the Woldrepresentation:()22122221102,...tttttttqtqttEuuσεεεεαεααε−=Ω−=++++=−−−0...11=−−−qqzazα()Σ=−=⇒−−−+=qiitqqttELLu1021021...1ααεαααεΣ=<qii11αShocks to volatility (u)The unconditional variance is finite, positive and constant (i.e., homoscedastic) if:If this condition holds then volatility is a constantin the long-run. If the process is stationarythenshocks to volatility do not persist⇒the conditional variance returnseventually to it’s long run level.This condition must hold if the process is stationary.Warwick Business School 11
GARCH(p,q) (GeneralisedAutoregressive Conditional Heteroscedasticity)
In practice q may need to be set high to capture all the non-linear dependence in returns.Also with a lot of lags the non-negativity constraints are likely to be violated.Bollerslev(1986) proposed the GARCH model as a parsimoniousalternative to ARCHFor GARCH(p,q):The non-negativity constraints (sufficient restrictions) are: Typically p=q=1 is adequate in most empirical applications.{}2212211,...,,,...,pttqttt−−−−−=ΩσσεεΣΣ=−=−++=pjjtjqiitit121202σβεαασpjqiji,..,1 ,,..,1 ,0,0,00==≥≥>βααWarwick Business School 12#p#分页标题#e#
GARCH(1,1)The GARCH(1,1) has an ARMA(1,1) representation in the squared residuals. The unconditional/long-run variance is:()()ttttttttttttuuuu+−++=⇒−++=−⇒++=−−−−−−−1121110212112110221121102βεβααεεβεααεσβεαασ111<+βα()()11021βααε+−=tE111=+βαVolatility is stationaryif:But ifthen shocks to volatility have a permanent effect:⇒INTEGRATED GARCH(IGARCH) process.Apparent IGARCH behaviouris found quite often in empirical work (see below).The long-run variance converges to a constantiff: 111<+βαAn implication of IGARCH is that investors should be frequently altering their portfolios following shocks to reflect permanentchanges in risk. Since this kind of behaviour isn’t observed, IGARCH is incompatible with volatility in the ‘real world’. It’s possible that shocks to volatility are just highly persistent if not permanent.Warwick Business School 13
Multivariate GARCH (MGARCH)
Generalization of GARCH to systems of n-asset returns.The conditional volatility is an n×nvariance-covariance matrix:A widely used formulation of MGARCH is the BEKK model:⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=2,,1,12,1tntntnttHσσσσKMOMKConditional varianceson the diagonalConditional covariancesoff the diagonalBBAHAWWHtttt111−−−Σ′Σ′+′+′=⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛=Σ−−−1,1,11tnttεεMW, A and B are n×nmatrices of parameters.H is positive definite (because the RHS terms are quadratic forms):⇒The variances are positive⇒The off-diagonal terms are symmetric:()jiijσσ=Warwick Business School 14
ARCH-GARCH processes and fat-tail distributions
Another nice feature of ARCH-GARCH models is that they generate fat-tailed unconditional returns distributions (which are observed empirically –see Seminar 1/2).
For example an ARCH(1) model produces an unconditional return distribution with kurtosis coefficient
(recall the normal distribution has a kurtosis coefficient=3).
()3311321214>−−=ααmWarwick Business School 15
Financial applications of volatility/GARCH modelling
Value at Risk (VaR)VaRmeasures the £value of market risk on an asset/portfolio of marketable assets.The maximumthe investor can expect to lose in 19/20 days = VaR(at a 5% critical value) ⇒expect to lose more thanVaRin 1/20 days.In the case of a single asset if the return is normally distributed then a 90% confidence interval for the return is Returns will be less than μ-1.65σon 1 in every 20 days (5% of the time).Assuming μ≅0 (reasonable for daily returns) then the downside riskwith 5% probability is 1.65σσμ65.1±NOTTO BE CONFUSED WITH VAR(VECTOR-AUTOREGRESSIVE PROCESS) see lecture 9Warwick Business School 16#p#分页标题#e#
Financial applications of volatility/GARCH modelling
VaR
If the value of the asset is £V then:
A forecast of volatility is needed to calculate VaR.
•GARCH provides one option for making this forecast.
•A more commonly used model for VaRvolatility is an exponentially weighted moving average(EWMA):
tttVVaRσ65.1£×=()212121−−−+=tttrλλσσ
EWMA used widely by practitioners
e.g., JP Morgan (who recommend
using λ=0.94)Weighted average of lagged ex-ante/forecasted volatility and lagged ex-post/realized volatility (assuming μ=0).21−tσ21−tr()Σ=−−−+=⇒tjjtjttr1212021λλσλσ
Weights attached to previous volatility
decline geometrically/exponentially with
the lag.Warwick Business School 17
Financial applications of volatility/GARCH modelling
Dynamic hedge ratios
A common risk-management practice is to take opposite positions in spot and futures markets (a futures hedge).
The finance director’s job is to determine the optimal hedge ratio: θ≡number of futures contracts/number of spot contracts.
The optimal value of θwill minimize the risk on the spot-futures portfolios. Choose θto minimize the portfolio variance:
()()2,,,2,,2,22,(FOC) 222varvartftsfttsftfttsfttfttsstfttfttstrrrrσσθσσθσθσθσθθ=⇒=⇒−+=−=−Short hedgeLong hedgefsrron Simplest estimate of θis a statichedge:Regress θ=estimated slope coefficientIf the variances/covariancesare time-varyingestimate θusing an MGARCH model (n=2) for the spot and futures returns.See Brooks 8.28/8.29 for applications of MGARCH to estimating dynamic hedge ratios and time-varying CAPM betas.Warwick Business School 18
ARCH-GARCH ModellingStrategy
Analogous to ARMA modeling (Box-Jenkins technique: see lecture 5), ARCH-GARCH modeling involves:
1. Identification of a suitable ARCH-GARCH model
2. Estimation (using Maximum Likelihood)
3. Testing/diagnostic checking of the model to ensure it provides an adequate representation of the actual DGP (see Seminar 5).Warwick Business School 19
Identification of ARCH-GARCH models
First perform a simple ARCH-test(see Appendix) to test for ARCH effects.
If there are ARCH effects present, identify a particular ARCH-GARCH model as follows:
i) The AR and ARMA representations for the squared residuals suggest the ACF and PACFsof the squared residuals can be used to identify a specific ARCH-GARCH model.
An ARCH(q) model is indicated by:
a) An infinite decay in the ACF of the squared residuals.
b) q spikes in the PACF of the squared residuals.
A GARCH(p,q) model is indicated by an infinite decay in both the ACF and the PACF of the squared residuals.#p#分页标题#e#
ii) In practice its more common to use information criteria such as the Schwarz Criterionto help select a model (see lecture 5).
iii) Many authors simply assumea GARCH(1,1) specification.Warwick Business School 19
Identification of ARCH-GARCH models
First perform a simple ARCH-test(see Appendix) to test for ARCH effects.
If there are ARCH effects present, identify a particular ARCH-GARCH model as follows:
i) The AR and ARMA representations for the squared residuals suggest the ACF and PACFsof the squared residuals can be used to identify a specific ARCH-GARCH model.
An ARCH(q) model is indicated by:
a) An infinite decay in the ACF of the squared residuals.
b) q spikes in the PACF of the squared residuals.
A GARCH(p,q) model is indicated by an infinite decay in both the ACF and the PACF of the squared residuals.
ii) In practice its more common to use information criteria such as the Schwarz Criterionto help select a model (see lecture 5).
iii) Many authors simply assumea GARCH(1,1) specification.Warwick Business School 20
Maximum Likelihood (ML) (see Brooks Chp8, Appendix)
Suppose we have a sample of independent observations drawn from a knowndensityHowever the parameters are unknownand there is a given sample of data. Therefore re-interpretthe joint distribution as the likelihood function:()Tyy,...,1()()()θθθTTyfyfyyf××=...,...,11()()()θθθTTyfyfyyL××≡...,...,11()()Σ==TttyfL1loglogθθThe probability of observing different realizations of y for givenparameters θ.The likelihood of observing the (given) data for different values of θML estimator found by maximizingthe log-likelihood functionwith respect to θML chooses θto maximize the likelihood of observing the sample data. ML estimators are:1) Consistent.2) Asymptotically efficient.3) Asymptotically normally distributed.Warwick Business School 21
ML estimation of ARCH-GARCH model
The ARCH-GARCH likelihood function involves the conditional error density. If v~NID(0,1) then:The log likelihood function is:()()θσσεπσθεttttttvff2222121exp21=⎭⎬⎫⎩⎨⎧−=()()()()()ΣΣΣ===−−−=⎥⎦⎤⎢⎣⎡−=TtttTttTtttTvfL122121221log212log2log21loglogσεσπσθθ()22με−=ttr2tσis a function of the unknown parameters of the conditional variance.is a function of the unknown parameters of the conditional mean.The ML estimates of the conditionalmean/variance parameters are the values which maximize log L(θ).Warwick Business School 22
Alternative conditional distributions (see Seminar 5)
What happens if the standardized residuals, v,are non-normal (but normality is assumed)?
The ML estimator is still consistent and asymptotically normalbut the standard errorsare inconsistent.
Eviewsgives an option to estimate ARCH-GARCH models with a conditional normal distribution but with a var-covmatrix which is robust to non-normality. This is known as Quasi-ML (QML)#p#分页标题#e#
Alternatively we can choose a different conditional distribution. Eviewsallows the choice of a:
i) Student’s tdistribution
ii) Generalized error distributionWarwick Business School 23
Extensions to ARCH-GARCH
Asymmetric GARCHStandard GARCH models force a symmetric response of the conditional variance to shocks (since depends on lagged squaredresiduals).However, typically bad news (negative shocks) may be expected to increase volatility more than good news (positive shocks) of the same magnitude.In the context of equity returns this may be due to leverage effects⇒Negative shocks result in a fall in the value of the firm which increases the debt-equity ratio.As a result stockholders perceive the firm as being more risky ⇒volatility increases.2tσWarwick Business School 24
Asymmetric GARCH models : Threshold ARCH (TARCH) –Glosten, Jagannathanand Runkle(GJR) model
where: When (good news) the ARCH effect is .When (bad news) the ARCH effect is .If leverage effects are present then expect γ>0..0 if 00 if 1111>=<=−−−tttIεε12121121102−−−−+++=tttttIγεσβεαασ01>−tε1α01<−tεγα+1Dummy variableWarwick Business School 25
Asymmetric GARCH models: Exponential GARCH (EGARCH)
The log transformation ensures that the conditional variance is positive regardless of the parameter values⇒no need for non-negativity constraints.Also the effect of past shocks is exponential rather than quadratic(as in GARCH/TARCH)⎥⎦⎤⎢⎣⎡−+++=−−−−−πσεδσεγσβασ2loglog1111212tttttt0 0 and 00 0 and 0111111<⇒><>⇒<<−−−−−−ttttttσεγεγσεγεγBad newsGood newsIn the EGARCH modelleverage effects ⇒γ<0**Asymmetry coefficient hasopposite sign from TARCH**Warwick Business School 26
The News Impact Curve (NIC) .000.001.002.003.004.005.006.007-.15-.10-.05.00.05.10.15.20.25.30Value of lagged residualSIG2_GARCHSIG2_EGARCH.00.02.04.06.08.10.12.14-1.2-0.8-0.40.00.40.81.2Value of lagged residualSIG2_GARCHSIG2_TARCHGARCHTARCHGARCHEGARCHThe NIC describes the response of volatility to past shocks:•In the GARCH model the impact of shocks is symmetricabout the origin. •In the asymmetric models (TARCH/EGARCH) negative shocks have a biggerimpact on volatility than positiveshocks of the same magnitude.2tσThe asymmetry is more pronounced for EGARCH than TARCH due to the exponential relationship between volatility and past shocksWarwick Business School 27
GARCH-in-Mean (GARCH-M) (see Seminar 5)
CAPM applied to the market portfolio
gives:
The market return therefore varies
directlywith the conditional variance
(which is usually modelled as a GARCH(1,1) process as above).#p#分页标题#e#
This is an example of a GARCH-M model: the conditional variance enters the mean equation.
GARCH-M is a model of a time-varying risk premium.
Recall that mistakenly assuming a constantrisk premium may give rise to rejection of the EMH (see lecture 2)
ttftrrελσ++=22121102tttεβσαασ++=−Applying CAPM to the market portfolio λIs the MARKET PRICE OF RISK:()()[]()()[]()mmmfmmmfmfrrrrrrErrrrErrE≡==−=−=− ,,cov,cov22λσλσβ()()2mfmrrEσ−Warwick Business School 28
Conclusion
GARCH models have assumed a central role in empirical finance for modelling time varying volatility.
There is a proliferation of variations on the ‘plain vanilla’GARCH model (including models for absolute/power returns, multivariate models…)
Numerous applications in finance discussed well in Brooks Chp8 and Cuthbertson& NitzscheChp29).Warwick Business School 29
References
Bollerslev(1986), A generalized autoregressive conditional heteroscedasticity, Journal of Econometrics, 31, 307-327.
Brooks (2002), Introductory econometrics for finance, CUP: Cambridge. Chp8**
http://www.ukthesis.org/thesis_sample/Cuthbertsonand Nitzsche(2004) Quantitative financial economics: stocks, bonds and foreign exchange, Wiley: Chichester. Chps29.1-3**
Engle (1982), Autoregressive conditional heteroscedasticitywith estimates of the variance of UK inflation, Econometrica, 50, 987-1008.
**Key referencesWarwick Business School 30
Appendix: A simple test for ARCH effects
ARCH LM TestStep 1: Estimate the mean equationStep 2: Regress on qlagged values of itself Test using the LM statisticReject the null (no ARCH effects) if the LM stat exceeds the 5% critical value of the distribution. ttrεμˆˆ+=2ˆtεtqtqttuˆˆ...ˆˆ221102++++=−−εγεγγε0...:10===qHγγ22~qaTRχ2qχ