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16;⎝⎛=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
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. Ident