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VOLATILITY MODELS

VOLATILITY MODELS. GARCH. GENERALIZED- more general than ARCH model AUTOREGRESSIVE-depends on its own past CONDITIONAL-variance depends upon past information HETEROSKEDASTICITY- fancy word for non-constant variance. HISTORY. I DEVELOPED THE ARCH MODEL WHEN I WAS VISITING LSE IN 1979

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VOLATILITY MODELS

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  1. VOLATILITY MODELS

  2. GARCH • GENERALIZED- more general than ARCH model • AUTOREGRESSIVE-depends on its own past • CONDITIONAL-variance depends upon past information • HETEROSKEDASTICITY- fancy word for non-constant variance

  3. HISTORY • I DEVELOPED THE ARCH MODEL WHEN I WAS VISITING LSE IN 1979 • IT WAS PUBLISHED IN 1982 WITH MACRO APPLICATION - THE VARIANCE OF UK INFLATION • TIM BOLLERSLEV DEVELOPED THE GARCH GENERALIZATION AS MY PHD STUDENT - PUBLISHED IN 1986

  4. The GARCH Model • The variance of rt is a weighted average of three components • a constant or unconditional variance • yesterday’s forecast • yesterday’s news

  5. PARAMETER ESTIMATION • Historical data reveals when volatilities were large and the process of volatility • Pick parameters to match the historical volatility episodes • Maximum Likelihood is a systematic approach: • Max

  6. DIAGNOSTIC CHECKING • Time varying volatility is revealed by volatility clusters • These are measured by the Ljung Box statistic on squared returns • The standardized returns no longer should show significant volatilty clustering • Best models will minimize AIC and Schwarz criteria

  7. THEOREMS • GARCH MODELS WITH GAUSSIAN SHOCKS HAVE EXCESS KURTOSIS • FORECASTS OF GARCH(1,1) ARE MONOTONICALLY INCREASING OR DECREASING IN HORIZON

  8. FORECASTING WITH GARCH • GARCH(1,1) can be written as ARMA(1,1) • The autoregressive coefficient is • The moving average coefficient is

  9. GARCH(1,1) Forecasts

  10. Monotonic Term Structure of Volatility

  11. FORECASTING AVERAGE VOLATILITY • Annualized Vol=square root of 252 times the average daily standard deviation • Assume that returns are uncorrelated.

  12. TWO YEARS TERM STRUCTURE OF PORT

  13. Variance Targeting • Rewriting the GARCH model • where is easily seen to be the unconditional or long run variance • this parameter can be constrained to be equal to some number such as the sample variance. MLE only estimates the dynamics

  14. The Component Model • Engle and Lee(1999) • q is long run component and (h-q) is transitory • volatility mean reverts to a slowly moving long run component

  15. GARCH(p,q)

  16. The Leverage Effect -Asymmetric Models • Engle and Ng(1993) following Nelson(1989) • News Impact Curve relates today’s returns to tomorrows volatility • Define d as a dummy variable which is 1 for down days

  17. NEWS IMPACT CURVE

  18. Other Asymmetric Models

  19. PARTIALLY NON-PARAMETRICENGLE AND NG(1993)

  20. EXOGENOUS VARIABLES IN A GARCH MODEL • Include predetermined variables into the variance equation • Easy to estimate and forecast one step • Multi-step forecasting is difficult • Timing may not be right

  21. EXAMPLES • Non-linear effects • Deterministic Effects • News from other markets • Heat waves vs. Meteor Showers • Other assets • Implied Volatilities • Index volatility • MacroVariables or Events

  22. WHAT IS THE BEST MODEL? • The most reliable and robust is GARCH(1,1) • For short term forecasts, this is good enough. • For long term forecasts, a component model with leverage is often needed. • A model with economic causal variables is the ideal

  23. Component with Leverage

  24. Procter and Gamble Daily Returns 2/89-3/99Variance Equation Garch(1,1) C 5.11E-06 1.24E-06 4.112268 ARCH(1) 0.047402 0.006681 7.094764 GARCH(1) 0.927946 0.011114 83.49235 AIC -5.7086 , SCHWARZ CRITERION -5.699617

  25. CORRELOGRAM OF SQUARED RESIDUALS AC PAC Q-Stat Prob 1 0.030 0.030 2.3720 0.124 2 -0.018 -0.019 3.1992 0.202 3 -0.010 -0.009 3.4859 0.323 4 0.018 0.018 4.3016 0.367 5 -0.011 -0.013 4.6317 0.462 6 -0.002 -0.001 4.6408 0.591 7 0.013 0.013 5.0865 0.649 8 -0.016 -0.017 5.7164 0.679 9 0.010 0.012 5.9973 0.740 10 0.003 0.002 6.0183 0.814

  26. P&G TARCH C 0.0000 0.0000 4.6121 0.0000 ARCH(1) 0.0269 0.0093 2.9062 0.0037 (RESID<0)*ARCH(1)0.0520 0.0141 3.6976 0.0002 GARCH(1) 0.9123 0.0139 65.8060 0.0000 AIC -5.7114 SCHWARZ CRITERION -5.7002

  27. P&G EGARCH C -0.3836 0.0672 -5.7052 |RES|/SQR[GARCH](1) 0.1186 0.0153 7.7645 RES/SQR[GARCH](1) -0.0392 0.0103 -3.8195 EGARCH(1) 0.9656 0.0070 137.9063 AIC -5.7114 SCHWARZ CRITERION -5.7002

  28. P&G Asymmetric Component Perm: C 0.0002 0.0000 14.1047 Perm: [Q-C] 0.9835 0.0049 201.5877 Perm: [ARCH-GARCH] 0.0335 0.0079 4.2577 Tran: [ARCH-Q] -0.0361 0.018 -2.0045 Tran: (RES<0)*[ARCH-Q] 0.0910 0.0213 4.2838 Tran: [GARCH-Q] 0.8063 0.0819 9.8403 AIC -5.7132, SCHWARZ CRITERION -5.6974

  29. P&G VOLATILITIES

  30. DISCUSS GAUSSIAN ASSUMPTION • EMPIRICAL EVIDENCE INDICATES THAT INNOVATIONS ARE LEPTOKURTIC • GAUSSIAN GARCH IS QMLE • CONSISTENT BUT NEEDS BOLLERSLEV WOOLDRIDGE STANDARD ERRORS • T-DISTRIBUTION MAY BE MORE EFFICIENT • CAN DO SEMI-PARAMETRIC ESTIMATOR OF ENGLE AND GONZALES-RIVERA

  31. Bollerslev Wooldridge Standard Errors ROBUST TO NON-NORMAL ERRORS Perm: C 0.0002 0.0000 8.3857 Perm: [Q-C] 0.9835 0.0081 121.9762 Perm: [ARCH-GARCH] 0.0335 0.0107 3.1264 Tran: [ARCH-Q] -0.036 0.0242 -1.4898 Tran: (RES<0)*[ARCH-Q] 0.0910 0.0429 2.1220 Tran: [GARCH-Q] 0.8063 0.1223 6.5919

  32. RISK PREMIA • WHEN RISK IS GREATER, EXPECTED RETURNS SHOULD BE GREATER • HOW MUCH? • WHAT COUNTS AS RISK? • CAPM GIVES AN ANSWER • MULTI-BETA GIVES ANOTHER • PRICING KERNEL COVERS ALL

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