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392 MULTIVARIATE VOLATILITY MODELSfrml gdet = -0.5*(log(h1(t) = v1(t))+ log(h2(t)=v2(t)) $+log(h3(t)=v3(t)))frml garchln = gdet-0.5*(a1t(t)**2/h1(t)+bt(t)**2/h2(t) $+b1t(t)**2/h3(t))smpl 8 2275compute c1 = 0.07, c2 = 0.33, c3 = 0.19, p1 = 0.1, p3 = -0.04compute p21 =0.2, p22 = -0.1, p31 = -0.26, p33 = 0.06compute a0 = .01, a1 = 0.05, a2 = 0.94compute t0 = 0.28, t1 =0.82, t2 = -0.035compute b0 = .17, b1 = 0.08, b2 = 0.89compute u0= 0.04, u1 = 0.97, u2 = 0.01compute w0 =0.006, w1=0.98, w2=0.004compute d0 =1.38, d1 = 0.06, d2 = 0.64, d5 = -0.027nlpar(criterion=value,cvcrit=0.00001)maximize(method=bhhh,recursive,iterations=250) garchlnset fv1 = v1(t)set resi1 = a1t(t)/sqrt(fv1(t))set residsq = resi1(t)*resi1(t)* Checking standardized residuals *cor(qstats,number=12,span=4) resi1* Checking squared standardized residuals *cor(qstats,number=12,span=4) residsqset fv2 = v2(t)+q1t(t)**2*v1(t)set resi2 = a2t(t)/sqrt(fv2(t))set residsq = resi2(t)*resi2(t)* Checking standardized residuals *cor(qstats,number=12,span=4) resi2* Checking squared standardized residuals *cor(qstats,number=12,span=4) residsqset fv3 = v3(t)+q2t(t)**2*v1(t)+q3t(t)**2*v2(t)set resi3 = a3t(t)/sqrt(fv3(t))set residsq = resi3(t)*resi3(t)* Checking standardized residuals *cor(qstats,number=12,span=4) resi3* Checking squared standardized residuals *cor(qstats,number=12,span=4) residsq* print standardized residuals and correlation-coefficientsset rho21 = q1t(t)*sqrt(v1(t)/fv2(t))set rho31 = q2t(t)*sqrt(v1(t)/fv3(t))set rho32 = (q2t(t)*q1t(t)*v1(t) $+q3t(t)*v2(t))/sqrt(fv2(t)*fv3(t))print 10 2275 resi1 resi2 resi3print 10 2275 rho21 rho31 rho32print 10 2275 fv1 fv2 fv3EXERCISES1. The file “m-spibmge.dat” contains the monthly log returns in percentages of S&P500 index, IBM stock, and General Electric stock from January 1926 to December1999. The returns include dividends.
REFERENCES 393(a) Compute the sample mean, sample covariance matrix, and sample correlationmatrix of the three return series.(b) Compute the lag-1 and lag-2 cross-correlation matrixes of the three series.Draw inference concerning the linear relationships between the three series.Is there any lead-lag relation?(c) Use the multivariate Ljung–Box statistics to test the null hypothesis that H 0 :ρ 1 =···=ρ 4 = 0 at the 5% significance level. Draw your conclusion.2. Focus on the monthly log returns in percentages of GE stock and the S&P 500index. Build a constant correlation GARCH model for the bivariate series. Checkthe adequacy of the fitted model, and obtain 1-step ahead forecast of the covariancematrix at the forecast origin December 1999.3. Focus on the monthly log returns in percentages of GE stock and the S&P 500index. Build a time-varying correlation GARCH model for the bivariate seriesusing a logistic function for the correlation coefficient. Check the adequacy ofthe fitted model, and obtain 1-step ahead forecast of the covariance matrix at theforecast origin December 1999.4. Focus on the monthly log returns in percentages of GE stock and the S&P 500index. Build a time-varying correlation GARCH model for the bivariate seriesusing the Cholesky decomposition. Check the adequacy of the fitted model, andobtain 1-step ahead forecast of the covariance matrix at the forecast origin December1999. Compare the model with the other two models built in the previousquestions.5. Consider the three-dimensional return series jointly. Build a multivariate timevaryingcorrelation volatility model for the data, using the Cholesky decomposition.Discuss the implications of the model and compute 1-step ahead volatilityforecast at the forecast origin t = 888.6. An investor is interested in daily Value at Risk of his position on holding long$0.5 million of Dell stock and $1 million of Cisco Systems stock. Use 5% criticalvalues and the daily log returns from January 2, 1990 to December 31, 1999 todo the calculation. The data are in the file “d-dellcsco9099.dat.” Apply the threeapproaches to volatility modeling in Section 9.5 and compare the results.REFERENCESBollerslev, T. (1990), “Modeling the coherence in short-term nominal exchange rates: A multivariategeneralized ARCH approach,” Review of Economics and Statistics, 72, 498–505.Chib, S., Nardari, F., and Shephard, N. (1999), “Analysis of high dimensional multivariatestochastic volatility models,” Working paper, Washington University, St Louis.Harvey, A., Ruiz, E., and Shephard, N. (1995), “Multivariate stochastic variance models,” inARCH Selected Readings, ed. R. F. Engle, pp. 253–276, Oxford University Press: Oxford,UK.
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Analysis of FinancialTime SeriesFin
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ContentsPrefacexi1. Financial Time
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CONTENTSix6.6 Black-Scholes Pricing
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PrefaceThis book grew out of an MBA
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Analysis of Financial Time Series.
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ASSET RETURNS 3Multiperiod Simple R
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ASSET RETURNS 5r t [k] =ln(1 + R t
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DISTRIBUTIONAL PROPERTIES OF RETURN
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10 FINANCIAL TIME SERIES AND THEIR
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REFERENCES 21Federal Reserve Bank o
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CORRELATION AND AUTOCORRELATION FUN
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CORRELATION AND AUTOCORRELATION FUN
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WHITE NOISE AND LINEAR TIME SERIES
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SIMPLE AUTOREGRESSIVE MODELS 29whic
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SIMPLE AUTOREGRESSIVE MODELS 31wher
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SIMPLE AUTOREGRESSIVE MODELS 33wher
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SIMPLE AUTOREGRESSIVE MODELS 35expa
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SIMPLE AUTOREGRESSIVE MODELS 37Tabl
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SIMPLE AUTOREGRESSIVE MODELS 39Mode
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SIMPLE AUTOREGRESSIVE MODELS 41The
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SIMPLE MOVING-AVERAGE MODELS 43wher
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SIMPLE MOVING-AVERAGE MODELS 45s-rt
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SIMPLE MOVING-AVERAGE MODELS 47The
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SIMPLE ARMA MODELS 49A time series
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SIMPLE ARMA MODELS 51operator, the
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SIMPLE ARMA MODELS 53Table 2.5. Sam
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SIMPLE ARMA MODELS 55This represent
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UNIT-ROOT NONSTATIONARITY 57ˆp h (
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UNIT-ROOT NONSTATIONARITY 59log-pri
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SEASONAL MODELS 61which is commonly
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SEASONAL MODELS 63Series : xSeries
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SEASONAL MODELS 65models use the sa
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• ••• •••••••
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REGRESSION MODELS WITH TIME SERIES
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REGRESSION MODELS WITH TIME SERIES
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LONG-MEMORY MODELS 73=(k − d −
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APPENDIX A: SOME SCA COMMANDS 75est
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EXERCISES 77relation. Draw your con
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STRUCTURE OF A MODEL 813.2 STRUCTUR
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THE ARCH MODEL 83corrected asset re
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THE ARCH MODEL 85Series : xACF0.0 0
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THE ARCH MODEL 87sample mean from t
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THE ARCH MODEL 89where Ɣ(x) is the
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THE ARCH MODEL 91Series : resiSerie
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THE GARCH MODEL 93other softwares a
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THE GARCH MODEL 95ahead forecast ca
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THE GARCH MODEL 97The fitted AR(3)
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THE GARCH MODEL 99Series : stdatACF
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THE GARCH-M MODEL 101two models. Th
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THE EXPONENTIAL GARCH MODEL 103To b
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THE EXPONENTIAL GARCH MODEL 105eros
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THE CHARMA MODEL 107ˆσ 2 864 (3)
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RANDOM COEFFICIENT AUTOREGRESSIVE M
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THE LONG-MEMORY STOCHASTIC VOLATILI
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AN ALTERNATIVE APPROACH 113where Va
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APPLICATION 115(a) IBMlog-rtn-30 -1
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APPLICATION 117equationThe fitted m
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KURTOSIS OF GARCH MODELS 119where K
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SOME RATS PROGRAMS FOR ESTIMATING V
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EXERCISES 123• Is there evidence
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REFERENCES 125Melino, A., and Turnb
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NONLINEAR TIME SERIES 127To put non
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NONLINEAR MODELS 129Example 4.1. Co
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NONLINEAR MODELS 131jumps when it b
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NONLINEAR MODELS 133the series and
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NONLINEAR MODELS 135els to the mont
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NONLINEAR MODELS 137growth-2 -1 0 1
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NONLINEAR MODELS 139average of y t
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NONLINEAR MODELS 141indicating that
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NONLINEAR MODELS 143s T,2t=1T∑K h
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NONLINEAR MODELS 145f i (.) of Eq.
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NONLINEAR MODELS 1474.1.9.1 Feed-Fo
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NONLINEAR MODELS 1494.1.9.2 Trainin
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NONLINEAR MODELS 151prob0.0 0.2 0.4
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NONLINEARITY TESTS 153idea behind v
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NONLINEARITY TESTS 155C l (δ, T )
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NONLINEARITY TESTS 157and obtain th
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NONLINEARITY TESTS 159• Step 2: C
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FORECASTING 161returns. In summary,
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FORECASTING 163in m 11 and m 22 ind
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APPLICATION 165unem. rate4 6 8 10
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APPLICATION 167Table 4.4. Out-of-Sa
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APPENDIX A 169compute a0 = 0.1, a1
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REFERENCES 171P(s t = 2 | s t−1 =
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REFERENCES 173Fan, J. (1993), “Lo
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NONSYNCHRONOUS TRADING 177t − k t
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BID-ASK SPREAD 179The lag-1 autocov
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EMPIRICAL CHARACTERISTICS 181The ma
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n(trades)0 20406080 1200 1000 2000
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EMPIRICAL CHARACTERISTICS 185after-
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MODELS FOR PRICE CHANGES 187deep un
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MODELS FOR PRICE CHANGES 189σ 2i (
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MODELS FOR PRICE CHANGES 191A i = 1
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MODELS FOR PRICE CHANGES 193( )piln
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DURATION MODELS 195For the IBM data
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DURATION MODELS 197(a) is the avera
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DURATION MODELS 199actions might no
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DURATION MODELS 201Series : xACF0.0
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DURATION MODELS 203reduces to that
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DURATION MODELS 205Series : xACF0.0
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THE PCD MODEL 207where w(α) denote
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THE PCD MODEL 209(a) All transactio
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THE PCD MODEL 2110 10 20 30 40 500
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THE PCD MODEL 213⎧⎪⎨1f (x |
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THE PCD MODEL 215Generalized Gamma
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THE PCD MODEL 217smpl 2 3534compute
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REFERENCES 219(a) Use the data to c
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STOCHASTIC PROCESSES 223write a con
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STOCHASTIC PROCESSES 2253. stationa
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ITO’S LEMMA 2276.3.2 Stochastic D
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ITO’S LEMMA 229This result shows
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DISTRIBUTIONS OF PRICE AND RETURN 2
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BLACK-SCHOLES EQUATION 233and G t =
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BLACK-SCHOLES FORMULA 235risk prefe
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BLACK-SCHOLES FORMULA 237The stock
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BLACK-SCHOLES FORMULA 239(a) Call o
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AN EXTENSION OF ITO’S LEMMA 2410
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STOCHASTIC INTEGRAL 243using the It
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JUMP DIFFUSION MODELS 245where w t
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JUMP DIFFUSION MODELS 247where it i
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JUMP DIFFUSION MODELS 249sudden jum
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APPENDIX A 251Pricing formulas for
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EXERCISES 253which involves the CDF
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REFERENCES 255Bakshi, G., Cao, C.,
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VALUEATRISK 257log return-0.2 -0.1
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RISKMETRICS 259we use log returns r
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RISKMETRICS 261investor is$10,000,0
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ECONOMETRIC APPROACH TO VAR 263Cons
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ECONOMETRIC APPROACH TO VAR 265The
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QUANTILE ESTIMATION 267Using the fo
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QUANTILE ESTIMATION 269ˆx 0.01 = p
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EXTREME VALUE THEORY 271= 1 −= 1
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EXTREME VALUE THEORY 273shows that
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EXTREME VALUE THEORY 275{ [] }r n(i
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EXTREME VALUE THEORY 2777.5.3 Appli
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EXTREME VALUE TO VAR 2797.6 AN EXTR
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EXTREME VALUE TO VAR 281VaR =−2.5
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EXTREME VALUE TO VAR 283stock. The
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A NEW APPROACH TO VAR 285series (e.
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A NEW APPROACH TO VAR 287In Eq. (7.
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A NEW APPROACH TO VAR 289Example 7.
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A NEW APPROACH TO VAR 291feature is
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A NEW APPROACH TO VAR 293(a) Homoge
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A NEW APPROACH TO VAR 295Table 7.4.
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REFERENCES 297a Gaussian GARCH(1, 1
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CROSS-CORRELATION 301correlation co
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CROSS-CORRELATION 303where ̂D is t
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CROSS-CORRELATION 305Table 8.1. Sum
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30yr20yr10yr5yr1yr-10 0 510-10 0 51
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VAR MODELS 3098.2 VECTOR AUTOREGRES
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VAR MODELS 311where G = Cov(b t ).
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VAR MODELS 313where φ 0 and a t ar
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VAR MODELS 315To specify the order
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VAR MODELS 317the S&P 500 index. Th
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VECTOR MA MODELS 319[ ] [ ] [ ] [r1
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VECTOR MA MODELS 321turn used to co
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VECTOR ARMA MODELS 323For the VAR(1
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VECTOR ARMA MODELS 325log-rate0.0 0
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VECTOR ARMA MODELS 327interest-rate
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CO-INTEGRATION 329Series : x1ACF0.0
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CO-INTEGRATION 331Stock Exchange; o
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THRESHOLD CO-INTEGRATION 333ously b
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PRINCIPAL COMPONENT ANALYSIS 3358.6
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PRINCIPAL COMPONENT ANALYSIS 337(c
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PRINCIPAL COMPONENT ANALYSIS 339(a)
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Page 351 and 352: FACTOR ANALYSIS 343andCov(r, F) = E
Page 353 and 354: FACTOR ANALYSIS 345matrix P to tran
Page 355 and 356: FACTOR ANALYSIS 347Example 8.9. In
Page 357 and 358: APPENDIX A. REVIEW OF VECTORS AND M
Page 359 and 360: APPENDIX A. REVIEW OF VECTORS AND M
Page 361 and 362: APPENDIX B. MULTIVARIATE NORMAL DIS
Page 363 and 364: REFERENCES 355(b) Build a bivariate
Page 365 and 366: Analysis of Financial Time Series.
Page 367 and 368: REPARAMETERIZATION 359To illustrate
Page 369 and 370: REPARAMETERIZATION 361where ⊥ den
Page 371 and 372: GARCH MODELS FOR BIVARIATE RETURNS
Page 385 and 386: VECTOR VOLATILITY MODELS 377using t
Page 387 and 388: VECTOR VOLATILITY MODELS 379large-c
Page 389 and 390: VECTOR VOLATILITY MODELS 381(a) S&P
Page 391 and 392: FACTOR-VOLATILITY MODELS 383conside
Page 393 and 394: APPLICATION 385[ ]σ11,t=σ 22,t⎡
Page 395 and 396: MULTIVARIATE t DISTRIBUTION 387σ 1
Page 397 and 398: APPENDIX A. SOME REMARKS ON ESTIMAT
Page 399: APPENDIX A. SOME REMARKS ON ESTIMAT
Page 403 and 404: Analysis of Financial Time Series.
Page 405 and 406: GIBBS SAMPLING 397mean adding auxil
Page 407 and 408: BAYESIAN INFERENCE 399distributions
Page 409 and 410: BAYESIAN INFERENCE 401normal with m
Page 411 and 412: ALTERNATIVE ALGORITHMS 403distribut
Page 413 and 414: ALTERNATIVE ALGORITHMS 40510.4.2 Me
Page 415 and 416: REGRESSION WITH SERIAL CORRELATIONS
Page 417 and 418: REGRESSION WITH SERIAL CORRELATIONS
Page 419 and 420: MISSING VALUES AND OUTLIERS 411y h
Page 421 and 422: MISSING VALUES AND OUTLIERS 413Cons
Page 423 and 424: MISSING VALUES AND OUTLIERS 415we h
Page 425 and 426: MISSING VALUES AND OUTLIERS 417c3t-
Page 427 and 428: STOCHASTIC VOLATILITY MODELS 419tha
Page 429 and 430: STOCHASTIC VOLATILITY MODELS 421whe
Page 431 and 432: STOCHASTIC VOLATILITY MODELS 423den
Page 433 and 434: STOCHASTIC VOLATILITY MODELS 425The
Page 435 and 436: STOCHASTIC VOLATILITY MODELS 427(a)
Page 437 and 438: •••MARKOV SWITCHING MODELS 42
Page 439 and 440: MARKOV SWITCHING MODELS 431(a) GARC
Page 441 and 442: MARKOV SWITCHING MODELS 433β i ∼
Page 443 and 444: MARKOV SWITCHING MODELS 435(a) mont
Page 445 and 446: MARKOV SWITCHING MODELS 437α ij ar
Page 447 and 448: FORECASTING 439garchrtn-sq0 200 400
Page 449 and 450: EXERCISES 441eter uncertainty in pr
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REFERENCES 443Justel, A., Peña, D.
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446 INDEXData (cont.)equal-weighted
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448 INDEXPoisson process, 244inhomo
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