"Frontmatter". In: Analysis of Financial Time Series

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tn-0.2 0.0 0.2 0.4••• •••••••••• •• •••• • • •• • •••••• •• ••• • •••• •••• •• ••• •• ••• •••••••• ••••• • •• ••••••• • • • • •••• • • • • • • ••• • •••• • •••• • • • ••• • •• • • • • • • • • •• • ••• •• ••••• • • •• • ••••• • •• ••• • •• • ••• •• ••• •• • • •• • •• •• •••• ••••• ••••• • • ••• • • ••• • • •• ••• •••••• • • •• • •• •• • • ••• • • • •••• •• ••••• ••• • •• • • •• • ••• • •• • • • • • • • •••• ••• • •••• • •••••••• •••0 200 400 600 800time indexFigure 3.5. Time series plot of the monthly excess returns of S&P 500 index.Series : spACF0.0 0.4 0.80 5 10 15 20 25LagSeries : spsqPartial ACF-0.1 0.0 0.1 0.2 0.30 5 10 15 20 25LagFigure 3.6. (a) Sample ACF of the monthly excess returns of S&P 500 index, and (b) samplePACF of the squared monthly excess returns.96
THE GARCH MODEL 97The fitted AR(3) model, under the normality assumption, isr t = 0.088r t−1 − 0.023r t−2 − 0.123r t−3 + 0.0066 + a t , ˆσ 2 a= 0.00333. (3.17)For the GARCH effects, we use the GARCH(1, 1) modela t = σ t ɛ t , σ 2t = α 0 + β 1 σ 2t−1 + α 1a 2 t−1 .A joint estimation of the AR(3)-GARCH(1, 1) model givesr t = 0.021r t−1 − 0.034r t−2 − 0.013r t−3 + 0.0085 + a tσ 2t = 0.000099 + 0.8476σ 2t−1 + 0.1219a2 t−1 .From the volatility equation, the implied unconditional variance of a t is0.0000991 − 0.8476 − 0.1219 = 0.00325,which is very close to that of Eq. (3.17). However, t ratios of the parameters inthe mean equation suggest that all AR coefficients are insignificant at the 5% level.Therefore, we refine the model by dropping all AR coefficients. The refined modelisr t = 0.0065 + a t , σ 2t = 0.00014 + 0.8220σ 2t−1 + 0.1352a2 t−1 . (3.18)The standard error of the parameter in the mean equation is 0.0015, whereas thoseof the parameters in the volatility equation are 0.00002, 0.0208, and 0.0166, respectively.The unconditional variance of a t is 0.0001/(1 − 0.822 − 0.1352) = 0.00324.This is a simple stationary GARCH(1, 1) model. Figure 3.7 shows the estimatedvolatility process and the standardized shocks ã t = a t /σ t for the GARCH(1, 1)model in Eq. (3.18). The ã t series looks like a white noise process. Figure 3.8 providesthe sample ACF of the standardized shocks ã t and the squared process ãt 2.These ACFs fail to suggest any significant serial correlations in the two processes.More specifically, we have Q(10) = 10.32(0.41) and Q(20) = 22.66(0.31) for ã t ,and Q(10) = 8.83(0.55) and Q(20) = 15.82(0.73) for ãt 2 , where the number inparentheses is the p value of the test statistic. Thus, the model appears to be adequate.Note that the fitted model shows ˆα 1 + ˆβ 1 = 0.9572, which is close to 1. Thisphenomenon is commonly observed in practice and it leads to imposing the constraintα 1 + β 1 = 1inaGARCH(1, 1) model, resulting in an integrated GARCH (orIGARCH) model; see Section 3.5.Finally, to forecast the volatility of monthly excess returns of S&P500 index, wecan use the volatility equation in Eq. (3.18). For instance, at the forecast origin h,wehave σh+1 2 = 0.00014 + 0.822σ h 2 + 0.1352a2 h. The 1-step ahead forecast is then
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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
Page 53 and 54: SIMPLE MOVING-AVERAGE MODELS 45s-rt
Page 55 and 56: SIMPLE MOVING-AVERAGE MODELS 47The
Page 57 and 58: SIMPLE ARMA MODELS 49A time series
Page 59 and 60: SIMPLE ARMA MODELS 51operator, the
Page 61 and 62: SIMPLE ARMA MODELS 53Table 2.5. Sam
Page 63 and 64: SIMPLE ARMA MODELS 55This represent
Page 65 and 66: UNIT-ROOT NONSTATIONARITY 57ˆp h (
Page 67 and 68: UNIT-ROOT NONSTATIONARITY 59log-pri
Page 69 and 70: SEASONAL MODELS 61which is commonly
Page 71 and 72: SEASONAL MODELS 63Series : xSeries
Page 73 and 74: SEASONAL MODELS 65models use the sa
Page 75 and 76: • ••• •••••••
Page 77 and 78: REGRESSION MODELS WITH TIME SERIES
Page 79 and 80: REGRESSION MODELS WITH TIME SERIES
Page 81 and 82: LONG-MEMORY MODELS 73=(k − d −
Page 83 and 84: APPENDIX A: SOME SCA COMMANDS 75est
Page 85 and 86: EXERCISES 77relation. Draw your con
Page 87 and 88: Analysis of Financial Time Series.
Page 89 and 90: STRUCTURE OF A MODEL 813.2 STRUCTUR
Page 91 and 92: THE ARCH MODEL 83corrected asset re
Page 93 and 94: THE ARCH MODEL 85Series : xACF0.0 0
Page 95 and 96: THE ARCH MODEL 87sample mean from t
Page 97 and 98: THE ARCH MODEL 89where Ɣ(x) is the
Page 99 and 100: THE ARCH MODEL 91Series : resiSerie
Page 101 and 102: THE GARCH MODEL 93other softwares a
Page 103: THE GARCH MODEL 95ahead forecast ca
Page 107 and 108: THE GARCH MODEL 99Series : stdatACF
Page 109 and 110: THE GARCH-M MODEL 101two models. Th
Page 111 and 112: THE EXPONENTIAL GARCH MODEL 103To b
Page 113 and 114: THE EXPONENTIAL GARCH MODEL 105eros
Page 115 and 116: THE CHARMA MODEL 107ˆσ 2 864 (3)
Page 117 and 118: RANDOM COEFFICIENT AUTOREGRESSIVE M
Page 119 and 120: THE LONG-MEMORY STOCHASTIC VOLATILI
Page 121 and 122: AN ALTERNATIVE APPROACH 113where Va
Page 123 and 124: APPLICATION 115(a) IBMlog-rtn-30 -1
Page 125 and 126: APPLICATION 117equationThe fitted m
Page 127 and 128: KURTOSIS OF GARCH MODELS 119where K
Page 129 and 130: SOME RATS PROGRAMS FOR ESTIMATING V
Page 131 and 132: EXERCISES 123• Is there evidence
Page 133 and 134: REFERENCES 125Melino, A., and Turnb
Page 135 and 136: NONLINEAR TIME SERIES 127To put non
Page 137 and 138: NONLINEAR MODELS 129Example 4.1. Co
Page 139 and 140: NONLINEAR MODELS 131jumps when it b
Page 141 and 142: NONLINEAR MODELS 133the series and
Page 143 and 144: NONLINEAR MODELS 135els to the mont
Page 145 and 146: NONLINEAR MODELS 137growth-2 -1 0 1
Page 147 and 148: NONLINEAR MODELS 139average of y t
Page 149 and 150: NONLINEAR MODELS 141indicating that
Page 151 and 152: NONLINEAR MODELS 143s T,2t=1T∑K h
Page 153 and 154: 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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FACTOR ANALYSIS 341(a) 5 stock retu
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FACTOR ANALYSIS 343andCov(r, F) = E
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FACTOR ANALYSIS 345matrix P to tran
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FACTOR ANALYSIS 347Example 8.9. In
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APPENDIX A. REVIEW OF VECTORS AND M
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APPENDIX A. REVIEW OF VECTORS AND M
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APPENDIX B. MULTIVARIATE NORMAL DIS
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REFERENCES 355(b) Build a bivariate
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REPARAMETERIZATION 359To illustrate
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REPARAMETERIZATION 361where ⊥ den
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GARCH MODELS FOR BIVARIATE RETURNS
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VECTOR VOLATILITY MODELS 377using t
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VECTOR VOLATILITY MODELS 379large-c
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VECTOR VOLATILITY MODELS 381(a) S&P
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FACTOR-VOLATILITY MODELS 383conside
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APPLICATION 385[ ]σ11,t=σ 22,t⎡
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MULTIVARIATE t DISTRIBUTION 387σ 1
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APPENDIX A. SOME REMARKS ON ESTIMAT
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APPENDIX A. SOME REMARKS ON ESTIMAT
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REFERENCES 393(a) Compute the sampl
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GIBBS SAMPLING 397mean adding auxil
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BAYESIAN INFERENCE 399distributions
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BAYESIAN INFERENCE 401normal with m
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ALTERNATIVE ALGORITHMS 403distribut
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ALTERNATIVE ALGORITHMS 40510.4.2 Me
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REGRESSION WITH SERIAL CORRELATIONS
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REGRESSION WITH SERIAL CORRELATIONS
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MISSING VALUES AND OUTLIERS 411y h
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MISSING VALUES AND OUTLIERS 413Cons
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MISSING VALUES AND OUTLIERS 415we h
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MISSING VALUES AND OUTLIERS 417c3t-
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STOCHASTIC VOLATILITY MODELS 419tha
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STOCHASTIC VOLATILITY MODELS 421whe
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STOCHASTIC VOLATILITY MODELS 423den
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STOCHASTIC VOLATILITY MODELS 425The
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STOCHASTIC VOLATILITY MODELS 427(a)
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•••MARKOV SWITCHING MODELS 42
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MARKOV SWITCHING MODELS 431(a) GARC
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MARKOV SWITCHING MODELS 433β i ∼
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MARKOV SWITCHING MODELS 435(a) mont
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MARKOV SWITCHING MODELS 437α ij ar
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FORECASTING 439garchrtn-sq0 200 400
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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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