"Frontmatter". In: Analysis of Financial Time Series

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196 HIGH-FREQUENCY DATATime], and f 7 (t i ) is the indicator for the last 30 minutes of daily trading [i.e., f 7 (t i ) =1 if and only if the trade occurred between 3:30 pm and 4:00 pm Eastern Time].Figure 5.5 shows the plot of f i (.) for i = 1,...,4, where the time scales in thex-axis is in minutes. Note that f 3 (43,200) = f 4 (43,200), where 43,200 correspondsto 12:00 noon.The coefficients β j of Eq. (5.32) are obtained by the least squares method of thelinear regressionThe fitted model isln(t i ) = β 0 +7∑β j f j (t i ) + ɛ i .j=1ln(̂t i ) = 2.555 + 0.159 f 1 (t i ) + 0.270 f 2 (t i ) + 0.384 f 3 (t i )+ 0.061 f 4 (t i ) − 0.611 f 5 (t i ) − 0.157 f 6 (t i ) + 0.073 f 7 (t i ).Figure 5.6 shows the time plot of average durations in 5-minute time intervals overthe 63 trading days before and after adjusting for the deterministic component. Part(a)(b)ave-dur10 20 30 400 10 20 30 40 50 60 70 805-minute intervalsave-dur2.0 2.2 2.4 2.6 2.8 3.0 3.20 10 20 30 40 50 60 70 805-minute intervalsFigure 5.6. IBM transactions data from 11/01/90 to 1/31/91: (a) The average durations in 5-minute time intervals, and (b) the average durations in 5-minute time intervals after adjustingfor the deterministic component.
DURATION MODELS 197(a) is the average durations of t i and, as expected, it exhibits a diurnal pattern. Part(b) is the average durations of t ∗ i(i.e., after the adjustment), and the diurnal patternis largely removed.5.5.1 The ACD ModelThe autoregressive conditional duration (ACD) model uses the idea of GARCH modelsto study the dynamic structure of the adjusted duration ti ∗ of Eq. (5.31). For easein notation, we define x i = ti ∗.Let ψ i = E(x i | F i−1 ) be the conditional expectation of the adjusted durationbetween the (i − 1)th and ith trades, where F i−1 is the information set available atthe (i − 1)th trade. In other words, ψ i is the expected adjusted duration given F i−1 .The basic ACD model is defined asx i = ψ i ɛ i , (5.33)where {ɛ i } is a sequence of independent and identically distributed non-negative randomvariables such that E(ɛ i ) = 1. In Engle and Russell (1998), ɛ i follows a standardexponential or a standardized Weibull distribution, and ψ i assumes the formψ i = ω +r∑γ j x i− j +j=1s∑ω j ψ i− j . (5.34)Such a model is referred to as an ACD(r, s) model. When the distribution of ɛ i isexponential, the resulting model is called an EACD(r, s) model. Similarly, if ɛ i followsa Weibull distribution, the model is a WACD(r, s) model. If necessary, readersare referred to Appendix A for a quick review of exponential and Weibull distributions.Similar to GARCH models, the process η i = x i − ψ i is a Martingale differencesequence [i.e., E(η i | F i−1 ) = 0], and the ACD(r, s) model can be written asj=1max(r,s) ∑x i = ω + (γ j + ω j )x i− j −j=1s∑ω j η i− j + η j , (5.35)which is in the form of an ARMA process with non-Gaussian innovations. It is understoodhere that γ j = 0for j > r and ω j = 0for j > s. Such a representation canbe used to obtain the basic conditions for weak stationarity of the ACD model. Forinstance, taking expectation on both sides of Eq. (5.35) and assuming weak stationarity,we haveE(x i ) =j=1ω1 − ∑ max(r,s)j=1(γ j + ω j ) .
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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
Page 153 and 154: NONLINEAR MODELS 145f i (.) of Eq.
Page 155 and 156: NONLINEAR MODELS 1474.1.9.1 Feed-Fo
Page 157 and 158: NONLINEAR MODELS 1494.1.9.2 Trainin
Page 159 and 160: NONLINEAR MODELS 151prob0.0 0.2 0.4
Page 161 and 162: NONLINEARITY TESTS 153idea behind v
Page 163 and 164: NONLINEARITY TESTS 155C l (δ, T )
Page 165 and 166: NONLINEARITY TESTS 157and obtain th
Page 167 and 168: NONLINEARITY TESTS 159• Step 2: C
Page 169 and 170: FORECASTING 161returns. In summary,
Page 171 and 172: FORECASTING 163in m 11 and m 22 ind
Page 173 and 174: APPLICATION 165unem. rate4 6 8 10
Page 175 and 176: APPLICATION 167Table 4.4. Out-of-Sa
Page 177 and 178: APPENDIX A 169compute a0 = 0.1, a1
Page 179 and 180: REFERENCES 171P(s t = 2 | s t−1 =
Page 181 and 182: REFERENCES 173Fan, J. (1993), “Lo
Page 183 and 184: Analysis of Financial Time Series.
Page 185 and 186: NONSYNCHRONOUS TRADING 177t − k t
Page 187 and 188: BID-ASK SPREAD 179The lag-1 autocov
Page 189 and 190: EMPIRICAL CHARACTERISTICS 181The ma
Page 191 and 192: n(trades)0 20406080 1200 1000 2000
Page 193 and 194: EMPIRICAL CHARACTERISTICS 185after-
Page 195 and 196: MODELS FOR PRICE CHANGES 187deep un
Page 197 and 198: MODELS FOR PRICE CHANGES 189σ 2i (
Page 199 and 200: MODELS FOR PRICE CHANGES 191A i = 1
Page 201 and 202: MODELS FOR PRICE CHANGES 193( )piln
Page 203: DURATION MODELS 195For the IBM data
Page 207 and 208: DURATION MODELS 199actions might no
Page 209 and 210: DURATION MODELS 201Series : xACF0.0
Page 211 and 212: DURATION MODELS 203reduces to that
Page 213 and 214: DURATION MODELS 205Series : xACF0.0
Page 215 and 216: THE PCD MODEL 207where w(α) denote
Page 217 and 218: THE PCD MODEL 209(a) All transactio
Page 219 and 220: THE PCD MODEL 2110 10 20 30 40 500
Page 221 and 222: THE PCD MODEL 213⎧⎪⎨1f (x |
Page 223 and 224: THE PCD MODEL 215Generalized Gamma
Page 225 and 226: THE PCD MODEL 217smpl 2 3534compute
Page 227 and 228: REFERENCES 219(a) Use the data to c
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Page 231 and 232: STOCHASTIC PROCESSES 223write a con
Page 233 and 234: STOCHASTIC PROCESSES 2253. stationa
Page 235 and 236: ITO’S LEMMA 2276.3.2 Stochastic D
Page 237 and 238: ITO’S LEMMA 229This result shows
Page 239 and 240: DISTRIBUTIONS OF PRICE AND RETURN 2
Page 241 and 242: BLACK-SCHOLES EQUATION 233and G t =
Page 243 and 244: BLACK-SCHOLES FORMULA 235risk prefe
Page 245 and 246: BLACK-SCHOLES FORMULA 237The stock
Page 247 and 248: BLACK-SCHOLES FORMULA 239(a) Call o
Page 249 and 250: AN EXTENSION OF ITO’S LEMMA 2410
Page 251 and 252: STOCHASTIC INTEGRAL 243using the It
Page 253 and 254: 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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