"Frontmatter". In: Analysis of Financial Time Series

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210 HIGH-FREQUENCY DATAµ i = ω 0 + ω 1 D i−1 + ω 2 ln(t i )∣ 4∑ ∣∣∣∣ln(σ i ) = βD i− j = β| D i−1 + D i−2 + D i−3 + D i−4 |.∣j=1In other words, D i is governed by the sign of a normal random variable with mean µ iand variance σi 2.A special characteristic of the prior model is the function for ln(σ i).For intraday transactions, a key feature is the price reversal between consecutiveprice changes. This feature is modeled by the dependence of D i on D i−1 in themean equation with a negative ω 1 parameter. However, there exists occasional localtrend in the price movement. The previous variance equation allows for such a localtrend by increasing the uncertainty in the direction of price movement when the pastdata showed evidence of a local trend. For a normal distribution with a fixed mean,increasing its variance makes a random draw have the same chance to be positiveand negative. This in turn increases the chance for a sequence of all positive or allnegative draws. Such a sequence produces a local trend in price movement.To allow for different dynamics between positive and negative price movements,we use different models for the size of a price change. Specifically, we haveS i | (D i =−1, N i ,t i , F i−1 ) ∼ p(λ d,i ) + 1, with (5.52)ln(λ d,i ) = η d,0 + η d,1 N i + η d,2 ln(t i ) + η d,3 S i−1S i | (D i = 1, N i ,t i , F i−1 ) ∼ p(λ u,i ) + 1, with (5.53)ln(λ u,i ) = η u,0 + η u,1 N i + η u,2 ln(t i ) + η u,3 S i−1 ,where p(λ) denotes a Poisson distribution with parameter λ, and 1 is added to thesize because the minimum size is 1 tick when there is a price change.The specified models in Eqs. (5.48)–(5.53) can be estimated jointly by either themaximum likelihood method or the Markov Chain Monte Carlo methods. Basedon Eq. (5.47), the models consist of six conditional models that can be estimatedseparately.Example 5.5. Consider the intraday transactions of IBM stock on November21, 1990. There are 194 price changes within the normal trading hours. Figure 5.15shows the histograms of ln(t i ), N i , D i ,andS i . The data for D i are about equallydistributed between “upward” and “downward” movements. Only a few transactionsresulted in a price change of more than 1 tick; as a matter of fact, there were sevenchanges with two ticks and one change with three ticks. Using Markov Chain MonteCarlo (MCMC) methods (see Chapter 10), we obtained the following models for thedata. The reported estimates and their standard deviations are the posterior meansand standard deviations of MCMC draws with 9500 iterations. The model for thetime duration between price changes isln(t i ) = 4.023 + 0.032 ln(t i−1 ) − 0.025S i−1 + 1.403ɛ i ,
THE PCD MODEL 2110 10 20 30 40 500 50 100 1500 2 4 6log(duration)1.0 1.5 2.0 2.5 3.0size, in ticks0 20 40 60 80 1000 20 40 60 80 120-1.0 -0.5 0.0 0.5 1.0direction0 5 10 15 20number of tradesFigure 5.15. Histograms of intraday transactions data for IBM stock on November 21, 1990:(a) log durations between price changes, (b) direction of price movement, (c) size of pricechange measured in ticks, and (d) number of trades without a price change.where standard deviations of the coefficients are 0.415, 0.073, 0.384, and 0.073,respectively. The fitted model indicates that there was no dynamic dependence in thetime duration. For the N i variable, we havePr(N i > 0 | t i , F i−1 ) = logit[−0.637 + 1.740 ln(t i )],where standard deviations of the estimates are 0.238 and 0.248, respectively. Thus,as expected, the number of trades with no price change in the time interval (t i−1 , t i )depends positively on the length of the interval. The magnitude of N i when it ispositive isN i | (N i > 0,t i , F i−1 ) ∼ 1 + g(λ i ), λ i = exp[0.178 − 0.910 ln(t i)]1 + exp[0.178 − 0.910 ln(t i )] ,where standard deviations of the estimates are 0.246 and 0.138, respectively. Thenegative and significant coefficient of ln(t i ) means that N i is positively related tothe length of the duration t i because a large ln(t i ) implies a small λ i , whichin turn implies higher probabilities for larger N i ; see the geometric distribution inEq. (5.27).
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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
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
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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 and 204: DURATION MODELS 195For the IBM data
Page 205 and 206: DURATION MODELS 197(a) is the avera
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: THE PCD MODEL 209(a) All transactio
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
Page 255 and 256: JUMP DIFFUSION MODELS 247where it i
Page 257 and 258: JUMP DIFFUSION MODELS 249sudden jum
Page 259 and 260: APPENDIX A 251Pricing formulas for
Page 261 and 262: EXERCISES 253which involves the CDF
Page 263 and 264: REFERENCES 255Bakshi, G., Cao, C.,
Page 265 and 266: VALUEATRISK 257log return-0.2 -0.1
Page 267 and 268: 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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