"Frontmatter". In: Analysis of Financial Time Series

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430 MCMC METHODS(a) GARCH with time-varying correlationvol20 40 60 80 120 1601970 1980 1990 2000year(b) Stochastic volatilityvol20 40 60 80 120 1601970 1980 1990 2000yearFigure 10.8. Time plots of fitted volatilities for monthly log returns of IBM stock from 1962to 1999: (a) a GARCH model with time-varying correlations, and (b) a bivariate stochasticvolatility model estimated by Gibbs sampling with 300 + 1000 iterations.Tsay (1994) discuss a Gibbs sampling procedure to estimate such a model when thevolatility in each state is constant over time. These authors applied the procedureto estimate a Markov switching model with different dynamics and mean levels fordifferent states to the quarterly growth rate of U.S. real gross national product, seasonallyadjusted, and obtained some interesting results. For instance, the dynamicsof the growth rate are significantly different between periods of economic “contraction”and “expansion.” Since this chapter is concerned with asset returns, we focuson models with volatility switching.Suppose that an asset return r t follows a simple two-state Markov switchingmodel with different risk premiums and different GARCH dynamics:r t ={β1√ht + √ h t ɛ t , h t = α 10 + α 11 h t−1 + α 12 a 2 t−1if s t = 1β 2√ht + √ h t ɛ t , h t = α 20 + α 21 h t−1 + α 22 a 2 t−1if s t = 2,(10.40)where a t = √ h t ɛ t , {ɛ t } is a sequence of Gaussian white noises with mean zeroand variance 1, and the parameters α ij satisfy some regularity conditions so thatthe unconditional variance of a t exists. The probability transition from one state to
MARKOV SWITCHING MODELS 431(a) GARCH with time-varying correlationvol0 20 40 60 80 1001970 1980 1990 2000year(b) Stochastic volatility modelvol0 20 40 60 80 1001970 1980 1990 2000Figure 10.9. Time plots of fitted volatilities for monthly log returns of the S&P 500 indexfrom 1962 to 1999: (a) a GARCH model with time-varying correlations, and (b) a bivariatestochastic volatility model estimated by Gibbs sampling with 300 + 1000 iterations.yearanother is governed byP(s t = 2 | s t−1 = 1) = e 1 , P(s t = 1 | s t−1 = 2) = e 2 , (10.41)where 0 < e i < 1. A small e i means that the return series has a tendency to stay in theith state with expected duration 1/e i . For the model in Eq. (10.40) to be identifiable,we assume that β 2 >β 1 so that State 2 is associated with higher risk premium. Thisis not a critical restriction because it is used to achieve uniqueness in labeling thestates. A special case of the model results if α 1 j = α 2 j for all j so that the modelassumes a GARCH model for all states. However, if β i√ht is replaced by β i , thenmodel (10.40) reduces to a simple Markov switching GARCH model.Model (10.40) is a Markov switching GARCH-M model. For simplicity, weassume that the initial volatility h 1 is given with value equal to the sample varianceof r t . A more sophisticated analysis is to treat h 1 as a parameter and estimate it jointlywith other parameters. We expect the effect of fixing h 1 will be negligible in mostapplications, especially when the sample size is large. The “traditional” parametersof the Markov switching GARCH-M model are β = (β 1 ,β 2 ) ′ , α i = (α i0 ,α i1 ,α i2 ) ′for i = 1 and 2, and the transition probabilities e = (e 1 , e 2 ) ′ . The state vector
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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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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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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
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Page 401 and 402: REFERENCES 393(a) Compute the sampl
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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: •••MARKOV SWITCHING MODELS 42
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
Page 451 and 452: REFERENCES 443Justel, A., Peña, D.
Page 453 and 454: 446 INDEXData (cont.)equal-weighted
Page 455: 448 INDEXPoisson process, 244inhomo
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