"Frontmatter". In: Analysis of Financial Time Series

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402 MCMC METHODSResult 2: Suppose that x 1 ,...,x n form a random sample from a normal distributionwith a given mean µ and an unknown precision η. If the prior distribution of ηis a gamma distribution with positive parameters α and β, then the posterior distributionof η is a gamma distribution with parameters α+(n/2) and β+ ∑ ni=1 (x i −µ) 2 /2.A random variable θ has a beta distribution with positive parameters α and β ifits probability density function isf (θ | α, β) =Ɣ(α + β)Ɣ(α)Ɣ(β) θ α−1 (1 − θ) β−1 ,0
ALTERNATIVE ALGORITHMS 403distribution of λ is a beta distribution with positive parameters α and β, then the posteriordistribution of λ is a beta distribution with parameters α+mn and β + ∑ ni=1 x i .Next we consider the case of a normal distribution with an unknown mean µand an unknown precision η. The two-dimensional prior distribution is partitioned asP(µ, η) = P(µ | η)P(η).Result 7: Suppose that x 1 ,...,x n form a random sample from a normal distributionwith an unknown mean µ and an unknown precision η. Suppose also that theconditional distribution of µ given η = η o is a normal distribution with mean µ oand precision τ o η o and the marginal distribution of η is a gamma distribution withpositive parameters α and β. Then the conditional posterior distribution of µ givenη = η o is a normal distribution with mean µ ∗ and precision η ∗ ,µ ∗ = τ oµ o + n ¯xτ o + nand η ∗ = (τ o + n)η o ,where ¯x = ∑ ni=1 x i /n is the sample mean, and the marginal posterior distribution ofη is a gamma distribution with parameters α + (n/2) and β ∗ ,whereβ ∗ = β + 1 2n∑(x i −¯x) 2 + τ on( ¯x − µ o ) 2.2(τ o + n)i=1When the conditional variance of a random variable is of interest, inverted chisquareddistribution (or inverse chi-squared) is often used. A random variable Yhas an inverted chi-squared distribution with v degrees of freedom if 1/Y followsa chi-squared distribution with the same degrees of freedom. The probability densityfunction of Y isf (y | v) = 2−v/2Ɣ(v/2) y−(v/2+1) e −1/(2y) , y > 0.For this distribution, we have E(Y ) = 1/(v − 2) if v>2andVar(Y ) = 2/[(v −2) 2 (v − 4)] if v>4.Result 8: Suppose that a 1 ,...,a n form a random sample from a normal distributionwith mean zero and variance σ 2 . Suppose also that the prior distribution of σ 2is an inverted chi-squared distribution with v degrees of freedom [i.e., (vλ)/σ 2 ∼ χv 2,where λ>0]. Then the posterior distribution of σ 2 is also an inverted chi-squareddistribution with v + n degrees of freedom—that is, (vλ + ∑ ni=1 ai 2)/σ 2 ∼ χv+n 2 .10.4 ALTERNATIVE ALGORITHMSIn many applications, there are no closed-form solutions for the conditional posteriordistributions. But many clever alternative algorithms have been devised in the statis-
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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
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 and 400: APPENDIX A. SOME REMARKS ON ESTIMAT
Page 401 and 402: REFERENCES 393(a) Compute the sampl
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: BAYESIAN INFERENCE 401normal with m
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
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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