CS代考 Models of Volatility Dynamics

Models of Volatility Dynamics
. Lochstoer
UCLA Anderson School of Management
Winter 2022

. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

1 Stylized facts of volatility clustering
2 Realized Variance
3 ARCH models
4 GARCH models I GARCH(1,1)
I I-GARCH, GARCH-M, EGARCH, GJR
5 Value-at-Risk
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022

so far, we have focused on modeling the conditional mean: rt =E[rtjrt1,rt2,…;θ]+εt
the benchmark models for the conditional mean are ARMA models
rt = φ0 +φ1rt1 +…+φprtp +εt θ1εt1 +…θqεtq
given estimates, φˆ0,φˆ1,…,θˆq1,θˆq, we can calculate the residuals εˆt = vt (0) recursively from the initial condition.
the key modeling goal is to make sure that the residuals fεˆt g are white noise I No additional predictable components left
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 3 / 91

Volatility Clustering
suppose we have a good model for the conditional mean μt (e.g. ARMA(p, q))
now we look at the squared residuals fεˆ2t g to test for conditional heteroskedasticity
for returns on most Önancial assets, there is a lot more autocorrelation in conditional second moments than in the conditional Örst moments.
volatility is predictable!
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022

Volatility?
In most asset markets, volatility varies dramatically over time. episodes of high volatility seem to be clustered.
we donít simply observe volatility
how do we measure vol (vol is short-hand for volatility)?
1 implied volatility (back out volatility from option prices)
2 (non-parametric) realized volatility (e.g., realized volatility of stock returns over one-month using daily data)
3 (parametric) model volatility
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022

VIX: Option Implied Market Volatility
Annualized measure of 30-day vol. VIX, designed to measure the marketís expectation of 30-day volatility implied by at-the-money S&P 100 Index (OEX) option prices. Monthly data. 1990-2009. VIX white paper: https://www.cboe.com/micro/vix/vixwhite.pdf.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 6 / 91

Why do we care about Volatility?
why do we care about modeling vol?
1 portfolio allocation
2 risk management
3 option pricing
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022

ARCH model of Engle (1982) DeÖnition
The ARCH(m) model is given by:
εt = σtηt, σ2t = α0 +α1ε2t1 +…+αmε2tm
where ηt is i.i.d., mean zero, has variance of one , α0 > 0 and αi 0 for i > 0
the standard normal is a common choice for ηt
the (standardized) Studentís t is another option for ηt
allows for large (small) shocks to be followed by more large (small) shocks: volatility clustering
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022

Building a Volatility Model
1 estimate a model for the conditional mean (e.g. ARMA model)
2 use the residuals εˆt from the mean equation to test for ARCH e§ects
3 specify a volatility model for ARCH e§ects
4 check the Ötted model
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022

Stylized Facts on Volatility Clustering
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 10 / 91

Squared Monthly Log Market Returns
1916 1930 1943
1984 1998 2012 2026
log returns on market squared
log stock returns on market -squared. Monthly data. 1926-2012.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

(P)ACFís of absolute value, squared, and regular market returns; Long sample
0.4 0.3 0.2 0.1
0.4 0.3 0.2 0.1
ACF f or log returns
ACF f or |log returns|
0.4 0.3 0.2 0.1
0 5 10 15 20 25
ACF f or squared returns
0.4 0.3 0.2 0.1
0 5 10 15 20 25
PACF f or squared returns
-0.1 -0.1 0 5 10 15 20 25
0 5 10 15 20 25
log stock returns on CRSP-VW. Monthly data. 1926-2012.
Winter 2022
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics

Dynamics of Variance: Monthly Returns, postwar Sample
ACF f or log returns
0.1 0.3 0.05 0.2 0 0.1 -0.05 0
-0.1 -0.1 0 5 10 15 20 25
ACF f or squared returns
0.4 0.3 0.3 0.2 0.2 0.1 0.1 0
-0.1 -0.2 0 5 10 15 20 25
ACF f or |log returns|
0 5 10 15 20 25
PACF f or squared returns
log stock returns on CRSP-VW. Monthly data. 1945-2012.
0 5 10 15 20 25
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

Dynamics of Variance: Monthly Returns, Short Sample
0.1 0.05 0
-0.05 -0.1
0.15 0.1 0.05
-0.05 -0.1
0 5 10 15 20 25
ACF f or squared returns
0.25 0.2 0.15 0.1 0.05 0
-0.05 -0.1
0 5 10 15 20 25
PACF f or squared returns
ACF f or log returns
ACF f or |log returns|
0.15 0.1 0.05 00
0 5 10 15 20 25
-0.05 -0.1
0 5 10 15 20 25
log stock returns on CRSP-VW. Monthly data. 1970-2012.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

Daily Market Returns Squared
1971 1984 1998 2012 2026
log dailyreturns on market squared
log stock returns on market -squared. Daily data. 1988-2012.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

Dynamics of Variance: Daily Returns, Long Sample
0. 08 0. 06 0. 04 0. 02
0.4 0.3 0.2 0.1
ACF f or log returns
ACF f or |log returns|
0 -0.02 -0.04
0 5 10 15 20 25
ACF f or squared returns
0 5 10 15 20 25
PACF f or squared returns
0.3 0. 25 0.2 0. 15 0.1 0. 05
0.3 0. 25 0.2 0. 15 0.1 0. 05 00
0 5 10 15 20 25
0 5 10 15 20 25
log stock returns on CRSP-VW. Daily data. 1926-2012.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

Dynamics of Variance: Daily Returns, Short Sample
0. 06 0. 04 0. 02
0 -0.02 -0.04
0 5 10 15 20 25
ACF f or squared returns
0.4 0.3 0.2 0.1
ACF f or log returns
ACF f or |log returns|
0.3 0. 25 0.2 0. 15 0.1 0. 05
0.3 0. 25 0.2 0. 15 0.1 0. 05 00
0 5 10 15 20 25
PACF f or squared returns
0 5 10 15 20 25
0 5 10 15 20 25
log stock returns on CRSP-VW. Daily data. 1970-2012.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0
1993 1995 1998
2006 2009 2012
log returns on IBMsquared
log stock returns on IBM -squared. Monthly data. 1988-2009.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

ACF f or log returns
ACF f or |log returns|
0 5 10 15 20 25
0 5 10 15 20 25
ACF f or squared log returns
0 5 10 15 20 25
PACF f or squared log returns
0.1 00 -0.1 -0.1
log stock returns on IBM. Monthly data. 1988-2009.
0 5 10 15 20 25
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

Yen/USD Squared Log Changes: Daily Data
0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001
1993 1998 2004 2009
log changes in Y en/USD – squared
log changes in Yen/USD squared. Daily data. 1971-2009.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

Yen/USD (P)ACFís
0. 15 0.1 0. 05
0.3 0.2 0.1
ACF f or log changes
ACF f or |log changes|
-0.05 -0.1 0 5 10 15 20 25
0 5 10 15 20 25
ACF f or squared log changes
PACF f or squared log changes
0.2 0. 15 0.1 0. 05
-0.05 -0.05 0 5 10 15 20 25
0.2 0. 15 0.1 0. 05 00
log changes in Yen/USD. Daily data. 1971-2009.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022
0 5 10 15 20 25

Testing for ARCH
estimate a model for the conditional mean: e.g. ARMA(p,q) use the squared residuals to test for ARCH
specify a volatility model if ARCH e§ects are documented
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 22 / 91

Testing for ARCH(1)
we can test for autoregressive conditional heteroscedasticity using a LB Q-test:
suppose the model for the conditional mean is ARMA(p,q)
estimate the model
run a Q-test on the estimated squared residuals fεˆ2t g I test the null that:
H0 :ρ1 =ρ2 =…=ρm =0
where ρi denotes the autocorrelation of the squared residuals
I if you cannot reject the null, no need for ARCH machinery
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

Testing for ARCH(2)
we can test for autoregressive conditional heteroscedasticity using a LM test: suppose the model is ARMA(p, q)
estimate the model
run a LM-test on the estimated squared residuals fε2t g
I test the null that αi = 0,i = 1,2,…,m:
ε2t =α0 +α1ε2t1 +…+αmε2tm +et,t =m+1,…,T
I test the null that:
H0 :α1 =α2 =…=αm =0
I use the usual F-test, reject the null if F exceeds χ2 (α) ñthe (1 α)-th upper
percentile
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 24 / 91

Leverage E§ects: Monthly Returns on Market Squared
0.14 0.12 0.1 0.08 0.06 0.04 0.02 0
0 -0.05 -0.1 -0.15 -0.2 -0.25 -0.3 -0.35
1998 2012 2026
1998 2012 2026
log returns on market squared
negative log returns on market
log stock returns on market -squared. Monthly data. 1926-2012.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

Volatility Summary
volatility clusters
volatility is stationary
leverage e§ect (asymmetry)
I negative returns seem to be followed by larger increases in volatility than positive returns
I Örst emphasized by Black (1976)
I Black (1976) suggested that negative returns (decreases in price) change a
companyís debt/equity ratio, increasing their leverage.
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 26 / 91

Realized Variance
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 27 / 91

High-Frequency Data
let rt denote log of gross returns over a period of time t t may denote 1 day, 1 week, 1 month.
suppose over the time t, we observe equally spaced log-returns frt,igni=1 at a higher frequency. Example: t is monthly and i are days in the month
rt = ∑rt,i
for log returns, the variance of the return over time t is given by: n
V (rtjFt1)= ∑V (rt,ijFt1)+2∑Cov(rt,i,rt,jjFt1) i=1 i0andαi 0fori>0
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 33 / 91

the unconditional mean is zero:
E [εt] = E(E [εtjFt1]) = E(σtE [ηtjFt1]) = 0
the unconditional variance is:
V [ ε t ] = E h ε 2t i = E ( E h ε 2t j F t 1 i ) = E ( α 0 + α 1 ε 2t 1 )
because of stationarity, we get that:
V [εt] = α0 +α1V [εt]
as a result, the variance is:
V[εt]= α0 1α1
werequirethat0α1 <1 . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 Fourth Moment the unconditional fourth moment is : E h ε 4t i = E ( E h ε 4t j F t 1 i ) = E 3 σ 4t = 3 h E ( α 0 + α 1 ε 2t 1 ) i 2 using stationarity, this implies m4 = 3α20(1+α1) (1 α1)(1 3α21) the unconditional kurtosis is given by: which means 0 α21 < 1/3 E ε 4t = 3 1 α 21 > 3
V [ ε t ] 2 1 3 α 21
positive kurtosis even though innovations are conditionally Gaussian
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 35 / 91

ARCH(1) process is conditionally normal if ηt N(0, 1) ARCH(1) process is not unconditionally normal
I time variation in the variance generates fat tails
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 36 / 91

1-step ahead forecast
the ARCH(m) model is given by:
σ 2t + 1 = α 0 + α 1 ε 2t + . . . + α m ε 2t m + 1
hence the forecast is simply:
σ2t(1)=α0+α1ε2t +…+αmε2tm+1
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics
Winter 2022

2-step ahead forecast
the ARCH(m) model is given by:
σ2t+2 = α0 +α1ε2t+1 +…+αmε2tm+2
hence the forecast is simply:
σ2t(2)=α0 +α1σ2t(1)+α2ε2t …+αmε2tm+2
note that we have used the following:
Et [ε2t+1jFt ] = Et [σ2t+1η2t+1jFt ] = σ2t+1Et [η2t+1jFt ]
. Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 38 / 91

h-step ahead forecast
the ARCH(m) model is given by:
σ2t+h = α0 +α1ε2t+h +…+αmε2tm+h
General expression:
σ2t(h)=α0 +α1σ2t(h1)+…+αmσ2t(hm)
whereσ2t(hj)=ε2t+hj ifhj<0 . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 Building an ARCH model we can pick the order of an ARCH model by looking at PACF at fε2t g; you might need a large number of lags for a well-speciÖed ARCH model, the standardized residual: ηˆ t = εˆ t σˆ t should be white noise this can be tested by examining ηˆt 1 check the volatility spec by doing a Q-test on ηˆ2t 2 check the mean spec by doing a Q-test on ηˆt . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 Weakness of ARCH Models 1 symmetry in e§ects of positive and negative shocks on vol 2 ARCH model: mechanical description of volatility (no economics) 3 sometimes many lags are needed to describe vol dynamics . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 41 / 91 Parameter óóóñ Constant ARCH1 ARCH2 ARCH3 ARCH4 ARCH5 Table: ARCH(5) Value s.e. t-stat óóóñ óóóó óóóñ 3.09E-05 6.65E-07 0.099307 0.005071 0.144208 0.008707 0.141901 0.009809 0.168116 0.01012 0.150002 0.010459 46.472 19.5838 16.5623 14.4668 16.6119 14.3424 ARCH(5) estimated on daily CRSP-VW stock returns 1971-2012. . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 Parametric Volatility σt 140 Conditional Annualized Volatility ar c h( 5) 1976 1982 1987 1993 1998 2004 2009 2015 ARCH(5). Daily data. CRSP-VW.1971-2012. . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 Parametric Volatility against Realized Vol. 1976 1982 1987 1993 1998 2004 2009 2015 Conditional Annualized Volatility ARCH(5). Daily data. CRSP-VW. 1971-2012. . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 Standardized Residuals (εt/σt)2 Standardized Residuals squared 1976 1982 1987 1993 1998 2004 2009 2015 ARCH(5). Daily data. CRSP-VW.1971-2012. . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 Standardized Residuals εt/σt ACF for standardized residuals ACF for | stand. residuals| 0.15 0.1 0.05 0 -0.02 -0.04 0 5 10 15 20 25 ACF for squared stand. residuals 0.06 0.04 0.02 0 -0.02 -0.04 0 5 10 15 20 25 PACF for squared standard residuals 0 5 10 15 20 25 -0.02 -0.04 0 5 10 15 20 25 ARCH(5). Daily data. CRSP-VW.1971-2012. . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 Parameter óóóñ Constant ARCH1 ARCH2 ARCH3 ARCH4 ARCH5 ARCH6 ARCH7 ARCH8 ARCH9 Value óóóñ 2.04E-05 0.083715 0.113967 0.080502 0.095266 0.107659 0.075962 0.052205 0.083885 0.073778 0.049606 standard error t-stat óóóó óóóñ Table: ARCH(10) ARCH(10) estimated on daily CRSP-VW stock returns 1971-2012. 7.56E-07 0.004496 0.008698 0.009147 0.00937 0.011581 0.009292 0.008305 26.927 18.6207 13.1022 8.80109 10.1667 9.29659 8.1747 6.28631 9.56551 0.00877 0.008694 0.008884 8.4864 5.58371 . Lochstoer UCLA Anderson School of Management ()Lecture 8 Models of Volatility Dynamics Winter 2022 Parametric Volatility σt 120 Conditional Annualized Volatility 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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