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Financial Econometrics – Slides-09: Volatility Modelling

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

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Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Financial Econometrics
Slides-09: Volatility Modelling

School of Economics1

1©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be
removed from this material.

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Lecture Plan

• Motivation for modeling return volatility
• Measures of return volatility
• Conditional volatility via smoothing

• Conditional variance is a function of info set;
• It captures “clustering” in return series;
• It explains non-normality of return, to some extent;
• It can be used to improve interval forecasts and VaR (Value at Risk);
• Estimation and testing.

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Introduction and Motivation

eg. Volatility in NYSE Composite index return

• Clustering.
• Squared returns are strongly autocorrelated.

Topic 5. Modelling Return Volatility: ARCH

• Motivation
eg. Volatility in NYSE Composite index return

• Clustering.
• Squared returns are strongly autocorrelated.

School of Economics, UNSW Slides-7, Financial Econometrics 3

250 500 750 1000 1250 1500 1750

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Motivation

eg. Volatility in NYSE Composite index return

• Monthly realised variance:
RV =sample mean of squared daily returns in a month

• RV is negatively correlated to lagged monthly return.
Corr(RV,Return(−1)) = −0.419.

Topic 5. Modelling Return Volatility: ARCH

• Motivation
eg. Volatility in NYSE Composite index return

• Monthly realised variance:
RV = sample mean of squared daily returns in a month

• RV is negatively correlated to lagged monthly return.
Corr(RV, Return(-1)) = −0.419

School of Economics, UNSW Slides-7, Financial Econometrics 4

1996 1998 2000 2002

Return(-1)

-15 -10 -5 0 5

Comp. Return(-1)

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Motivation

I Importance of return volatility
Asset pricing, risk management and portfolio selection

Substantial dependence structure in volatility

I Clustering:
– strong autocorrelations in squared returns,
– large variations tend to be followed by large variations

I Asymmetry:
– negative returns tend to cause more volatility than positives

I ARMA are unable to capture these features
Conditional variance is constant in ARMA.

Amend ARMA with a suitable conditional variance: ARCH and GARCH

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Volatility

Measures of return volatility (tendency of variation)

• Historical volatility: Sample variance or Stddev

Topic 5. Modelling Return Volatility: ARCH

• Volatility
– Measures of return volatility

(tendency of variation)
• Historical volatility: Sample variance or Stddev
eg. NYSE composite return: Sample Stddev

School of Economics, UNSW Slides-7, Financial Econometrics 7

250 500 750 1000 1250 1500 1750

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Realized Volatility

Measures of return volatility

• Realised volatility: Realised variance = Sample mean of squared higher
frequency returns

(eg. daily RV = Sample mean of squared 5-min returns)

Topic 5. Modelling Return Volatility: ARCH

• Volatility
– Measures of return volatility

• Realised volatility: Realised variance =
Sample mean of squared higher frequency returns
(eg. daily RV = Sample mean of squared 5-min returns)

eg. NYSE composite return: Monthly realised variance
RV = Sample mean of squared daily returns in a month

School of Economics, UNSW Slides-7, Financial Econometrics 8
1996 1998 2000 2002

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Realized Volatility

Measures of return volatility

• Range (high/low):
100× ln(high/low) in a time interval (eg, a day)

Topic 5. Modelling Return Volatility: ARCH

• Volatility
– Measures of return volatility

• Range (high/low) :
100∙ln(high/low) in a time interval (eg, a day)

daily return and range

School of Economics, UNSW Slides-7, Financial Econometrics 9

2004 2006 2008 2010 2012

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Realized Volatility

Implied Volatility

Implied volatility:

standard deviation derived from options prices
– Option of an asset: the right to buy/sell the asset at a future time

(maturity) at a fixed price (strike).
– Given theprice of an option, maturity, strike and risk-free interest rate, the

std deviation can be recovered from Black-Scholes formula, known as IV.
– IV represents market’s opinions on the return’s std deviation.

Black-Scholes formula:
price of an option =f(stdev, maturity,strike, rf−rate)

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Realized Volatility

Implied Volatility

Topic 5. Modelling Return Volatility: ARCH

• Volatility
– Measures of return volatility

• Implied volatility:
eg. VIX: index of IVs of a set of options on the SP500 index
SP500 daily return & VIX

School of Economics, UNSW Slides-7, Financial Econometrics 11

1990 1995 2000 2005 2010

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Conditional Volatility

Topic 5. Modelling Return Volatility: ARCH

• Conditional Volatility
– Conditional variance of return

• 𝜎𝜎𝑡𝑡+1|𝑡𝑡
2 = Var(𝑟𝑟𝑡𝑡+1|Ω𝑡𝑡) ,

where 𝑟𝑟𝑡𝑡+1 = 100ln (𝑃𝑃𝑡𝑡+1/𝑃𝑃𝑡𝑡) is the return and
Ω𝑡𝑡 is the information set at the end of period 𝑡𝑡 .
• It should capture “clustering” or autocorrelations in

squared returns, and facilitate predicting the return
volatility

• Knowing it helps to
– assess the risk of an asset via value-at-risk;
– price options;
– form mean-variance efficient portfolios.

School of Economics, UNSW Slides-7, Financial Econometrics 12

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

Conditional Volatility

Exponentially weighted moving average (EWMA)

• The squared returns {r2t , r2t−1, · · · , r21} carry info about the volatility as
E(r2t ) ≡ variance.
• A weighted average of squared returns is an approximation to the

conditional variance. Recent observations should weigh more.

• EWMA: for 0 < λ < 1, t+1|t = (1− λ)(r t−2 + · · · ) - weights decay exponentially; - weights sum up to 1. - RiskMetrics recommend λ = 0.94 Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 5. Modelling Return Volatility: ARCH • Conditional Volatility • EWMA: alternative formulation  𝜎𝜎𝑡𝑡+1|𝑡𝑡 2 = 1 − 𝜆𝜆 𝑟𝑟𝑡𝑡2 + 𝜆𝜆𝜎𝜎𝑡𝑡|𝑡𝑡−1 2 , for 𝑡𝑡 = 1,2,3, … – Quick and easy; – Can be used as 1-step ahead prediction. eg. NYSE Composite return: School of Economics, UNSW Slides-7, Financial Econometrics 14 1600 1650 1700 1750 1800 1850 1900 Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ARCH (autoregressive conditional heteroskedasticity) Engle (1982) – Nobel price winner 1993 ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Models allowing for non-constant volatility ARCH models ARCH models Autoregressive conditional heteroscedasticity (ARCH) models are a class of models where the conditional variance evolves according to an autoregressive process. First define the conditional variance of the error term ut to be σ2t = var (µt |µt−1, µt−2, ...) = E (µt − E (µt))2 |µt−1, µt−2, ... As it is usually assumed that E (µt) = 0 σ2t = var (µt |µt−1, µt−2, ...) = E µ2t |µt−1, µt−2, ... The ARCH(1) model assumes σ2t = Et−1 = α0 + α1µ The conditional variance captures ’clustering’: large past shock leads to large conditional variance. Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ARCH (autoregressive conditional heteroskedasticity) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Models allowing for non-constant volatility ARCH models Extensions I An ARCH(q) model is given by σ2t = α0 + α1µ t−2 + ...+ αqµ I Under ARCH, the conditional mean equation can take any form. An example of a full model would be yt = β1 + β2x2t + β3x3t + β4x4t + µt µt ∼ N σ2t = α0 + α1µ Alternative notation yt = β1 + β2x2t + β3x3t + β4x4t + µt µt = νtσt νt ∼ N (0, 1) σ2t = α0 + α1µ Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Properties of ARCH Properties of ARCH(1) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material • ARCH(1): µt|Ωt−1 ∼ N(0, σ2t ), Ωt−1 = {yt−1, µt−1, yt−2, µt−2 · · · } is the info set at the end of period t− 1: σ2t = α0 + α1µ t−1, α0 > 0, 0 ≤ α1 < 1 • Its conditional variance is time varying: Var(µt|Ωt−1) = σ2t , CI(95%) =?• It is WN:(Use LIE) E(µt) = 0, Var(µt) = α01−α1 , Cov(µt, µt−j) = 0 But it is NOT independent WN or iid WN. Why? Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Properties of ARCH Proof of properties ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Definition (Law of Iterated Expectations) For a random variable Y and information sets Ω1 and Ω2, the the LIE states that E (Y |Ω1) = E (E (Y |Ω2) |Ω1) , where information set Ω1 is included in information set Ω2. Example: E (Yt|Ωt−2) = E (E (Yt|Ωt−1) |Ωt−2) Special Case: If Ω1 is empty set, then E (Y ) = E (E (Y |Ω2)) . µt = νtσt = νt t−1, where νt is N(0, 1) 1 Unconditional Expectation of µt. We have that µt|Ωt−1 ∼ N(0, σ2t ): E (µt) = E [E [µt|Ωt−1]] (1) E [µt|Ωt−1] = 0 (2) E (µt) = 0. (3) Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Properties of ARCH Proof of properties ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material µt = νtσt = νt t−1, where νt is N(0, 1) 2 Unconditional variance of µt. We have that = α0 + α1E = · · · = α0 1 + α1 + α 1 + · · ·+ α As t→∞, the unconditional variance converges if α1 < 1 to: . −→ Unconditionally, the process µt is homoskedastic. Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Properties of ARCH Properties of ARCH(1) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material • It can be alternatively expressed as: µt = σtvt, vt ∼ iidN(0, 1), where vt = µt/σt is the standardised shock. • When model is correct,v2t should have no autocorrelation • The unconditional distribution of µt is NOT normal, with heavy tails (kurtosis > 3).

Slides-09 UNSW

Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model

ML Estimation

MLE of ARCH(1)

• An example: AR(1)−ARCH(1)
yt = c+ φ1yt−1 + µt, µt|Ωt−1 ∼ N(0, σ2t ), (10)
σ2t = α0 + α1µ

α0 > 0, 0 ≤ α1 < 1. (12) • Likelihood of {y1, y2, · · · , yT−1, yT }: L(Θ) = f (yT |ΩT−1) f (yT−1|ΩT−2) · · · f (y2|Ω1) f(y1) f (yt|Ωt−1) = (2πσ2t )−1/2exp{− (yt − c− φ1yt−1)2 • ML Estimator maximises the Log likelihood function lnL(Θ) = −T ln(σ2t ) + (yt − c− φ1yt−1)2 Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ML Estimation MLE of ARCH(1) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material • ML estimators are generally consistent with an asymptotic normal distribution. • The above holds even when the conditional normality µt|ωt−1 ∼ N(0, σ2t ) is incorrectly assumed, as long as the conditional mean and conditional variance are correctly specified. • With robust quasi ML standard errors, inference is standard. Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ML Estimation ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material eg. NYSE composite return: AR(1)-ARCH(5) Topic 5. Modelling Return Volatility: ARCH • Conditional Volatility – ML Estimation of ARCH(1) eg. NYSE composite return: AR(1)-ARCH(5) School of Economics, UNSW Slides-7, Financial Econometrics 19 -6 -4 -2 0 2 4 Sample 1 1931 Observations 1929 Mean -0.033709 Median -0.037415 Maximum 5.392314 Minimum -6.773783 Std. Dev. 1.004962 Skewness -0.198864 Kurtosis 7.131158 Jarque-Bera 1384.431 Probability 0.000000 -5.00 -3.75 -2.50 -1.25 0.00 1.25 2.50 Sample 1 1931 Observations 1929 Mean -0.048030 Median -0.043219 Maximum 3.427925 Minimum -5.188789 Std. Dev. 0.999109 Skewness -0.413633 Kurtosis 4.558816 Jarque-Bera 250.3099 Probability 0.000000 Type in Eviews upper panel: arch(5,0,h) rc c ar(1) Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model ML Estimation ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material eg. NYSE composite return: AR(1)-ARCH(5) • Squared residuals (E2) of AR(1) have strong autocorrelation. Squared standardised residuals (V2) are not autocorrelated • Residuals (E) of AR(1) have larger kurtosis. Standardised residuals (V) larger negative skewness. • Normality is rejected for both E and V. Two essential checks for the ’adequacy’ of a model I Adequate mean equation: E (residuals) has no autocorrelation; I Adequate variance equation: V2 has no autocorrelation Slides-09 UNSW Measures of Volatility Conditional Volatility ARCH Comments on ARCH Model Comments and limitations of ARCH ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Advantages of ARCH • It is able to capture ’clustering’ in return series or the autocorrelation in squared • It facilitates volatility forecasting. • It explains, partially, non-normality in return series. Limitations of ARCH I In ARCH(q), the q may be selected by AIC, SIC or LR test. The correct value of q might be very large. The model might not be parsimonious. (eg. ARCH(1) would not work for the composite return) I The conditional variance σ2t cannot be negative: Requires non-negativity constraints on the coefficients. Sufficient (but not necessary) condition is: αi ≥ 0 for all i = 0, 1, 2, · · · q. Especially for large values of q this might be violated Slides-09 UNSW Measures of Volatility Realized Volatility Conditional Volatility Properties of ARCH ML Estimation Comments on ARCH Model 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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