1 Ka-fu Wong University of Hong Kong Volatility Measurement, Modeling, and Forecasting.

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Ka-fu WongUniversity of Hong Kong

Volatility Measurement,Modeling, and Forecasting

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Importance of volatility

Good volatility forecasts are crucial for the implementation and evaluation of asset and derivative pricing theories as well as trading and hedging strategies.

Two assets: an risky and a riskless (i.e., volatility = 0) Risky asset generally has a higher expected return than

the riskless assets.

We would like to invest in a portfolio consisting of the two assets. When the risky asset has a very high volatility, the

portfolio will consist of the riskless asset only. When the risky asset has a very low volatility, the

portfolio will consist of more risky assets.

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Importance of volatility

The variance of inflation may have impact on various macro and investment decisions.

High variance in inflation may also imply welfare loss.

Previous studies have tried to measure the time-varying variance of inflation.

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Clustering of volatility

It is a well-established fact, dating back to Mandelbrot (1963) and Fama (1965), that financial returns display pronounced volatility clustering.

Therefore, models of volatility should allow such clustering.

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Example: AR(1)

yt = φ yt-1 + t t ~ WN(0, 2)AR(1):

yt = φ (φ yt-2 + t-1) + t

= φ2 yt-2 + φ t-1 + t

= φ2 (φ yt-3 +t-2)+ φ t-1 + t

= φ3 yt-3 + φ2 t-2+ φ t-1 + t

= t + φt-1 + φ2t-2 + φ3t-3 + φ4t-4 + φ5t-5+ …

…

E(t) = 0, E(yt) = 0

Var(t) = E[(t – E(t))2] = 2

Var(yt) = E[(yt – E(yt))2] = 2(1+ φ + φ2 + φ3 + φ4 +…)

Repeatd substitution:

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Homoskedasticity vs. Heteroskedasticity

So far, innovation are assumed to be i.i.d.

It is possible to allow variance to change across observations, i.e., Heteroskedasticity.

Information available at time t-1

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A general linear process

Consider a general linear process:

Need not be i.i.d.

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Two examples

Consider a general linear process:

Need not be i.i.d.

yt = φ yt-1 + t

yt = t + φt-1 + φ2t-2 + φ3t-3 + φ4t-4 + …

bi = φi

AR(1)

MA(2)

yt = t + θ1t-1 + θ 2t-2

b0=1, b1= θ1, b2= θ2, b3=b4=…=0

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b0=1, b1= θ1, b2= θ2, b3=b4=…=0

Unconditional means and variances

Consider a general linear process:

yt = φ yt-1 + t

yt = t + φt-1 + φ2t-2 + φ3t-3 + φ4t-4 + …

bi = φi

AR(1)

yt = t + θ1t-1 + θ 2t-2

MA(2)

E(yt)= E(t) + φE(t-1) + φ2E(t-2) + … = 0

V(yt)= V(t) + φ2V(t-1) + φ4V(t-2) + …

E(yt)= E(t) + θ1E(t-1) + θ2E(t-2) = 0

V(yt)= V(t) + θ12V(t-1) + θ2

2V(t-2)

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Conditional variances change with horizon of forecast but are not time-varying given a horizon.

Consider a general linear process:

Conditional mean is time-varying :

h-step ahead forecast is time-varying:

Conditional information

b0=1, b1= θ1, b2= θ2, b3=b4=…=0

yt = t + θ1t-1 + θ 2t-2

MA(2)

E(yt|t-1)= θ1t-1 + θ2t-2

E(yt+1|t)= θ1t + θ2t-1

E(yt+2|t+1)= θ1t+1 + θ2t

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Conditional variances change with horizon of forecast but are not time-varying given a horizon.

Consider a general linear process:

Conditional variance is not time-varying:

Conditional prediction error variance:

Conditional information

b0=1, b1= θ1, b2= θ2, b3=b4=…=0

yt = t + θ1t-1 + θ 2t-2

MA(2)

E[(yt-E(yt|t-1) )2|t-1]= E(t

2 |t-1) = 2

Non-time-varying!

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ARCH(p) process

Examples:(1)ARCH(1): t

2 = + 1 t-12

(2) ARCH(2): t2 = + 1 t-1

2+ 2 t-22

ARCH(p)

AutoRegressive Conditional Heteroskedasticy of order p

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ARCH(p) process

Examples:(1)ARCH(1): t

2 = + 1 t-12

(2) ARCH(2): t2 = + 1 t-1

2+ 2 t-22

ARCH implies volatility clustering. That is, large changes tend to be followed by large changes and small by small, of either sign.

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ARCH(p) process

Examples:(1)ARCH(1): t

2 = + 1 t-12

(2) ARCH(2): t2 = + 1 t-1

2+ 2 t-22

(1) Unconditional mean

(2) Unconditional variance

(3) Conditional variance

Some properties

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ARCH(1)

t2 = + 1 t-1

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Note that E[t

2] = E[ E(t2|t-1) ] = E(t

2) = 2

E[(t-E(t))2] = ?

E[t2] = + 1 E[t-1

2]

2 = + 1 2

2 = / (1- 1)

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How to simulation ARCH(1)?

Suppose we are interested in generating T observations of t that has the property of ARCH(1). t ~ N(0,t

2), wheret2 = + 1 t-1

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(1) Fixed the parameters. Compute the unconditional variance of t.

2 = / (1- 1)

(2) Generate T+1 observations of standard normal random variables, v0, v1, …., vT

(3) Generate t recursively

For t=0, t2 = 2, t = vt t

For t=1, t2 = + 1 t-1

2, and t = vt t

For t=2, t2 = + 1 t-1

2, and t = vt t

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The inflation example of Engle (1982)

First difference of the log of the quarterly consumer price index

log of the quarterly manual wage rates

Lagged 4 periods

Engle, Robert F. (1982): “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation,” Econometrica, 50(4): 987-1007.

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The inflation example of Engle (1982)OLS regression

Restriction imposed.

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The inflation example of Engle (1982)ML estimation with ARCH(1)

The ARCH model comes closer to truly random residuals afterstandardizing for their conditional distributions.

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GARCH(p,q)

Backward substitution on t2 yields

A infinite-order ARCH process with some restriction in the coefficients.(Analogy: An ARMA(p,q) process can be written as MA(∞) process.)

GARCH can be viewed as a parsimonious way to approximate a high order ARCH process

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Important properties of GARCH(p,q)(1) Unconditional variance is fixed but conditional variance is time-varying

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Important properties of GARCH(p,q)(2) Unconditional distribution of conditionally Gaussian GARCH is symmetric and leptokurtic.

Real-world financial asset returns, are often found to symmetrically distributed and have a fatter tail than Gaussian distribution.

Ordinary Gaussian distribution does not provide a good approximation of the asset returns, but the Gaussian distribution with GARCH does.

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Important properties of GARCH(p,q)(3) Conditional prediction error variance varies with conditional information set.

unbiased forecast

Conditional variance of the prediction error

Conditional variance approaches unconditional variance

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Important properties of GARCH(p,q)(3) t follows GARCH implies t

2 follows an ARMA .

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Extension of ARCH and GARCH ModelsThreshold GARCH

When the lagged return is positive (good news yesterday), D=0, so the effect of the lagged squared return on the current conditional variance is simply .

When the lagged return is negative (negative news yesterday), D=1, so the effect of the lagged squared return on the current conditional variance is simply .

Allowance for asymmetric response has proved useful for modeling “leverage effects” in stock returns, which occur when < 0.

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Extension of ARCH and GARCH Modelsexponential GARCH

Volatility is drive by both the size and sign of shocks (both positive and negative). Hence, the model allows for asymmetric response depending on the sign of news.

When the shock is positive, the impact of (t-1/t-1) on ln(t2) is

+ When the shock is negative, the impact of (t-1/t-1) on ln(t

2) is +

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Extension of ARCH and GARCH ModelsGARCH with exogenous variables

Financial market volume, for example, often helps to explain market volatility.

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Extension of ARCH and GARCH ModelsGARCH-in-Mean (i.e., GARCH-M)

High risk, high return.

Conditional mean regression

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Estimating, Forecasting, and Diagnosing GARCH Models

Diagnostic: Estimate the model without GARCH in the usual way. Look at the time series properties of the squared residuals.

Correlogram, AIC, SIC, etc. ARMA(1,1) in the squared residuals implies GARCH(1,1).

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Estimating, Forecasting, and Diagnosing GARCH Models

Estimation: Usually use maximum likelihood with the assumption of normal distribution. Maximum likelihood estimation finds the parameter values

that maximize the likelihood function

Forecast: In financial applications, volatility forecasts are often of direct interest.

1-step-ahead conditional variance

Better forecast confidence interval

vs.

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Application: Stock Market Volatility

Objective: Model and forecast the volatility of daily returns on the New York Stock Exchange

Data: Daily returns on the New York Stock Exchange (NYSE)

form January 1, 1988, through December 31, 2001. Excluding holidays, there are 3531 observations.

Estimation: 1-3461 Forecast: 3462-3531.

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Time Series Plot, NYSE Returns

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Histogram and Related Diagnostic Statistics, NYSE Returns

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Correlogram, NYSE Returns

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Time Series Plot, Squared NYSE Returns

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Correlogram, Squared NYSE Returns

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AR(5) Model, Squared NYSE Returns

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ARCH(5) Model, NYSE Returns

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Correlogram, Squared Standardized ARCH(5) residuals, NYSE Returns

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GARCH(1,1) Model, NYSE Returns

t-12

t-12

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Correlogram, Squared Standardized GARCH(1,1) residuals, NYSE Returns

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Estimated Conditional Standard Deviation, GARCH(1,1) Model, NYSE Returns

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Estimated Conditional Standard Deviation, Exponential Smoothing, NYSE Returns

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Conditional Standard Deviation, History and Forecast, GARCH(1,1) Model

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Conditional Standard Deviation, Extended History and Extended Forecast, GARCH(1,1) Model

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Is GARCH(1,1) enough most of the time?

330 GARCH-type models are compared in terms of their ability to forecast the one-day-ahead conditional variance.

The models are evaluated out-of-sample using six different loss functions, where the realized variance is substituted for the latent conditional variance.

Hansen, Peter R. and Asger Lunde (2005): “A Forecast Comparison Of Volatility Models: Does Anything Beat A GARCH(1,1)?” Journal of Applied Econometrics, 20: 873-889.

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Is GARCH(1,1) enough most of the time?

Data: DM–$ spot exchange rate data,

the estimation sample spans the period from October 1, 1987 through September 30, 1992 (1254 observations) and

the out-of-sample evaluation sample spans the period from October 1, 1992 through September 30, 1993 (n = 260).

IBM stock returns, the estimation period spans the period from January

2, 1990 through May 28, 1999 (2378 days) and the evaluation period spans the period from June 1,

1999 through May 31, 2000 (n = 254).

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Specifications of the conditional variance

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Loss functions for forecast evaluation

MSE2 and R2Log are similar to R2 of the MZ regressions.

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The test

Giving benefits of the doubt to the benchmark, i.e., GARCH(1,1).

Loss of GARCH(1,1)

Loss of alternatiave GARCH models.

The maintained hypothesis is that GARCH(1,1) is better unless there is strong evidence against it.

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Superior Predictive Ability and Reality Check for data snooping

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The test resultssuperior predictive ability (SPA)

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IBM data:superior predictive ability and reality check for data snooping

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Does Anything Beat A GARCH(1,1)?

No. So, use GARCH(1,1) if no other information is available.

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End