CS代写 ECON3206/5206 Financial Econometrics - Slides-11: GARCH, VaR and Extensions

ECON3206/5206 Financial Econometrics – Slides-11: GARCH, VaR and Extensions

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

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Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

ECON3206/5206 Financial Econometrics
Slides-11: GARCH, VaR and Extensions

School of Economics1

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

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

Lecture Plan

• Forecasting Volatility with GARCH
• Volatility and Risk: VaR
• Typical estimates of GARCH parameters

A measure of volatility persistence

• Integrated GARCH and EWMA

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

Forecasting volatility with GARCH(1,1)

Extra topics MBF: Modelling volatility

Forecasting volatility

At first sight, forecasting the volatility in the error terms may not
seem very useful.
However, keep in mind that

var (yt |yt−1, yt−2, …) = var (µt |µt−1, µt−2, …)

Therefore, these models are very useful as they can add a model
for the volatility of a time series to traditional ARMA models.

I forecasting the volatility of stock returns is useful e.g. in
option pricing as this requires the expected volatility of the
underlying asset over de lifetime of the option as an input

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

Forecasting volatility with GARCH(1,1)

Extra topics MBF: Modelling volatility

Forecasting volatility

Consider the following GARCH(1,1) model

yt = µ+ µt µt ∼ N

σ2t = α0 + α1µ

Generate one-, two- and three-step-ahead forecasts for the
conditional variance of yt at time T .

I First update the equations for the conditional variance:

σ2T+1 = α0 + α1µ

σ2T+2 = α0 + α1µ

σ2T+3 = α0 + α1µ

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

Forecasting volatility with GARCH(1,1)

Extra topics MBF: Modelling volatility

Forecasting volatility

I Then let σ2f1,T be the one-step-ahead forecast for σ
2 at time T

σ2f1,T = ET

= α0 + α1µ

σ2f2,T = ET

= α0 + α1ET

= α0 + (α1 + β1)σ

= α0 + (α1 + β1)

σ2f3,T = ET

= α0 + (α1 + β1)σ

= α0 + α0 (α1 + β1) + (α1 + β1)

σ2fs,T = ET

+ (α1 + β1)

I For s →∞ σ2fs,T = α0 /(1− (α1 + β1)) if |α1 + β1| < 1 Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Example 1:Forecasting with GARCH(1,1) Example: Forecasting volatility with GARCH(1,1) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Extra topics MBF: Modelling volatility Forecasting volatility Example: forecasting mean and variance from an AR(1)-GARCH(1,1) for returns on the S&P500 index in a hold-out sample of 100 observations. EViews: in the Equation Window select Forecast Note that volatility is highly persistent! I forecasted volatility converges only slowly to the unconditional mean, which is equal to 0.000000792 1− 0.068012− 0.923437 = 0.000093 I there is a great deal of predictability in volatility! Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Example 1:Forecasting with GARCH(1,1) Forecasting volatility with GARCH(1,1) ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Figure 20: Forecasting mean and volatility Extra topics MBF: Modelling volatility Forecasting volatility Figure 20: Forecasting mean and volatility Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Volatility and Risk: Risks of Large Losses ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Amaranth h/f ($6.5 billion in one week in September 2006) • Credit Lyonnais ($5.0 billion in 1990) • Long-Term Capital Management h/f ($4.6 billion in 1998) • ($3.9 billion in September 2008) • Orange County ($2 billion in 1994) • Barings ($1.4 billion in 1995) • Daiwa Bank ($1.1 billion in 1995) • Allied Irish Bank ($0.7 billion in 2002) • China Aviation Oil ($0.6 billion in 2004) Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Value at Risk VaR ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Risk managers/regulators are often interested in the following statement: ”I am 99% certain that my portfolio of assets will not lose more than $V over the next period and have sufficient reserves to cover losses lower than this level. ” period is often one day, but can be a month, quarter, year (1− α)100% VaR : VaR1−α = F−1(α)× Value of Investment VaR is the maximum portfolio loss in a given period (eg, 1 day) with a given probability (eg, 0.99). 99% Value at Risk VaR0.99 $ PORTFOLIO RETURNS ( ) ( ) 0.01 Pdf , f(x), of the next period portfolio returns School of Economics, UNSW Slides-08, Financial Econometrics 15 Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Conditional Value at Risk Conditional Value at Risk ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Consider AR(1)−GARCH(1, 1) for the portfolio return yt yt = c+ φ1yt−1 + µt, where µt|Ωt−1 ∼ N(0, σ2t ) σ2t = α0 + α1µ � νt = µtσt = ∼ N(0, 1), where yt|t−1 = E(yt|Ωt−1) � P (νt < −2.326) = 0.01 = 1− 0.99 implies: P (yt < yt|t−1 − 2.326σt) = 0.01 � VaR0.99 = 1100 (yt|t−1 − 2.326σt)× Portfolio Value Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Conditional Value at Risk ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 5. Modelling Return Volatility: GARCH – Conditional value at risk (VaR) eg. NYSE composite return (continued) Portfolio valued at $1m at T = 2002-08-29. AR(1)-GARCH(1,1): 𝜎𝜎𝑇𝑇+1 =1.64196, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.05132. 𝑦𝑦𝑇𝑇+1|𝑇𝑇 − 2.326𝜎𝜎𝑇𝑇+1 ×$1m = −$37,678 If using the sample mean, sample variance and normality, [.0353 – 2.326(1.0062)] ×$1m = − $23,051. • But, normality is strongly rejected! School of Economics, UNSW Slides-08, Financial Econometrics 17 Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Empirical Quantile VaR using Empirical Quantile ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material BUT normality is often rejected? � GARCH is able to account for clustering, such that the standardised shock (νt) can be viewed as iid. � To compute VaR, we only need the lower quantile of νt, which can be estimated by the empirical quantile of the standardised residuals. • Instead of using the N(0, 1) to find F−1(α), we need to use the distrubution of the the estimated standardised residuals νt • νt = µtσt = ∼ iid(0, 1) • P (νt < Q0.01) = 0.01 = 1− 0.99 implies P (yt < yt|t−1 −Q0.01σt) = 0.01 = 1− 0.99 (yt|t−1 −Q0.01σt)× Portfolio Value Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Empirical Quantile ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material Topic 5. Modelling Return Volatility: GARCH – Conditional value at risk (VaR) eg. NYSE composite return (continued) Portfolio valued at $1m at T = 2002-08-29. AR(1)-GARCH(1,1): 𝜎𝜎𝑇𝑇+1 =1.64196, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.05132. The 1% quantile of 𝑣𝑣𝑡𝑡: 𝑞𝑞0.01 = −2.873 𝑦𝑦𝑇𝑇+1|𝑇𝑇 − 2.873𝜎𝜎𝑇𝑇+1 ×$1m = −$46,660 For ARCH(5): 𝜎𝜎𝑇𝑇+1 = 1.25322, 𝑦𝑦𝑇𝑇+1|𝑇𝑇 =0.05037, 𝑞𝑞0.01 = −2.774, VaR = −$34,260 School of Economics, UNSW Slides-08, Financial Econometrics 20 Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH A measure of persistence: half-life time ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material • Let ωt = µ2t − σ2t , then µ2t has an ARMA(1,1) representation:µ2t = α0 + (α1 + β1)µ t−1 + ωt − β1ωt−1 • When the shocks are zero, ie, ω = 0 for all t, by substitution: 1 + · · ·+ (α1 + β1)t−1 + (α1 + β1) The impact of µ20 on µ t is (α1 + β1) t, ceteris paribus. I Half-life time, tH , is defined as the number of periods required for the impact to be halved 0, or tH = ln(α1 + β1) eg. Composite return: α1 + β1 = 0.996, tH = 172.9 (days). Slides-11 UNSW Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH Integrated GARCH: iGARCH ©Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this material I What happens if α1 + β1 = 1? (known as iGARCH) I When α0 > 0, the unconditional variance is NOT finite and grows with t:
E(σ2t ) = α0t+ E(σ

True because E(σ2t ) = α0 + (α1 + β1)E(σ
t−1) = α0 + E(σ

We may write α0 = (1− α1 − β1)ω, where ω is the unconditional variance
of µt for α1 + β1 = 1.

I When α1 + β1 = 1 and α0 = 0, the conditional variance is an EWMA of

t = (1− β1)µ

which, as an EWMA, is not mean-reverting.

eg. NYSE composite return: The above explains why GARCH is very slow to
revert to the average level

Slides-11 UNSW

Forecasting Volatility in GARCH Volatility and Risk Half Time IGARCH

• Forecasting with the GARCH follows similar recurssive structure as an
autoregressive model.

• The long run forcast of volatility converges to the unconditional variance
of the process.
• Application to VaR: measures the risk exposure and the maximum amount

of loss in dollar value forecast for the next period:
• The VaR involves the mean, and variance of the distribution of

returns/payoffs of investment,
• GARCH/ARCH models allow us to compute Conditional VaR,
• The unconditional mean and variance underestimates the VaR: conditional

VaR bigger in absolute value than Unconditional VaR (based on the mean
and sample variance)

• The normal distribution quantile leads to underestimating the VaR
compared to using the empirical quantile.

• The Half-life time measures the amount of persistence in the GARCH.

Slides-11 UNSW

Forecasting Volatility in GARCH
Example 1:Forecasting with GARCH(1,1)

Volatility and Risk
Conditional Value at Risk
Empirical Quantile

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