GARCH models Magnus Wiktorsson SW-[?]ARCH An advanced extension is - - PowerPoint PPT Presentation

GARCH models

Magnus Wiktorsson

SW-[?]ARCH

An advanced extension is the switching ARCH model.

▶ The conditional variance is given by a standard

ARCH, GARCH or EGARCH (the later two are non-trivial, due to their non-Markovian structure)

▶ The model is given by

rt = √ g(St)σ2

t zt, ▶ where g(1) = 1 and (g(n), n ≥ 2) are free

parameters.

Fractionally Integrated GARCH

▶ Recall the ARMA representation of the GARCH

model (1 − ψ(B)) ϵ2

t = ω + (1 − β(B)) νt

(1) and the IGARCH representation is given by Φ(B)(1 − B)ϵ2

t = ω + (1 − β(B)) νt.

(2)

▶ Can we have something in-between?

Yes, that is the FIGARCH model B 1 B d 2

t

1 B

t

(3) with the process having finite variance if 5 d 0 5. The fractional differentiation can be computed as 1 B d

k

k d k 1 d Bk (4)

Fractionally Integrated GARCH

▶ Recall the ARMA representation of the GARCH

model (1 − ψ(B)) ϵ2

t = ω + (1 − β(B)) νt

(1) and the IGARCH representation is given by Φ(B)(1 − B)ϵ2

t = ω + (1 − β(B)) νt.

(2)

▶ Can we have something in-between? Yes, that is

the FIGARCH model Φ(B)(1 − B)dϵ2

t = ω + (1 − β(B)) νt

(3) with the process having finite variance if −.5 < d < 0.5. The fractional differentiation can be computed as 1 B d

k

k d k 1 d Bk (4)

Fractionally Integrated GARCH

▶ Recall the ARMA representation of the GARCH

model (1 − ψ(B)) ϵ2

t = ω + (1 − β(B)) νt

(1) and the IGARCH representation is given by Φ(B)(1 − B)ϵ2

t = ω + (1 − β(B)) νt.

(2)

▶ Can we have something in-between? Yes, that is

the FIGARCH model Φ(B)(1 − B)dϵ2

t = ω + (1 − β(B)) νt

(3) with the process having finite variance if −.5 < d < 0.5.

▶ The fractional differentiation can be computed

as (1 − B)d =

∞

∑

k=0

Γ(k − d) Γ(k + 1)Γ(−d)Bk. (4)

GARCH in Mean

Asset pricing models may include variance terms as explanatory factors (think CAPM). This can be captured by GARCH in Mean models. rt = µt + δf(σ2

t ) +

√ σ2

t zt.

Multivariate models

What about multivariate models? Returns: Rt = H1/2

t

Zt

▶ Huge number of models.

▶ VEC-MVGARCH (1988) ▶ BEKK-MVGARCH (1995) ▶ CCC-MVGARCH (1990) ▶ DCC-MVGARCH (2002) ▶ STCC-MVGARCH(2005)

Most are overparametrized. I recommend starting with the CCC-MVGARCH

Multivariate models

What about multivariate models? Returns: Rt = H1/2

t

Zt

▶ Huge number of models.

▶ VEC-MVGARCH (1988) ▶ BEKK-MVGARCH (1995) ▶ CCC-MVGARCH (1990) ▶ DCC-MVGARCH (2002) ▶ STCC-MVGARCH(2005)

▶ Most are overparametrized. ▶ I recommend starting with the CCC-MVGARCH

log-Likelihood

The log-likelihood for a general Multivariate GARCH model is given by ℓT(θ) = −1 2

T

∑

t=1

ln | det(2πHt)| − 1 2

T

∑

t=1

rT

t H−1 t rt.

(5)

VEC-MVGARCH

Uses the vech operator. Model given by vech Ht C

p j 1

Ajvech rt

jrT t j q j 1

Bjvech Ht

j

Cons: Large number of parameters! Difficult to impose positive definiteness on Ht

VEC-MVGARCH

Uses the vech operator. Model given by vech(Ht) = C +

p

∑

j=1

Ajvech(rt−jrT

t−j) + q

∑

j=1

Bjvech(Ht−j) Cons: Large number of parameters! Difficult to impose positive definiteness on Ht

VEC-MVGARCH

Uses the vech operator. Model given by vech(Ht) = C +

p

∑

j=1

Ajvech(rt−jrT

t−j) + q

∑

j=1

Bjvech(Ht−j) Cons:

▶ Large number of parameters! ▶ Difficult to impose positive definiteness on Ht

BEKK-MVGARCH

Uses Cholesky decomposition Model given by Ht = CCT +

p

∑

j=1 K

∑

k=1

AT

k,jrt−jrT t−jAk,j + q

∑

j=1 K

∑

k=1

BT

k,jHt−jBk,j

Pros: Fewer parameters (but still too many!) Positive definite by construction

BEKK-MVGARCH

Uses Cholesky decomposition Model given by Ht = CCT +

p

∑

j=1 K

∑

k=1

AT

k,jrt−jrT t−jAk,j + q

∑

j=1 K

∑

k=1

BT

k,jHt−jBk,j

Pros:

▶ Fewer parameters (but still too many!) ▶ Positive definite by construction

CCC-MVGARCH

Constant Conditional Correlation

▶ Ht = ∆tPc∆t where

▶ ∆ = diag(σt,k) ▶ Pc is a constant correlation matrix.

Here

t k is any standard univariate [X]ARCH

model. Easy to optimize the likelihood for the CCC-MVGARCH, not so easy for other models.

CCC-MVGARCH

Constant Conditional Correlation

▶ Ht = ∆tPc∆t where

▶ ∆ = diag(σt,k) ▶ Pc is a constant correlation matrix.

▶ Here σt,k is any standard univariate [X]ARCH

model. Easy to optimize the likelihood for the CCC-MVGARCH, not so easy for other models.

STCC-GARCH and DCC-GARCH

The main limitation of the CCC-GARCH is the fixed correlation.

▶ The DCC-GARCH uses

Pt = (I ⊙ Qt)−1/2Qt(I ⊙ Qt)−1/2 (6) where Qt = (1−a−b)S+aϵt−1ϵT

t−1+bQt−1, a+b < 1. (7) ▶ An alternative is the STCC-GARCH

Pt = (1 − G(st))P(1) + G(st)P(2), (8) where P(1), P(2) are correlation matrices and G(·) is some smooth transition function.

Some wellknown Swedish assets

On the second computer exercise you will try to fit a CCC-MVGARCH model to this.

2005 2006 2007 2008 2009 2010 100 200 300 400 500 600 ABB AstrazenecaB Boliden InvestorB Lundin MTGB Nordea Tele2

B

Related concepts

▶ Recursive parameter estimation methods

transforms fixed parameters into variable quantities (ex RLS)

▶ This was taken one step further (?) in the GAS -

Generalized Autoregressive Score framework (Creal et al., 2013)

Generalized Autoregressive Score

Assume the data is given by rt = σtzt (9) How does σt vary?

▶ GARCH/EGARCH/... ▶ Or updating the parameters such that the

likelihood is increasing.

Generalized Autoregressive Score

Let the data be generated from the observation density yt ∼ p(yt|ft, θ, Ft−1) (10) where

▶ ft are time varying parameters ▶ θ are static parameters ▶ Ft−1 is some set of information (ex. lagged

values of y)

Furthermore, assume that the time varying parameters have an autoregressive dynamics ft+1 = ω +

p

∑

i=1

Aist−i+1 +

q

∑

j=1

Bjft−j+1 (11) with ω, Ai, Bj being parameters and st some function

• f the past, typically

st = St ∂ log p(yt|ft, θ, Ft−1) ∂ft (12) with St being some matrix, e.g. St = (IF)−1.

Gaussian example

Let the model by heteroscedastic white noise yt = σtzt, and define the time varying parameter as ft = σ2

t .

That leads to log p yt ft

t 1

ft 1 2 y2

t

ft f2

t

(13) IF 1 2f2

t

Gaussian example

Let the model by heteroscedastic white noise yt = σtzt, and define the time varying parameter as ft = σ2

t .

That leads to ∂ log p(yt|ft, θ, Ft−1) ∂ft = 1 2 (y2

t − ft

f2

t

) (13) IF = 1 2f2

t

That leads to the parameter dynamics ft+1 = ω + A1 ( 1 2f2

t

)−1 1 2 (y2

t − ft

f2

t

) + B1ft This simplifies into ft+1 = ω + A1y2

t + (B1 − A1)ft

Does this look familiar? Yes, it is the GARCH(1,1) model! However, the student-t version is not identical to the GARCH(1,1) version

That leads to the parameter dynamics ft+1 = ω + A1 ( 1 2f2

t

)−1 1 2 (y2

t − ft

f2

t

) + B1ft This simplifies into ft+1 = ω + A1y2

t + (B1 − A1)ft

Does this look familiar? Yes, it is the GARCH(1,1) model! However, the student-t version is not identical to the GARCH(1,1) version

That leads to the parameter dynamics ft+1 = ω + A1 ( 1 2f2

t

)−1 1 2 (y2

t − ft

f2

t

) + B1ft This simplifies into ft+1 = ω + A1y2

t + (B1 − A1)ft

Does this look familiar? Yes, it is the GARCH(1,1) model! However, the student-t version is not identical to the GARCH(1,1) version

Stochastic Volatility (SV)

Let rt be a stochastic process.

▶ The log returns (observed) are given by

rt = exp(Vt/2)zt.

▶ The volatility Vt is a hidden AR process

Vt = α + βVt−1 + et.

▶ Or more general

A(·)Vt = et.

▶ More flexible than e.g. EGARCH models! ▶ Multivariate extensions.

A simulation of Taylor (1982)

100 200 300 400 500 600 700 800 900 1000 0.1 0.2 0.3 0.4

exp(x/2)

100 200 300 400 500 600 700 800 900 1000

• 0.5

0.5 1

returns

With α = −0.2, β = 0.95 and σ = 0.2.

Long Memory Stochastic Volatility (LMSV)

The autocorr. of volatility decays slower than exp. rate

▶ The returns (observed) are given by

rt = exp(Vt/2)zt.

▶ The volatility Vt is a hidden, fractionally

integrated AR process A(·)(1 − q−1)bVt = et, where b ∈ (0, 0.5).

▶ This gives long memory!

Long Memory Stochastic Volatility (LMSV)

▶ The long memory model can be approximated

by a large AR process.

▶ It can be shown that

(1 − q−1)b =

∞

∑

j=0

πjq−j, where πj = Γ(j − b) Γ(j + 1)Γ(−b).

Quasi Likelihood inference

▶ The parameters in the SV model can be found by

studying yt = log(r2

t ) and xt = Vt.

This leads to (with

t

log z2

t )

yt log r2

t

log exp Vt log z2

t

xt

t

xt xt

1

et Estimate volatility and parameters using a Kalman filter! Practical consideration: rt 0 in the real world.

Quasi Likelihood inference

▶ The parameters in the SV model can be found by

studying yt = log(r2

t ) and xt = Vt. ▶ This leads to (with ηt = log(z2 t ))

yt = log(r2

t ) = log(exp(Vt)) + log(z2 t ) = xt + ηt

xt = α + βxt−1 + et.

▶ Estimate volatility and parameters using a

Kalman filter! Practical consideration: rt 0 in the real world.

Quasi Likelihood inference

▶ The parameters in the SV model can be found by

studying yt = log(r2

t ) and xt = Vt. ▶ This leads to (with ηt = log(z2 t ))

yt = log(r2

t ) = log(exp(Vt)) + log(z2 t ) = xt + ηt

xt = α + βxt−1 + et.

▶ Estimate volatility and parameters using a

Kalman filter!

▶ Practical consideration: P (rt = 0) > 0 in the real

world.

Stochastic Volatility in continuous time

A popular application of stoch. volatility models is

• ption valuation.

▶ Several parameterizations. ▶ The (Heston, 1993) model is the most used

model, mainly due to computational properties dSt = µStdt + √ VtStdW(S)

t

dVt = κ(θ − Vt)dt + σ √ VtdW(V)

t

dW(S)

t dW(V) t

= ρdt

▶ Note that the drift and squared diffusion have

affine form in V.

▶ This reduces the task of computing prices to

inversion of a Fourier integral.

References

▶ Creal, D., Koopman, S. J., & Lucas, A. (2013).

Generalized autoregressive score models with

• applications. Journal of Applied Econometrics,

28(5), 777-795.

▶ Heston, S. L. (1993). A closed-form solution for

• ptions with stochastic volatility with

applications to bond and currency options. Review of financial studies, 6(2), 327-343.

▶ Bauwens, L., Laurent, S., & Rombouts, J. V.

(2006). Multivariate GARCH models: a survey. Journal of applied econometrics, 21(1), 79-109.

▶ Silvennoinen, A., & Teräsvirta, T. (2009).

Multivariate GARCH models. Handbook of financial time series, 201-229.