Leture 5: Multivariate Volatility Modelling – Part II
FM321: Risk Management and Zhu
25 October 2022
LSE Finance
Copyright By PowCoder代写 加微信 powcoder
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 1 / 10
Review and Preview
In multivariate volatility models, a vector of returns are written as:
r =Σ1/2z ttt
Previously, we wrote down models for Σt by directly generalizing univariate models.
EWMA: Σt = (1 − λ)rt−1rt′−1 + λΣt−1 BEKK: Σt = Ω′Ω + Art−1rt′−1A′ + BΣt−1B′
Now we will look at alternative approaches particularly suited for multivariate models.
Correlation models: Σt = DtCtDt, where Dt = diag{σ1,t,…,σN,t} and Ct correlation.
Factor models: rt = βft + εt so that Vart−1(rt ) = βΣf ,t β′ + Σε,t .
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 2 / 10
Conditional Correlation Models
If define the matrices Dt and Ct (which contain information about individual asset volatility and correlations, respectively) by:
σ1,t 0 … 0 1 ρ12,t … Dt = 0 σ2,t … 0 Ct =ρ21,t 1 … . . . . . . . . . . . . . . . . . . . . . 0 0 … σN,t ρN1,t ρN2,t …
then a simple algebraic verification shows that
Σt =DtCtDt
This allows us to model volatilities (Dt) and correlations (Ct)
separately.
Ct can be modeled as:
ρ1N,t ρ2N,t
. . . 1
a constant matrix → Constant Conditional Correlation (CCC).
a time-varying matrix → Dynamic Conditional Correlation (DCC).
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 3 / 10
Conditional Correlation Models: estimation
Estimate a univariate volatility model for each asset (e.g., GARCH).
For each asset, use univariate model to compute estimates for σˆt for each asset.
Use conditional volatilities to standardize the returns (that is, compute qˆt = rt .
Use the series for qˆt for all assets to model Ct.
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 4 / 10
Constant Conditional Correlation
In CCC, we assume that the true correlation matrix is a constant matrix, Ct = C , and estimate it using all available data:
1 ρˆ12,t … ρˆ1N,t
. . . ρˆ N 2 , t
. . . . . . . . .
ρˆ P t − 1 qˆ i , s qˆ j , s 2N,t s=1
ρˆ N 1 , t
ρˆ =r ij,t
. . . P t − 1 qˆ 2 P t − 1 qˆ 2 1 s = 1 i , s s = 1 j , s
The complexity of the model depends on what assumptions we place on C:
If no restrictions are placed, the number of correlations to estimate is still of the order N2.
If we say that the pairwise correlation between assets is constant (that is, ρij = ρ for every pair of securities), the curse of dimensionality disappears; but should we believe this assumption?
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 5 / 10
Dynamic Conditional Correlation
In DCC, we specify the evolution of Ct. The elements of Ct are given by
hij ,t ρij,t = phii,thjj,t
hij,t: latent covariance between qˆi,t and qˆj,t. Two common choices for the evolution of hij,t:
hij,t = (1 − λ)qi,t−1qj,t−1 + λhij,t−1
GARCH(1,1):
hij,t = (1 − α − β)h ̄ij + αqi,t−1qj,t−1 + βhij,t−1
Ct symmetric and positive semidefinite (if we make sure h ̄ij = h ̄ji . Number of parameters fixed regardless of N.
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 6 / 10
Large Problems
Managers deal with large universes.
MSCI World All-Country Index has over 2400 stocks.
Around 2,800 stocks trade in the NYSE.
Financial institutions can have tens of thousands of assets, if not more.
None of the models we have discussed so far can be scaled to that number of securities.
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 7 / 10
Factor Models
Basic Idea: find a few sources of risk (factors) that account for a sizeable portion of the common risk across securities, and model risk around those sources.
Market risk
Country risk
Sector risk
Macroeconomic risk factors
Risk factors derived from anomalies (value, size, momentum) In yield curve contexts: level, slope and convexity
In currency contexts: carry trade factor
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 8 / 10
Factor Models – General Formulation
In general, the formulation is
rt =βft +εt
r is an N × 1 vector of (excess) returns.
β is an N × F matrix of factor exposures (containing the exposure of
each security to each factor).
f is an F × 1 vector of factor returns.
ε is an N × 1 vector of residual returns for the securities.
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 9 / 10
Factor Models
From these assumptions, we conclude that
Σt = Vart−1(rt) = βΣf ,tβ′ + Σε,t The number of parameters to estimate is therefore:
The N × F factor exposures in β (can be constant or time-varying); The 12F(F +1) elements of Σf,t; and
the N non-zero elements of Σε,t.
The total number of parameters is N(F +1)+ 12F(F +1) , which is linear in N.
LSE FM321 Leture 5: Multivariate Volatility Modelling – Part II 10 / 10
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com