Leture 4: Multivariate Volatility Modelling – Part I
FM321: Risk Management and Zhu
18 October 2022
LSE Finance
Copyright By PowCoder代写 加微信 powcoder
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 1 / 15
Motivation: a hedge fund’s problem
Imagine you run a quantitative equity fund.
6,000+ US stocks to choose from (around 1,000 considered very liquid, Russell 1000)
If the fund trades international equity, 60,000+ stocks to choose from, though some not accessible from abroad.
A global macro strategy reviews securities in other asset classes (fixed income, currency, commodities etc.) although most likely you will trade index futures rather than individual securities.
How would you decide which securities to invest in and how much?
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 2 / 15
Motivation: a hedge fund’s problem
Security return vector: rt = (r1,t,…,rN,t)′.
Conditional variance-covariance matrix of rt (given information up to
Analogous to the univariate case, we can write our model as
r =Σ1/2z ttt
where zt = (z1,t,…,zN,t), zi,t ∼ iid (0,1) and zi,t’s are independent across i.
Portfolio weight vector: wt = (w1,t,…,wN,t)′.
Portfolio return: rP,t = wt′rt = PNi=1 wi,tri,t
Portfolio variance: σ2 = w′Σ w P,t ttt
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 3 / 15
Motivation: a hedge fund’s problem
A simplest way to frame a hedge fund’s problem at t − 1 is given below.
max E [r ]−λσ2 t−1 P,t P,t
wt 2 s.t. rP,t=wt′rt
σ2 = w′Σ w P,t ttt
This is called the quadratic optimization problem and is often used as the benchmark problem for portfolio construction. The fund chooses λ depending on its risk preference.
To solve the problem, we need an estimate of Σt .
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 4 / 15
What do we need for a good estimate of Σt?
First, we must have σ2 ≥ 0 for any vector wt , which implies that P,t
the matrix Σt must be positive semidefinite.
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 5 / 15
What do we need for a good estimate of Σt?
Second, all elements of Σt, including covariance terms, should be
easily estimated.
As the number N of securities grows, the number of distinct elements of their covariance matrix grows quadratically.
For N = 3, we need to estimate 6 distinct elements of the covariance matrix:
t 12,t 2,t
σ σ σ2 13,t 23,t 3,t
(Note that, by symmetry, σij = σji )
With N assets, the number of distinct elements is N(N+1). 2
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I
The curse of dimensionality
Depending on what model we’re considering, the number of parameters to be estimated can grow at even higher rates than quadratically in N.
Recall that ideally we’d work with many more observations than parameters to be estimated.
Financial institutions often work with hundreds or thousands of securities in their universe (if not more).
Models that suffer from the Curse of Dimensionality will suffer from a great degree of statistical uncertainty for portfolios with typical number of securities, and won’t be useful in practice.
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 7 / 15
Challenges of multivariate volatility models
Challenge 1: Σˆt must be positive semidefinite.
Challenge 2: need to deal with the curse of dimensionality
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 8 / 15
Multivariate moving averages: EWMA
The exponentially-weighted version is given by
Σt = (1 − λ)rt−1rt′−1 + λΣt−1
Individual element:
σij,t =(1−λ)ri,t−1rj,t−1 +λσij,t−1 i,j =1,…,K
EWMA ensures that Σˆt is positive semidefinite.
Suffers from the same problems as in the univariate case.
A single λ would lead to excessive movements of covariance terms.
Multiple λ’s would lead to problem of the curse of dimensionality. The number of parameters is quadratic in N.
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 9 / 15
BEKK model
Recall the univariate GARCH model:
σ2 =ω+αr2 +βσ2
A generalization to the multivariate case (where rt−1 becomes a vector and σt become a matrix Σt):
Σt = ΩΩ′ + Art−1rt′−1A′ + BΣt−1B′ Guarantees positive definite estimates.
The number of parameters is still quadratic in N. Developed by Baba, Engle, Kraft and Kroner (1990).
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 10 / 15
BEKK model
The specification for N = 2 and 1 lag is: ω11 0 ω11 ω12
Σt = ω12 ω22 0 ω22 | {z }| {z }
11 12 1,t−1
ααrr r2 αα
Ω Ω′ α α r2
r1,t−1r2,t−1α α 11 21
1,t−1 2,t−1 {z
r t − 1 r t′− 1 σ12,t−1β
β β σ σ2 β β
Leture 4: Multivariate Volatility Modelling – Part I
Appendix (will not be tested in ICA)
Vector GARCH
Usually expressed in compact form to avoid repeated elements For example, with N = 2 we denote
σ2σ r Σ=1,t12,t r=1,t
tσσ2 tr 12,t 2,t 2,t
The model formulation for a V-GARCH(1,1) in this case is
σ2 ω A A A r2 1,t 11 11 12 13 1,t−1
σ12,t = ω22 + A21 A22 A23 r1,t−1r2,t−1 σ2 ω A A A r2
33 31 32 33 B B Bσ2
11 12 13 1,t−1 + B21 B22 B23 σ122,t−1
B31 B32 B33 σ2
Leture 4: Multivariate Volatility Modelling – Part I
Vector GARCH
In the general formulation, every element of Σt is allowed to depend on every element of Σt−1 and rt−1rt′−1.
The number of parameters in the model is of the order of N4.
In addition to severe dimensionality problems, any estimates from V-GARCH are not guaranteed to be positive definite.
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 14 / 15
Diagonal V-GARCH
Simplification of V-GARCH that restricts the coefficient matrices to be diagonal.
The model formulation for a V-GARCH(1,1) in this case is σ2ωA 0 0r2
1,t 11 11 1,t−1 σ12,t=ω22+ 0 A22 0 r1,t−1r2,t−1
σ2 ω 00A r2
2,t 33 33 2,t−1
B 0 0σ2 11 1,t −1
+ 0 B22 0 σ122,t−1 0 0 B33 σ2
While this limits the dimensionality problem (the number of parameters is of the order N2), resulting estimates are still not guaranteed to be positive definite.
LSE FM321 Leture 4: Multivariate Volatility Modelling – Part I 15 / 15
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com