Leverage, asymmetry and heavy tails in the high-dimensionalfactor stochastic volatility model

Mengheng Li, m.li@vu.nl, +316 141 519 03Department of Econometrics, VU University Amsterdam, NLMarcel ScharthDiscipline of Business Analytics, The University of Sydney Business School, AU

AbstractWe propose a flexible high-dimensional factor stochastic volatility (SV)

model with leverage effect based on the generalised hyperbolic skew Stu-dent’s t-error to model asymmetry and heavy tails. With shrinkage, the modelleads to different parsimonious forms, and thus is able to disengage system-atic leverage effect and skewness from asset-specific ones. We also developa highly efficient Markov chain Monte Carlo estimation procedure to analysethe univariate version of the model based on efficient importance sampling.With marginalisation of factors, extension to high-dimensional factor modelis achieved with computational complexity shown to be linear in the numberof factors and assets.

IntroductionIn the literature of univariate SV models, parameter-driven SV modelsand observation-driven (G)ARCH type of models (Kim et al., 1998)are mainly used to model the following “stylised facts” of asset re-turns:•Volatility clustering (consecutive volatile periods);•Heavy tails of asset returns (extreme events);•Asymmetry or skewness of asset returns (loss bigger than gains);• Leverage effect (loss correlates with higher volatility).

When modelling a portfolio of asset returns, “stylised facts” of in-dividual asset returns are linked via the hypothetical market portfo-lio (Fama and French, 1993), leading to non-diagonal higher-momentmatrices. This makes modelling high-dimensional portfolios expo-nentially complex due to the “curse of dimensionality”.

Factor SV modelA straightforward remedy is through a factor model. A factor modelnaturally delivers systematic interpretation for the multivariate dy-namics of a vector of time series, which can help track the sourcesof observed “stylised facts”. In the spirit of Chib et al. (2006), wepropose the following factor SV model

yt = Λft + ut, t = 1, ..., T,

fj,tTt=1 ∼ Model (1), ∀j ∈ 1, ..., p,ui,tTt=1 ∼ Model (1), ∀i ∈ 1, ..., n.

Model (1) is sufficiently flexible for capturing “stylised facts”. A goodcandidate of such a model is the following SV model of Nakajima andOmori (2012),

yt = νt exp(ht/2), t = 1, ..., T,

νt = α + βWt +√Wtεt, t = 1, ..., T,

ht+1 = µ(1− φ) + φht + ηt, t = 1, ..., T − 1,[εtηt

]∼ N

( [00

],

[1 ρσ

ρσ σ2

] ), t = 1, ..., T,

Wt ∼ IG(ζ

2,ζ

2), t = 1, ..., T,

(1)

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Figure 1: Different density shapes of generalised hyperbolic skew Student’s t-distribution. Left: varying β with ζ = 10; Right: varying ζ with β = −2.

MCMC sampler for the factor SV modelThe sampling scheme is straightforward, and with marginalisation offactors ft, the loadings Λ can be sampled efficiently. Notice the factorSV model gives

yi,t =

p∑j=1

Λij(αfj + βfjWj,t +

√Wj,tξj,t)e

hj,t/2 + (αui+

βuiQi,t +√Qi,tεi,t)e

li,t/2.

The MCMC sampler iterates over• Sample ft|y1:T ,Λ, θ;•Marginalise fj,t = fj,t + ξj,t; Sample Λ|y1:T , fj,1:T

pj=1, θ;

•Obtain ut = yt − Λft;• Sample SV ht and lt, mixture series Wt and Qt, as well as other

hyperparameters θ using sampling scheme for Model (1).

MCMC sampler for the univariate SV modelThe last step of the sampler for the factor SV model is to apply anMCMC sampler for the p + n individual univariate SV model (1).Thus our model is linearly scalable in number of assets n and numberof factors p. The MCMC sampler has two parts.1. Sampling (h1:T ,W1:T )|y1:T , θ;2. Sampling θ|y1:T , h1:T ,W1:T .

PGAS-EIS sampler

We term our sampler particle Gibbs with ancestor sampling using effi-cient importance sampling (EIS-PGAS). EIS stems from particle effi-cient importance sampling (PEIS) of Scharth and Kohn (2016) whichbuilds a globally optimal importance density. Based on the fact thatexponential family kernels are closed under multiplication, a sequen-tial Monte Carlo sampler with EIS is given by

ht,Wt|ht−1,Wt−1, y1:T ∼ N(µt, vt) · IG(ζ

2+ st,

ζ

2+ rt),

where

νt =(1− ρ2)σ2

1 + (1− ρ2)σ2ct, µt = νt

(bt +

µ(1− φ) + φht−1 + ρσεt−1

(1− ρ2)σ2

),

and εt−1 = (yt−1e−ht−1/2 − α − βWt−1)/

√Wt−1. bt, ct, st and rt

are importance parameters determined by a sequence of simple OLS,minimising the χ2-divergence between the importance density and theconditional posterior.

Figure 2: Posterior estimate of the SV series h1:T . Left: Particle Gibbs with abootstrap filter; Right: Particle Gibbs with EIS importance density. As is seen, withthe constructed globally optimal importance density particles receive higher weightsand the degeneration of particle system is mitigated.

The sampler is augmented with the ancestor sampling devise of Lind-sten et al. (2014), leading to improved mixing.

Lindsten, Jordan and Schon

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2

Time (t)

State

(xt)

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State

(xt)

Figure 2: Particle systems generated by the PG algorithm (left) and by the PGAS algorithm(right), for the same reference trajectory x′1:T (shown as a thick blue line in the leftpanel, partly underneath the red line). The gray dots show the particle positionsand the thin black lines show the ancestral dependencies of the particles. Theextracted trajectory x?1:T is illustrated with a red line. In the right panel, AShas the effect of breaking the reference trajectory into pieces, causing the particlesystem to degenerate toward something different than x′1:T . (This figure is bestviewed in color.)

blue lines are again used to illustrate the reference particles, but now with updated ancestor

indices. That is, the blue line segments are drawn between xaNtt−1 and x′t for t ≥ 2. It can

be seen that the effect of AS is that, informally, the reference trajectory is broken intopieces. It is worth pointing out that the particle system still collapses; AS does not preventpath degeneracy. However, it causes the particle system to degenerate toward somethingdifferent than the reference trajectory. As a consequence, x?1:T (shown as a red line in thefigure) will with high probability be substantially different from x′1:T , enabling high updaterates and thereby much faster mixing.

4. Theoretical Justification

In this section we investigate the invariance and ergodicity properties of the PGAS kernel.

4.1 Stationary Distribution

We begin by stating a theorem, whose proof is provided later in this section, which showsthat the invariance property of PG is not violated by the AS step.

Theorem 1. For any N ≥ 1 and θ ∈ Θ, the PGAS kernel PNθ leaves γθ,T invariant:

γθ,T (B) =

∫PNθ (x′1:T , B)γθ,T (dx′1:T ), ∀B ∈ X T .

2152

Figure 3: Particle degeneration or impoverishment is inevitable. Left: Without an-cestor sampling, particle Gibbs repeats the previous draw with higher probabilityas t increases; Right: ancestor sampling breaks the reference trajectory into pieces,which means new draw differs from previous draw with high probability.

Theorem 1 (Invariance). The EIS-PGAS kernel KMθ parametrised byθ ∈ Θ with any M ≥ 0 leaves the posterior probability density func-tion p(x1:T |y1:T ) invariant:∫Bp(x1:T |y1:T )dx1:T =

∫KMθ (x?1:T , B)p(x?1:T |y1:T )dx?1:T , ∀B ∈ F1:T .

Theorem 2 (Ergodicity). Suppose for any t = 1, ..., T and θ ∈ Θ,given xi1:t−1

M+1i=1 , B ∈ F1:T , and supxt

(maxi ω

iθ,t

)≤ ωθ < ∞.

Then for any M ≥ 1 and θ ∈ Θ, there exists some ϕ ∈ [0, 1) such that∣∣∣∣∣∣∣∣(KMθ )n(x?1:T , B)−∫Bp(x1:T |y1:T )dx1:T

∣∣∣∣∣∣∣∣TV≤ p(y1:T )

(N − 1

Nωθ

)Tϕn.

Shrinkage on leverage effect and asymmetry

To investigate the systematic content of factors, we modify the conju-gate normal prior for p + n skewness parameters βk, k = 1, ..., p + nusing a Bayesian shrinkage prior akin to Bayesian variable selection.This results in a shrinkage posterior

βk|· ∼ ∆βkD0(βk) + (1−∆βk)N(µβk, σ2βk

),

where

∆βk =1−∆β

∆βσ2βk

+ 1−∆β, with σ2

βk=σβkvβ

exp(µ2βk

2σ2βk

).

The shrinkage probability ∆β has a conjugate beta prior. The leverageeffect parameter ρk can be shrunken similarly via a conjugate normal-inverse-gamma prior.

Empirical applications

Some results

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Figure 4: Sorted posterior estimate of ρ and β. Left / Middle: posterior mean esti-mate of leverage effect ρ / skewness parameter β with 95% credible interval; Right:posterior zero probability of β against that of ρ. Coloured dots indicate the parame-ters corresponding to four factors.

We can calculate the implied time-varying correlation as

Corrij,t =

∑4k=1 ΛikΛjk exp(hk,t)

σi,tσj,t.

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The Dow Chemical CompanyWalgreens Boots Alliance, Inc.PayPal Holdings

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The Dow Chemical vs. Walgreens BootsThe Dow Chemical vs. PayPalWalgreens Boots vs. PayPal

Figure 5: Posterior mean of factor and stochastic volatility process. Left: SVexp(hj,t/2) from the j = 1, 4-th factors; Middle: Volatility σi,t of three chosen as-set returns; Right: Implied time-varying correlations Corrij,t among the three assetreturns.

In the year of financial crisis the three correlation series start climbingup, one of which even shoots up to over 0.4. Yet outside the crisis pe-riod the correlation can be low in absolute value. So equicorrelationmodels are suboptimal in diversification.

Dynamic portfolio management

The solution of this MVP problem is given by ωt+h|t =

Ω−1t+h|t/(′Ω−1

t+h|t). So, the value-at-risk (VaR) is

V aRp,t+1|t(α) =√ω′t+1|tΩt+1|tωt+1|tF

−1yp,t+1|t(α).

Sharpe ratio and information ratio are defined similarly using filteredestimate of covariance matrix.

Figure 6: The table shows p-values of coverage ratio tests for the US portfolio. Port-folio weights are updated weekly based on one-step ahead forecast of covariancematrix. α is the nominal level of VaR. Shaded cells indicate rejection of coverageratio test at 10% level.

Figure 7: The table shows the average weekly MVP returns and variances for theU.S. portfolio under different models. EquWgt denotes a equal weighted portfolio.SR and IR are also reported where the latter is relative to S&P 100 index return.One- and two-week rebalancing policies are considered. Shaded cells indicate thebest performer with lowest variance, highest mean, highest SR, or highest IR.

Conclusion

We propose a high-dimensional factor SV model with leverage ef-fect using the generalised hyperbolic skew Student’s t-error to addressasymmetry and heavy tails of equity returns, as well as a highly effi-cient MCMC algorithm for Bayesian inference. The model is shownto be flexible enough to distinguish asset-specific mean and volatilitydynamics from common factors. With shrinkage, the model helps an-swer whether leverage effect and return asymmetry are systematic oridiosyncratic.

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variate stochastic volatility models. Journal of Econometrics, 134(2):341–371.

Fama, E. F. and French, K. R. (1993). Common risk factors in the returns on stocksand bonds. Journal of financial economics, 33(1):3–56.

Kim, S., Shephard, N., and Chib, S. (1998). Stochastic volatility: likelihood in-ference and comparison with ARCH models. The Review of Economic Studies,65(3):361–393.

Lindsten, F., Jordan, M. I., and Schon, T. B. (2014). Particle gibbs with ancestorsampling. Journal of Machine Learning Research, 15(1):2145–2184.

Nakajima, J. and Omori, Y. (2012). Stochastic volatility model with leverage andasymmetrically heavy-tailed error using GH skew Student’s t-distribution. Com-putational Statistics & Data Analysis, 56(11):3690–3704.

Scharth, M. and Kohn, R. (2016). Particle efficient importance sampling. Journalof Econometrics, 190(1):133–147.