M01439

A Comparison of MCMC Samplers for

Estimating Leveraged Stochastic Volatility

Model

Didit B. Nugroho1,2 _{and Takayuki Morimoto}1

1_{Department of Mathematical Sciences, Kwansei Gakuin University, Japan} 2_{Department of Mathematics, Satya Wacana Christian University, Indonesia}

December 23, 2013, Hiroshima University of Economics

Abstract

1. Introduction: SV and MCMC methods

Stochastic volatility (SV) model is among the most important tools for modeling volatility of a financial time series.

A problem in parametric SV models is that it is not possible to obtain an explicit expression for the likelihood function of some unknown parameters.

An approach has become very attractive is Markov Chain Monte Carlo (MCMC) method proposed by Shephard (1993) and Jacquier et al (1994).

Omori and Watanabe (2008)proposed an efficient multi-move (block) Metropolis-Hastings sampler for sampling high-dimensional latent volatil-ity in the leveraged SV model.

2. Survey on MCMC Comparison

Kim et al. (1998) andJacquier-Polson (2011)compared the perfor-mance of the single-move and multi-move MH (MM-MH) samplers to estimate basic SV model. The MM-MH sampler has been proposed to reduce sample autocorrelations effectively. Jacquier-Polson par-ticularly note that the single-move and multi-move samplers deliver almost the same output.

Takaishi (2009) compared the performance of the single-move MH and HMC samplers and concluded that HMC sampler is superior to the single-move sampler.

3. Our Purpose

First purpose of this study is to extend the HMC and RMHMC sam-pling procedures proposed byGirolami and Calderhead (2011)for the leveraged SV model.

4. HMC and RMHMC Samplers

HMC-based methods are based on Hamiltonian dynamics system:

H₍θ_{,ω) =−L(}θ_{) +}1

2log

n

(2π)D|M|o+1₂ω′M−1ω,

where L(θ_{) is the logarithm of the joint probability distribution for}

the parameters θ_∈RD_{,Mis the covariance matrix, andω∈}RD_is

the independent auxiliary variable.

In the RMHMC sampling,Mdepends on the variableθ_{and is chosen}

to be the metric tensor, i.e.

4. HMC and RMHMC Samplers (Cont’ed)

The full algorithm for HMC or RMHMC can then be summarized in the following three steps.

(1) Randomly draw a sample momentum vector ω∼ N(ω|0,M). (2) Run the leapfrog algorithm for N_{L steps with step size}∆_{τ to}

generate a proposal (θ∗_,ω∗_{)according to the Hamiltonian equations}

d_θ

d_τ = ∂H ∂ω and

d_ω

d_τ =− ∂H

∂θ.

over a fictitious time τ.

(3) Accept (θ∗_,ω∗_{)with probability}

5. Leveraged SV Model

Omori and Watanabe (2008) proposed a leveraged SV model with normal errors (LSV-N model, hereafter) formulated as

Rt = σre

using MH algorithm separately and the latent volatilityht_using

6. Comparison on Real Data

6. Comparison on Real Data (Cont’ed)

Tabel: Tuning parameters for the HMC and RMHMC implementations.

Parameter of model Sampler Parameter of

sampler h φ (σr,σh,ρ)

HMC N_∆L 100 100 100

τ 0.01 0.0125 0.0125

RMHMC

NL 50 6 6

∆_{τ 0.1 0.5 0.5}

N_FPI - 5 5

6. Comparison on Real Data (Cont’ed)

Tabel: Posterior summary statistics.

Parameter Mean (SD) 95%Interval IACT Time (sec) Panel A: Using MM-MH sampler

σr 1.259(0.070) [1.121, 1.398] 20.8

−

φ 0.945(0.019) [0.902, 0.974] 118.2

σh 0.193(0.033) [0.138, 0.267] 206.7

ρ −0.442(0.103) [−0.630,−0.231] 92.7 Panel B: Using HMC sampler

σr 1.241(0.070) [1.098, 1.377] 53.0

465.88

φ 0.954(0.014) [0.925, 0.980] 132.7

σh 0.169(0.025) [0.120, 0.219] 279.9

ρ −0.461(0.097) [−0.648,−0.272] 83.9 Panel C: Using RMHMC sampler

σr 1.238(0.070) [1.098, 1.378] 19.4

6. Comparison on Real Data (Cont’ed)

Tabel: Acceptance rates.

Acceptance rates Parameter

MM-MH HMC RMHMC

h 0.856 0.889 0.975

φ 0.955 0.910 0.901

7. Conclusion

Based on the empirical results, we conclude:

The RMHMC sampler give the best performance in terms of autocor-relation time.

In particular, the HMC sampler exhibits slightly slower convergence than MM-HMC sampler except for the leverage parameter.

Some References

Girolami, M. and Calderhead, B.(2011). Riemann manifold Langevin and

Hamiltonian Monte Carlo methods.Journal of the Royal Statistical Society, Series B,73(2), 1–37.

Jacquier, E. and Polson, N. G. (2011). Bayesian methods in finance. In J.

Geweke, G. Koop, and H. van Dijk (Eds.),Handbook of Bayesian Econometrics. Oxford University Press.

Kim, S. and Shephard, N. and Chib, S.(1998). Stochastic volatility: likelihood

inference and comparison with ARCH models. In N. Shephard (Ed.),Stochastic Volatility: Selected Readings. Oxford University Press.

Omori, Y. and Watanabe, T.(2008). Block sampler and posterior mode

esti-mation for asymmetric stochastic volatility models.Computational Statistics and Data Analysis,52, 2892–2910.