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A triple-threshold leverage stochastic volatility model

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Published/Copyright: November 25, 2014
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Studies in Nonlinear Dynamics & Econometrics
From the journal Studies in Nonlinear Dynamics & Econometrics Volume 19 Issue 4

Abstract

In this paper we introduce a triple-threshold leverage stochastic volatility (TTLSV) model for financial return time series. The main feature of the model is to allow asymmetries in the leverage effect as well as mean and volatility. In the model the asymmetric effect is modeled by a threshold nonlinear structure that the two regimes are determined by the sign of the past returns. The model parameters are estimated using maximum likelihood (ML) method based on the efficient importance sampling (EIS) technique. Monte Carlo simulations are presented to examine the accuracy and finite sample properties of the proposed methodology. The EIS-based ML (EIS-ML) method shows good performance according to the Monte Carlo results. The proposed model and methodology are applied to two stock market indices for China. Strong evidence of the mean and volatility asymmetries is detected in Chinese stock market. Moreover, asymmetries in the volatility persistence and leverage effect are also discovered. The log-likelihood and Akaike information criterion (AIC) suggest evidence in favor of the proposed model. In addition, model diagnostics suggest that the proposed model performs relatively well in capturing the key features of the data. Finally, we compare models in a Value at Risk (VaR) study. The results show that the proposed model can yield more accurate VaR estimates than the alternatives.

Keywords: asymmetry; efficient importance sampling; leverage effect; stochastic volatility; threshold effect; Value at Risk
Corresponding author: Xin-Yu Wu, School of Finance, Anhui University of Finance and Economics, Bengbu, 233030, P.R. China, Phone: +86 18655276957, e-mail:

Acknowledgments

We would like to thank the editor, Bruce Mizrach, and an anonymous referee for their insightful and helpful comments and suggestions that greatly improved the paper. This work was supported by the National Natural Science Foundation of China under Grant No. 71101001, the MOE (Ministry of Education in China) Project of Humanities and Social Sciences under Grant No. 14YJC790133, the Natural Science Foundation of Anhui Province of China under Grant No. 1408085QG139, and the Anhui Province College Excellent Young Talents Fund of China under Grant No. 2013SQRW025ZD.

Appendix A

Assuming that EIS density mt(Vt|Xt,Vt–1,at) is normally distributed with mean ωat and variance σat2, its log density is given by

(A.1)lnmt(Vt|Xt, Vt−1, at)=−12ln2πσat2−12(Vt−ωatσat)2=−12ln2πσat2−Vt22σat2+ωatσat2Vt−ωat22σat2 (A.1)

Also, from Eqs. (17) and (19), we have

(A.2)lnmt(Vt|Xt, Vt−1, at)=lnkt(Vt|Xt, Vt−1, at)−lnχt(Xt, Vt−1, at) =−12ln2πσt2−(Vt−ωt)22σt2+a1,tVt+a2,tVt2−lnχt(Xt, Vt−1, at) =−12ln2πσt2+(a2,t−12σt2)Vt2+(a1,t+ωtσt2)Vt−ωt22σt2−lnχt(Xt, Vt−1, at) (A.2)

where ωt and σt2 are the mean and variance of the NIS density p(Vt∣Xt, Vt–1, Θ), respectively, which are defined in Eqs. (10) and (11).

Comparing (A.1) and (A.2), we have

(A.3)ωat=σat2(a1,t+ωtσt2) (A.3)
(A.4)σat2=σt21−2a2,tσt2 (A.4)

and

(A.5)lnχt(Xt, Vt−1, at)=12lnσat2σt2+ωat22σat2−ωt22σt2. (A.5)

Appendix B

Let ℱt denote the information set generated by the observations up to time t, i.e., {X0,…,Xt}. The objective is to recursively obtain a sample of draws from (Vt∣ℱt, Θ) for t=1,…,T. Formally, suppose that p(Vt–1∣ℱt–1, Θ) is known and we want to obtain p(Vt∣ℱt, Θ). Note that

(B.1)p(Vt|ℱt, Θ)=∫p(Vt, Vt−1|ℱt, Θ)dVt−1=∫p(Vt, Vt−1|ℱt, Θ)p(Vt−1|ℱt−1, Θ)dP(Vt−1|ℱt−1, Θ) (B.1)

Also, from p(Xt, Vt, Vt–1∣ℱt–1, Θ)=p(Vt, Vt–1∣ℱt, Θ)p(Xt∣ℱt–1, Θ), we get

(B.2)p(Vt, Vt−1|ℱt, Θ)=p(Xt, Vt, Vt−1|ℱt−1, Θ)p(Xt|ℱt−1, Θ)=p(Xt|Vt, Vt−1, Θ)p(Vt|Vt−1, Θ)p(Vt−1|ℱt−1, Θ)p(Xt|ℱt−1, Θ) (B.2)

where

p(Xt|ℱt−1, Θ)=∫∫p(Xt|Vt, Vt−1, Θ)p(Vt|Vt−1, Θ)p(Vt−1|ℱt−1, Θ)dVtdVt−1

and p(Xt∣Vt, Vt–1, Θ) is the normal density of Xt with the conditional mean μt−1+ρstσXst−1exp(Vt−1/2)σV(Vt−ϕstVt−1) and the conditional variance σXst−12exp(Vt−1)(1−ρst2) and p(Vt∣Vt–1, Θ) is the normal density of Vt with the conditional mean ϕstVt−1 and the conditional variance σV2.

Plugging (B.2) into (B.1), we get

(B.3)p(Vt|ℱt, Θ)=∫p(Xt|Vt, Vt−1, Θ)p(Vt|Vt−1, Θ)p(Xt|ℱt−1, Θ)dP(Vt−1|ℱt−1, Θ) (B.3)

In summary, the particle filter algorithm for the TTLSV model as follows:

Step 1: Given a set of random samples {Vt−1(1), …, Vt−1(N)} from the probability density function p(Vt–1∣ℱt–1, Θ).

Step 2: Draw samples {Vt(1∗), …, Vt(N∗)} from the probability density function p(Vt∣Vt–1, Θ).

Step 3: Compute the normalized weight for each sample

qj=p(Xt|Vt(j∗), Vt−1(j), Θ)∑i=1Np(Xt|Vt(i∗), Vt−1(i),Θ), j=1, …, N

Thus define a discrete distribution over {Vt(1∗), …, Vt(N∗)}, with probability mass {q1,…,qN}.

Step 4: Resample N times from the discrete distribution defined above to generate samples {Vt(1), …, Vt(N)}.

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Supplemental Material

The online version of this article (DOI: 10.1515/snde-2014-0044) offers supplementary material, available to authorized users.

Published Online: 2014-11-25
Published in Print: 2015-9-1

©2015 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. A video interview of James Stock
  3. More powerful cointegration tests with non-normal errors
  4. Asset pricing with flexible beliefs
  5. Improving model performance with the integrated wavelet denoising method
  6. Noncausality and inflation persistence
  7. A triple-threshold leverage stochastic volatility model
  8. Estimating dynamic copula dependence using intraday data
https://doi.org/10.1515/snde-2014-0044
Keywords for this article
asymmetry; efficient importance sampling; leverage effect; stochastic volatility; threshold effect; Value at Risk

Articles in the same Issue

  1. Frontmatter
  2. A video interview of James Stock
  3. More powerful cointegration tests with non-normal errors
  4. Asset pricing with flexible beliefs
  5. Improving model performance with the integrated wavelet denoising method
  6. Noncausality and inflation persistence
  7. A triple-threshold leverage stochastic volatility model
  8. Estimating dynamic copula dependence using intraday data
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