Modelling asset returns under price limits with mixture of truncated Gaussian distribution

ABSTRACT

In this paper, we propose to use the mixture of truncated Gaussian distribution in modelling the financial asset return distribution under price limits. Theoretically, while retaining many convenient statistical properties of the Gaussian distribution, the proposed model assumes a flexible structure accommodating some important special features of the return data under price restrictions, such as the clusters near the bounds (due to the ‘bound effect’) and peaked shape around zero (due to the minimum-tick size effect). It can also allow for a wide range of variances and kurtosis even with a bounded domain. These are the salient features that the conventional models (such as the Normal, Truncated Normal and Censored Normal) do not have. For empirical illustration, we apply our proposed model to stocks under different price restrictions. Some common interesting features have been found. Furthermore, in our Value-at-Risk analysis, we find that ignoring the bound cluster in the tail of the distribution could lead to a significant overestimation of the number of violations and produce unreliable Value-at-Risk measures. In addition, we also find that the proposed model has a better empirical performance when the data are highly asymmetric and heavy-tailed.

KEYWORDS:

• Price limits
• truncated Gaussian distribution
• mixture models
• magnet effect
• bound effect

JEL CLASSIFICATION:

• C01
• C58
• G15

Acknowledgments

We would like to thank the comments/suggestions from the participants in the Econometrics Seminar at the Ohio State University, Dalhousie University, the Econometric Society China Meeting 2019. We would like to acknowledge the International Research Partnership Grant at the University of Waterloo and National Natural Science Foundation (NSF) China for the financial support of this project.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ In the long run, returns are often found to be approximately normally distributed. The Gaussian distribution is statistically convenient. It has a simple and well-defined probability density function of two parameters (mean and variance). It also has closed-form expression for moment generating function and characteristic function.

² We also apply our model to the high frequency data (for example, 1 minute). Consistent with Chen et al. (2019), we also detect significant magnet effect on the right tails in the intra daily data on China stock markets (with 10% price restrictions).

³ The censored Gaussian can only capture the clusters on the bounds rather than the clusters within a range.

⁴ $l$ and $u$ are equal in most of cases in practice, indicating symmetric bounds on both ends. In this paper, we do not impose this restriction in our theoretical framework for generality.

⁵ Given the MGF defined in ((3)), it is straightforward to get analytical solutions for any order of moment conditions. In Proposition 1, we only provide the mean and variance expressions. We can achieve, for example, the skewness and kurtosis in the similar way. However, the higher order moments tend to have long expressions. To save space, we do not report them in this paper.

⁶ To save space, we do not investigate one-side truncation in this paper. The derivation is straightforward.

⁷ As a note, the base distribution in Figure 1 is a standard truncated normal with bound $[-3,3]$, i.e. $TN(0,1,[-3,3])$. When we do the plot in each scenario, we fix other parameters as constants.

⁸ The conventional censored Gaussian distribution also has restrictive second and fourth moments. To save space, we do not provide detailed illustration in this paper.

⁹ Please refer to Figure 2 on Page 731 in Friedmann and Sanddorf-Kohle (2007)) for the specific parameter set up in the corresponding mixed beta specifications.

¹⁰ In this paper, we acknowledge the Matlab codes from Lee and Scott (2012) at http://www.eecs.umich.edu/~cscott/code/tcem.zip. Our implementation is a modified version based on their Mablab codes.

¹¹ $C_{1,k}$ is the negative first moment of the truncated Gaussian. $C_{2,k}$ is the difference between the second central moment and uncentered moment of the truncated Gaussian. For the detailed derivations, please see Lee and Scott (2012).

¹² After June 2015, the price limits have been increased to 30% from 15% on the Korean markets.

¹³ In this paper, we do not include the mixed Beta distribution in our empirical application. The comparison of the performance between the proposed MTRN and mixed beta distribution is beyond the scope of this paper. As a note, given the Beta distribution involves the Beta function (in an integration form) in the density, the corresponding risk measures such as the Value-at-Risk and Expected Shortfall are not straightforward to achieve. Thanks to one anonymous referee to point out this interesting direction to us. We will leave this for future study.

¹⁴ To further investigate the relationships between the bound effects and price limits, ideally one should utilize the whole market data for a complete analysis. This is not the focus of this paper. We will leave this for further research.

¹⁵ We would like to thank one anonymous referee pointing out the important asymmetric bound effects. In general, we believe that the interesting asymmetric bound effects are closely related to the short-sell constraints of the stock exchanges. The short-selling usually creates high risks on the market. Many stock exchanges have various forms of policy/restrictions/bans on short-selling (especially during the crisis periods).

¹⁶ We would like to thank the suggestion from one anonymous referee on extending our empirical analysis to a larger data set. We have applied our proposed MTRN to the stocks on Shanghai Stock Exchange with $\pm 10\mathrm{\%}$ price-limit restriction. To save space, we only report the results based on 20 stocks in this paper. Other results are available upon request.

¹⁷ The data is downloaded from the Yahoo Finance. We use the most recent data up to April 13th, 2020. For each stock, we choose the max-length option when downloading the data.

¹⁸ We would like to thank one anonymous referee to point out this extended empirical analysis based on highly-skewed data.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China [71771087]; University of Waterloo [International Research Partnership Grant].