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Taking the power-law behavior of human activities into consideration, we conduct an empirical study on the distribution of jump intervals after using BNS nonparametric method to detect jumps in 5 min closing data of HIS. Our result shows that there is a “power law” in jump intervals, and Fokker-Planck distribution is the more suitable distribution to describe jump intervals than the traditional Poisson process. So the jump diffusion model of power law can depict the movement of stock price more accurately.

The price of underlying assets is not only the foundation for financial derivatives pricing, but also plays an important role in the investment decision-making of the investors. Many classical theories assume that asset prices have continuous-time paths, e.g. Black & Scholes’ option pricing formula (hereafter BS formula) [

Lots of studies have showed that human activities are quite uneven: high-frequency outbreaks in short time separate a long time of silence, whose distribution of human activity’s interval presents a deviation from the Poisson process, and is characterized by the power-law distribution with a fat tail. Barabási [

The movement of asset price is the result of many individual investors’ or institutes’ decision-making activities. So there is sufficient reason to believe that the characteristics of the asset price volatility should be consistent with the general statistical law of human activities.

Cao Hong-duo et al. [

The distribution of HSI’s jump intervals is empirically researched in this paper in order to further verify the universality of the power law to jump interval.

The rest of the article is organized as follows. Section 2 introduces the candidate models. Section 3 empirically examines the distribution of jump intervals. Finally, we conclude in Section 4.

Merton’s model [

where

Although Kou’s model can explain the skewness of asset returns, it fails to give a description of the volatility clustering. The key to the problem is the assumption to the counting process

If we use the Poisson process to describe the jump intervals, then the probability density of intervals between two adjacent jumps obey exponential distribution,

where and hereinafter x is the interval between two adjacent jumps,

For the renewal process with power-law nature, we focus on the Fokker-Planck distribution and power law with exponential cutoff. Fokker-Planck distribution is used to describe the time evolution of the probability density function of the velocity of a particle, and its probability density function is

The power law with exponential cutoff is one of the mixture distributions, and its probability density function is

ARCH class of models and SV class of models are two classical parametric methods in depicting the volatility of asset price. However, non-parametric method, which is based on facts and all sample data and doesn’t have complicated parameter estimations, has an unparalleled advantage over parametric method. For the outcomes of parametric method are just approximations of the historical data. The non-parametric methods that are often used in previous studies to describe the volatility of asset price are quadratic variation and bi power variation (hereafter BPV). And it has been proved that the realized BPV can be used to consistently estimate the integrated volatility, even there are jumps in return processes [

The BPV which is put forward by Barndorff-Nielsen and Shephard [

The jump detecting method is as follows:

Dividing a day-trading-time into M intervals, M + 1 datum will be observed as

We use 5 min closing data of HIS (Hang Seng Index in Hong Kong stock) as our research object. Because of the constraint of data availability, the time span of our data is 13 weeks or 64 trading days (Hong Kong stock market closed for one day on 2 July 2012 (Monday) because of the 15^{th} Hong Kong Special Administrative Region Establishment Day), from 09:35 on 28 May 2012 to 16:00 on 24 August 2012. The total sample size is 4224 and they are all collected from tdx.com.cn Corporation.

Although BNS suggests

According to BNS, if the ratio jump statistic is smaller than its corresponding 95% critical value, then we reject the hypothesis of no jump in this unit of time. We detect 86 jumps altogether, as shown in

Make a subtraction between two adjacent jumps’ numbers, we can get a jump interval. There are 85 jump intervals in this study, and the smallest one is 1 unit of time, while the largest one is 16 units, see

Unit of times | Number of units | Frequency | Cumulative frequency | Unit of times | Number of units | Frequency | Cumulative frequency |
---|---|---|---|---|---|---|---|

1 | 17 | 0.2000 | 0.2000 | 9 | 1 | 0.0118 | 0.8941 |

2 | 14 | 0.1647 | 0.3647 | 10 | 1 | 0.0118 | 0.9059 |

3 | 9 | 0.1059 | 0.4706 | 11 | 1 | 0.0118 | 0.9176 |

4 | 12 | 0.1412 | 0.6118 | 12 | 1 | 0.0118 | 0.9294 |

5 | 13 | 0.1529 | 0.7647 | 13 | 2 | 0.0235 | 0.9529 |

6 | 3 | 0.0353 | 0.8000 | 14 | 1 | 0.0118 | 0.9647 |

7 | 4 | 0.0471 | 0.8471 | 15 | 2 | 0.0235 | 0.9882 |

8 | 3 | 0.0353 | 0.8824 | 16 | 1 | 0.0118 | 1.0000 |

According to the data in

Fit distribution | R Square | Adjusted R square | SSE | Degrees of freedom | RMSE | #Coefficient |
---|---|---|---|---|---|---|

Fokker-Planck distribution | 0.8155 | 0.7871 | 3.3794 | 13 | 0.2600 | 3 |

Power law with exponential cutoff | 0.7870 | 0.7542 | 3.9012 | 13 | 0.3001 | 3 |

Exponential distribution | 0.7513 | 0.7336 | 4.5536 | 14 | 0.3253 | 2 |

Fit distribution | Coefficient | Value | 95% Confidence interval |
---|---|---|---|

Fokker-Planck distribution | A | 2.3763 | (0.2191, 25.7711) |

1.9210 | (0.9992, 2.8429) | ||

2.4609 | (−0.5597, 5.4815) | ||

Power law with exponential cutoff | A | 0.3091 | (0.1246, 0.7668) |

0.7601 | (−0.3536, 1.8737) | ||

0.0830 | (−0.1016, 0.2676) | ||

Exponential distribution | a | 0.1965 | (0.1035, 0.3732) |

b | −0.2012 | (−0.2675, −0.1348) |

According to

tegral result is 1 so that it satisfies the nature of probability density function. Similarly, we get the power law with exponential cutoff:

We also get the exponential distribution:

This paper focuses on the counting process N(t) of Kou’s asymmetric double exponential jump-diffusion model [

diffusion model of power law, it can portray the phenomenon of stock price movement more precisely that has intensive occurrence of jumps, and meanwhile allow the disappearance of jumps over an extended period. This amendment comes into line with the general statistical law of human activities. As the result of this paper, on the one hand, the new “anomaly phenomena” is given out for the already study; on the other hand, human dynamics is introduced into financial mathematical modeling.

The research is supported by National Natural Science Foundation of China (No. 71371200 and No. 71071168) and Fundamental Research Funds for the Central Universities (No. 31650017).

Cao, H.D., Li, Y., He, H.P. and He, Z. (2016) Jump Intervals of Stock Price Have Power-Law Distribution: An Empirical Study. Journal of Mathematical Finance, 6, 770-777. http://dx.doi.org/10.4236/jmf.2016.65053