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The impact of successive jumps in price process on volatility is very important. We study the nature of self-motivation in price process using data from China’s stock market. Our empirical results suggest that: 1) Price jumps in China’s stock market are generally self-motivated, <i>i.e.</i>, price jumps are clustering. 2) The jump intensity of China’s stock market is time-varying, and follows log-normal distribution, which indicates that the jump intensity is asymmetrical. 3) The jump intensities’ sequence exhibits typical long memory.

With the availability of high-frequency data, the study of dynamic behavior of volatility based on high-frequency data has become a focus of econophysics. A large number of studies have shown that volatility has remarkable clustering and long memory [

In this paper, our contribution is empirical. We use the jump self-motivation test [

The remainder of the paper is organized as follows. Section 2 formally introduces the jump detection method and self-motivation detection method. Section 3 presents the empirical analysis. Section 4 concludes. Appendix details the kernel database of statistics generated in our framework.

One-dimensional asset price process defined on probability space ( Ω , F t , P ) could be presented as

log S ( t ) = log S ( 0 ) + ∫ 0 t μ s d s + ∫ 0 t σ s d W s (1)

where S ( t ) is asset price at time t, μ ( t ) for drift process, σ ( t ) for realized market volatility, and W ( t ) is standard Brownian process. The jump component is introduced to obtain the following formula:

log S ( t ) = log S ( 0 ) + ∫ 0 t μ s d s + ∫ 0 t σ s d W s + h ( δ ) ∗ s ∗ ( b s − v s ) t + ( δ − h ( δ ) ) ∗ b t s (2)

where h is truncation function, b s and v s are poisson distribution estimate in ℝ × E , δ is predictable function. For Equation (2), assumptions follow:

Assumption 1: b and σ are locally bounded.

Assumption 2: v s ( ω , d t , d s ) = d t ⊗ F s ( ω , d s ) , where F s is a predictable random measure. Set three non-random numbers ( β ∈ ( 0 , 2 ) , β ′ ∈ [ 0 , β ) and γ > 0 ) and a locally bounded process ( L t ≥ 1 ) to have F t = F ′ t + F ″ t for all ( ω , d t ) , where

1) F ′ t ( ω , d x ) = f ′ t ( x ) λ t − ( ω ) d x , 0 < λ t ( ω ) ≤ L t , f ′ t ( x ) = 1 + | x | γ h t ( x ) | x | 1 + β , 1 + | x | γ h t ( x ) ≥ 0 , | h t ( x ) | ≤ L t ;

2) F ″ t is a singular measure and F ′ t satisfies ∫ ( | x | β ′ ∧ 1 ) F ″ t ( d x ) ≤ L t ;

3) λ t is a positive Itô semimartingale process, λ t = λ 0 + ∫ 0 t μ ′ s d s + ∫ 0 t σ ′ s d W s + ∫ 0 t σ ″ s d B ˜ s + δ ′ × μ t x + δ ″ × μ t ⊥ x , where B ˜ is a standard Brownian process independent from W, μ t x is orthogonal to μ t ⊥ x , and δ ′ , δ ″ are predictable.

On the basis of asset price process above, define asset return at time t i as r t i :

r t i = log S ( t i ) − log S ( t i − 1 ) (3)

When applying to discrete data, T refers to total number of minutes ( T ∈ ℕ ). Let sampling frequency be Δ n = T / n , where sample size is n and t = i ∗ Δ n ( i ∈ ℕ ) . Let X i = log ( S i ∗ Δ n ) , asset return can be defined as:

r Δ n , i = log ( S i ∗ Δ n ) − log ( S ( i − 1 ) ∗ Δ n ) ≜ Δ i n X (4)

In this paper, we use the self-motivation test method of price jump to make empirical analysis [

∑ 0 ≤ t ≤ T Δ λ t 1 { Δ λ t ≥ 0 , | Δ X t | ≠ 0 } > 0 (5)

where T stands for a total length of the sample, X t = log ( S t ) , and λ t is density of jump arrivals att.

Equation (5) shows, within the data range, the jump arrival density also changes in a non-negative way when the price changes, which means there’s self-motivation in the jumps.

Choose a proper window size k n , the jump intensity λ ^ ( k n ) i in ith window size

λ ^ ( k n ) i = Δ n ϖ β ^ k n Δ n ∑ j = i + 1 i + k n g ( | r Δ n , j | α Δ n ϖ ) α β ^ C β ^ ( 1 ) (6)

In Equation (6), 0 < ϖ < 1 2 , α is the standard deviation of the income distribution and function g ( ⋅ ) has following 3 forms:

g 0 ( x ) = 1 { x > 1 } (7)

g 1 ( x ) = { | x | p , | x | ≤ 1 1 , | x | > 1 (8)

g 2 ( x ) = { c − 1 | x | p , | x | ≤ a c − 1 ( a p + p a p − 1 2 ( b − a ) ( ( b − a ) 2 − ( | x | − b ) 2 ) ) , a ≤ | x | ≤ b 1 , | x | ≥ b (9)

In Equation (9), c = a p + p a p − 1 ( b − a ) 2 , and when a = b = 1 , g 2 ( x ) = g 1 ( x ) . In applications, g ( ⋅ ) can be taken as any of the three forms.

C β ^ ( k ) is defined as

C β ( k ) = ∫ 0 ∞ ( g ( x ) ) k / x 1 + β d x (10)

where the estimation of β is

β ^ n ( t , ϖ , α , α ′ ) = log V ( ϖ , α , g ) t n V ( ϖ , α ′ , g ) t n / log α ′ α (11)

where α ′ = 2 ∗ α , and V ( ϖ , α , g ) :

V ( ϖ , α , g ) t n = ∑ i = 1 n g ( r Δ n i α Δ n ϖ ) (12)

According to Equation (5), statistics U ( H , k n ) T and U ( G , k n ) T :

U ( H , k n ) T = ∑ i = k n + 1 [ T / Δ n ] − k n H ( X ( i − 1 ) Δ n , X i Δ n , λ ^ ( k n ) i − k n − 1 , λ ^ ( k n ) i ) × 1 { | r Δ n , i | > α Δ n ϖ } × 1 { | r Δ n , i | ∈ E \ { 0 } } (13)

where ϵ = r ¯ θ Δ n ϖ . The H function is expressed as following:

H ( x 1 , x 2 , y 1 , y 2 ) = | x 2 − x 1 | p h 2 ( y 1 , y 2 ) (14)

In Equation (14), p is usually 2 or 4. And h 2 ( y 1 , y 2 ) could be expressed as:

h 2 ( y 1 , y 2 ) = { exp ( − 1 / ( y 2 − y 1 ) ) if y 2 > y 1 0 if y 2 ≤ y 1 (15)

Statistic U ( G , k n ) T is showed in the following form

U ( G , k n ) T = ∑ i = k n + 1 [ T / Δ n ] − k n G ( X ( i − 1 ) Δ n , X i Δ n , λ ^ ( k n ) i − k n − 1 , λ ^ ( k n ) i ) × 1 { | r Δ n , i | > α Δ n ϖ } × 1 { | r Δ n , i | ∈ E \ { 0 } } (16)

where G function is

G ( x 1 , x 2 , y 1 , y 2 ) = α β C β ( 2 ) ( C β ( 1 ) ) 2 ( y 1 H ′ 3 ( x 1 , x 2 , y 1 , y 2 ) 2 + y 2 H ′ 4 ( x 1 , x 2 , y 1 , y 2 ) 2 ) (17)

where H ′ 3 ( x 1 , x 2 , y 1 , y 2 ) and H ′ 4 ( x 1 , x 2 , y 1 , y 2 ) are the first partial derivatives of function H ( x 1 , x 2 , y 1 , y 2 ) with respect to y 1 , y 2 . Statistics U ( H , k n ) T and U ( G , k n ) T are continuous parameters with finite variance, and have the following asymptotic properties:

t n : = k n Δ n Δ n ϖ β U ( H , k n ) T U ( G , k n ) T { → ℙ − ∞ ω ∈ Ω T − → L s t N ( 0 , 1 ) ω ∈ Ω T 0 → ℙ + ∞ ω ∈ Ω T ÷ (18)

Under the null hypothesis that price is not self-motivated, we could apply the above statistics to do hypothesis testing. However, under the null hypothesis that price is self-motivated, U ( H , k n ) could not be estimated as a finite sample error. Therefore, we need to reconstruct relative error statistic as following:

R n = U ( H , ω k n ) T − U ( H , k n ) T U ( H , k n ) T (19)

Relative error R n has following asymptotic properties:

{ R n → ℙ 0 ω ∈ Ω T s e l f R n → L s t U ¯ ′ T U ¯ T − 1 ≠ 0 ω ∈ Ω T 0 (20)

where L s t denotes stable convergence in law.

And its rejection region is

C n = { k n Δ n Δ n ϖ β U ( H , k n ) T U ( G , k n ) T > z a ˜ } (21)

where z a is quantile of the standard normal distribution, and statistic V n is shown as following

V n = Δ n ϖ β k n Δ n ( ω − 1 ) U ( G , k n ) T ω ( U ( H , k n ) T ) 2 (22)

In empirical analysis, we selected the CSI300 index and its constituent stocks of China stock market from January to December 2015. Twenty-three stocks were randomly sampled from CSI300 index (SH000300). The names and codes of 23 stocks were shown in

Resample 1-minute price into 5-minute price and smooth the return series:

X i = log S 5 ∗ i , i ∈ ( 0 , n ) (23)

r i = 1 5 ∑ k = 1 5 ( log S 5 + k − log S k ) (24)

Since we connect all the 1-minute price series, the overnight effect and weekend effect should be taken into consideration, which could have a negative impact on our result of detected jumps. We applied such method to get rid of the unexpected effects:

r G t = r g t − μ g t σ g t σ + μ (25)

Name | Code | Name | Code | Name | Code |
---|---|---|---|---|---|

CSI300 | SH000300 | CGGC | SH600068 | PETROCHINA | SH601857 |

HD MEDICINE | SH000963 | NARI-TECH | SH600406 | JDCMOLY | SH601958 |

HPI | SH600011 | KWEICHOW MOUTAI | SH600519 | PAB | SZ000001 |

SIPG | SH600018 | XTC | SH600549 | TCL | SZ000100 |

BAO STEEL | SH600019 | COOEC | SH600583 | WEICHAI POWER | SZ000338 |

SEP | SH600021 | FYG | SH600660 | FINANCIAL STREET | SZ000402 |

PRE | SH600048 | DAQIN RAILWAY | SH601006 | YUNNAN BAIYAO | SZ000538 |

CHINA UNICOM | SH600050 | BANKCOMM | SH601328 | SHUANGHUI | SZ000895 |

where r g t , μ g t , σ g t represent overnight return, average overnight return and standard deviation of overnight return before operation. r G t is the overnight return after operation, σ is standard deviation of all non-overnight returns and μ is the average of non-overnight return. The weekend effect is removed in the same way.

After getting rid of the effects, we obtain the price and return series, two samples (CSI300 index, SH60048) is shown as follows.

In this section, we test the self-motivation of price jumps of CSI300 index and constituent stocks on 5-min price series. Set k n = 10 , ϖ = 1 5 , p = 4 , ω = 2 ,

SH000300 | SH000963 | |
---|---|---|

Mean | 0.0000286 | 0.0000368 |

Std. | 0.0000743 | 0.002618 |

Skew. | −0.034909 | −0.271355 |

Kurt. | 11.35313 | 13.91738 |

θ = 1 16 . Function g ( ⋅ ) is set in the simple form of Equation (8). CSI300 index and 23 constituent stocks all passed the test under the null hypothesis that price jump is self-motivated. The p-value for CSI300 index is 0.3879 and p-values of 23 constituent stocks are showed in

The estimates of jump arrival intensities λ ^ ’s of CSI300 index and constituent stocks are critical parameters in self-motivation test. The λ ^ ’s of CSI300 index and 3 stocks (SH000963, SH600011 and SH600018) in 2015 is shown as follows (the λ ^ ’s of other 20 stocks’ are displayed in Appendix 1).

In

Code | Hurst Index | Code | Hurst Index |
---|---|---|---|

SH000300 | 0.8710882 | SH000963 | 0.8849673 |

SH600011 | 0.9014997 | SH600018 | 0.8531737 |

SH600019 | 0.8835869 | SH600021 | 0.8617589 |

SH600048 | 0.7457267 | SH600050 | 0.8404174 |

SH600068 | 0.8452361 | SH600406 | 0.8815753 |

SH600519 | 0.877194 | SH600549 | 0.8455916 |

SH600583 | 0.8936722 | SH600660 | 0.8064734 |

SH601006 | 0.8731213 | SH601328 | 0.8386584 |

SH601857 | 0.8339011 | SH601958 | 0.852093 |

SZ000001 | 0.7710014 | SZ000100 | 0.8499707 |

SZ000338 | 0.8469565 | SZ000402 | 0.8098829 |

SZ000538 | 0.858791 | SZ000895 | 0.8585242 |

MEAN | 0.8892194 |

long (i.e. 09:35, 2015/01/05), transform the datetime into serial number ranging from 1 to 12,000. Since the sampling frequency is 5 minutes, each number on the abscissa represents a 5-minute return. From Equation (6), λ ^ ( k n ) i is calculated using k n returns in ith window. Combing Equation (7), ith window and (i+1)th window have only one different return value, so it’s probable that λ ^ ( k n ) i ≈ λ ^ ( k n ) i ＋ 1 . For this reason, it’s reasonable to have continuously equal λ ^ in

In

Each λ ^ is an estimate of jump arrival intensity and it’s time-varying. To study the nature of

In

In

6000, showing that China’s stock market has the highest jumping intensity in the average sense during this period. It corresponded with the fact that a crash happened in the middle of 2015 in China’s stock market.

Right subfigure in

We calculated the Hurst indices of

From

As an emerging market, China’s stock market shows more volatility. The jumping phenomenon in the high-frequency price process is one of the main causes

of the market volatility. In this paper, we choose the CSI300 index and 23 stocks that are randomly selected in the constituents of CSI300 index as the representative of China’s stock market, and take the 1-minute high-frequency price data of 2015, which is highly volatile in China’s stock market, as samples, and then resamples into 5-minute return series to conduct empirical analysis such as jump test and jump self-motivation test. Our empirical results show that:

1) It’s clear that the number of jumps of an individual stock is more than the one of an index in high-frequency dataset. This is consistent with many studies. When individual stocks jump, the index doesn’t jump much, showing that the probability that the index resonates with individual stocks is low. When the index jumps, the proportion of individual stocks that jump is not high, either. Within portfolio theory, the jumps of the index could be regarded as a co-jump contributed by individual stocks’ heterogeneous jumps after fully smoothing, which means the connections between stocks and the index could be implied in called co-jumps. However, co-jumps would also be covered by heterogenous jumps and random noises of individual stocks.

2) In the empirical analysis of high-frequency price series of the CSI300 index and selected 23 constituent stocks, the index and the stocks all passed the test under the null hypothesis that there is a self-motivation jump in the price process, indicating that the price jump in China’s stock market is generally self-motivated. And the fluctuations of jump intensity estimates used in the self-motivation test basically correspond to current situation in last decade.

3) In the analysis of time-varying jump intensity series, we found that the jump intensity series of asset price in China’s stock market is approximately lognormal distribution, which indicates that the change of jump intensity is asymmetric. At the same time, we also found that the price jump intensity series shows strong long-term memory, which indicates that the self-motivation phenomenon of price jump intensity is long-term related. This can provide a new perspective for us to understand the risk in China’s stock market.

Our work has shown a sound result that self-motivated jumps exist in China’s stock market, though we need more precise price stochastic process forms. There are still some facts needed focused in future:

1) How the log-normal distributed price processes interact and contribute to the index fluctuations. Reconsidering the price generation tools and theoretical derivations is advised.

2) China’s stock market has been experiencing subtle and swift adjustments in many ways in recent years, including new market trade instruments and more specific regulatory policies. The uncertainty comes from the structure of market and would lead to more unexpected fluctuations or severe changes, though it’s in belief that China’s stock market would keep open and integrate more with international financial markets. Finding a straightforward toolbox in dealing with the more common and frequent jumps is necessary and what we’re devoted to.

Based on the results of this paper, subsequent research will focus on the generating mechanism of the price jump process, so as to improve the price process and make the jump terms in the price process include the co-jumps and the jump self-motivation part. Then we could discuss the jumps and co-jumps in more detailed situations and provide a reasonable explanation to our interests.

This work is supported by the National Natural Science Foundation of China (Grant No. 71671017).

No potential conflict of interest was reported by the authors.

Tian, Y.Z., Shi, D.Y. and Li, H.D. (2021) The Long Memory of the Jump Intensity of the Price Process. Journal of Mathematical Finance, 11, 176-189. https://doi.org/10.4236/jmf.2021.112009