mrow> τ i and N t .

Suppose that on the finite and fixed time horizon [ 0 , T ] , there are n + 1 discrete realization p t 0 , p t 1 , , p t n 1 , p t n of process p t . 0 = t 0 < t 1 < < t n = T is an arbitrary partition of interval [ 0 , T ] . For simplicity, assume that the observations are equally spaced. Denote Δ n = T / n , then t i = i Δ n . In the presence of microstructure noise, at any given time t i , the actually observed log-price is Z t i other than p t i , which can be given as

Z t i = p t i + ε t i , (2)

where ε t is the noise term. Assume that the ε t s are i.i.d. and independent of W t and J t processes, and with E ε t = 0 , and E ε t 2 < . Although the noises are not necessary i.i.d, this assumption is only for the simplicity to prove the theoretical properties. See the studies in Yu et al. [19] , where we show that the estimation method for intraday jump used in this paper also performs well in the setting of correlated noises.

Our goal is to estimate the intraday jump X i , with these noisy observation data { Z t i , i = 0 , 1 , , n } . For the simplicity of notation, we denote V i n = V i Δ n , Δ i n V = V i n V i 1 n for any process V = ( V t ) in the following.

In this paper, we use the pre-averaging approach to diminish the effect of noise. Let Z ¯ i n denote the weighted average of k n observations of

Z i n , Z i + 1 n , , Z i + k n 1 n , where Z ¯ i n = j = 1 k n 1 g j n Δ i + j n Z , with weights g j n = g ( j / k n ) . We

require that the weighting function g ( x ) is continuous on [ 0 , 1 ] , piecewise C 1 with a piecewise Lipschitz derivative g , and satisfies g ( 0 ) = g ( 1 ) = 0 ,

0 1 g ( s ) 2 d s > 0 . We further require that the integer sequence k n satisfies

k n Δ n = θ + ο ( Δ n 1 / 4 ) for some constant θ > 0 .

Then we can use the threshold technique to identify the jump with these pre-averaging observations { Z ¯ i n } . The threshold function is required to satisfy the following assumption.

Assumption 1 The threshold function r ( Δ n ) is a deterministic function of the step length Δ n , such that

(a) lim Δ n 0 r ( Δ n ) = 0 ;

(b) lim Δ n 0 Δ n 1 / 2 ( log 1 Δ n ) 2 r ( Δ n ) = 0 .

Power functions r ( Δ n ) = β Δ n α for any α ( 0 , 1 / 2 ) and β R are possible choices. Under the Assumption 1, for P-almost all ω , Δ > 0 such that

Δ n Δ , we have that i = 0 , , n 1 , I { ( Z ¯ i n ) 2 r ( Δ n ) } = I { j = i + 1 i + k n 1 ( Δ j N = 0 ) } , which says

that the threshold function r ( Δ n ) can be used to asymptotically identify the intervals where no jump occurred; also see the literature on the noise- and jump- robust volatility estimation (Jing et al. [20] ). In other words, if #Math_73#, there exists jumps on interval ( t i , t i + k n 1 ] . Thus, we can use this threshold method to identify the intervals where jump occurs and further give a coarse estimation of the location of jumps. Let T τ denote the location set of jumps occurred on [ 0 , T ] , then

T ^ τ = { t i [ 0 , T ] : ( Z ¯ i n ) 2 > r ( Δ n ) , i = 0 , , j n } , (3)

where i = i ( k n 1 ) .

We now turn to estimate the jump size by a simple nonparametric method. Denote by Δ i i + k n 1 N = N ( i + k n 1 ) Δ n N i Δ n ; if Δ i i + k n 1 N 1 , by τ ( i ) the first instant a jump occurs within ( t i , t i + k n 1 ] , and X τ ( i ) the size of this first jump, also let X ¯ = min j = 1 , , N T | X τ j | . For the simplicity of notation, we denote j n = [ n / ( k n 1 ) ] in the following. For small Δ n , we have that a.s. in any time interval ( t i , t i + k n 1 ] , at most only one jump can occur. Moreover, we can obtain that the pre-averag- ing observation Z ¯ 0 i n of continuous diffusion process without jump satisfies sup Z ¯ 0 i n = O ( Δ n 1 / 4 log 1 / Δ n ) , while the pre-averaging observation of jump process J ¯ i n is greater than X ¯ multiplying some constant, which is not negligible. So we propose the following estimator for jump size X τ ( i ) ,

X ^ τ ( i ) = Z ¯ i n I { ( Z ¯ i n ) 2 > r ( Δ n ) } . (4)

Yu et al. [19] demonstrated the theoretical properties of estimator (4). The results shows that for each i , X ^ τ ( i ) estimates the product of some constant g ¯

and the size of the first jump occurs within ( t i , t i + k n 1 ] , g ¯ X τ ( i ) I { Δ i i + k n 1 N 1 } , where

g ¯ = 0 1 g ( s ) d s .

2.2. Intraday Jump Tail Risk Measurement

In this subsection, we present how to model the intraday jump tail and then to measure the jump tail risk, i.e. VaR (Value-at-Risk) and ES (Expected Shortfall) based on extreme value theory (EVT). Extreme value theory provides simple parametric models to capture the extreme tails of distribution and to forecast risk. There are mainly two methods of applying EVT: the first is known as the Block Maxima (Minima) (BMM) method based on the generalized extreme value distribution (GEV), while the second is known as the peaks-over-threshold (POT) approach based on the generalized Pareto distribution (GPD). Since the POT method uses GPD to fit the exceedances over a given threshold and hence it doesn’t require a large data set as BMM, it is considered more efficient in modelling limited data (McNeil, Frey and Embrecht [21] ). Thus, in the following, we use the POT method to model the tail distribution of the identified intraday jump series.

Suppose that the jump series { X i } are identically distributed random variables with unknown underlying distribution function F ( x ) = P ( X i x ) . The excess distribution F u over a threshold u is given by

F u ( y ) = P ( X u y | X > u ) = F ( y + u ) F ( u ) 1 F ( u ) = F ( x ) F ( u ) 1 F ( u ) (5)

for 0 < y < x F u , where x F is the right endpoint of F , and y = x u .

In EVT framework, there is a key result that for a large class of underlying distributions F (containing all the common continuous distributions in statistics, such as normal, lognormal, t, gamma, exponential, beta, etc.), as the threshold u progressively increases, the excess distribution F u converges to a generalized Pareto distribution. In the sense of this result, the GPD is the natural model for the excess distribution above sufficiently high thresholds. That is the excess distribution function F u can be approximated by GPD for a certain u :

F u ( y ) G ξ , σ ( y ) , (6)

where G ξ , σ is the generalized Pareto distribution (GPD), which is given by

G ξ , σ ( y ) = { ( 1 + ξ σ y ) 1 / ξ if ξ 0 1 e y / σ if ξ 0 (7)

for y [ 0 , ( x F u ) ] if ξ 0 , and y [ 0 , σ ξ ] if ξ < 0 . Here ξ is the

shape parameter and σ is the scale parameter for GPD.

Hence, for x u , replacing the F u by GPD,

F ¯ ( x ) = P ( X > u ) P ( X > x | X > u ) = F ¯ ( u ) P ( X u > x u | X > u ) = F ¯ ( u ) F ¯ u ( x u ) = F ¯ ( u ) ( 1 + ξ x u σ ) 1 / ξ . (8)

This gives a formula for tail probabilities. The inverse of (8) gives the high quantile of the distribution or VaR. Thus, for α F ( u ) (i.e. tail probability is 1 α ), VaR is given by

V a R α = q α ( F ) = u + σ ξ ( ( 1 α F ¯ ( u ) ) ξ 1 ) . (9)

For ξ < 1 , the ES is given by

E S α = 1 1 α α 1 q x ( F ) d x = V a R α 1 ξ + σ ξ u 1 ξ . (10)

Equations (9) and (10) give the theoretical formulae to calculate the jump tail risk measure. In the following, we show that how to estimate the VaR and ES with the identified jump series.

For the identified jump series { X ^ i } , if there are total n observations and N u of observations above u , we get an empirical estimator N u / n of F ¯ ( u ) . Putting the maximum likelihood estimates of the parameters of the GPD together, we arrive an estimator for tail distribution F ( x ) ,

F ^ ( x ) = 1 N u n ( 1 + ξ ^ σ ^ ( x u ) ) 1 / ξ ^ . (11)

Also, we get the estimator of VaR

Math_140# (12)

and the estimator of ES

E S ^ α = V a R ^ α 1 ξ ^ + σ ^ ξ ^ u 1 ξ ^ . (13)

The estimation procedure presented above depends heavily on the important parameter u . In this paper, we will use the mean excess plot to choose a reasonable threshold. The idea behind this method is demonstrated as follows. Given a high threshold u 0 , suppose that the excess X u 0 follows a GPD with parameter ξ and σ . Then the mean excess over the threshold u 0 is

E ( X u 0 | X > u 0 ) = σ 1 ξ . (14)

For any u > u 0 , define the mean excess function e ( u ) as

e ( u ) = E ( X u | X > u ) = σ + ξ ( u u 0 ) 1 ξ . (15)

Thus, for a fixed ξ , the mean excess function is a linear function of u for u > u 0 . This result leads to simple graphical method to infer the appropriate threshold value u 0 for the GPD. Define the empirical mean excess function as

e ^ ( u ) = 1 N u i = 1 N u ( x i u ) . (16)

The scatter plot of e ^ ( u ) against u is called the mean excess plot, which should be linear in u for u > u 0 . Hence, we can choose a reasonable threshold according to the mean excess plot.

3. Empirical Example

In this section, we implement our procedure of measuring the intraday jump tail risk with actual high frequency data. We collect the transaction data for Microsoft Corporation (MSFT) shares carried out on NASDAQ from Jan 3, 2011 to Jul 29, 2011 from Wharton Research Data Services (WRDS). We use every ten seconds data to identify and estimate the intraday jumps in one minute return by implementing pre-averaging step with k n = 7 observations. Over this seven months time period, there were total 336,960 ten-seconds observations corresponding to daily 6.5 trading hours in valid 144 trading days excluding weekends and holidays. The return is calculated by r t i = ( log P t i log P t i 1 ) × 100 , where P t i denotes the transaction price at t i .

Firstly, we use the pre-averaging threshold method to estimate the intraday jump. Let g ( x ) = x ( 1 x ) , which is used in Jacod et al. [17] . In addition, choose the threshold function following the studies in Christensen et al. [22] . In order to study the intraday dynamic pattern of jumps, we summarize their frequencies at one-minute frequency of all trading days. Figure 1 presents the frequency distribution of the identified intraday jumps occurred in 6.5 trading hours. It’s obvious that the intraday jumps for MSFT from Jan 3, 2011 to Jul 29, 2011 take on “L”-type dynamics. It says that most jumps occurred around the market opening time. For example, there are over 40 trading days with jumps observed at 9:31 (i.e. one minute after the market opening). However, there are less than 10 trading days with jumps observed at half an hour after opening time. This “L”-type intraday pattern may be driven by the accumulations of news arrivals overnight.

Figure 2 presents the Q-Q plot of the estimated intraday jumps. The result shows that the intraday jump has fatter tails than normal distribution. This further demonstrates the reasonability of using the EVT to model the jump tails.

Figure 1. Frequency distribution of intraday jump.

Next, we use the POT method and generalized Pareto distribution (GPD) to fit the negative and positive jump tail respectively. The threshold u is chosen by the mean excess function. Figure 3 and Figure 4 present the mean excess plot for negative jump tail and positive jump tail respectively. Observing the plots, we choose u = 0.20 for negative jump, and u = 0.25 for positive jump.

Figure 2. QQ plot of intraday jump.

Figure 3. Mean excess function for negative jump tail.

Figure 4. Mean excess function for positive jump tail.

Based on the chosen threshold u , Table 1 presents the estimation results of intraday jump and jump tail. Firstly, we can see that there are 452 positive jumps and 437 negative jumps happened among the total one-minute return observations and the corresponding percentage is 0.81% and 0.78% respectively. The number of exceedances over threshold is 293 and 286 for positive and negative jump respectively. It seems that the number of jumps occurred or the intensity of jumps is symmetric for positive and negative jumps. Secondly, by comparing the results of jump tail distribution, we find that the shape parameter ξ for positive jump is −0.0803 and is not significant at the given 10%, 5%, 1% levels, which means that positive jump tail may follow exponential distribution. However, the shape parameter ξ for negative jump is 0.2176 and is significant at 1% level, which means that negative jump tail follows GPD with heavy tail. These results show that the positive and negative jump tail is asymmetric. In particular, the negative tail is heavier than the positive tail, which shows that there are more negative extreme events happened than positive events over the periods from Jan. 3, 2011 to Jul. 29, 2011 for MSFT.

We then calculate the VaR and ES for negative and positive jumps based on the above estimation results of jump tail distribution. The results of VaR and ES are presented in Table 2. We find that as the significance level (i.e. tail probability) decreases, the results of VaR and ES for negative jump becomes larger than positive jump as expected, which further demonstrates the asymmetry of negative and positive jump tails. Meanwhile, the values in parenthesis in Table 2 are the p values in testing the validity of VaRs and ESs by Kupiec test. Values smaller than a given significance level indicate that the risk measures are invalid. From the results, we can see that the risk measure are valid except the case of 10% significance level for positive jump, which further in turn shows the success of our measuring method for jump tail risk.

4. Conclusion

Jump component in asset price process is a very important source of financial

Table 1. Estimation results of intraday jump and jump tail.

Note: Values in parenthesis are the standard errors of the estimates, *, **, *** mean that the results are significant at 10%, 5%, 1% level respectively.

Table 2. Results of VaR and ES for intraday jump.

Note: Values in parenthesis are the p values in testing the validity of VaRs and ESs, *, **, *** mean that the risk measures are invalid at 10%, 5%, 1% level respectively.

extreme risk. With the availability of high frequency data, it has aroused wide attention of researchers in last two decades. However, with the frequency of data increases, the identification of jump and its relevant studies will run into the bias problem caused by market microstructure noise. In this paper, we propose a simple nonparametric method to identify the intraday jump and measure the intraday jump tail risk with noisy high frequency data. We use a two-step procedure to measure the jump tail risk. In first step, we use a pre-averaging approach to diminish the effects of noises, and then propose the pre-averaging threshold estimator of intraday jump. In second step, we fit the tail distribution of the identified jump series with POT method and GPD, and then to calculate the risk measure (VaR and ES) of jump tail. Finally, we show the power of our procedure by a real data study. The results show that our proposed procedure of measuring the jump tail risk is valid and is easy to implement. Moreover, the nonparametric identification of intraday jump can also be used to analyze the dynamics of intraday jump, which is useful to study the microstructure of the market. Further studies on risk management, such as analyzing the impactors of jump tail risk, dynamic jump tail risk forecasting are the future research directions.

Acknowledgements

This research was supported in part by the NSFC (71601048), and the Fundamental Research Funds for the Central Universities in UIBE (13QD09).

Cite this paper
Yu, C. , Zhao, X. and Zhang, F. (2017) Measuring the Intraday Jump Tail Risk of Financial Asset Price with Noisy High Frequency Data. Open Journal of Statistics, 7, 72-83. doi: 10.4236/ojs.2017.71006.
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