Testing Jumps for Individual Stock - Dissertation Example

?Literature review BNS 2004 – 2006 a) Theoretical background In order not to require a fully observed variable as is realized in Ait-Sahalia's approach in 2001, evidence is provided by Barndorff-Nielsen and Shephard (2004a, 2006) of the presence of jumps in higher frequency financial time series. In this test, emphasis is placed in the comparison of two measures of variance: the Bipower Variation which is robust to jump contribution and the Realized Variance which includes the contribution of jumps to the total variance. And based on a high frequency data set of exchange rates, a statistically significant test of the difference between these two measures of variance provides evidence on the presence of jumps. Through the approach employed by Barndorff-Nielson and Shephard in 2004a, there emerged two natural measures of realized within-day price variance; the Realized Variance computed as a sum of squared financial returns over the small time periods: and the Bipolar Variation computed as: Where According to Andersen, Bollerslev and Diebold's research conducted in 2001, the Realized Variance is expressed as a consistent estimator of the integrated variance plus the jump contribution: Where RVt: the Realized Variance at day t Nt: the number of jumps within day t : the jump size : the integrated variance In addition, Barndorff-Nielsen and Shephard's result in 2004a, together with extensions that were included in Barndorff-Nielsen, Graversen, Jacod and Shephard in 2005 define the Realized Bipower Variation as a consistent estimator of the integrated variance unaffected by jumps: Where BVt: the realized Bipower Variation at day t : the integrated variance Barndorff-Nielsen and Shephard (2004a, 2006) emphasized the difference RVt – BVt as a consistent estimator of the pure jump contribution and the basis of the jumps test allowing that the Relative Jump be measured as an indicator of the jumps contribution to the total variance within day t: In a sample of T days, the total realized variance can be measured as: subsequently, the total bi-power variation as: And the corresponding relative jump measure finally computed as: With the assumption of no jump coupled with other conditions , Barndorff-Nielsen and Shephard’s test in 2006 give the joint asymptotic distribution of BVt and RVt as M Where And using It can be seen that there is no coincidence of the fact that asymptotically similar to a situation encountered in Hausman’s test in 1978. Asymptotically, RVt is the most efficient estimate of the integrated variance and under the no jumps assumption, BVt is less efficient estimator, therefore the difference of RVt – BVt is independent of RVt on the volatility path following of the Hausman (1978) test. According on Huang and Tauchen (2005), the power of each absolute return should be less than 2 to be robust to jumps for the statistics. With the results from Barndorff-Nielsen and Shephard (2006), Andersen, Bollerslev and Diebold in 2004 used time series to test for daily jumps: Where on the assumption of no jumps: Another test for daily jumps is: The results of research conducted by Andersen, Bollerslev, Diebold and Labys (2001, 2003) and Barndorff-Nielsen and Shephard (2004a) show that the sample performance is improved by basing the test on the logarithm of the variation measures. Therefore the test is: And the maximum adjustment: The logarithmic adjustment to is: And the maximum adjustment is: The OP-versions of these tests are equivalent to the ratio jump of Barndorff-Nielsen and Shephard’s results in 2006. A simple t-test on the Relative Jump measure is: Where the classical estimate of the variance of the mean Another form is: Where : a HAC estimator of the variance of the mean. A bootstrap version is: Where : a bootstrap estimate of the variance of the mean. The Relative Jump can get a bootstrap confidence interval (tlow, tup) for the t test. b) Empirical results: The Monte Carlo findings developed z-tests for performing the jumps in a fairly realistic scenario and analyzed on daily basis in ratio form, the power and size are excellent and the test statistics identify the days of occurring jumps. They first consider the characteristics of the daily statistics computed over a long simulated realizations, and of length N=45000. The data generation process is the null case ?jmp = 0.00, with medium mean reversion (?v = -0.100) and 5-minute returns. Since large values of the z statistics discredit the null hypothesis of no jumps, they are only interested in the right-hand tail. With the exception of the statistic, the sampling frequency has a significant impact on the size. As the sampling frequency decreases and the sampling interval increases, the actual sizes of all statistics except increase above the Monte Carlo confidence band. The behavior of the size for the slow mean reversion and the fast cases are the same. Furthermore, a simulation of length 1400 days of the 5 versions of statistics under the null of no jumps and ?v = -0.100. Because of the time averaging effect, when the sampling interval increases from 1 minute to 30 minutes, the test statistics signal fewer instances of jumps when they occur; subsequently, an increase in jump intensity ?, has a positive effect on the detection rate. This is because the statistics detect jumps through the proportion of the total price variation attributable to jumps; when ? increases, the expected number of jumps per day increases which in turn increases the expected accumulated jump contribution per day, making it more likely that the statistics detect the jumps. Overall, jump intensity, jump size and sampling frequency have got a positive effect on power. High frequency data is essential for jump detection because a combination of the results in size, jump detection rate and power in the absence of market microstructure noise, for lower sampling frequency the statistics not only neglect true jumps when there is jumps (lower detection rate and lower power), but also signal more false jumps when there is no jump (larger size). As evidenced by the empirical work, there is strong support for jumps where they account for about 4.5 to 7 % of the total daily variance of the S&P Index, cash or futures. This allows for the prevalent assumption of local continuity, implying that financial econometricians will need to enrich considerably the class of parametric models used to model very high frequency price series. 2. ABD 2007 a) Theoretical background In order to describe observed asset return distributions, Andersen, Bollerslev and Dobrev (2007) developed and implemented a sequential procedure designed to test the adequacy of continuous-time jump-diffusion models. Their procedure relied on high-frequency intraday data and non-parametric realized volatility measures for transforming the observed returns, along with a new intraday jump detection technique and appropriate conditional moment tests for directly assessing the import of both jumps and leverage effects. Their approach allows the logarithmic asset price process to evolve according to a generic jump-diffusion with the sudden release of news or arrival of orders will often induce a distinct shift, or jump, in the asset price. The jump-diffusive setting is also theoretically appealing implying that under standard regularity, that the price process constitutes a special semi-martingale, thus ruling out arbitrage (Back, 1991). In addition, it derives the distribution of discretely observed returns at any frequency through aggregation, or integration, of the increments to the underlying process. Finally, it provides for a flexible setting with the potential to accommodate all major features of daily financial return series. For a pure diffusive process without leverage effects or jumps, the volatility process is independent of the innovation process. Returns normalized appropriately by the realized return variability are truly Gaussian, Evidence suggests that many markets, including those for equity indices, are characterized by a pronounced asymmetric relationship between return and volatility innovations; this is often labelled a “leverage level” although this asymmetry arguably has little, if anything, to do with underlying financial leverage (Cambell and Hentschel, 1992). When the volatility process, , is negatively correlated with the return innovation process, , then the daily integrated variance is informative and the above equations can no longer hold. Nonetheless, the Dambis-Dubins-Schwartz theorem (Dambis, 1965; Dublins and Schwartz, 1965) ensures that a suitably time-changed continuous martingale is a Brownian Motion. Hence, appropriately sampled returns will be Gaussian even in the case of leverage. The following distributional result holds, even in the case of leverage: where denotes for a fixed positive period of “financial” time Through a uniform decision rule that allows for simultaneous identification of multiple significant jumps on each trading day, Andersen, Bollerslev and Dobrev (2007) developed an alternative procedure which focuses directly on individual intraday returns. They consider whether a randomly selected intraday return, is subject to a jump, whether is an independently drawn index from the set {1,2,…,1/} and have conditional mean and variance given by and V, respectively. where denotes the daily mean conditional on the daily integrated variance. For the conditional intraday return variance, given the day integrated variance, The appropriately scaled version has a vanishing mean, and a variance approaching the integrated variance, Hence, for very frequent sampling the diffusive null implies that each scaled intraday return is approximately Gaussian, The jump detection procedure is formalized as follows. First, choose the size of the jump test at the daily level, and define as the level of the corresponding confidence interval for a randomly drawn intraday diffusive return distributed approximately Second, detect possibly multiple intraday jumps based on the rule: where refers to the corresponding critical value from the standard normal distribution. Andersen, Bollerslev and Dobrev (2007) maintain that this procedure tends to over-reject the diffusive null hypothesis whenever there is substantial intraday variation in volatility. This, therefore, suggests a conservative choice but they still find that it tends to outperform the existing BN-S procedure in term of identifying timing and number of jumps. b) Empirical results: An assumption that the underlying high-frequency returns are generated by a jump diffusion calibrated to parameter values obtained through recent empirical studies of the S&P 500 equity index is provided by Andersen, Bollerslev and Dobrev (2007) after providing evidence on the finite sample distribution of jump-adjusted and realized volatility standardized return series. The base scenario features pronounced volatility persistence, a strong leverage effect, and fairly frequent jumps which were explored to determine their separate impact by studying a pure diffusion and jump diffusion, both with and without leverage effects. Through the use of different jump intensities and jump sizes, results were obtained for a simulated sample size of 5000 days with 195 intraday return observations, corresponding to the use of two-minute returns over a 61/2 hour trading day, reflecting their actual implementation with the S&P 500 features data. Finally, both the original intraday series and the jump-adjusted series are converted into financial period return series. The BNS test is seen to be somewhat oversized at 0.16%, in terms of the evidence on the intraday jump-detection procedure, but the average daily imputed jump volatility is 0 to 3 decimals, reflecting the fact that the identified jumps are quite small, and the associated root mean squared error (RMSE) is also minimal at 0.009%. This test still retains more power than the larger sized BNS test and the relative size of the RMSE due to true jump volatility is largest in the moderate jump scenario compared with the rare large jump and the frequent small jump designs. The empirical analysis was conducted by the 2-minute transaction returns from the S&P 500 futures contract traded at the Chicago Mercantile Exchange (CME) from January 1, 1988 through July 26, 2004. The procedure eliminates 44 problematic trading days and leaves 4126 complete trading days, and indicates the presence of 382 jumps, which contributes about 4.4% to the overall return variability. According to Andersen, Bollerslev and Dobrev (2007) outliers in realized volatility will induce inliers in the corresponding standardized trading day return series, which are harder to identify in a small sample. It can be more difficult to detect relatively infrequent and small jump from the standardized returns than from the raw returns themselves. The exclusion of the generally less active morning period covering 8:30-9:30 eliminates the large jumps, which neglects the release of macroeconomic announcements taking place at 8:30am. It is however problematic to include this period for every trading day because this period appears very quiet in the absence of such news. Andersen et al. (2007) indicates that if the properties of the jump series per se are of primary concern, it may be advisable to use a lower significance level in the jump tests and to include the periods surrounding the macroeconomic news releases in the analysis. 3. LM 2008 a) Theoretical background: As a result of the difficulties that were realized in estimating jump parameters and the importance of incorporating market information into option pricing models, Lee and Mykland (2008; hereafter LM) introduced a new nonparametric test to detect jump arrival time and realized jump sizes in asset prices up to the intra-day level. Their test calculates the ratio of the return at each price observation to a measure of “instantaneous volatility” over a period preceding that observation. If this ratio exceeds a certain threshold, they identify the observation as jump. The statistic L(i) that tests for jump at time is defined as where The instantaneous volatility measure is similar to bipower variation, with a different scaling constant,therefore this test is robust to the presence of jumps in prior periods. According to Lee and Mykland (2008) that it is appropriate to choose the window size Kbetweenn and where n is the number of observations in a day. These window sizes are long enough, so that instantaneous volatility maintains the quality of being a jump-robust bipower variation and it effectively scales the test statistic for trends in volatility. The asymptotic distribution of the test statistic is as follow: where The constants and scale the L(i) statistic to be exponentially distributed. LM test is meant to be applicable to all kinds of financial time series, including equity returns and volatility, interest rates, and exchange rates. The presence of jumps makes incomplete market with the degree of market incompleteness depending largely on the size and intensity of jumps, which determines the magnitude of derivative hedging error (see Naik and Lee, 1990; and Bertsimas et al., 2001). Their technique gives both the direction and size of detected jumps, allowing the characterization of jump size distribution as well as stochastic jump intensity. Hence, hedging strategies can be developed, and detection of arrival times of jumps is important in maintaining the balance of hedging portfolio. b) Empirical results The Monte Carlo simulation proves that their test can precisely disentangle jump arrivals using high-frequency observations. Furthermore, their test is found to outperform BN-S test and JO test (Jiang and Oomen, 2005) by simulation. These two tests cannot distinguish two jumps a day with a low variance and one jump with high variance in terms of detection rates, the reason for this problem is these tests depend on integrated quantities. Even if the presence of jumps can be recognized, these two tests cannot determine how many jumps occurred, whether the jump(s) was (were) negative or positive, at what time of day the jump(s) occurred, and how large each jump was. However, these issues can be resolved by LM test. LM test does not use the conventional terms of size and power, but introduce the misclassification of jumps. In essence, the probability of global success in detecting actual jumps is the power of the test, and the probability of global spurious detection of jumps is the size of the test. The simulation results indicate that LM test outperforms BN-S and JO test for all jump sizes, number of jumps and frequencies. They examine jump dynamics and their distributions in the U.S. equity markets, three individual stock and S&P 500 Index prices transacted on the New York Stock Exchange (NYSE). They note the fact that jumps do not come to market regularly, but their arrivals tend to depend on market information. The empirical results provide evidence of strong association between jumps and news event in the U.S. equity markets. The detected jumps are related to news releases from Factiva, a real-time financial news database. Individual stock jumps occur with earnings announcement and other company-specific news releases, whereas the S&P 500 Index jumps with more overall market news such as FOMC meeting. They conclude that one should incorporate not only earnings announcements but also other company-specific news for individual equity option pricing. They also find more frequent jumps with greater mean and variance in individual equities than the index returns, explained to be negative risk-neutral skewness (Bakshi et al., 2003). 4. Jiang and Oomen test 2009 a) Theoretical background Following their previous analysis, Barndorff-Nielsen and Shephard, Jiang and Oomen (2009) proposed their own tests for the presence of jumps in asset prices that could be conceptualized as measuring the impact of jumps on high order moments of returns by taking the difference between the accumulated arithmetic and geometric returns over the course of a day. Jiang and Oomen (2008) proposed another approach to jump identification with the null of no jumps in the sample path between 0 and t. The test exploits the differences that can occur between arithmetic and logarithmic returns computed as follows: where Rj : the arithmetic return j-th :intraday return rj :log return. The absence of jumps makes the difference between SwVt and the realized variance equal to 0: where J u = exp(Ju ) ? Ju ? 1, with J the jump process. The test statistic is defined as: ?SwV is estimated using a realized multipower variation (Barndorff-Nielsen et al., 2003; Barndorff- Nielsen et al., 2006): where a suitable choice form is either 4 or 6, as suggested by the authors, and ?6 = E(|U |)6 , U ? N (0, 1). b) Empirical results The Monte Carlo simulation for the size of the tests for jumps with medium mean reversion revealed the largest distortion to be encountered in the JO test with a one second sampling frequency having a size equal to 6.5 % and increasing even more as the sampling frequency diminishes (Dumitru & Urga, 2011). The size of tests at 5% significance level reveals that for 1 second sampling frequency, the size is equal to 5.2% and 5.4% which increases at lower sampling frequencies and become over-sized more rapidly. Power of tests at varying intensities and jump sizes depicts the JO test to have a very high power (between 96% and 98%) at one second. However at lower frequencies, its power becomes slightly lower than the other tests except AJ. And as in the case of varying jump intensity, JO procedure exhibits a very high power at 1 second sampling frequency ranging between 89% and 98% but at low frequencies, this reduces in relation to all the other tests except AJ (Dumitru & Urga, 2011). The size and power of the tests in the presence of microstructure noise depicted that JO finds a bias correction for the realize bipower variation in the presence of i.i.d microstructure noise; all tests become severely undersized in the presence of microstructure noise The JO procedure generally displays a very high size in the presence of noise at 1 second, which increases with the variance of the noise. However, when sampling is done at lower frequencies (from 1 minute onward), size decreases abruptly in the beginning and then, moderately increases again. The large size at 1 second is due to the fact that the distribution of the test statistic shifts to the right in the presence of microstructure noise. This effect becomes more intense as the variance of the noise becomes larger. An empirical application based on high frequency data for five stocks listed in the New York Stock Exchange for five years for an average of 1250 days was subjected to a series of jump tests in which the transaction data was sampled at 1,5,10,15 and 30 minutes. At one minute, the Jiang and Omen test (2009) detected a higher percentage of jumps which decreased substantially at five minutes from which onwards the decrease in the percentage became much slower and stabilization occurred at between 10-15 minutes. This was attributed to microstructure noise which supported the concept that at higher frequencies, the procedure detected a high number of spurious jumps. The JO test seems only slightly affected by zero returns at one minute (37 % with jumps) and the percentage of detected jumps does not change very much with the frequency (Dumitru & Urga, 2011). 5. Podolskij and Ziggel test 2010 a) Theoretical background The Podolskij and Ziggel (2010) test (PZ henceforth) is based on comparison between a realized power variation and a robust to jumps estimator to detect jumps. Their choice for the robust to jumps estimator is Mancini (2009)’s threshold estimator.; however, since the derivation of a limiting theory for the simple differentiation between the two has proved particularly difficult, authors define the test statistics as a difference between a realized power variation estimator and a threshold estimator perturbed by some external positive i.i.d. random variables, (?j )1?j?[t/?] , with E[?j ] = 1 and finite variation: where 1{|rj |?c??w } is an indicator function for absolute returns lower than a threshold fixed to c ? ? w , v with c = 2.3 BVt and w = .4. The test statistic can be defined as follows: where V ar[?j ] is the variance of the ?j variables. For the perturbing variables, Podolskij and Ziggel (2010) recommend to sample them from the following distribution: where ? = Dirac measure ? = constant chosen relatively small, e.g. ? = 0.1 or 0.05. b) Empirical results The Monte Carlo findings denote a high distortion in all the sampling frequencies when the PZ test is employed; it displays a size close to the nominal one when sampling is performed every second but gets rapidly and highly over-sized (Dumitru & Urga, 2011). The PZ and intraday procedures display the poorest performance when compared to all the other tests, being severely over-sized even when a sample is taken every second (93.3% for the intraday tests and 70.1% for PZ). When the power of tests was evaluated, it was realized that for varying jump intensities, the PZ test, intraday and ABD_LM were the best; the corrected power for these tests is between 98-99% for a sampling frequency of 1 second which gradually diminishes as the sampling frequency decreases. For the PZ procedure we observe a very high power (around 98% and 99% at 1 sec) which decreases with the sampling frequency. It remains higher than the other procedures (except the intraday tests) for data sampled at 1, 5 and 15 minutes and at 30 minutes the power of PZ is very close to 0 in all cases, even if the actual power (not reported here) ranges between 50% and 60%. This is due to the fact that the PZ statistic tends to become extremely large at very low frequencies under both the null and the alternative hypotheses. Microstructure noise does not seem to affect the PZ test because it is able to prove the validity of the test even in the presence of two types of noise, such as i.i.d. and i.i.d. plus rounding processes (Dumitru & Urga, 2011). It is the least affected by noise and at the highest sampling frequency, displays a size close to the nominal one even for the highest values of ?noise . This is a consequence of its higher and rapidly increasing size, which turns out to be an advantage in this case, as it compensates the downward bias caused by the presence of noise. An application of the PZ test to real financial data of stocks listed in the New York Stock Exchange revealed that at one minute, there was a high percentage of jumps which then substantially decreased at 5 minutes after which the decrease was much slower with stabilization being achieved from 10-15 minutes onwards PZ identified at 97% and at lower frequencies the percentage drastically dropped. It was realized that at higher frequencies, the test detected a high number of spurious jumps possibly as a result of the microstructure noise with one minute producing 57% at higher frequencies (Dumitru & Urga, 2011). When multipower variations are computed as a sum of the adjacent returns, they tend to be downward biased in the presence of many zero returns; as seen when BNS is calculated as the difference between Rvt and Bvt. The Bvt becomes smaller denoting that the presence of the threshold makes the multi-power variation even smaller leading to an over-rejection of the null. This effect is noticed in the PZ procedure with the result being an increase in the test statistic. References Andersen, T., Bollerslev, T. and Dorbev, D. (2007). No-Arbitage Semi-Martingale Restrictions for Continuous-Time Volatility Models subject to Leverage effects, Jumps and i.i.d. Noise Theory and Testable Distributional Implications. Journal of Econometrics 138(1), 125-180. Corsi, F., Piriono, D. and Reno, R. (2010). Threshold bipower variation and The Impact of Jumps on Volatility Forecasting. Journal of Econometrics 159(2), 276-288. Dumitru, A. and Urga, G. (2011). Identifying Jumps in Financial Assets: a Comparison between Nonparametric Jump Tests. Journal of Business and Economic Statistics, Forthcoming. Huang, X. and Tauchen, G. (2005). The Relative Contribution of Jumps to Total Price Variance. Journal of Financial Econometrics 3(4), 456-499. Nielsen, O. and Shephard, N. (2003a). Econometrics of testing for jumps in financial economics using bipower variation. Nielsen, O. and Shephard, N. (2003b). Power and bipower variation with stochastic volatility and jumps. Read More