From that figure we understand that, on average, the cross-sectional distribution of the trading volume is close to mesokurticity actually slightly above. The behaviour of the kurtosis significantly augments between semester 11 and semester 16, which includes the aforementioned period for which both the concavity and the symmetry of the cross-sectional variance boosted. In the first three semesters of that spell, General Motors was not traded for the reasons we have already explained.
Yet, the larger values of the kurtosis occur when GM. NY trading had resumed. In that span, the DJIA rocketed from In order words, the period spanning between semester 11 and 16 is typified by a hefty rise in the index. Therefrom, we see that in respect of the trading volume, the DJIA set of stocks starts strongly non-Gaussian, relaxing to Gaussianity after about 1 hour of trading and remaining around there for the next hour; from that time on, we observe a surge in the kurtosis until circa h.
For mere comparison purposes, we present using the grey symbols as well. The dashed red line represents the value of the kurtosis for Gaussian distributed variables. To that, we have used 1-minute sampling rate data of blue chip Dow Jones Industrial Average equities spanning the years between and Trading volume is widely viewed as a proxy for information in a financial market, a feature that utterly justifies the relevance of our work. We have done so twofold; in first place, by studying the intraday behaviour—specifically the profile—of the cumulants of the trading volume up to order four and then by looking to the nonstationary features of those profiles.
Due to the latter analysis, we have split our data into contiguous patches of six months.
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Additionally, we have considered two kinds of approach: i performing the statistics for each company and then averaging over them individual analysis and ii carrying out the computation of the statistical moments of the trading volume over the companies cross-sectional analysis. In both cases, we have confirmed that the trading volume is significantly nonstationary, at least for the period we have studied.
For the individual analysis, using an analogy between the decrease in the average trading volume after the opening of the market and the relaxation observed after earthquakes and avalanches in complex systems, we have adjusted the initial part of the average trading volume curve with a power-law and verified that the market changed its early trading behaviour after the second semester of , i. Explicitly, we have found that was decaying more slowly before 2S08 than it started taking place after that semester. That claim is bolstered by the analysis of the behaviour towards the ringing of the bell at ; by adjusting the last part of the with a power-law we have found a monotonic increase in the value of the exponent followed by a crossover in the first semester of In other words, the changes in the rules for short-selling are likely to have affected the closing of the session and had negligible impact on the opening.
The study made on price fluctuations has shown that their intra-day time dependent standard deviation decays in the first part of the session similarly to a power-law with an exponent around 0. The closeness of that exponent with the value we have computed for the relaxation of hints at a relation between volatility and trading volume that would fit for the Mixture of Distribution Hypothesis. Accordingly, we expect that ongoing work over the nonstationarity intraday relation between the trading volume and the volatility will shed light on this matter [ 41 ].
In addition, we have verified there is an asymmetry in the curve of the average trading volume as a function of the intraday time; yet, in this case it has been impossible to specify any trend whatsoever. We have also confirmed that the form of the curve plays a major role in the increase decrease of the trading activity, i. We have endeavoured to connect the properties of the average trading volume curve with quantities like the price fluctuation and the volatility in the respective period; however, we could not find any clear relation between these quantities.
This fact is at odds with the results on the relaxation of the average trading volume, which emphasises the need for additional analysis on the relation between v and the volatility. Yet, our analysis has showed that its impact goes beyond the mere return to trading. Note that apart some from a very located spikes, the effect does not appear in nor , but gets utterly clear for the kurtosis instead. Along this view, we have comprehended that at the beginning of the trading session, there is a large uncertainty in the value of the trading volume that is kept up with a large value of the surprise as well.
That relation is in contrast with the last part of the session, where the hiking up of the fluctuations is not that surprising and the kurtosis decreases. Pairing the kurtosis with the average of the trading volume, we have noticed that in the morning their relation is robustly parabolic whereas in the afternoon period we verified that the parabolic second order coefficient dwindled until the crisis semester 2S08 and effectively vanished from that semester onwards.
Inspecting the behaviour of the stocks as elements of a group cross-sectional statistics , we have understood the same qualitative behaviour we found for the individual variance and the kurtosis. Quantitatively, the values of the cross-sectional analysis are smaller than those obtained for the individual approach. In the first part of the morning, we have observed that both the dispersion and the kurtosis decay; this can be read as follows: mainly due to overnight episodes, there is a different level of activity among the stocks with some of them—eventually linked to relevant information disclosed when the market was closed—having abnormal values of trading.
That flustering is then dissipated—i. Leading up to the beginning of the afternoon part of the session, the kurtosis increases, a fact that we associate with the dual effect of contrarian strategies or intraday traders willing to secure some profit on the stocks that have shown abnormal activity at the opening. Afterwards, there is a relaxation to the Gaussian and the stocks behave among them as expected—i.
We also would like to thank M. Naeeni for reading the manuscript. Conceptualization: SMDQ. Data curation: MBG.
Funding acquisition: SMDQ. Methodology: SMDQ. Project administration: SMDQ. Resources: MBG. Software: MBG. Supervision: SMDQ. Visualization: MBG. Writing — original draft: MBG. Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field. Abstract We study the intraday behaviour of the statistical moments of the trading volume of the blue chip equities that composed the Dow Jones Industrial Average index between and Download: PPT.
Main formulae and definitions Let us consider a general quantity, , to introduce the notation we will employ hereinafter. The Mann-Whitney-Wilcoxon statistical test The Mann-Whitney-Wilcoxon test is a rank-based fully nonparametric test whose null hypothesis assumes that the two sets of observations come from the same population whereas the alternative hypothesis states that one of the sets has larger values than the other one. Results and Discussion Time dependent statistics Mean trading volume.
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Fig 1. Average trading volume, v intraday time minutes. Fig 2. Fig 3. Fig 4. Concavity of the average traded volume v semester s. Fig 5. Symmetry of the traded volume v semester s. Fig 6. Fig 7. Variance of the trading volume, v t.
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Fig 8. Relation between the variance and the average of the trading volume in 1S Fig Relaxation exponents of the kurtosis in Eq 18 as a function of the semester. Relation between the kurtosis and the mean trading volume. Cross-sectional statistics In the previous section, we have introduced the results concerning the statistics of the trading volume across intraday time conditioned to semester s and presented them carrying out averages over trading dates.
Cross-sectional variance v intra-day time in 2S Individual and Cross-sectional Concavity left and Symmetry right v semester s. Skewness and Kurtosis. Intraday cross-sectional kurtosis v semester s. References 1. Cambridge: Cambridge University Press. Bouchaud JP, Potters M. Focus on Complex networks in finance. Nature Phys 9. View Article Google Scholar 4. Complexity in quantitative finance and economics. Chaos Solitons Fractals.
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View Article Google Scholar 5. Varian HR. Northampton—MA: Edgar Elgar. Garman MB. Market Microstructure. J Financ Econ. View Article Google Scholar 7. Karpoff JM. The relation between price changes and trading volume: a survey. J Financ Quart Anal. View Article Google Scholar 8. Clark PK.
An Introduction to Global Financial Markets
However, the course is quantitative and students must be willing to learn and work with new concepts in mathematics and statistics. Course work will involve a significant amount of algebra and numerical exercises. Students should be comfortable with 'High School' level Mathematics and simple algebra, e. This course is a self-contained introduction to finance and it covers roughly the same topics as FM The course explores the way that firms and the capital market function to channel savings toward productive investments.
From the investor's perspective it considers characteristics of the major financial contracts and the principles used in their valuation.
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Overview of the Financial Markets Introduction to financial markets Concept of disintermediation Explain how banking and financial markets co-exist Identifying the key players and their impact on the market Inside a financial institution Case study - how the changes in regulation are impacting the market 2. Debt Markets Long-term and short-term debt Discount and interest-bearing products Interest rate and credit products Deposits - bank to customer and interbank markets LIBOR, EURIBOR and overnight indexed rates Introduction to credit debt - commercial paper Introduction to interest rate debt - treasury bills Role of primary dealers Calculating proceeds Features of bond markets Interest rate and credit products in the long-term market Government debt in different countries Corporate bonds Understanding credit spreads Valuing bonds - accrued interest Case study - new issues in the market 3.
Derivatives Interaction between cash and derivative markets Key uses of derivatives - risk management and risk replication OTC and exchange-traded derivatives Impact of legislation such as EMIR and Dodd-Frank Review of currency forwards Developing the model for interest rates and credit Structure of the futures market The role of a futures exchange Process of margining Options - comparison with forwards and futures The training is delivered over two and a half days with an exam taking place on the afternoon of day three.