Stylized facts in finance and data nonstationarities

SC, S. M. D. Queirós,

C. Anteneodo

Introduction

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• Nonstationary quantities can be considered as a juxtaposition of intervals of length L characterized by few parameters.

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• At the scale L, the parameters are assumed constant, but in the long-term follow a certain probability density function.

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Data

• Price fluctuations:

r_i(t) = s_i(t)- s_i(t - 1)

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• Normalized trading volume:

v_i = V_i(t)/<V_i>

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For 30 blue chip companies defining DJIA recorded at every 1 minute during the 2^nd semester of 2004

Nonstationarities

• Considering that a nonstationary time series can be split into stationary segments, the aim of segmentation is to find the optimal positions to separate the time series in such segments
• From the segmentation procedure* we are able to introduce a quantitative description of statistical features of these two quantities

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*PRE 84, 046702 (2011)

Stylized facts

• Tails of distribution of trading volume and price fluctuations
• A dynamics compatible with the U-shaped profile of the volume in a trade section
• Slow decay of the autocorrelation function

Heterogeneities in trading volume

Statistics of the patches

Typical exponential decay of the complementary cumulative distribution of segments length (trading volume)

Shortest and longest characteristic scale are indicated

Size of the segment versus local average (trading volume)

Local adjustment given by a locally weighted regression

Conditional probability of having a change of local regime which lasts

for L minutes averaged over all companies

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Longer segments have higher probability of starting during the first hours of a trading session

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Smaller segments show almost the same probability during the first 5 h of trading, which increases as the terminus of the session comes up (cleaning the order book)

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To what extent segments distribute within the trading session re-

gardless their length?

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Changes of local

stationarity occur less in the middle of the session

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Averaged correlation function of the local mean

value of the trading volume vs. the lag measured in segments

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Noise level: 18 segments

Intraday signature: trading close to 4 segments of average length

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Heterogeneities in trading volume

Long-term behavior from local statistics

Describing trading volume

After some work...

Describing trading volume

Long term distribution: empirical data (dots), and numerical results (lines) from log-normal distribution description of the volume

Trading volume and price fluctuations

Probability of having a positive (negative) price fluc-

tuation vs. trading volume

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Probability of the magnitude of price fluctuations, Π (|r|), vs. magnitude of the price fluctuations, |r|: empirical (dots) and numerical results (line)

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Distribution of the segments of the

segmentation of the absolute values of the price fluctuations.

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Discussion

• Log-normal distribution provides the best agreement with the long term distribution of trading volume
• Changes in the statistics of price fluctuations occur at a fast scale than in the case of trading volume
• Capture and identify the U -shape of trading volume within each session

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Discussion

• From the statistical properties of trading volume we were capable of obtaining a fair representation of the central part of the distribution of the magnitude of price fluctuations
• For magnitude of price fluctuations, we got a very clear exponential decay of the distribution of segments of local stationarity without the slower tail exhibited by the distribution trading volume segments.

Discussion

• Quantitatively, we found a typical scale around 75 min, which is substantially smaller than the 115 min found in the segmentation of trading volume

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Thanks!