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Quantifying Stock Return Distributions in Financial Markets.

Botta F, Moat HS, Stanley HE, Preis T - PLoS ONE (2015)

Bottom Line: Being able to quantify the probability of large price changes in stock markets is of crucial importance in understanding financial crises that affect the lives of people worldwide.Large changes in stock market prices can arise abruptly, within a matter of minutes, or develop across much longer time scales.Our findings may inform the development of models of market behavior across varying time scales.

View Article: PubMed Central - PubMed

Affiliation: Centre for Complexity Science, University of Warwick, Coventry, CV4 7AL, United Kingdom; Data Science Lab, Behavioural Science, Warwick Business School, University of Warwick, Coventry, CV4 7AL, United Kingdom.

ABSTRACT
Being able to quantify the probability of large price changes in stock markets is of crucial importance in understanding financial crises that affect the lives of people worldwide. Large changes in stock market prices can arise abruptly, within a matter of minutes, or develop across much longer time scales. Here, we analyze a dataset comprising the stocks forming the Dow Jones Industrial Average at a second by second resolution in the period from January 2008 to July 2010 in order to quantify the distribution of changes in market prices at a range of time scales. We find that the tails of the distributions of logarithmic price changes, or returns, exhibit power law decays for time scales ranging from 300 seconds to 3600 seconds. For larger time scales, we find that the distributions tails exhibit exponential decay. Our findings may inform the development of models of market behavior across varying time scales.

No MeSH data available.


Related in: MedlinePlus

Relationship between Δt and the scaling exponent for the empirical tails of return distributions.(a) We investigate the relationship between the time lag between price observations used to build the returns distribution and the scaling exponents of the tails of distributions. We consider here the tails of the positive component of the distributions obtained when analyzing all trading days present in our dataset. We find that the mean scaling exponent increases as Δt increases (Adjusted R2 = 0.802, N = 12, p < 0.001, ordinary least squares regression) (b) In a similar fashion, we observe that when analyzing all trading days the mean scaling exponent for the tail of the negative component of the distributions increases with the time lag (Adjusted R2 = 0.839, N = 12, p < 0.001, ordinary least squares regression) (c) We now restrict our analysis to trading days on which the prices of stocks have changed by more than 1%. We find that the mean scaling exponent of positive tails consistent with power law behavior increases with Δt (Adjusted R2 = 0.856, N = 12, p < 0.001, ordinary least squares regression) (d) Under 1% stress, an increase in the time lag Δt results again in an increase of the mean scaling exponent for the tails of the negative returns distributions (Adjusted R2 = 0.729, N = 12, p < 0.001, ordinary least squares regression) (e) We now perform the same analysis for days on which the prices of stocks have changed by more than 2%. The mean scaling exponent for the tails of the positive component of the distributions again shows an increase with increasing Δt (Adjusted R2 = 0.782, N = 12, p < 0.001, ordinary least squares regression) (f) Similarly, the mean scaling exponent for the tails of negative returns distributions at the 2% stress level increases as the time lag Δt between price observations increases (Adjusted R2 = 0.836, N = 12, p < 0.001, ordinary least squares regression).
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pone.0135600.g003: Relationship between Δt and the scaling exponent for the empirical tails of return distributions.(a) We investigate the relationship between the time lag between price observations used to build the returns distribution and the scaling exponents of the tails of distributions. We consider here the tails of the positive component of the distributions obtained when analyzing all trading days present in our dataset. We find that the mean scaling exponent increases as Δt increases (Adjusted R2 = 0.802, N = 12, p < 0.001, ordinary least squares regression) (b) In a similar fashion, we observe that when analyzing all trading days the mean scaling exponent for the tail of the negative component of the distributions increases with the time lag (Adjusted R2 = 0.839, N = 12, p < 0.001, ordinary least squares regression) (c) We now restrict our analysis to trading days on which the prices of stocks have changed by more than 1%. We find that the mean scaling exponent of positive tails consistent with power law behavior increases with Δt (Adjusted R2 = 0.856, N = 12, p < 0.001, ordinary least squares regression) (d) Under 1% stress, an increase in the time lag Δt results again in an increase of the mean scaling exponent for the tails of the negative returns distributions (Adjusted R2 = 0.729, N = 12, p < 0.001, ordinary least squares regression) (e) We now perform the same analysis for days on which the prices of stocks have changed by more than 2%. The mean scaling exponent for the tails of the positive component of the distributions again shows an increase with increasing Δt (Adjusted R2 = 0.782, N = 12, p < 0.001, ordinary least squares regression) (f) Similarly, the mean scaling exponent for the tails of negative returns distributions at the 2% stress level increases as the time lag Δt between price observations increases (Adjusted R2 = 0.836, N = 12, p < 0.001, ordinary least squares regression).

Mentions: A power law probability distribution is a probability distribution in which the probability of an event decays as a negative power of the event. The distribution function is characterized by a scaling exponent. Distributions of returns typically exhibit power law decay in the tail of the distribution. Here, we want to understand how the exact nature of power law behavior depends on the time lag between price observations. We analyze all 25 stock price time series and use a time lag Δt ranging from 300 to 3,600 seconds. We investigate how the scaling exponent changes as a function of the time lag between price observations. We depict the exponent for the tails of the positive (denoted as α+; Fig 3a) and negative (denoted as α−; Fig 3b) returns distributions obtained when analyzing data from all trading days. For both positive and negative tails, we find that the mean scaling exponent increases with the time lag Δt (α+: Adjusted R2 = 0.802, N = 12, p < 0.001, ordinary least squares regression; α−: Adjusted R2 = 0.839, N = 12, p < 0.001, ordinary least squares regression):α+=0.010(±0.001)Δt+3.54(±0.05)α-=0.012(±0.001)Δt+3.42(±0.06)We find similar slopes for the positive and negative tails, which suggests that both exponents α+ and α− vary in a similar fashion as a function of the time lag Δt. Our results suggest that the probability of finding large price changes is underestimated by a Gaussian distribution and better quantified by a power law distribution, in line with a range of findings reported in the field of econophysics [30–42]. Previous findings for US markets have highlighted that stock returns may follow an inverse cubic law [31]. The analysis of different stock markets, such as the Warsaw Stock Exchange in Poland or the Australian Stock Exchange, has uncovered different power law regimes deviating from the inverse cubic law [38, 39]. By selecting appropriate cutoff values in the distributions under analysis, stocks from the Mexican Stock Market index exhibit a power law decay close to an inverse cubic law [40]. Analogous results have also been observed when analysing daily returns in Chinese stock markets [46, 47].


Quantifying Stock Return Distributions in Financial Markets.

Botta F, Moat HS, Stanley HE, Preis T - PLoS ONE (2015)

Relationship between Δt and the scaling exponent for the empirical tails of return distributions.(a) We investigate the relationship between the time lag between price observations used to build the returns distribution and the scaling exponents of the tails of distributions. We consider here the tails of the positive component of the distributions obtained when analyzing all trading days present in our dataset. We find that the mean scaling exponent increases as Δt increases (Adjusted R2 = 0.802, N = 12, p < 0.001, ordinary least squares regression) (b) In a similar fashion, we observe that when analyzing all trading days the mean scaling exponent for the tail of the negative component of the distributions increases with the time lag (Adjusted R2 = 0.839, N = 12, p < 0.001, ordinary least squares regression) (c) We now restrict our analysis to trading days on which the prices of stocks have changed by more than 1%. We find that the mean scaling exponent of positive tails consistent with power law behavior increases with Δt (Adjusted R2 = 0.856, N = 12, p < 0.001, ordinary least squares regression) (d) Under 1% stress, an increase in the time lag Δt results again in an increase of the mean scaling exponent for the tails of the negative returns distributions (Adjusted R2 = 0.729, N = 12, p < 0.001, ordinary least squares regression) (e) We now perform the same analysis for days on which the prices of stocks have changed by more than 2%. The mean scaling exponent for the tails of the positive component of the distributions again shows an increase with increasing Δt (Adjusted R2 = 0.782, N = 12, p < 0.001, ordinary least squares regression) (f) Similarly, the mean scaling exponent for the tails of negative returns distributions at the 2% stress level increases as the time lag Δt between price observations increases (Adjusted R2 = 0.836, N = 12, p < 0.001, ordinary least squares regression).
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pone.0135600.g003: Relationship between Δt and the scaling exponent for the empirical tails of return distributions.(a) We investigate the relationship between the time lag between price observations used to build the returns distribution and the scaling exponents of the tails of distributions. We consider here the tails of the positive component of the distributions obtained when analyzing all trading days present in our dataset. We find that the mean scaling exponent increases as Δt increases (Adjusted R2 = 0.802, N = 12, p < 0.001, ordinary least squares regression) (b) In a similar fashion, we observe that when analyzing all trading days the mean scaling exponent for the tail of the negative component of the distributions increases with the time lag (Adjusted R2 = 0.839, N = 12, p < 0.001, ordinary least squares regression) (c) We now restrict our analysis to trading days on which the prices of stocks have changed by more than 1%. We find that the mean scaling exponent of positive tails consistent with power law behavior increases with Δt (Adjusted R2 = 0.856, N = 12, p < 0.001, ordinary least squares regression) (d) Under 1% stress, an increase in the time lag Δt results again in an increase of the mean scaling exponent for the tails of the negative returns distributions (Adjusted R2 = 0.729, N = 12, p < 0.001, ordinary least squares regression) (e) We now perform the same analysis for days on which the prices of stocks have changed by more than 2%. The mean scaling exponent for the tails of the positive component of the distributions again shows an increase with increasing Δt (Adjusted R2 = 0.782, N = 12, p < 0.001, ordinary least squares regression) (f) Similarly, the mean scaling exponent for the tails of negative returns distributions at the 2% stress level increases as the time lag Δt between price observations increases (Adjusted R2 = 0.836, N = 12, p < 0.001, ordinary least squares regression).
Mentions: A power law probability distribution is a probability distribution in which the probability of an event decays as a negative power of the event. The distribution function is characterized by a scaling exponent. Distributions of returns typically exhibit power law decay in the tail of the distribution. Here, we want to understand how the exact nature of power law behavior depends on the time lag between price observations. We analyze all 25 stock price time series and use a time lag Δt ranging from 300 to 3,600 seconds. We investigate how the scaling exponent changes as a function of the time lag between price observations. We depict the exponent for the tails of the positive (denoted as α+; Fig 3a) and negative (denoted as α−; Fig 3b) returns distributions obtained when analyzing data from all trading days. For both positive and negative tails, we find that the mean scaling exponent increases with the time lag Δt (α+: Adjusted R2 = 0.802, N = 12, p < 0.001, ordinary least squares regression; α−: Adjusted R2 = 0.839, N = 12, p < 0.001, ordinary least squares regression):α+=0.010(±0.001)Δt+3.54(±0.05)α-=0.012(±0.001)Δt+3.42(±0.06)We find similar slopes for the positive and negative tails, which suggests that both exponents α+ and α− vary in a similar fashion as a function of the time lag Δt. Our results suggest that the probability of finding large price changes is underestimated by a Gaussian distribution and better quantified by a power law distribution, in line with a range of findings reported in the field of econophysics [30–42]. Previous findings for US markets have highlighted that stock returns may follow an inverse cubic law [31]. The analysis of different stock markets, such as the Warsaw Stock Exchange in Poland or the Australian Stock Exchange, has uncovered different power law regimes deviating from the inverse cubic law [38, 39]. By selecting appropriate cutoff values in the distributions under analysis, stocks from the Mexican Stock Market index exhibit a power law decay close to an inverse cubic law [40]. Analogous results have also been observed when analysing daily returns in Chinese stock markets [46, 47].

Bottom Line: Being able to quantify the probability of large price changes in stock markets is of crucial importance in understanding financial crises that affect the lives of people worldwide.Large changes in stock market prices can arise abruptly, within a matter of minutes, or develop across much longer time scales.Our findings may inform the development of models of market behavior across varying time scales.

View Article: PubMed Central - PubMed

Affiliation: Centre for Complexity Science, University of Warwick, Coventry, CV4 7AL, United Kingdom; Data Science Lab, Behavioural Science, Warwick Business School, University of Warwick, Coventry, CV4 7AL, United Kingdom.

ABSTRACT
Being able to quantify the probability of large price changes in stock markets is of crucial importance in understanding financial crises that affect the lives of people worldwide. Large changes in stock market prices can arise abruptly, within a matter of minutes, or develop across much longer time scales. Here, we analyze a dataset comprising the stocks forming the Dow Jones Industrial Average at a second by second resolution in the period from January 2008 to July 2010 in order to quantify the distribution of changes in market prices at a range of time scales. We find that the tails of the distributions of logarithmic price changes, or returns, exhibit power law decays for time scales ranging from 300 seconds to 3600 seconds. For larger time scales, we find that the distributions tails exhibit exponential decay. Our findings may inform the development of models of market behavior across varying time scales.

No MeSH data available.


Related in: MedlinePlus