13 research outputs found
Modeling record-breaking stock prices
We study the statistics of record-breaking events in daily stock prices of
366 stocks from the Standard and Poors 500 stock index. Both the record events
in the daily stock prices themselves and the records in the daily returns are
discussed. In both cases we try to describe the record statistics of the stock
data with simple theoretical models. The daily returns are compared to i.i.d.
RV's and the stock prices are modeled using a biased random walk, for which the
record statistics are known. These models agree partly with the behavior of the
stock data, but we also identify several interesting deviations. Most
importantly, the number of records in the stocks appears to be systematically
decreased in comparison with the random walk model. Considering the
autoregressive AR(1) process, we can predict the record statistics of the daily
stock prices more accurately. We also compare the stock data with simulations
of the record statistics of the more complicated GARCH(1,1) model, which, in
combination with the AR(1) model, gives the best agreement with the
observational data. To better understand our findings, we discuss the survival
and first-passage times of stock prices on certain intervals and analyze the
correlations between the individual record events. After recapitulating some
recent results for the record statistics of ensembles of N stocks, we also
present some new observations for the weekly distributions of record events.Comment: 20 pages, 28 figure
Records statistics beyond the standard model - Theory and applications
In recent years, there was a surge of interest in the statistics of record-breaking events, not only from scientists, but also from the general public. In sports and in climatology, but also in nature and in economy, observers are interested in the setting and breaking of new records. This cumulative dissertation is dedicated to the study of record-breaking events. It concludes a series of published and hitherto unpublished articles on theory and applications of record statistics. This work mainly consists of five parts. The first part is about the statistics of records in uncorrelated random variables sampled from time-dependent distributions. In particular, we present a simple model of random variables with a linearly growing mean value and discuss its record statistics thoroughly. Furthermore, the effects of rounding on the occurrence of records in series of independent and identically distributed random variables are considered. Then, in part two, these results are applied to explain and model record temperatures in the context of climatic change. Using our minimal model of random variables with a linear drift, we show that global warming has in fact a significant effect on the occurrence and the values of heat and cold records. The third part focuses on records in correlated processes, in particular random walks. A number of different random walk processes are presented and analyzed. We find that their record statistics are surprisingly interesting and manifold. The results derived in this part are important to understand the occurrence of records in financial data, which will be discussed in the fourth part. There it is demonstrated that random walks are helpful to model records in stock data, nevertheless we find significant deviations from the analytical results and propose an alternative model, which describes the stock data more accurately. The final, fifth part is a general review of the recent developments in the study of record-breaking events in time series of time-dependent and correlated random variables
Record occurrence and record values in daily and monthly temperatures
We analyze the occurrence and the values of record-breaking temperatures in
daily and monthly temperature observations. Our aim is to better understand and
quantify the statistics of temperature records in the context of global
warming. Similar to earlier work we employ a simple mathematical model of
independent and identically distributed random variables with a linearly
growing expectation value. This model proved to be useful in predicting the
increase (decrease) in upper (lower) temperature records in a warming climate.
Using both station and re-analysis data from Europe and the United States we
further investigate the statistics of temperature records and the validity of
this model. The most important new contribution in this article is an analysis
of the statistics of record values for our simple model and European reanalysis
data. We estimate how much the mean values and the distributions of record
temperatures are affected by the large scale warming trend. In this context we
consider both the values of records that occur at a certain time and the values
of records that have a certain record number in the series of record events. We
compare the observational data both to simple analytical computations and
numerical simulations. We find that it is more difficult to describe the values
of record breaking temperatures within the framework of our linear drift model.
Observations from the summer months fit well into the model with Gaussian
random variables under the observed linear warming, in the sense that record
breaking temperatures are more extreme in the summer. In winter however a
significant asymmetry of the daily temperature distribution hides the effect of
the slow warming trends. Therefore very extreme cold records are still possible
in winter. This effect is even more pronounced if one considers only data from
subpolar regions.Comment: 16 pages, 20 figures, revised version, published in Climate Dynamic
Correlations between record events in sequences of random variables with a linear trend
The statistics of records in sequences of independent, identically
distributed random variables is a classic subject of study. One of the earliest
results concerns the stochastic independence of record events. Recently,
records statistics beyond the case of i.i.d. random variables have received
much attention, but the question of independence of record events has not been
addressed systematically. In this paper, we study this question in detail for
the case of independent, non-identically distributed random variables,
specifically, for random variables with a linearly moving mean. We find a rich
pattern of positive and negative correlations, and show how their asymptotics
is determined by the universality classes of extreme value statistics.Comment: 19 pages, 12 figures; some typos in Sections 3.1 and 3.3. correcte
Record statistics and persistence for a random walk with a drift
We study the statistics of records of a one-dimensional random walk of n
steps, starting from the origin, and in presence of a constant bias c. At each
time-step the walker makes a random jump of length \eta drawn from a continuous
distribution f(\eta) which is symmetric around a constant drift c. We focus in
particular on the case were f(\eta) is a symmetric stable law with a L\'evy
index 0 < \mu \leq 2. The record statistics depends crucially on the
persistence probability which, as we show here, exhibits different behaviors
depending on the sign of c and the value of the parameter \mu. Hence, in the
limit of a large number of steps n, the record statistics is sensitive to these
parameters (c and \mu) of the jump distribution. We compute the asymptotic mean
record number after n steps as well as its full distribution P(R,n). We
also compute the statistics of the ages of the longest and the shortest lasting
record. Our exact computations show the existence of five distinct regions in
the (c, 0 < \mu \leq 2) strip where these quantities display qualitatively
different behaviors. We also present numerical simulation results that verify
our analytical predictions.Comment: 51 pages, 22 figures. Published version (typos have been corrected
Records and sequences of records from random variables with a linear trend
We consider records and sequences of records drawn from discrete time series
of the form , where the are independent and identically
distributed random variables and is a constant drift. For very small and
very large drift velocities, we investigate the asymptotic behavior of the
probability of a record occurring in the th step and the
probability that all entries are records, i.e. that . Our work is motivated by the analysis of temperature time series in
climatology, and by the study of mutational pathways in evolutionary biology.Comment: 21 pages, 7 figure
Record statistics for biased random walks, with an application to financial data
We consider the occurrence of record-breaking events in random walks with
asymmetric jump distributions. The statistics of records in symmetric random
walks was previously analyzed by Majumdar and Ziff and is well understood.
Unlike the case of symmetric jump distributions, in the asymmetric case the
statistics of records depends on the choice of the jump distribution. We
compute the record rate , defined as the probability for the th
value to be larger than all previous values, for a Gaussian jump distribution
with standard deviation that is shifted by a constant drift . For
small drift, in the sense of , the correction to
grows proportional to arctan and saturates at the value
. For large the record rate approaches a
constant, which is approximately given by
for .
These asymptotic results carry over to other continuous jump distributions with
finite variance. As an application, we compare our analytical results to the
record statistics of 366 daily stock prices from the Standard & Poors 500
index. The biased random walk accounts quantitatively for the increase in the
number of upper records due to the overall trend in the stock prices, and after
detrending the number of upper records is in good agreement with the symmetric
random walk. However the number of lower records in the detrended data is
significantly reduced by a mechanism that remains to be identified.Comment: 16 pages, 7 figure
Record Statistics for Multiple Random Walks
We study the statistics of the number of records R_{n,N} for N identical and
independent symmetric discrete-time random walks of n steps in one dimension,
all starting at the origin at step 0. At each time step, each walker jumps by a
random length drawn independently from a symmetric and continuous distribution.
We consider two cases: (I) when the variance \sigma^2 of the jump distribution
is finite and (II) when \sigma^2 is divergent as in the case of L\'evy flights
with index 0 < \mu < 2. In both cases we find that the mean record number
grows universally as \sim \alpha_N \sqrt{n} for large n, but with a
very different behavior of the amplitude \alpha_N for N > 1 in the two cases.
We find that for large N, \alpha_N \approx 2 \sqrt{\log N} independently of
\sigma^2 in case I. In contrast, in case II, the amplitude approaches to an
N-independent constant for large N, \alpha_N \approx 4/\sqrt{\pi},
independently of 0<\mu<2. For finite \sigma^2 we argue, and this is confirmed
by our numerical simulations, that the full distribution of (R_{n,N}/\sqrt{n} -
2 \sqrt{\log N}) \sqrt{\log N} converges to a Gumbel law as n \to \infty and N
\to \infty. In case II, our numerical simulations indicate that the
distribution of R_{n,N}/\sqrt{n} converges, for n \to \infty and N \to \infty,
to a universal nontrivial distribution, independently of \mu. We discuss the
applications of our results to the study of the record statistics of 366 daily
stock prices from the Standard & Poors 500 index.Comment: 25 pages, 8 figure