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

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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

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 $X_{n}=Y_{n}+cn$, where the $Y_{n}$ are independent and identically distributed random variables and $c$ is a constant drift. For very small and very large drift velocities, we investigate the asymptotic behavior of the probability $p_n(c)$ of a record occurring in the $n$th step and the probability $P_N(c)$ that all $N$ entries are records, i.e. that $X_1 < X_2 < ... < X_N$. 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 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