Record statistics of financial time series and geometric random walks

The study of record statistics of correlated series in physics, such as random walks, is gaining momentum, and several analytical results have been obtained in the past few years. In this work, we study the record statistics of correlated empirical data for which random walk models have relevance. We obtain results for the records statistics of select stock market data and the geometric random walk, primarily through simulations. We show that the distribution of the age of records is a power law with the exponent $\alpha$ lying in the range $1.5\le \alpha \le 1.8$. Further, the longest record ages follow the Fréchet distribution of extreme value theory. The records statistics of geometric random walk series is in good agreement with that obtained from empirical stock data.

Figure 1

(a) Record ages (in days) calculated from IBM stock data. (b) The (logarithmically binned) distribution of record ages for three stocks. The solid line in (b) has slope $-1.623\pm 0.081$.

Figure 2

The (logarithmically binned) distribution of record ages obtained from (a) GRW simulations for three values of $N$ and (b) stock data other than those shown in Fig. 1. The solid line in (a) has slope $-1.652\pm 0.006$ and in (b) has slope $-1.611\pm 0.051$. See text for details.

Figure 3

Maximum likelihood estimate for the power-law exponent $\alpha$ for the stocks studied in this work. The error bars represent the uncertainty in the estimated value of $\alpha$ [21]. Full name of the stocks in $x$ axis is given in the Appendix.

Figure 4

The autocorrelation function of the record ages obtained from three different stock data and GRW simulations. For GRW results, we have used the same value of $N$ as for IBM stock, $\langle {\mu }_{\text{emp}}\rangle =0.00031$ and $\langle {\sigma }_{\text{emp}}\rangle =0.015$. See text for details.

Figure 5

Scaled distribution of longest record age obtained from GRW simulations with (a) $N=15000$ and (b) $N=85000$. The solid curve is the Fréchet distribution with shape parameter $k>0$. (c) The location parameter $a_N$ and (d) the scale parameter $b_N$ of the Fréchet distribution shown as a function of $N$. The solid line in (c, d) is the logarithmic fit for $N>30000$.

Figure 6

The distribution of scaled longest record ages computed from stock data (solid circles), GRW simulations (histogram) with $N=1000$. The solid curve is the Fréchet distribution with parameters $a_N$ and $b_N$ corresponding to $N=1000$.