Market dynamics immediately before and after financial shocks: Quantifying the Omori, productivity, and Bath laws

We study the cascading dynamics immediately before and immediately after 219 market shocks. We define the time of a market shock $T_c$ to be the time for which the market volatility $V\left(T_c\right)$ has a peak that exceeds a predetermined threshold. The cascade of high volatility “aftershocks” triggered by the “main shock” is quantitatively similar to earthquakes and solar flares, which have been described by three empirical laws—the Omori law, the productivity law, and the Bath law. We analyze the most traded 531 stocks in U.S. markets during the 2 yr period of 2001–2002 at the 1 min time resolution. We find quantitative relations between the main shock magnitude $M\equiv {\text{log}}_{10}V\left(T_c\right)$ and the parameters quantifying the decay of volatility aftershocks as well as the volatility preshocks. We also find that stocks with larger trading activity react more strongly and more quickly to market shocks than stocks with smaller trading activity. Our findings characterize the typical volatility response conditional on $M$, both at the market and the individual stock scale. We argue that there is potential utility in these three statistical quantitative relations with applications in option pricing and volatility trading.

Figure 1

(Color online) Typical volatility curves on 01/11/2002 with market shock at $T_c=256\text{min}$. (a) The cumulative volatility $N^j\left(t\right)$ for the stock of several large companies has varying behavior before $T_c$, but each stock shown begins to cascade soon after $T_c$. The market average $N\left(t\right)$ over all $S=531$ stocks analyzed demonstrates a distinct change in curvature at $t=T_c$. (b) The average market volatility $V\left(t\right)$ demonstrates a sharp peak at $T_c$, and also two precursor events at $t\approx 190$ and $\approx 230\text{min}$.

Figure 2

(Color online) (a) An illustration of $N_b\left(\tau \right)$ and $N_a\left(\tau \right)$ for the same set of curves plotted in Fig. 1. The displaced time $\tau =\left|t-T_c\right|$ is defined symmetrically around $T_c=256\text{min}$ on 01/11/2002. (b) ${\text{log}}_{10}{N}_b\left(\tau \right)$ and ${\text{log}}_{10}{N}_a\left(\tau \right)$ are linear with ${\text{log}}_{10}\tau$ over two orders of magnitude on a logarithmic scale. The Omori parameters in Eq. (5) calculated from $N\left(t\right)$ are ${\Omega }_b=0.09\pm 0.01$, ${\alpha }_b=0.21\pm 0.01$ and ${\Omega }_a=0.32\pm 0.01$, ${\alpha }_a=0.81\pm 0.01$.

Figure 3

(Color online) The fraction $n\left(t\right)$ of the market above the volatility threshold $q$ is nonstationary through the trading day. We plot in (a) the average daily trading pattern $\overline{n}\left(t\right)$ for $S=136$ stocks and in (b) the corresponding standard deviation to demonstrate the trends we remove in the normalized quantity $n^{\prime }\left(t\right)$. In practice, we use the smoothed average of these curves in order to diminish statistical fluctuations on the minute-to-minute scale. For comparison, we compute ${\overline{n}}_{sh}\left(t\right)\approx 0.23$ and ${\sigma }_{sh}\approx 0.09$ for shuffled $v_i\left(t\right)$. The values of $\overline{n}\left(t\right)$ provide an estimate for the background market comovement that can be attributed to random fluctuations.

Figure 4

(Color online) Example of market comovement $n\left(t\right)$ in both price volatility and total volume, and the qualitative relationship between the quantities $n\left(t\right)$, $n^{\prime }\left(t\right)\equiv \left[n\left(t\right)-\overline{n}\left(t\right)\right]/\sigma \left(t\right)$, and $x\left(t\right)\equiv n\left(t\right)n^{\prime }\left(t\right)$. The market shock on 01/11/2002 occurred at $T_c=256$ in response to a public comment by the Fed chairman A. Greenspan concerning economic recovery [20].

Figure 5

(Color online) Using the volatility threshold $q=3$ and $S=136$ stocks, we determine the market comovement threshold $x_c$ from the pdf of $x\left(t\right)\equiv n\left(t\right)n^{\prime }\left(t\right)$. (a) The pdf for the $190000$ min analyzed of the volatility rate $n\left(t\right)$ corresponding to the fraction of the market with volatility $v_i\left(t\right)>q$. (b) The pdf of $x\left(t\right)$, where in the quantity $x\left(t\right)$ we have removed the average daily trend of $n\left(t\right)$, so that $x\left(t\right)$ is relatively large when market comovement is large and significant. For comparison, we also plot the pdf of $x_{sh}\left(t\right)$ computed from randomly shuffled volatility time series $v_i\left(t\right)$. We find a divergence between the pdf’s of $x\left(t\right)$ and of $x_{sh}\left(t\right)$ for $x>1.0$, which we define as the comovement threshold $x_c\equiv 1$ in our analysis.

Figure 6

(Color online) (a) and (b) Comparison of the probability density functions $P\left(\alpha \right)$ and $P\left(\Omega \right)$ of Omori parameters $\alpha$ and $\Omega$ computed from the average market response $N_{a,b}\left(\tau \right)$. (c) and (d) The analogous pdf plots computed from individual stock response $N_{a,b}^j\left(\tau \right)$. The average and standard deviations of each data set are (a) ${\Omega }_a=0.09\pm 0.07$, ${\Omega }_b=0.06\pm 0.07$; (b) ${\alpha }_a=0.35\pm 0.11$, ${\alpha }_b=0.28\pm 0.09$; (c) ${\Omega }_a=0.08\pm 0.20$, ${\Omega }_b=0.03\pm 0.22$; and (d) ${\alpha }_a=0.53\pm 0.25$, ${\alpha }_b=0.46\pm 0.24$. Values of both ${\Omega }_a$ and ${\alpha }_a$ are consistently larger than ${\Omega }_b$ and ${\alpha }_b$, indicating that the response time after $T_c$ is shorter than the activation time leading into $T_c$. However the response cascade after $T_c$ has, generally, a larger amplitude.

Figure 7

(Color online) In order to account for the dispersion in the pdf’s plotted in Figs. 6c, 6d for individual stocks, we compare the average values ${\overline{\alpha }}_{a,b}$ and ${\overline{\Omega }}_{a,b}$ computed from all $N_{a,b}^j\left(\tau \right)$ with ${\alpha }_{a,b}$ and ${\Omega }_{a,b}$ computed from the corresponding average market response $N_{a,b}\left(\tau \right)$ for each of the 219 $T_c$’s. The visually apparent correlation indicates that the parameters quantifying $N_{a,b}\left(\tau \right)$ are a good representation of the average $N_{a,b}^j\left(\tau \right)$. The correlation coefficient $r$ for each linear regression is provided in each panel.

Figure 8

(Color online) The relation between the magnitude $M\equiv {\text{log}}_{10}V\left(T_c\right)$ and the Omori exponents ${\Omega }_{a,b}$. In (a) and (c) we compare values calculated from the average market response $N_{a,b}\left(\tau \right)$, and in (b) and (d) we compare values calculated from individual stock response $N_{a,b}^j\left(\tau \right)$. (a) Weak relation before $T_c$, where we validate the linear regression model at $p=0.001$ significance level, but with correlation coefficient $r=0.22$. The dispersion may result from the variability in anticipation preceding the market shock at $T_c$. (c) The relation between ${\Omega }_a$ and $M$ is stronger after $T_c$ than before $T_c$, with linear regression significance $p\approx 0$, correlation $r=0.40$, and regression slope $m=0.19\pm 0.03$. The increasing trend demonstrates that a faster response, quantified by larger ${\Omega }_a$, follows a larger $M$. Data points in (a) and (c) denoted by the symbol $\times$ correspond to values of ${\Omega }_{a,b}$ calculated for randomly selected $T_c$ on those 118 days analyzed without a single value of $x\left(t\right)>x_c$. In (b) and (d) there is much dispersion in the $\Omega$ values of individual stocks for given $V\left(T_c\right)$. However, the average trends demonstrate a significant crossover at $M_{\mathsf{x}}\approx 0.5$ from ${\Omega }_{a,b}<0$ to ${\Omega }_{a,b}>0$. The case of $\Omega <0$ can occur when there is more volatility clustering for large $\tau$ than for small $\tau$, whereas the case of $\Omega >0$ occurs for large volatility cascading around $\tau \gtrsim 0$. This crossover could result from the difference between anticipated and surprise shocks at $T_c$.

Figure 9

(Color online) The relation between the magnitude $M\equiv {\text{log}}_{10}V\left(T_c\right)$ and the Omori amplitudes ${\alpha }_{a,b}$. In (a) and (c) we compare the values calculated from the average market response $N\left(\tau \right)$, and in (b) and (d) we compare values calculated from individual stock response $N_{a,b}^j\left(\tau \right)$. (a) The increasing relation between ${\alpha }_b$ and $M$ is statistically stronger than the relation between ${\Omega }_b$ and $M$ in Fig. 8a, with significance $p\approx 0$, correlation coefficient $r=0.52$, and regression slope $m=0.35\pm 0.04$. (c) The relation between ${\alpha }_a$ and $M$ is strong, with significance $p\approx 0$, $r=0.84$, and regression slope $m=0.68\pm 0.03$. Data points in (a) and (c) denoted by the symbol $\times$ correspond to values of ${\alpha }_{a,b}$ calculated for randomly selected $T_c$ on those 118 days analyzed without a single value of $x\left(t\right)>x_c$. The result that $\alpha$ increases with increasing $V\left(T_c\right)$ holds even for random times. In (b) and (d) there is much dispersion in the $\alpha$ values of individual stocks for given $V\left(T_c\right)$. However, the average trends demonstrate a significant crossover at $M_{\mathsf{x}}\approx 0.5$ from ${\alpha }_{a,b}\approx 0.2$ for $M<0.5$ to ${\alpha }_{a,b}>0.2$ for $M>0.5$. This crossover occurs at a similar location as the crossover observed in Figs. 8b, 8d for ${\Omega }_{b,a}$. The average amplitude value $\overline{\alpha }$ increases sharply for $M>M_{\mathsf{x}}$, consistent with first-order phase-transition behavior.

Figure 10

(Color online) The increasing relation between the productivity $P_{a,b}\left(\Delta t\right)$ of each market shock and the size of the main shock $M\equiv {\text{log}}_{10}V\left(T_c\right)$ with $\Delta t\equiv 90\text{min}$. As is found in earthquakes, we find a power-law relationship between $M$ and $V\left(T_c\right)$ described by a productivity exponent ${\Pi }_b$ before and exponent ${\Pi }_a$ after the market shock. Data points in (a) and (c) denoted by the symbol $\times$ correspond to values of $P_{a,b}\left(\Delta t\right)$ calculated for randomly selected $T_c$ on those 118 days analyzed without a single value of $x\left(t\right)>x_c$. The result that $P\left(\Delta t\right)$ increases with increasing $V\left(T_c\right)$ holds even for random times. For the average market response $N_{b,a}\left(\Delta t\right)$, we find (a) ${\Pi }_b=0.38\pm 0.07$ and (c) ${\Pi }_a=0.48\pm 0.04$. For the productivity of individual stocks corresponding to $N_{b,a}^j\left(\Delta t\right)$ we find (b) ${\Pi }_b=0.23\pm 0.01$ and (d) ${\Pi }_a=0.25\pm 0.01$. For comparison, the power-law exponent value pertaining to earthquake aftershocks is ${\Pi }_a\approx 0.7–0.9$ [38].

Figure 11

(Color online) A comparison of Omori parameters before and after $T_c$ for $N\left(\tau \right)$ and varying $\Delta t$ indicate that ${\alpha }_b$ and $P_b\left(\Delta t\right)$ are better conditional estimators for the dynamics after $T_c$. (a) Weak relationship between ${\Omega }_b$ and ${\Omega }_a$ for $\Delta t=90$ and 120. (b) Strong relationship between ${\alpha }_b$ and ${\alpha }_a$ for $\Delta t=90$ and 120, with both linear regressions passing the ANOVA (analysis of variance) F test at the $p<0.001$ confidence level. (c) Strong relationship between $P_b\left(\Delta t\right)$ and $P_a\left(\Delta t\right)$ for $\Delta t=90$ and 120 min at the $p<0.001$ confidence level.

Figure 12

(Color online) The increasing relation between the size of the main shock $V\left(T_c\right)$ and the size of the second largest aftershock (or preshock) $V_2\left(\Delta t\right)$ within $\Delta t$ minutes of $T_c$ demonstrate that the volatility of the largest aftershock (or preshock) increases with main shock volatility. As with the Bath law for earthquakes, we observe a proportional relation $V_{2,a}\left(\Delta t\right)\equiv C_{B}V\left(T_c\right)$, which corresponds to a Bath parameter $B=-{\text{log}}_{10}{C}_B$. For the average market response $N_{b,a}\left(\Delta t\right)$ we calculate $C_B$ for (a) the dynamics before, $C_B=0.81$, with correlation coefficient $r=0.70$ and ${\chi }^2=212$, and for (c) the dynamics after, $C_B=0.9$, with $r=0.87$ and ${\chi }^2=109$. For the Bath law corresponding to individual stocks we find that a linear function best fits the relation between $V\left(T_c\right)$ and the average value ${\overline{V}}_2\left(\Delta t\right)$ calculated for equal-sized bins as indicated by circles with one standard deviation error bars. We calculate that the regression slope for the Bath law (b) before is $m=0.65\pm 0.02$ and (d) after is $m=0.40\pm 0.01$.

Figure 13

(Color online) Relations between individual stock trading activity and dynamic response parameters (a)–(d) after $T_c$ and (e)–(h) before $T_c$, averaged over all the days with a market shock. We measure the trading activity $⟨\omega ⟩$ for each stock, defined as the average number of transactions per minute over the 2 yr period of 2001–2002. We find that stocks with large trading activity react both more strongly [larger $\alpha$ and larger $P\left(\Delta t\right)$] and quickly (larger $\Omega$) to market shocks. However, (d) shows that there is little relation between $⟨\omega ⟩$ and the average size of the largest aftershock $⟨v_2⟩$.