Linear response theory in stock markets

Linear response theory relates the response of a system to a weak external force with its dynamics in equilibrium, subjected to fluctuations. Here, this framework is applied to financial markets; in particular we study the dynamics of a set of stocks from the NASDAQ during the last 20 years. Because unambiguous identification of external forces is not possible, critical events are identified in the series of stock prices as sudden changes, and the stock dynamics following an event is taken as the response to the external force. Linear response theory is applied with the log-return as the conjugate variable of the force, providing predictions for the average response of the price and return, which agree with observations, but fails to describe the volatility because this is expected to be beyond linear response. The identification of the conjugate variable allows us to define the perturbation energy for a system of stocks, and observe its relaxation after an event.


INTRODUCTION
The analysis presented in the manuscript has been performed on NASDAQ stocks. In order to provide further support to the conclusions, we extended the analysis to other stock markets. In all cases, we take the log-return as the variable conjugate to the external force, as concluded in the paper, and test the predictions for the evolution of the log-price and log-return after an event, and for the linear relationship between the log-return and log-price.
Two sets of stocks are considered. In the first case, the New York Stock Exchange (NYSE), the biggest market in terms both of capitalization and trade volume, is considered.
In the second one, a set of European stocks from different national floors is conformed and studied. All data for both sets have been taken from Yahoo! Finance, with a time resolution of 1 day.

NEW YORK STOCK EXCHANGE
This set comprises 1084 stocks, from 2001 to 2020, and is comparable to the set of stocks from the NASDAQ.
In Fig. S1 the average response of the log-price and log-return after an event is presented, together with the calculations from the correlation functions, as studied previously for the set of stocks from the NASDAQ. Around 11000 events can be identified, ca. 5500 positive events and 5700 negative events. Following Figs. 1 and 2 in the paper, positive and negative events are averaged, and the absolute log-price and log-return variations are presented. However, different from the case of NASDAQ, the overshoot after the event in the log-price, and the subsequent relaxation is much less important (below 10%).
For the analysis of the log-price evolution, the integral of the log-return autocorrelation function (ACF) has been used. Again, the predicted response shows an almost unnoticiable relaxation to equilibrium, in agreement with the observed response. For the log-return, on the other hand, its normalized ACF C(v 2 , v 2 ) gives the response. The agreement of both quantities is comparable to the case of NASDAQ stocks, although less impressive due to the absence of memory. to the external force. The dashed line in the figure gives the prediction from the theory, ∆v ∞ = k v /k x ∆x ∞ , which describes approximately the experimental data close to the origin.
The transport coefficients, as calculated within linear response are: In Fig. S3 the response of the log-price and log-return after an event are presented, together with the calculations from the correlation functions, as studied previously. To improve the statistics, events are defined with a different threshold: an event occurs for a given stock when the one day log-return is more than 2.5 times the root mean square deviation of log-returns of this stock. Even so, less than 600 events are identified, or less than three events per stock in ten years. Again, positive and negative events are averaged, and the absolute log-price and log-return variation are presented. In agreement with the theory expectations, the correlation function C(v, v) gives the response of the log-return, and its integral corresponds to the response of the log-price. The agreement of both quantities is similar to the previous cases, although the noise of the data here is increased.
The relationship between the log-return change and log-price variation provoked by an event is studied in Fig. S4. The data, close to the origin, can be described by the straight line ∆v ∞ = k v /k x ∆x ∞ , with k v and k x calculated from LRT: Note that, due to poor statistics of this system, these data are strongly affected by numerical errors, despite the agreement shown in Fig. S4.