Figure 3
Gaussian Models. The lefthand side shows the \(S_1^\text {Gauss}\)-values computed via Monte Carlo simulation as functions of Gaussian correlation \(\rho\), with thresholds taken as alpha multiplied by the respective volatility (i.e. \(r_A = \alpha \sigma _A\) and \(r_B = \alpha \sigma _B\), with \(\alpha\) chosen in the range from 1 to 3). The righthand side shows a heatmap of the positive excess \(\Delta = S_1^\text {emp} - S_1^\text {Gauss}\) from daily closing price changes of S&P-500 stocks using the same time period, GICS sector classification and ordering as in Fig. 1. For each pair of stocks, \(\Delta\) was computed by setting the threshold, that separates strong from weak days, to the stock’s daily volatility, as it was observed over the entire time period. Equation (1) was used to compute \(S_1^\text {emp}\) from the time series of historic stock prices, while the \(S_1^\text {Gauss}\)-value used was based on Monte Carlo simulations (as illustrated in Fig. 3 for positive values), with \(\rho\) set to the historic correlation between the daily returns of the two stocks under consideration.
