Description of multiple testing problems with many random processes

ou_testing.R

# discretised ornstein-uhlenbeck
ou <- function(steps, theta = 1.0, mu = 1.2, sigma = 0.3, dt = 0.01) {
  x <- rep(NA, steps+1)
  x[1] = 0 
  for(i in 2:(steps+1)) { 
    x[i] = x[i-1] + theta * (mu - x[i-1]) * dt + sigma * sqrt(dt) * rnorm(1);
  }
  return(x)
}
# ~~2 years of simulated data if you account for weekends and holidays
snp = replicate(ou(520, 0.3), n = 500)
# compute correlations
corrs = cor(snp, method = 'spearman')
# drop diagonal elements since those are all 1
corrs = corrs[c(lower.tri(corrs), upper.tri(corrs))]
# show 10 lowest 
print(sort(corrs)[1:10])
# show 10 highest
print(-sort(-corrs)[1:10])

So if you carry out this sort of multiple testing just on 2 years of S&P, you will get very high positive and negative correlations, driven by nothing but randomness (if you accept that individual stocks can be modeled by a OU-like stochastic process).