Overlap Bias

overlap-bias.jl

using Random, Distributions, Statistics, DataFrames, Plots

# parameters
Random.seed!(1)
mu = 0.005             # drift per 30d step
sigma = 0.08           # volatility per step
steps = 12 * 45 * 250  # number of 30d steps
S0 = 1.0               # initial price
ν = 3                  # degrees of freedom for Student-t (ν > 2)

lps = cumsum(rand(TDist(ν), steps) .* sigma .+ mu)
lrs30   = [lps[i] - lps[i-1] for i in 2:length(lps)]
lrs360o = [lps[i] - lps[i-12] for i in 13:length(lps)]
lrs360  = [lps[i] - lps[i-12] for i in 13:12:length(lps)]

p_tail(x) = begin
  tq, n = 0.01, length(x) # tail quantile
  nq = floor(Int, n * tq)

  x = sort(x)
  rtail = x[end-nq+1:end]

  rp = reverse((1:length(rtail)) ./ n)

  rtail, rp
end

rtailo, rpo = p_tail(lrs360o)
rtail, rp = p_tail(lrs360)

plot(
  rtail, rp;
  # rtailc, rpc;
  xscale = :log10,
  yscale = :log10,
  label = "Overlapping",
  xlabel = "Return (log scale)",
  ylabel = "P(X ≥ x) (log scale)",
  legend = :topright,
  seriestype = :scatter,
  markersize = 4,
  markerstrokewidth = 0 # ← smaller points
)

plot!(
  rtailo, rpo;
  label = "Non-overlapping",
  seriestype = :scatter,
  markersize = 2,
  markerstrokewidth = 0    # ← same here
)

overlap-bias.py

import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import t

# Parameters
np.random.seed(1)
mu = 0.005
sigma = 0.08
steps = 10 * 12 * 45 * 250  # number of 30d steps
S0 = 1.0
nu = 3  # degrees of freedom for Student-t

# Simulate log-price path from Student-t
lps = np.cumsum(t.rvs(df=nu, size=steps) * sigma + mu)

# Log returns
lrs30 = lps[1:] - lps[:-1]                        # 30d returns (no overlap)
lrs360o = lps[12:] - lps[:-12]                    # 360d overlapping
lrs360  = lps[12::12] - lps[:-12:12]              # 360d non-overlapping

# Tail extraction function
def p_tail(x, tq=0.01):
    x = np.sort(x)
    n = len(x)
    nq = int(np.floor(n * tq))
    rtail = x[-nq:]
    rp = np.flip(np.arange(1, nq + 1) / n)
    return rtail, rp

# Compute right tails
rtailo, rpo = p_tail(lrs360o)  # overlapping
rtail, rp   = p_tail(lrs360)   # non-overlapping

# Log-log scatter plot
plt.figure(figsize=(7, 5))
plt.scatter(rtail, rp, s=30, color="red", label='Non-overlapping', edgecolors='none')  # larger markers
plt.scatter(rtailo, rpo, s=4, label='Overlapping', color="black", edgecolors='none')
plt.xscale('log')
plt.yscale('log')
plt.xlabel("log r (log scale)")
plt.ylabel("P(X ≥ x) (log scale)")
plt.legend(loc='upper right')
plt.grid(True, which='both', linestyle='--', linewidth=0.5)
plt.tight_layout()
plt.show()