Search N trading rules on one price history and record the best in-sample Sharpe ratio. Even when every rule is worthless, that maximum grows with N. The quantity of interest is therefore not the winner's Sharpe but how far it sits above what pure search would produce anyway.

E[max Sharpe | no skill] ≈ σ_SR · [ (1 − γ)·Z⁻¹(1 − 1/N) + γ·Z⁻¹(1 − 1/(N·e)) ]

σ_SR is the cross-sectional standard deviation of trial Sharpe ratios, γ ≈ 0.5772 is the Euler–Mascheroni constant, Z⁻¹ is the inverse standard normal CDF, and N is the number of trials. This is the Gumbel approximation of the maximum of N independent normal draws.

PSR(SR*) = Φ( ( (SR − SR*) · √(n − 1) ) / √( 1 − skew·SR + ((kurt − 1)/4)·SR² ) )

• SR and SR* are expressed per observation (annualized Sharpe ÷ √252) so the n-scaling is correct.
• n is the number of daily observations in the window being judged.
• skew is the sample skewness, kurt the raw kurtosis (normal = 3) of the strategy's daily returns.
• Negative skew and fat tails reduce PSR at the same Sharpe ratio — the estimate is less trustworthy.
• The denominator is checked at run time; if it is non-positive the metric is reported as unavailable rather than guessed.

DSR = PSR( SR* = E[max Sharpe | no skill] )

The winner must beat the level that search alone would deliver. A DSR of 0.95 or more is treated here as passing. Assumptions made visible: trial Sharpe ratios are treated as independent and normally distributed around zero, and σ_SR is estimated from the realized dispersion of this run's trials. Overlapping parameter grids make trials correlated, so the effective number of independent trials is lower than N and this correction is conservative in that direction.

1. Split the post-warm-up history into S equal chronological folds (S is configurable).
2. Take every combination of S/2 folds as a training set, with the remaining folds as the test set (capped at 140 combinations for responsiveness).
3. Pool the daily returns of the chosen folds and pick the strategy with the highest pooled Sharpe ratio in the training set.
4. Rank that strategy's Sharpe ratio among all strategies on the complementary test folds.
5. PBO is the share of combinations where the training winner landed at or below the test-set median.

PBO = (1/C) · Σ_c  1{ rank_c ≤ 0.5 }      rank_c = #{ SR_test ≤ SR_test(winner_c) } / (M + 1)

A PBO near 0.5 means in-sample ranking is no better than a coin flip out of sample. Folds are chronological and not purged or embargoed, so serial dependence at fold boundaries is not removed — a documented simplification relative to full combinatorially purged cross-validation.

• Returns are simple daily close-to-close returns: r_t = P_t / P_(t−1) − 1.
• The position decided from information up to close t−1 earns r_t, so there is no lookahead.
• Costs: each unit of absolute position change is charged the configured bps, which also penalizes high turnover.
• Annualization uses 252 trading days; volatility uses the sample standard deviation.
• An indicator warm-up window (up to 260 days) is excluded from both train and test so long-window rules are not handicapped.
• Edge cases: zero-volatility tracks report Sharpe 0, degenerate variants are discarded and counted, and runs are rejected when the test window would be under 120 observations or the history under 400 closes.