METHODOLOGY

The fraction of bankroll that maximises expected log-growth on a binary bet at decimal odds b with win probability p. Full Kelly extracts the most growth but with brutal variance; most pros run ¼ or ½ Kelly to survive model error. Negative f★ ⇒ the edge is on the other side and you should not take the position.

For a continuous-return asset (perp), the analogue is f★ = μ / σ² (Merton continuous-time). hypo.markets shows both the empirical argmax of g(f) and the parametric μ/σ² as side-by-side markers on the perp Kelly card.

The Brier score is the mean squared error of probabilistic forecasts: BS = mean((fᵢ − oᵢ)²). Lower is better; zero is perfect; the climatology baseline is ō·(1 − ō).

Murphy's decomposition splits BRIER into three meaningful parts: REL (reliability — how miscalibrated the model is; lower is better), RES (resolution — how decisively the model separates outcomes; higher is better), and UNC (uncertainty — the irreducible base-rate term).

The reliability diagram bins forecasts and plots the observed frequency against the stated probability. On-diagonal points mean perfectly calibrated. The sharpness histogram below the diagonal shows how often each forecast level is used.

ROC plots true positive rate against false positive rate as the decision threshold sweeps from 1 to 0. AUC is the area underneath: 0.5 = coin-flip, 1.0 = perfect ranker. It measures ranking power — independent of calibration. A model with AUC = 0.9 but bad calibration can be rescued by Platt scaling; a model with AUC = 0.55 cannot.

Conjugate prior for a Bernoulli likelihood. We model your stated probability as the mean of a Beta posterior with concentration κ = q(1−q)/σ² − 1 driven by your stated 1σ uncertainty.

The 95% credible interval is the highest-density region containing 95% of the posterior mass — computed via cumulative trapezoidal integration over a 1000-point grid. If the market price falls outside this band, the disagreement is statistically meaningful.

Shannon's binary entropy peaks at 1 bit when p = 0.5 (maximal doubt) and falls to 0 at the boundaries (certainty).

KL divergence D_KL(q ‖ p) = q·ln(q/p) + (1−q)·ln((1−q)/(1−p)) measures the information your belief q adds beyond the market price p— the theoretical upper bound on exploitable edge. Zero KL ⇒ you know nothing the crowd doesn't. The branch decomposition splits KL into the YES contribution and the NO contribution.

Volatility-clustering model. The conditional variance at time t is a weighted sum of yesterday's squared shock and yesterday's variance. Persistence α + β near 1 ⇒ shocks decay slowly; calm and turbulent regimes cluster. Position sizing must track the conditional σ, not the unconditional average.

The Augmented Dickey-Fuller (ADF) test has H₀: the series has a unit root (non-stationary, behaves like a random walk). Reject at the 5% level if the t-statistic is below ≈ −2.86.

The Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test has the opposite null: H₀: the series is level-stationary. Reject at 5% if the statistic exceeds 0.463.

The cleanest verdict comes from running both: rejecting one and failing to reject the other gives an unambiguous answer. Rejecting both ⇒ the series sits in a grey zone.

Under a random walk, VR(q) ≈ 1 for all q. VR > 1 means positive serial correlation (trending — long horizons are riskier than the IID assumption suggests); VR < 1 means mean reversion (long horizons are tamer). We use Lo-MacKinlay's asymptotic z-statistic for the significance test.

Rescaled-range analysis estimates long-memory: H > 0.5 ⇒ persistent / trending, H < 0.5 ⇒ anti-persistent / mean-reverting, H ≈ 0.5 ⇒ memoryless random walk. Estimated via OLS on the log-log plot of R/S vs window size n.

Two non-stationary series can still have a stationary linear combination — they are cointegrated. Step 1: OLS regression y = α + β·x + ε. Step 2: ADF on the residuals. Reject H₀ (no cointegration) at 5% if the residual ADF statistic is below ≈ −3.34.

Useful for pairs-trading: when the cointegration relationship spreads, you can short one leg and buy the other expecting reversion.

VaR_α is the α-quantile loss: on the worst (1−α)% of bars you lose at least VaR_α. CVaR_α (also called Expected Shortfall) is the average loss conditional on being in that tail. CVaR is always ≥ VaR and is the number that actually ruins accounts. Under normality, ES₉₅ / VaR₉₅≈ 1.25; ratios > 1.5 signal fat tails.

For binary positions we sample N parallel careers of K bets each, drawing the true probability from N(q, σ²) each bet, then resolving against the market price at deployed-Kelly leverage. For continuous returns (perps) we bootstrap with replacement from the observed return distribution — this preserves the fat tails the parametric Gaussian misses.

Outputs include the percentile fan (5/25/50/75/95), VaR₉₅, CVaR₉₅, maximum drawdown of the median path, terminal-wealth distribution, and the ruin rate (fraction of paths that ever crossed the 50% bankroll floor).