quant-math

name

quant-math

description

Quantitative math reference for DaxAlgo Terminal's signal/regime work — Ornstein-Uhlenbeck mean-reversion (SDE, half-life, MLE/OLS calibration), linear algebra (correlation/covariance, PCA, Cholesky, EWMA), 3D geometry for Helix cube/surface viz, microstructure stats (VPIN/toxicity, cumulative delta, imbalance, Kyle's lambda), Markov transition matrices, and volatility estimators. Load when a strategy/regime/correlation agent touches the math, so the formula and its numerically-stable form are correct before coding.

Quant math — formulas + stable implementations

Reference for the math the strategy, regime, and correlation agents need. Each block gives the formula, the numerically stable form to actually code, and where it lands in the repo. Math in Core/MarketData/Indicators + Microstructure; calibration in the engine-side strategy.

Ornstein–Uhlenbeck (mean reversion) — Strategies.OrnsteinUhlenbeck

Continuous SDE: dXₜ = θ(μ − Xₜ)dt + σ dWₜ (θ>0 reversion speed, μ long-run mean, σ vol).

• Discrete AR(1) form (sample step Δt): Xₜ = a + b·Xₜ₋₁ + ε, fit by OLS, then
  • θ = −ln(b) / Δt μ = a / (1 − b) σ² = Var(ε)·(−2 ln b) / (Δt (1 − b²))
• Half-life of a shock: t½ = ln(2) / θ — the headline tradeable number (entry only when half-life is short enough to exit within the horizon).
• Entry signal: z-score z = (Xₜ − μ) / σ_eq, σ_eq = σ / √(2θ) (stationary std). Enter when |z| exceeds a band; size ∝ −z (fade the deviation).
• Stability: reject the fit if b ≥ 1 (non-stationary — no mean reversion) or b ≤ 0.

Linear algebra — correlation, regime PCA, correlated sims

• Pearson correlation ρ = cov(x,y)/(σₓσᵧ). Compute cov with a single-pass Welford accumulator, not E[xy]−E[x]E[y] (catastrophic cancellation on large prices).
• Spearman = Pearson on ranks — prefer it for fat-tailed returns / outlier robustness.
• EWMA covariance (RiskMetrics): Σₜ = λΣₜ₋₁ + (1−λ) rₜ rₜᵀ, λ≈0.94 daily. Use for a responsive correlation matrix instead of an equal-weight window.
• PSD repair: a sample correlation matrix can be non-positive-semidefinite (missing data, shrinkage). Clip negative eigenvalues to 0 and renormalize the diagonal before using it.
• PCA for regime axes: eigendecompose the correlation matrix; PC1 loadings ≈ "market factor". Sort eigenvalues desc; explained-variance ratio = λᵢ/Σλ.
• Cholesky Σ = L Lᵀ to generate correlated normals (x = L z) for Monte-Carlo / synth feeds. Falls over if Σ isn't PSD — do the repair above first.

3D geometry — Helix cube/surface viz — Strategies.OrderFlowCube / OrderFlowSurfaceSpike / IndexKScoreSurface

• Axis normalization: each raw axis (price-Δ, delta, time, toxicity…) has wildly different units. Map to a common [0,1] (or [−1,1]) cube via robust min/max (5th–95th pctile, not true min/max — one outlier shouldn't flatten the cube). Keep the scale factors in the VM so hover tooltips can invert back to real units.
• Color = 4th dimension: map the scalar (e.g. intensity) through a perceptually-uniform ramp; clamp to the same robust percentile range.
• Surface meshing: a value grid z = f(x,y) becomes a Helix MeshGeometry3D — vertices on the (x,y) lattice, two triangles per cell, shared vertices (don't duplicate or normals break). Recompute normals on update or lighting looks flat.
• Performance: rebuild the mesh off the UI thread, assign the finished Geometry3D on the dispatcher once. Never mutate vertex collections per-tick on the UI thread.

Microstructure — Strategies.OrderFlowToxicity / CumulativeDelta / ImbalanceHeatFront

• Trade sign (no quote-at-trade): tick rule (uptick=+1, downtick=−1, carry on equal) or Lee–Ready (compare to prevailing mid; at-mid → tick rule). Document which one.
• Cumulative delta = Σ signed volume; the slope and divergence vs price is the signal, not the level (which drifts).
• Order-book imbalance (Σbid − Σask)/(Σbid + Σask) over N levels — bounded [−1,1].
• VPIN (toxicity): bucket trades into equal-volume buckets (not equal time); per bucket |Vbuy − Vsell| / V; VPIN = rolling mean over n buckets. Volume buckets are the whole point — don't use time buckets.
• Kyle's λ (price impact): regress Δprice on signed order flow; slope = λ (illiquidity).

Markov regimes — markovregime

• Transition matrix Pᵢⱼ = count(i→j)/count(i→·); rows sum to 1. Laplace-smooth (+α) so an unseen transition isn't probability 0.
• Stationary distribution π solves πP = π (left eigenvector for eigenvalue 1, normalized).
• Forward / Viterbi for HMM regime inference — work in log space (sum of log-probs) to avoid underflow over long sequences.

Volatility — Strategies.VolatilityTargeted, vol-of-vol

• Realized vol = √(Σ rᵢ² · annualization). EWMA vol σ²ₜ = λσ²ₜ₋₁ + (1−λ)r²ₜ.
• Range estimators (tighter than close-to-close): Garman–Klass uses OHLC; Parkinson uses high-low. Use when you have bars, not just closes.
• Vol targeting: position scale = target_vol / realized_vol, clamped to a max leverage.
• Vol-of-vol = stdev of a rolling vol series — the regime axis for unstable markets.

Numerical hygiene (applies everywhere)

• Single-pass Welford for mean/variance; never Σx²−(Σx)²/n.
• Guard every divisor (σ, denominators, 1−b) — return a neutral signal, never NaN, on degenerate windows.
• Work in log-price returns ln(pₜ/pₜ₋₁), not raw price differences, for anything statistical.
• Warm-up: emit no signal until the window/estimator has enough samples; say how many.