CQF cheat sheet

Black-Scholes call price: C = S·N(d1) − K·e^(−rT)·N(d2), with d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T) and d2 = d1 − σ√T; put via put-call parity P = C − S + K·e^(−rT).

Risk-neutral pricing: derivative value = e^(−rT)·E^Q[payoff]; under Q the discounted asset price is a martingale, so dS/S = r·dt + σ·dW^Q for a non-dividend stock.

Itô's lemma: for f(S,t), df = (∂f/∂t + μS·∂f/∂S + ½σ²S²·∂²f/∂S²)dt + σS·(∂f/∂S)dW; the ½σ²S²·∂²f/∂S² term is the convexity correction absent from ordinary calculus.

Greeks: delta = ∂V/∂S = N(d1) for a call; gamma = φ(d1)/(Sσ√T); vega = S·φ(d1)·√T; theta for a call = −S·φ(d1)·σ/(2√T) − rK·e^(−rT)·N(d2); rho = K·T·e^(−rT)·N(d2).

Put-call parity: C − P = S − K·e^(−rT) (no dividends); with continuous dividend yield q, C − P = S·e^(−qT) − K·e^(−rT). Violations imply a static arbitrage: buy the cheap side, sell the rich side, and lend/borrow at r.

Value at Risk: for normal returns VaR_α = −(μ + z_α·σ) over the horizon (z_0.95 ≈ 1.645, z_0.99 ≈ 2.326); parametric VaR understates tail risk when returns are fat-tailed, so use historical simulation or EVT for the tail.

Expected shortfall (CVaR) = E[L | L > VaR_α] ≥ VaR_α; it is subadditive and coherent, whereas VaR can violate subadditivity, so ES is preferred under Basel FRTB.

GARCH(1,1): σ²_t = ω + α·ε²_(t−1) + β·σ²_(t−1); persistence = α + β, long-run variance = ω/(1 − α − β), and α + β < 1 is required for stationarity.

Binomial CRR tree: u = e^(σ√Δt), d = 1/u, p = (e^(rΔt) − d)/(u − d); price by backward induction, and the tree converges to Black-Scholes as Δt → 0.

Bias-variance decomposition: E[(y − f̂)²] = bias² + variance + irreducible noise; regularisation (L1/L2) trades a small bias increase for a larger variance reduction, lowering test error.

Cross-validation: k-fold CV estimates generalisation error with lower variance than a single hold-out; use time-series splits (expanding or rolling windows) for financial data to avoid look-ahead leakage.

L1 (lasso) adds λ·Σ|w_j| and drives coefficients exactly to zero (sparse selection); L2 (ridge) adds λ·Σw_j² and shrinks smoothly; elastic net combines both with mixing parameter α.

Logistic regression models p = 1/(1 + e^(−wᵀx)); the log-odds are linear in x, and coefficients are estimated by maximising the log-likelihood (equivalently minimising cross-entropy).

Random forest: bagging of decorrelated decision trees using bootstrap samples and random feature subsets at each split; it reduces variance but not bias, unlike boosting, which reduces both.

Gradient boosting fits each new tree to the negative gradient (pseudo-residuals) of the loss; shrinkage (learning rate) and tree depth control overfitting, and early stopping uses a validation set.

Classification metrics: precision = TP/(TP+FP), recall = TP/(TP+FN), F1 = 2·(precision·recall)/(precision + recall); ROC-AUC is threshold-independent but misleading under severe class imbalance, where PR-AUC is preferred.

Feature leakage occurs when information unavailable at prediction time (future returns, revised data, target-derived features) enters training; it inflates backtest performance and is the most common cause of live-vs-backtest decay.