CQF flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Black-Scholes formula

Call price C = S·N(d1) − K·e^(−rT)·N(d2), where d1 = [ln(S/K) + (r + σ²/2)T]/(σ√T) and d2 = d1 − σ√T; assumes lognormal prices, constant volatility, no arbitrage and continuous trading.

Risk-neutral measure

A probability measure Q under which every discounted asset price is a martingale; derivative prices equal e^(−rT) times the Q-expectation of the payoff, so no real-world drift is needed.

Put-call parity

For European options on a non-dividend stock, C − P = S − K·e^(−rT). Any deviation creates a static arbitrage: buy the underpriced portfolio and sell the overpriced one, locking in a riskless profit.

Delta of a call

The partial derivative ∂C/∂S equals N(d1) for a Black-Scholes call, ranging from 0 to 1. It measures the change in option value per unit change in the underlying and is the hedge ratio.

Gamma

The second derivative ∂²V/∂S², equal to φ(d1)/(S·σ·√T) for a Black-Scholes option. It measures how delta changes as the underlying moves and is largest for at-the-money options near expiry.

Vega

The sensitivity ∂V/∂σ of option value to volatility, equal to S·φ(d1)·√T for a call. Vega is positive for long options and peaks for at-the-money options with long time to expiry.

Theta

The sensitivity ∂V/∂t of option value to the passage of time. For a long call, theta is typically negative: value erodes as expiry approaches, all else equal, and the erosion accelerates near expiry.

Itô's lemma

The stochastic chain rule: for f(S,t) with dS = μS·dt + σS·dW, df = (f_t + μS·f_S + ½σ²S²·f_SS)dt + σS·f_S·dW. The f_SS term is the convexity correction.

Geometric Brownian motion

The process dS = μS·dt + σS·dW, whose solution is S_t = S_0·exp[(μ − σ²/2)t + σW_t]. It keeps prices positive and underlies the Black-Scholes model.

Value at Risk

The loss threshold not exceeded with probability α over a horizon. For normal returns VaR_α = −(μ + z_α·σ); it is not subadditive, so portfolio VaR can exceed the sum of component VaRs.

Expected shortfall

The conditional mean loss given that the loss exceeds VaR_α, i.e. E[L | L > VaR_α]. It is a coherent risk measure (subadditive) and captures tail severity better than VaR.

GARCH(1,1)

A volatility model σ²_t = ω + α·ε²_(t−1) + β·σ²_(t−1). Persistence α + β close to 1 implies slow volatility mean reversion; the unconditional variance is ω/(1 − α − β).

Implied volatility

The volatility input that makes the Black-Scholes price equal the observed market price. It summarises option prices across strikes and maturities and exhibits the volatility smile or skew.

Binomial option pricing

A discrete-time tree with up factor u = e^(σ√Δt), down factor d = 1/u, and risk-neutral probability p = (e^(rΔt) − d)/(u − d); price by backward induction, and the tree converges to Black-Scholes.

Monte Carlo pricing

Simulate risk-neutral paths of the underlying, average the discounted payoffs, and estimate the standard error as sample standard deviation divided by √n; variance reduction uses antithetic or control variates.

Duration and convexity

For a bond, modified duration approximates the percentage price change per unit yield change, and convexity captures curvature: ΔP/P ≈ −D·Δy + ½·C·(Δy)². Convexity is positive for plain bonds.

Bias-variance tradeoff

Expected squared prediction error decomposes into bias², variance and irreducible noise. More complex models reduce bias but raise variance; regularisation and ensembling lower variance at a small bias cost.

Cross-validation

A resampling method that splits data into k folds, trains on k−1 and validates on the held-out fold, averaging the scores. For time series, use expanding or rolling windows to prevent look-ahead bias.

Lasso regression

Linear regression with an L1 penalty λ·Σ|w_j| added to the loss. It shrinks coefficients and sets some exactly to zero, producing sparse models useful for feature selection in high-dimensional financial data.

Ridge regression

Linear regression with an L2 penalty λ·Σw_j². It shrinks coefficients smoothly toward zero, stabilising estimates under multicollinearity, but does not perform variable selection because coefficients rarely reach exactly zero.

Logistic regression

A classifier modelling p = 1/(1 + e^(−wᵀx)), so log-odds are linear in the features. Coefficients are fitted by maximising the log-likelihood, equivalent to minimising binary cross-entropy loss.

Random forest

An ensemble of decision trees trained on bootstrap samples with random feature subsets at each split. Averaging decorrelated trees reduces variance without materially increasing bias, improving generalisation over a single tree.

Gradient boosting

A sequential ensemble where each new tree fits the negative gradient of the loss (pseudo-residuals). A small learning rate, limited depth and early stopping on validation data control overfitting.

Precision and recall

Precision = TP/(TP+FP) is the fraction of predicted positives that are correct; recall = TP/(TP+FN) is the fraction of actual positives found. F1 is their harmonic mean.

ROC-AUC

The area under the receiver-operating-characteristic curve, equal to the probability that a random positive is scored above a random negative. It is threshold-independent but can look optimistic under class imbalance.

Look-ahead bias

Using information in a backtest that was not available at the decision time, such as future prices or restated fundamentals. It inflates measured performance and typically causes live results to fall short of backtests.

Survivorship bias

Constructing a dataset only from entities that survived to the present, excluding delisted or failed firms. It overstates historical returns and understates risk, especially in equity and credit backtests.

Feature leakage

Including predictors that contain information about the target unavailable at prediction time, such as target-derived ratios or future data. It produces unrealistically strong validation scores that do not persist out of sample.

Regularisation

Adding a penalty on model complexity, such as L1 or L2 norms, to the training objective. It constrains coefficients, reduces variance and overfitting, and improves out-of-sample performance on noisy financial data.

Confusion matrix

A table of true positives, false positives, true negatives and false negatives for a classifier at a chosen threshold. Accuracy, precision, recall and F1 are all derived from its four entries.

Walk-forward validation

A time-series validation scheme that repeatedly trains on all data up to a point and tests on the next period, rolling forward. It mimics real deployment and avoids using future information.

Principal component analysis

A linear dimensionality-reduction method projecting data onto orthogonal components ordered by explained variance. In finance, the first components of a yield-curve panel typically represent level, slope and curvature.