CQF: Quantitative Finance: Risk-Neutral Pricing and Ito Calculus

The Quantitative Finance domain tests whether you can move fluently between three equivalent descriptions of a derivative's value: a risk-neutral expectation, a partial differential equation, and a numerical scheme. CQF questions typically probe the links between these views rather than routine formula substitution.

Core rules and identities

Under the risk-neutral measure Q, the discounted asset price is a martingale: e^{-rt}S_t = E^Q[e^{-rT}S_T | F_t]. Pricing is an expectation under Q, not a forecast under the real-world measure P.
For geometric Brownian motion, dS = μS dt + σS dW under P. Under Q the drift becomes r - q, where q is the continuous dividend yield; volatility is unchanged by the measure change.
Ito's lemma for f(S,t): df = (f_t + μS f_S + ½σ²S² f_SS) dt + σS f_S dW. The second-order term survives because (dW)² = dt.
The Black-Scholes PDE is f_t + ½σ²S² f_SS + (r-q)S f_S - r f = 0, with terminal condition f(S,T) = payoff. Feynman-Kac identifies its solution with the discounted Q-expectation.
Delta is f_S and Gamma is f_SS. A continuously rebalanced delta hedge removes the dW term, so the hedged portfolio earns the riskless rate.

Worked example

Take S=100, K=100, r=5%, q=0, σ=20%, T=1. Then d1 = [ln(100/100) + (0.05 + 0.5×0.04)×1]/(0.2×1) = 0.07/0.2 = 0.35, and d2 = 0.35 - 0.20 = 0.15. Using N(0.35)=0.6368 and N(0.15)=0.5596, the call is C = 100×0.6368 - 100×e^{-0.05}×0.5596 = 63.68 - 53.23 = 10.45. The delta is 0.6368. Reasoning: discount the strike at the continuous rate, and note that N(d2)=0.5596 is the risk-neutral exercise probability, not the delta. Put-call parity gives P = C - S + Ke^{-rT} = 10.45 - 100 + 95.12 = 5.57.

Traps that cost marks

Inserting the real-world drift μ into the pricing formula. μ affects probabilities under P, not the no-arbitrage price.
Omitting q: replace r by r-q in d1 and discount the strike by e^{-qT} in the general formula.
Dropping the ½σ²S²f_SS term from Ito's lemma; without it the PDE is wrong.
Mixing up N(d1) and N(d2): delta uses N(d1), exercise probability uses N(d2).
Simulating GBM with a naive Euler step that can produce negative prices; use the exact lognormal update S_{t+Δt}=S_t exp[(r-q-½σ²)Δt + σ√Δt Z].

Machine Learning: Generalization, Validation, and Evaluation

The Machine Learning domain tests whether you can build models that generalize out of sample and detect when a backtest is misleading. CQF focuses on validation design, regularization, and evaluation under class imbalance or dependent financial observations.

Core rules

Bias-variance: expected squared error = bias² + variance + irreducible noise. Increasing model capacity usually lowers bias and raises variance; the best model minimizes their sum, not training error.
Ridge regression minimizes RSS + λ||β||² and shrinks coefficients toward zero; lasso minimizes RSS + λ||β||₁ and can set coefficients exactly to zero. Standardize features before penalizing because the penalty is scale-dependent.
Cross-validation: k-fold estimates generalization error. For time series, use walk-forward validation or purged K-fold with an embargo, because random folds leak future information into the past.
Data leakage occurs when information unavailable at prediction time enters training. Fit scalers, feature selectors, and target encoders inside each training fold only.
Evaluation: accuracy is misleading under class imbalance. Use precision = TP/(TP+FP), recall = TP/(TP+FN), F1, and PR-AUC or ROC-AUC as appropriate.

Worked example

A classifier produces TP=40, FP=10, FN=20, TN=130. Precision = 40/50 = 0.80; recall = 40/60 = 0.667; F1 = 2×0.80×0.667/(0.80+0.667) = 0.727; accuracy = (40+130)/200 = 0.85. Reasoning: 85% accuracy sounds strong, but the model misses one third of positives. If false negatives are expensive, lower the decision threshold to raise recall; if false positives are expensive, raise it. With rare positives, report PR-AUC rather than ROC-AUC.

Traps that cost marks

Tuning hyperparameters on the test set; use nested cross-validation or a held-out test set touched once.
Random K-fold on overlapping labels in finance; purge overlapping samples and embargo a buffer.
Selecting features or scaling before the CV split, which inflates scores.
Comparing models by accuracy when positives are 5% of the data.
Backtest overfitting from testing many strategies; correct for multiple testing with the deflated Sharpe ratio or a holdout.