Quantitative Finance Study Guide

The mathematical language for reasoning about uncertain outcomes.

Idealized models versus probabilities estimated from observed data.

Classical, frequentist, and Bayesian interpretations side by side.

All possible outcomes, and the subsets we actually care about.

Non-negativity, normalization, and additivity for disjoint events.

When events don't influence each other, and the long-run average outcome.

How far outcomes typically fall from the mean.

Whether two random variables move together, and by how much.

Markov, Chebyshev, and Jensen — bounding probabilities without the full distribution.

Union, intersection, and complement applied to events.

Permutations and combinations: how many ways can it happen?

Multinomials, Catalan numbers, inclusion–exclusion, and recurrences.

How likely is A, given that B already happened?

Flipping conditional probabilities — updating beliefs with evidence.

Stitching together cases that partition the sample space.

A function that maps outcomes to numbers we can do math with.

Countable outcomes versus a continuum of possible values.

Densities and cumulative probabilities, and how to move between them.

Summarizing a distribution with a few key numbers.

One-shot trials and the count of successes across many.

Rare events over time, and the wait until the first success.

Waiting for k successes, and sampling without replacement.

Equal-likelihood intervals and the bell curve at the center of statistics.

Inter-arrival times, sums of exponentials, and modeling probabilities themselves.

Pareto, Cauchy, and Student-t — when extreme events matter.

Why sample averages converge to expectations as n grows.

Why sums of independent variables look Gaussian, and when they don't.

Different senses in which random sequences settle down.

A range of plausible parameter values, with a stated coverage rate.

Testing claims about the world using sample data.

How surprising is the data under the null?

False alarms versus missed detections, and the trade-off between them.

Designing tests that can actually detect what you care about.

Approximating sampling distributions when theory is hard.

Computing expectations of functions of random variables.

A transform that uniquely identifies a distribution and makes sums easy.

MGFs for distributions where moments may not exist.

Memoryless transitions: where you go next depends only on where you are.

Long-run behavior of Markov chains and how to compute it.

Probability as degree of belief, not long-run frequency.

Combining initial belief with evidence to form an updated belief.

Pairs of distributions where the math stays in the family.

Sampling from posteriors when closed-form solutions don't exist.

The line that minimizes squared residuals.

Many predictors, one response — and the interpretations get subtle.

Residual plots, leverage, and the assumptions you should never trust blindly.

Ridge, Lasso, and Elastic Net — taming overfitting.

Regression when the response isn't normally distributed.

Three lenses on distributions over multiple variables.

The most useful multivariate distribution, end to end.

Encoding variance and pairwise relationships in one object.

What correlations can and cannot tell you.

Finding directions of maximum variance in high-dimensional data.

Discrete-time stochastic motion and its scaling limits.

Counting random arrivals over continuous time.

The continuous-time limit of a random walk and the workhorse of finance.

How to take derivatives of functions of stochastic processes.

Fair-game processes and the optional stopping theorem.

The stochastic differential equation behind Black–Scholes.

Estimating expectations by averaging random samples.

Antithetic variates, control variates, and importance sampling.

Resampling returns to estimate distributions and intervals.

Low-discrepancy sequences for faster integration in many dimensions.

When statistical properties hold across time, and when they don't.

Linear models for time-dependent data.

Modeling the autocorrelation of squared returns.

When non-stationary series move together in the long run.

Recursive estimation in dynamic linear systems with noise.

Measuring the uncertainty inside a distribution.

How different are two distributions, in nats?

Quantifying the information one variable carries about another.

Choosing the least-committal distribution consistent with constraints.

The geometry of vectors and what 'distance' means.

Multiplication, transposes, and matrix algebra essentials.

Special directions a matrix only stretches, not rotates.

The most useful matrix factorization in applied math.

Why covariance matrices have a special structure.

Conditioning, rank, and why naive code can silently lose precision.

Gradients, Jacobians, and Hessians for functions of many variables.

Constrained optimization, the elegant way.

Why convex problems are tractable and most others aren't.

First-order methods for high-dimensional optimization.

Differentiating in the presence of Brownian noise.

Learning from labels versus learning structure from data alone.

Why simple models underfit and complex models overfit.

Estimating out-of-sample performance honestly.

Decision trees, random forests, and gradient boosting.

Maximum-margin classifiers and the kernel trick.

Finding groups and compact representations without labels.

Precision, recall, ROC, and PR curves — and which to trust when.

Layers, activations, and universal approximation in practice.

How gradients flow backward through composed functions.

SGD, Adam, learning-rate decay, and warmup.

Dropout, weight decay, and early stopping.

MLPs for tabular data, RNNs and Transformers for sequences.

MSE, cross-entropy, Huber, and when to pick each.

What options pay, and the no-arbitrage relationship that ties calls and puts.

Pricing derivatives by changing measure — and why it works.

Discrete-time option pricing that converges to Black–Scholes.

The closed-form European option price and the assumptions behind it.

Delta, gamma, vega, theta, rho — sensitivities every trader watches.

Why implied volatility isn't a single number.

Asian, barrier, lookback, and other path-dependent payoffs.

The Markowitz frontier and the math behind diversification.

Decomposing returns into systematic exposures.

Risk-adjusted performance measures and how they differ.

Blending market-implied views with your own beliefs.

Allocating by risk contribution, not capital.

A quantile-based summary of portfolio downside.

The average loss in the tail beyond VaR.

Simulating a strategy on history without fooling yourself.

Inventory, adverse selection, and quoting around fair value.

Order books, slippage, and how prices actually form.

Mean-reverting baskets and the half-life of edge.