Normal / Gaussian

The original statcurves widget — generalized to a p-exponent bell curve — with a moments/MGF/quantile panel added underneath.

Poisson

Counts of independent events in a fixed interval, rate λ. Discrete PMF and step CDF.

Uniform (continuous)

Flat density on [a, b]. (Discrete uniform on {1,…,n} has mean (n+1)/2, variance (n²−1)/12 — same idea, without a slider here.)

Exponential

Memoryless waiting-time distribution, rate λ. Special case of Gamma(α=1, β=λ).

Gamma

Shape α, rate β. α = (E[X])² / Var(X) is the method-of-moments estimator used elsewhere in the roadmap for Heston-variance calibration.

Beta

Density on (0,1), shape parameters α and β — the natural prior for a probability or a proportion.

Lognormal

If ln X ~ N(μ, σ²), X is lognormal — the return distribution GBM produces. This page carries the explicit VaR = −Q(α) framing.

Joint distribution — bivariate normal

Heatmap of the joint density f(x,y) with correlation ρ, plus the two marginal densities. This is also the mechanism behind correlated Brownian motions (dW₁dW₂ = ρ dt) used later in the stochastic-vol tier.

LLN, CLT, σ-algebras & conditional expectation

Placed after the processes-adjacent intuition on purpose: filtrations mean more once you've seen a path you're conditioning on.

Histogram of the standardized sample mean Z = (X̄ₙ − μ)/(σ/√n) over 2000 repeated experiments, against the N(0,1) density it converges to.

A σ-algebra F on Ω is a collection of subsets closed under complement and countable union, containing Ω — it's "the set of questions you're allowed to ask." The Borel sets are the smallest σ-algebra containing all open intervals of ℝ, and an indicator function 1_A(ω) is measurable with respect to F exactly when A ∈ F. A filtration (F_t)ـ{t≥0} is an increasing family of σ-algebras — the information available by time t — and E[X|F_t] is "the best F_t-measurable prediction of X." Below, X(ω) = sin(2πω) + ω on Ω = [0,1]; the coarse partition is F_coarse, the fine partition is F_fine ⊃ F_coarse, and E[X|F] is computed as the cell-average of X — a real, checkable number, not just a diagram.

Wiener process & geometric Brownian motion

Stationary, independent Gaussian increments; quadratic variation [W]_t = t (not 0 — this is the fact Itô's lemma is built on).

Ornstein–Uhlenbeck & optional stopping

Mean-reverting SDE, then a Monte-Carlo check of the optional stopping theorem on a symmetric random walk.

A symmetric ±1 random walk is a martingale. Stop at τ = first time |X_t| hits the barrier. Optional stopping (for this bounded-increment, a.s.-finite τ) says E[X_τ] = X_0 = 0.

Discretization schemes & variance reduction

Strong convergence of Euler–Maruyama vs Milstein on GBM, then two variance-reduction techniques.

Estimating E[e^W] for W ~ N(0,1), true value e^0.5 ≈ 1.6487, using paired (Z, −Z) draws instead of independent draws.

Estimating π via the quarter-circle method: plain pseudo-random points vs a low-discrepancy Halton sequence (a simple stand-in for Sobol here).

Black–Scholes, Greeks & put–call parity

Closed-form price under the risk-neutral measure Q, live Greeks, and an implied-vol solver.

Illustrative synthetic curve only (a parabola-plus-skew around the ATM vol slider above) — not fetched market data. Real smiles need an actual options chain.

Metropolis–Hastings volatility estimation

Random-walk MH recovering a volatility parameter from returns — trace plot, posterior histogram, acceptance rate, and an autocorrelation diagnostic.

The "data" below is simulated in the browser from a chosen true σ — explicitly, not fetched market data — then MCMC tries to recover σ from the simulated returns alone.

Heston model & the 3D vol surface

Deliberately not rebuilt here.

This tier — CIR variance process, the Feller condition, correlated Brownians (dW₁dW₂ = ρ dt) via Cholesky, the Heston two-SDE system, its characteristic function, Gil-Pelaez Fourier inversion, least-squares calibration to implied vols, and the SVI/Dupire vol-surface work — is what the separate Heston regime dashboard already covers. Duplicating it inside this repo would fragment one working project into two half-finished ones.

What would land here if merged in: a κ/θ/ξ/ρ/v₀ slider panel driving both a live characteristic-function call price and a rotating 3D implied-vol surface, with a toggle comparing local vol (Dupire) against stochastic vol (Heston) on the same synthetic surface. Worth doing as a deliberate follow-up, not folded silently into this drop.

Roles, brainteasers & reading list

The part a pure math/pricing curriculum skips: what the roles actually are, and the kind of question a first-round interview actually asks.

The heaviest stochastic-calculus material in this site (Heston, jump diffusion) mostly maps to quant researcher/desk quant interviews at the later rounds — first rounds across all four roles lean on the brainteasers below far more than on any SDE.

102 questions across 8 categories — filter by category, or leave on "All" to browse everything grouped by topic.

—

• Pricing theory: Hull, Options, Futures, and Other Derivatives (the standard mechanics reference); Shreve, Stochastic Calculus for Finance I & II (the measure-theoretic rigor behind this site's Tier 1–2); Gatheral, The Volatility Surface (SVI, Heston calibration in practice)
• Time series & econometrics: Tsay, Analysis of Financial Time Series
• Market microstructure & pairs trading: Chan, Algorithmic Trading and Quantitative Trading; Harris, Trading and Exchanges (microstructure fundamentals)
• Risk & backtesting pitfalls: López de Prado, Advances in Financial Machine Learning (the standard reference on backtest overfitting specifically)
• Portfolio theory: Bodie, Kane & Marcus, Investments (CAPM, APT, and the classical portfolio-theory context)

This list points outward on purpose — each pillar page here is meant to give you the working intuition fast, then hand you off to the real textbook for depth, not replace it.