Method Tutorials

Each tutorial covers one solution method with a worked example, three practice problems (hints reveal one step at a time), and a reflection prompt.

88 available · 328 methods in the bank

No tutorials match that search. Try a different term.

Probability foundations (6)

Building blocks: events, conditional probability, Bayes, total probability, complement.

• Complement: when 'not the thing' is easier
  Easy
  When 'at least one' wrecks your enumeration, flip the question — counting the failure case is almost always faster.
  115 practice problems
  Start tutorial →
• Conditional probability: shrinking the sample space
  Conditioning on B is not multiplication — it's a brand-new sample space. Get that mental move right and 80% of brain teasers fall over.
  92 practice problems
  Start tutorial →
• Bayes' rule: inverting the conditioning
  Easy
  Why a 99%-accurate covid test can mean you have only a 1-in-6 chance of being sick. Bayes is the formula every quant interviewer reaches for first.
  65 practice problems
  Start tutorial →
• Law of total probability: average over the hidden case
  Split the world into mutually-exclusive cases, compute the conditional in each, weight by the case probability, sum.
  26 practice problems
  Start tutorial →
• Geometric probability: probability as a ratio of areas
  Probabilities become ratios of areas (or lengths or volumes). Sketch the region first; the algebra usually disappears.
  41 practice problems
  Start tutorial →
• Which distribution? — the decision tree
  The decision tree that picks one distribution from the question pattern. Then open the specific tutorial for the deep dive.
  3 practice problems
  Start tutorial →

Expectation & variance (3)

Linearity of expectation, variance of sums, indicator random variables.

• Linearity of expectation: decompose, then sum
  Easy
  $E[X+Y] = E[X] + E[Y]$ holds even when $X$ and $Y$ are dependent. This one identity collapses 70% of expectation problems.
  75 practice problems
  Start tutorial →
• Variance of sums + law of total variance
  $\mathrm{Var}(X+Y) = \mathrm{Var}(X) + \mathrm{Var}(Y)$ ONLY when $X,Y$ are uncorrelated. Otherwise you need Law of Total Variance — interviewers love testing the difference.
  7 practice problems
  Start tutorial →
• Indicator random variables: choose the indexing first
  Replace 'how many of these happen' with a sum of 0/1 indicators, then use linearity. Half the expectation tricks in finance interviews are exactly this.
  33 practice problems
  Start tutorial →

Distributions (12)

Discrete and continuous families. Open the Distribution Explorer to play with parameters live.

• Bernoulli: the atom of probability
  A single yes/no trial — the atom every other discrete distribution is built from. $E[X]=p$, $\mathrm{Var}(X)=p(1-p)$, maxed at $p=\tfrac12$.
  4 practice problems
  Start tutorial →
• Binomial: counting successes in n Bernoulli trials
  Sum of $n$ independent Bernoullis. Mean $np$, variance $np(1-p)$. The 'count successes in $n$ trials' distribution.
  11 practice problems
  Start tutorial →
• Geometric: trials until the first success
  How many trials until the first success? Memoryless — a fresh coin flip after $k$ failures has the same expected wait you started with.
  6 practice problems
  Start tutorial →
• Negative Binomial: trials until the k-th success
  Trials until the $k$-th success — Geometric extended. Used for over-dispersed counts where Poisson under-states the variance.
  3 practice problems
  Start tutorial →
• Exponential: time between Poisson events; memoryless
  Time between independent events arriving at a constant rate. The continuous memoryless distribution; gaps between Poisson arrivals.
  4 practice problems
  Start tutorial →
• Gamma: sum of k independent Exponentials
  Sum of $k$ independent Exponentials — total wait time for the $k$-th event in a Poisson process.
  3 practice problems
  Start tutorial →
• Beta: probability of a probability (conjugate to Bernoulli)
  A probability of a probability. Conjugate prior to Bernoulli — Beta$(a,b)$ updated by $s$ successes and $f$ failures becomes Beta$(a+s, b+f)$.
  3 practice problems
  Start tutorial →
• Normal: the CLT limit, the workhorse
  The CLT limit. Sums of many small independent things land here. Z-scores, confidence intervals, and Black-Scholes all live in the Normal world.
  5 practice problems
  Start tutorial →
• Lognormal: multiplicative Normal — stock prices
  Multiplicative Normal — $\log(X)$ is Normal. Stock prices are modelled lognormal because returns multiply, not add.
  3 practice problems
  Start tutorial →
• Gumbel: the limiting distribution of the max
  The limit distribution of the maximum of many independent draws. Extreme-value statistics for floods, market crashes, record events.
  3 practice problems
  Start tutorial →
• Lévy / stable: heavy-tailed, stable under sums
  Heavy-tailed, stable under sums — the only family where the sum stays in the family. CLT fails when variance is infinite; this is what replaces it.
  3 practice problems
  Start tutorial →
• Weibull: extreme value type III; failure modeling
  Extreme value type III. Failure modelling with non-constant hazard rate — wear-out vs infant mortality, depending on the shape parameter.
  3 practice problems
  Start tutorial →

Combinatorics & enumeration (9)

Inclusion-exclusion, pigeonhole, symmetry, parity, coloring, generating functions, invariants, recursion, info-theoretic lower bounds.

• Inclusion-exclusion: fix the over-counting by alternating
  Easy
  When events overlap, sum the singles, subtract the doubles, add the triples. Careful book-keeping that nothing is counted twice.
  29 practice problems
  Start tutorial →
• Pigeonhole: when collision is unavoidable
  More pigeons than holes means two pigeons share a hole. The interview trick is choosing the right pigeons and holes.
  103 practice problems
  Start tutorial →
• Generating functions: turn counting into coefficient extraction
  Turn a counting problem into 'what is the coefficient of $x^n$ in this polynomial?' — a multiplication that does the bookkeeping for you.
  15 practice problems
  Start tutorial →
• Recursion: relate the big problem to a smaller copy
  Solve a problem of size $n$ by relating it to size $n-1$. The hard part is finding the recurrence, never solving it.
  98 practice problems
  Start tutorial →
• Symmetry: when relabeling does the counting for you
  If two events are interchangeable by relabeling, they have the same probability. Saves you from the case-bashing interviewers love to test.
  113 practice problems
  Start tutorial →
• Coloring: when 2 colors aren’t enough — escalate to ≥3
  Two-color the board, count the imbalance, escalate to three or four colors when two won't separate the obstruction.
  70 practice problems
  Start tutorial →
• Invariants: the quantity the move can’t touch
  Some quantity stays fixed no matter what move you make. Find it, and the puzzle either has a clean parity argument or no solution at all.
  93 practice problems
  Start tutorial →
• Parity: when a 2-coloring decides the question
  Two-color the board and count what each move does to the imbalance. If it can't fix the imbalance, the goal is impossible.
  82 practice problems
  Start tutorial →
• Info-theoretic lower bounds: counting hypotheses vs. outcomes
  If you must distinguish $2^n$ hypotheses but each query has fewer than $n$ bits of resolution, you can't possibly succeed. Pure counting; no algorithm matters.
  3 practice problems
  Start tutorial →

Stochastic processes (8)

Markov chains, martingales, random walks, optional stopping, first-step analysis, reflection, coupling, Poisson approximation.

• Markov chains: stationary distributions and long-run fractions
  Med
  What you can predict next depends only on where you are now. Long-run fractions live in the stationary distribution — find it once and you're done.
  11 practice problems
  Start tutorial →
• Martingales + Optional Stopping: one line replaces the linear system
  If a process is fair on every step, its expected value at any reasonable stopping time equals its starting value. One line replaces solving a linear system.
  14 practice problems
  Start tutorial →
• Optimal stopping: the threshold rule
  The optimal rule is almost always 'stop when the current value beats this threshold'. The interview question is just figuring out the threshold.
  35 practice problems
  Start tutorial →
• Random walk: gambler's ruin and hitting probabilities
  On a finite line, the probability you hit the right wall before the left is just your distance from the left over the total distance. Symmetric and clean.
  19 practice problems
  Start tutorial →
• First-step analysis: condition on the first transition
  Condition on what happens on the first move. Everything after is a smaller copy of the same problem — solve recursively.
  35 practice problems
  Start tutorial →
• Reflection principle: count bad paths by mirroring them
  Count bad paths by mirroring them across a barrier. Maps gambler's ruin and ballot problems to clean combinatorial counts.
  9 practice problems
  Start tutorial →
• Coupling: one uniform, two processes, pointwise comparison
  Two processes share the same uniform-noise input — now compare their outcomes pointwise. Magic for tail bounds and convergence proofs.
  9 practice problems
  Start tutorial →
• Poisson approximation: rare events made mental
  Many independent rare events $\approx$ Poisson with mean equal to the expected count. Lets you estimate Birthday Problem variants in your head.
  14 practice problems
  Start tutorial →

Statistical inference (4)

MLE, hypothesis testing, concentration inequalities, the CLT.

• Maximum likelihood: write it, log it, differentiate, solve
  Write the likelihood, take the log, differentiate, solve. The estimator that maximises the data's plausibility is consistent and asymptotically efficient.
  12 practice problems
  Start tutorial →
• Hypothesis testing: assume $H_0$, measure the surprise
  Assume nothing's happening, then ask: how surprising is what I observed? A small p-value means the assumption is suspect.
  13 practice problems
  Start tutorial →
• Concentration inequalities: tail bounds without distribution
  Tail bounds (Markov, Chebyshev, Hoeffding) tell you how unlikely a deviation is — without knowing the distribution. Cheaper assumptions, looser bounds.
  3 practice problems
  Start tutorial →
• Central Limit Theorem: turn a sum into a Normal
  Sum a thousand small independent things and the average is approximately Normal — no matter what the things are. Why z-scores work everywhere.
  15 practice problems
  Start tutorial →

Finance & options (8)

Arbitrage, put-call parity, delta hedging, dynamic hedging, Greeks.

• Arbitrage: when prices have to agree, or you get paid
  Two portfolios with the same payoff in every state must have the same price. If they don't, you trade until they do, risk-free.
  31 practice problems
  Start tutorial →
• Put-call parity: one identity, five inputs, solve for the sixth
  Call - Put = Stock - PV(strike). One identity, five quantities — solve for whichever the interviewer hides.
  15 practice problems
  Start tutorial →
• Delta hedging: zero out the linear exposure
  Med
  Match your linear exposure to the underlying with a hedge that has the opposite delta. First-order risk vanishes; gamma is what you trade against.
  4 practice problems
  Start tutorial →
• Dynamic hedging: gamma slippage and the cost of convexity
  Med
  Continuous re-hedging eliminates first-order risk but pays a gamma cost when the underlying moves. The cost is exactly what the option premium pays for.
  3 practice problems
  Start tutorial →
• Greeks: option sensitivities as partial derivatives
  Easy
  The four levers that move every option book. Traders hedge by Greeks daily; interviewers test Greeks in every first-round screen. Know them cold before you walk in.
  3 practice problems
  Start tutorial →
• Black-Scholes: pricing European options closed-form
  Med
  The formula every trading-desk candidate is expected to walk through in a first-round interview. One closed-form expression for a vanilla call — and the replicating-portfolio argument that gets you there is the interview question.
  3 practice problems
  Start tutorial →
• Itô's lemma: where the half-sigma-squared comes from
  Med
  Every SDE derivation in a quant interview — Black-Scholes, GBM, CIR — uses this rule. The stochastic chain rule with one extra term: that $-\tfrac{1}{2}\sigma^2$ you see everywhere comes from here.
  0 practice problems
  Start tutorial →
• American options: backward induction on a binomial tree
  Med
  Early exercise turns option pricing from a one-shot expectation into a backward dynamic program. The binomial tree is the simplest model where you can see the early-exercise boundary appear.
  3 practice problems
  Start tutorial →

Volatility surfaces & stochastic vol (14)

Heston, SVI/SSVI, SABR, rough Bergomi, Fengler smoothing — what an equity vol desk calibrates daily.

• Heston: stochastic volatility's first closed form
  Hard
  Let variance itself follow CIR, correlate it with the spot, and the option price still has a closed-form characteristic function. The smile shape and the negative skew fall out of two SDEs and four parameters.
  0 practice problems
  Start tutorial →
• SVI (Stochastic Volatility Inspired): Gatheral's 5-parameter slice fit
  Hard
  Five parameters per maturity: ATM level, wing steepness, skew tilt, smile centre, smile curvature. Fits any reasonable single-slice equity smile in milliseconds.
  0 practice problems
  Start tutorial →
• SSVI: SVI extended to the whole surface
  Hard
  Replace five SVI parameters per slice with two surface parameters plus an ATM-variance curve. Gatheral-Jacquier (2014): SSVI is automatically calendar-arbitrage-free under a mild condition on the curvature function $\phi$.
  0 practice problems
  Start tutorial →
• SABR (Stochastic Alpha Beta Rho): joint diffusion of forward and vol
  Hard
  Hagan 2002: jointly stochastic forward and vol-of-forward, with one closed-form formula for implied vol covering the whole smile. Standard equity, rates, and FX skew calibration.
  0 practice problems
  Start tutorial →
• Rough Bergomi: when the kernel is fractional
  Hard
  Replace Bergomi's exponential kernel with a Volterra power-law kernel and you get rough vol — smiles that match short-dated SPX out to the wings, term-structure of ATM skew that obeys the empirical $T^{H - 1/2}$ scaling.
  0 practice problems
  Start tutorial →
• Bergomi: forward variance as the modelling primitive
  Hard
  Heston models the instantaneous variance; Bergomi models the *forward* variance curve directly — the same way HJM models the forward rate curve. Variance swaps price exactly by construction.
  0 practice problems
  Start tutorial →
• Fengler: non-parametric arb-free vol surface smoothing via Quadratic Programming (QP)
  Hard
  Given noisy quoted call prices, find the smoothest arb-free surface that's closest to them. A constrained Quadratic Program (QP) — minimise a quadratic objective subject to linear constraints — does the work; no parametric family is assumed.
  0 practice problems
  Start tutorial →
• Lee moment formula: how steep can a vol wing be?
  Hard
  Roger Lee (2004): the right-wing slope of total variance in log-moneyness is at most $2$, and is tied analytically to the largest finite moment of the underlying.
  0 practice problems
  Start tutorial →
• Variance swaps via static replication
  Med
  A variance swap pays realised variance minus a strike. Carr-Madan: that strike is replicable today by a *static* portfolio of OTM puts and calls weighted $2 dK / K^2$.
  0 practice problems
  Start tutorial →
• GARCH: vol clustering and the persistence parameter
  Med
  AR(1) on the conditional variance instead of the level. Big returns predict big returns; the persistence parameter $\alpha + \beta$ is one number that controls everything — long-run vol, shock half-life, calibration drama.
  0 practice problems
  Start tutorial →
• Vega-weighted calibration: why fit vols, not prices
  Equal-weighted least-squares on call prices overfits ATM strikes and ignores wings, because vega varies 3x across the surface. Vega-weighting converts price errors into vol errors.
  0 practice problems
  Start tutorial →
• Static arb-free constraints: butterfly, calendar, no-negative-density
  Three inequalities every option surface must satisfy: butterfly (positive density), calendar (calls monotone in maturity), and Lee-bound wings. Violate one and you've found a free trade.
  0 practice problems
  Start tutorial →
• fBM via Volterra kernel: where roughness comes from
  A fractional Brownian motion is a Brownian integral against a one-parameter kernel $(t - s)^{H - 1/2}$. The kernel is what makes the increments correlated and the paths rough or persistent.
  0 practice problems
  Start tutorial →
• Hurst exponent: measuring roughness of a process
  One number $H \in (0, 1)$ captures how a process's variance scales with time. $H = 0.5$ is Brownian; $H > 0.5$ is persistent; $H < 0.5$ is rough. Equity log-vol comes out at $H \approx 0.1$ — the rough-vol headline empirical fact.
  0 practice problems
  Start tutorial →

Rates & fixed income (5)

Short-rate models and the LIBOR Market Model — interview core for rates desks.

• Vasicek short rate: the first closed-form bond price
  Med
  Pick one number (today's overnight rate) and an SDE; the entire yield curve falls out as a closed-form expectation.
  0 practice problems
  Start tutorial →
• CIR short rate: no negative rates, by construction
  Med
  Replace Vasicek's constant diffusion with $\sigma\sqrt{r_t}$ and the diffusion vanishes at zero. Negative rates disappear, the bond price is still affine, and the math costs one extra auxiliary parameter.
  0 practice problems
  Start tutorial →
• Hull-White: Vasicek that matches today's curve exactly
  Med
  Take the Vasicek SDE, make the long-run mean a function of time, calibrate $\theta(t)$ to today's yield curve, and now your model reprices every market bond exactly. The one short-rate model production desks actually run.
  0 practice problems
  Start tutorial →
• LMM-SABR: the LIBOR Market Model with SABR vol for rates
  Hard
  Model the quoted forward rates directly as lognormal under their own forward measures (Black caplet pricing). Add SABR vol on each forward and you have the standard swaption-desk model.
  0 practice problems
  Start tutorial →
• Mean-reverting SDEs: the OU half-life trick
  Mean-reversion has one number that matters: how fast does a deviation die. The half-life formula collapses continuous OU and discrete AR(1) onto the same one-liner.
  0 practice problems
  Start tutorial →

Microstructure & market making (2)

Adverse selection, market-maker quoting, self-exciting order flow.

• Kyle / Glosten-Milgrom: how informed traders force market makers to widen spreads
  Med
  Bid-ask spreads aren't fees. They are the price a market maker charges to insure against trading with someone smarter than them. Kyle (1985) and Glosten-Milgrom (1985) derive the exact width.
  0 practice problems
  Start tutorial →
• Hawkes processes: when past events trigger future ones
  Add a kick to the intensity every time an event fires, and let it decay back. The result is a point process with realistic clustering — trade arrivals, order-book events, default cascades all look like this.
  0 practice problems
  Start tutorial →

Risk, inference & simulation (8)

VaR, expected shortfall, Monte Carlo variance reduction, ergodicity for backtests, pairs trading.

• VaR and Expected Shortfall: why ES is the coherent risk measure
  Med
  VaR is a quantile of your loss distribution. ES is the average of all worse-than-VaR losses. ES is *subadditive* (diversification reduces it); VaR isn't (you can construct portfolios where diversification penalises you under VaR).
  0 practice problems
  Start tutorial →
• Monte Carlo variance reduction: antithetic and control variates
  Med
  Plain MC error scales as $1/\sqrt{N}$. Two tricks — antithetic pairs (negatively correlated samples) and control variates (a related closed-form anchor) — can drop the variance by an order of magnitude at the same compute cost.
  0 practice problems
  Start tutorial →
• Cointegration: when two non-stationary series share a stationary spread
  Med
  Two stock prices each wander randomly. Their *combination* $X - \beta Y$ can be stationary, giving you a mean-reverting spread to trade. Engle-Granger's 1987 procedure makes the test mechanical.
  0 practice problems
  Start tutorial →
• Ergodicity and mixing: why your backtest can quietly lie
  Hard
  When you backtest a strategy on 20 years of data, you're implicitly assuming the market is ergodic — that one long path represents the full distribution. When ergodicity breaks, the backtest is meaningless. Mixing speed says how long the path needs to be for it to be safe.
  0 practice problems
  Start tutorial →
• Galton-Watson branching: the criticality threshold
  Hard
  Each individual produces $\mu$ children on average. Below $\mu = 1$: certain extinction. Above $\mu = 1$: positive survival probability. The boundary is everywhere in finance — default cascades, viral order flow, market-maker hedging chains all live on it.
  0 practice problems
  Start tutorial →
• Fisher information: how well can you know your parameter
  Curvature of the log-likelihood at its peak. Sharp curvature = small standard error = parameter well-identified. Flat curvature = big SE = parameter the data refuses to pin down.
  0 practice problems
  Start tutorial →
• Stationarity and invariant measures
  A process is stationary when its statistical properties stop drifting. Markov chains have a closed-form stationary distribution; AR(1) is stationary iff $|\phi| < 1$; cointegration is the trick that turns two non-stationary series into one stationary spread.
  0 practice problems
  Start tutorial →
• Coupon collector: when does the last coupon arrive?
  Easy
  Linearity of expectation turns a hard waiting-time problem into a sum of geometric expectations.
  6 practice problems
  Start tutorial →

Optimization & methods (3)

Dynamic programming, backwards induction, Kalman filter.

• Dynamic programming: memoize the overlapping subproblems
  When the optimal answer at size $n$ depends on optimal answers at sizes $<n$, memoize the subproblems and you're done.
  17 practice problems
  Start tutorial →
• Backwards induction: solve from the terminal state
  Set the value at the terminal state, then walk backwards one step at a time. Lets you price American options and solve finite-horizon games.
  23 practice problems
  Start tutorial →
• Kalman filter: inverse-variance weighting with memory
  Combine prior and noisy measurement using inverse-variance weighting. The Kalman gain is just 'how much do I trust the measurement?'.
  3 practice problems
  Start tutorial →

Applied math (3)

Real-world quantitative reasoning: relative-frame kinematics, work-rate problems, order-of-magnitude estimation.

• Relative motion: pick a frame, subtract speeds
  Pick the right reference frame and one of the moving things stops. The whole problem becomes a single object moving relative to a fixed point.
  3 practice problems
  Start tutorial →
• Work rates: reduce everyone to per-unit, then add
  Reduce everyone to per-unit (work per minute, water per second), then add. Combined rate equals the sum of individual rates.
  3 practice problems
  Start tutorial →
• Fermi estimation: order-of-magnitude from anchored factors
  Bracket the answer with order-of-magnitude factors that are easy to estimate. Geometric mean of low and high gives you within one order of magnitude.
  3 practice problems
  Start tutorial →

Other (3)

Methods that don't fit cleanly into one of the above buckets.

• Extremal principle: lean on the most extreme element
  Look at the largest, smallest, or most-something element first. Many combinatorial proofs collapse the moment you fix attention there.
  76 practice problems
  Start tutorial →
• P&oacute;lya&rsquo;s urn: exchangeability beats enumeration
  Drawing a ball and adding more of the same color builds a sequence that's exchangeable — order doesn't matter, even though trials aren't independent.
  12 practice problems
  Start tutorial →
• Entropy, cross-entropy, KL: cost of the wrong model
  How costly is it to use the wrong model? KL(P||Q) is the extra bits you pay encoding samples from $P$ with a code optimal for $Q$.
  3 practice problems
  Start tutorial →