Starting From Scratch

Two inequalities that show up in half of all quant interview maths. Cauchy-Schwarz caps how large a product-sum can be; AM-GM says an average never falls below a geometric average. Together they bound correlations, explain volatility drag, and solve optimisation problems without calculus.

A complete re-encoding of a distribution as a function of frequency. It turns the messy business of adding independent random variables into plain multiplication, always exists even when the moment generating function does not, and is the machinery behind the central limit theorem.

A way of measuring the size of sets and adding things up by sorting them into piles of equal value rather than sweeping left to right. It is the integral that makes expectation work for any random variable, mixes jumps and densities in one formula, and survives limits that break the Riemann integral.

Four different meanings of "the estimator settles down", almost sure, in probability, in mean square, and in distribution. They are not interchangeable, they sit in a strict hierarchy, and picking the wrong one is how proofs and simulations quietly go wrong.

A sigma-algebra is the list of questions your information can answer; a filtration is that list growing as time passes. Together they are how probability theory writes down "what a trader knows at 10:30 this morning", the machinery behind martingales, conditional expectation and every no-look-ahead rule in backtesting.

The exchange rate between two probability measures, a per-scenario weight that converts one set of beliefs into another. It is what makes risk-neutral pricing, Girsanov's theorem, importance sampling and likelihood ratios all the same operation.

The shape you get when quantities multiply instead of add. It keeps prices positive, pushes the average outcome above the typical one, and is the distribution sitting underneath Black-Scholes.

What the bell curve turns into when you have to estimate the spread from the same small sample you are testing. Fatter tails, wider intervals, and a much better model for daily returns than the normal.

The central limit theorem's counterpart for maxima. It says the worst outcome in a large sample has a predictable shape whatever the underlying data looks like, which is what lets you size a risk limit for a loss you have never actually seen.

One algebraic template that most named distributions secretly share. Recognise it and you get sufficient statistics, the maximum-likelihood equation, conjugate priors and generalised linear models handed to you for free.

Any smooth function looks like a straight line up close, a parabola if you zoom out slightly less, and a cubic if you keep going, the Taylor series is the recipe for building that approximation, one derivative at a time.

Any wiggly signal, however messy, is a sum of plain sine waves at different frequencies, the Fourier transform is the recipe for reading off exactly which frequencies are in it and how strong each one is.

The equation that describes heat spreading through a metal bar is, after a change of variables, exactly the Black-Scholes equation, both describe a quantity smoothing itself out over time at a rate set by curvature.

Every valid covariance matrix has a 'square root' that turns independent noise into correlated noise, the Cholesky decomposition is how simulations get correlated random numbers out of a stream of uncorrelated ones.

Fitting a line through data no line can pass through exactly is a shadow problem: the fitted values are literally the shadow of your data on the space of possible fits, cast straight down.

One number tells you whether solving a system of equations is safe or a numerical minefield: the condition number measures how much a matrix amplifies small errors, and a big one means your answer cannot be trusted.

Every symmetric matrix, however tangled it looks, is secretly just a stretch along a set of perpendicular directions, the spectral theorem guarantees those directions always exist and are always at right angles.

A compact way to describe a batch of numbers, minimum, first quartile, median, third quartile, maximum, and the standard picture (a box plot) that turns those five numbers into a shape you can compare across many groups at a glance.

The mistake of judging how likely something is from a positive test or signal alone, while ignoring how rare the thing being tested for was in the first place, a trap that makes even highly accurate signals produce mostly false alarms when what they're looking for is uncommon.

A practical workflow for picking which parametric distribution (normal, Student's t, lognormal, and so on) best describes a dataset, and estimating its parameters, instead of defaulting to the normal distribution out of habit.

A visual test for whether a sample of data plausibly came from a chosen distribution (like the normal), made by plotting the sample's sorted values against the values that distribution would predict at the same percentiles.

How to estimate a distribution's mean, variance, skewness and kurtosis from a finite sample, and, just as important, how uncertain each of those estimates is, since a moment computed from 60 data points is not the same thing as the true moment.

The short list of behaviours that show up in almost every traded market on earth, fat tails, volatility that clusters, direction that refuses to be predicted, and losses that hit harder than gains. Any model you build has to reproduce them, or it is describing a market that does not exist.

A way to measure how much knowing one variable tells you about another that, unlike correlation, catches any kind of dependence, curved, cyclical, or otherwise nonlinear, not just straight-line relationships.

An estimator is consistent if piling on more data drives it onto the true value and keeps it there. It is the minimum thing you should demand of a formula before you trust it, and it says nothing at all about whether the sample you actually have is big enough.

An estimation recipe for when you know some averages your model must reproduce but not the full distribution of the data. Write down the conditions, force the sample versions as close to zero as possible, and weight the reliable conditions more heavily.

A one-line rule for the error bar on a transformed estimate: multiply the original standard error by how steeply the transformation is rising at that point. It is how you get a standard error for a volatility, a ratio, or a Sharpe ratio without simulating anything.

A way out of the chicken-and-egg problem in models with hidden labels or missing values. Guess the fit, use it to fill in what you cannot see, refit on the filled-in data, and repeat until nothing moves.

Search hard enough and noise will hand you a beautiful result. This is how selection manufactures significance, why it happens even to honest analysts who ran only one test, and how to price the search you actually did.

Power is the resolution of your experiment, the chance you would actually notice a real effect if one were there. Work it out before you collect data, or you will spend months proving only that your instrument was too blunt to see anything.

A single rule for judging a simple explanation against a more elaborate one. Measure how much better the elaborate story fits the data you actually saw, then check whether that improvement is more than an extra free knob buys you for nothing.

The answer to a very practical question: given a fixed budget of false alarms, which test catches the most real signals? Rank the evidence by a likelihood ratio and act on the top of the list, nothing else can do better.

When some observations are far noisier than others, your regression coefficient is still right but its error bar is a lie. Robust standard errors let every point report its own noisiness instead of forcing one average on all of them.

Leave something out of a regression and the coefficient you did estimate quietly absorbs its influence. The result is not noisy but systematically wrong, and collecting more data only makes you more confident in the wrong number.

A regression always hands you numbers. The assumptions decide whether those numbers mean anything, some protect the estimate itself, others only protect the error bars, and knowing which is which tells you what to fix.

A two-pass method for testing whether a stock characteristic earns a return premium: run one cross-sectional regression per month, then treat the sequence of monthly slopes as an ordinary time series and t-test its average. It exists because pooling every stock-month into one regression produces standard errors that are far too small.

The recipe for fitting a straight-line-style model to outcomes that aren't shaped like a bell curve, counts, probabilities, yes/no events, by bending the line through a link function before it meets the data.

A way to strip out everything permanent and unobservable about each stock, trader, or country before you estimate the relationship you actually care about, by comparing each unit only to its own history.

A single security can carry four different identifiers at once, a global ISIN, a US CUSIP, a UK SEDOL and a market-specific ticker, because each was built for a different purpose, and confusing them is a routine source of trade breaks.

The last trade of the day and the exchange's official closing price are usually the same number, but not always, the closing auction exists precisely because a single last print can be an outlier, and most of the money in the market marks its books to the auction price, not the last tick.

A stock is quoted in dollars, a bond in yield or price-per-hundred, a swap in a spread, and an FX pair as a ratio, the same underlying idea of "what does it cost" takes a different quoting convention in every asset class, and mixing them up is a fast way to misread a screen.

Every corporate action a company can announce sorts into one of a handful of standard categories, mandatory, voluntary or mandatory-with-choice, and knowing which category an event falls into tells you immediately what a data feed, a back office and a shareholder each have to do about it.

A CCP steps into the middle of every cleared trade so that neither original party ever has to trust the other again, the mechanism that does it, novation, is what makes modern derivatives and repo markets survivable when a large member fails.

Index providers sell licences to a set of published rules, not to the securities themselves, a business model that turns "which stocks are in the S&P 500" into a recurring revenue stream and a source of real, tradeable market impact around every reconstitution.

Clicking buy is the easy part. Behind every trade is a chain of routing, matching, clearing and settlement that takes a verbal agreement to trade and turns it into an actual, legally final change of ownership.

A custodian is the institution that actually holds a fund's securities and cash, settles trades, collects income and keeps the official books, quiet infrastructure work that almost nobody outside operations ever thinks about until it fails.

A prime broker is the bank that lends a hedge fund cash and stock, holds its collateral, clears its trades away from a single executing broker, and in the process becomes one of the fund's largest single points of failure.

Different regulators have jurisdiction over different products and different geographies, a US equity option, a US futures contract and a UK-listed stock can each answer to a different regulator, and knowing which one matters the moment something goes wrong.

A central counterparty absorbs a defaulting member's losses through a strict, pre-agreed order of resources, and the CCP's own capital is deliberately placed in that queue so it shares the pain before its members do.

A share is not a claim on a company's buildings. It is a bundle of three things, a residual claim on whatever is left after everyone else is paid, a cap on your losses, and a vote, and the residual part is what makes equity behave the way it does.

An American Depositary Receipt is a US-listed claim on foreign shares held in a vault abroad, when the two prices drift apart, a depositary bank's conversion mechanism lets an arbitrageur create or cancel ADRs to pull them back together.

A stock's price drops by roughly the dividend on the ex-dividend date, a mechanical, predictable event that traders try to exploit around, and that options markets price in well before it happens.

The default way to build an index is to weight each stock by its market value, but that isn't the only choice, equal weighting and other alternative schemes trade off diversification, turnover and performance in very different ways.

An ETF tracks what it holds because a handful of large firms can always swap a basket of the underlying shares for brand-new ETF shares, or hand ETF shares back and take the basket. That swap is an arbitrage valve, and it is the whole reason an ETF is not just another closed-end fund.

An index is a rule, not a fact. The choice of which companies count and how much each one counts decides what "the market did today" means, and three sensible rules applied to the same four stocks can give three different answers.

Before an IPO prices, the underwriters spend a week collecting orders at different prices from investors, that order book, and who gets how much stock out of it, is how the offer price actually gets set.

A poorly-run public company is a standing invitation for someone else to buy it and replace management, takeover defences are the legal tools boards use to slow that down or price it up.

Every short sale starts with borrowing the stock first, a large, mostly invisible rental market where the fee is set by how scarce the shares are to borrow, not by anything happening in the price chart.

The extra return investors expect for holding stocks instead of a safe government bill. It is the single most important number in finance and nobody knows what it is, because it has to be estimated from data so noisy that a century of history still leaves the answer blurry.

When a stock is added to a major index like the S&P 500, trillions of dollars of passive money are forced to buy it on a fixed date, a predictable, price-insensitive demand shock that tends to push the price up before the buying even happens.

US stock trades used to settle two business days after the trade; since May 2024 they settle in one, a change that sounds administrative but compresses the entire window brokers, custodians and foreign investors have to move cash and shares.

Small-cap stocks often look cheap and undercovered on paper, but their thin trading volume means the cost of actually buying and later selling a meaningful position can quietly erase the edge.

An ETF's trading volume tells you almost nothing about how easily you can trade it in size. Real ETF liquidity is inherited from the liquidity of what it holds, not from how often its own ticker changes hands.

ETFs and mutual funds can hold identical portfolios and still behave completely differently, because one trades all day on an exchange and the other is priced once and bought directly from the fund company.

A strategy that returns 30% a year on $50 million rarely still returns 30% on $5 billion. Every strategy has a capacity limit, past which its own trading starts eating the edge it is trying to capture.

When money pours into or out of a fund, the manager has to buy or sell the underlying holdings to match, and that forced trading can move prices independent of any change in what the assets are actually worth.

Net asset value is the fund's total assets minus its liabilities, divided by shares outstanding. It is the single number that determines the price at which fund shares are created and redeemed.

Instead of buying the underlying stocks, a synthetic ETF holds a basket of collateral and enters a swap with a bank that promises to pay the index return. It tracks perfectly on paper, but swaps a tracking problem for a counterparty problem.

ETFs stay close to the value of what they hold because specialist trading firms can create or destroy shares on demand, trading away any gap between the ETF's market price and its underlying basket.

A 2x leveraged ETF that resets its exposure every day can lose money even in a flat, choppy market that goes nowhere over the long run. The daily reset, not the leverage itself, is what quietly bleeds value.

A fund takes money from many investors, invests it as one pool, and gives each investor a proportional slice. That simple idea underlies mutual funds, ETFs, hedge funds and everything in between.

The income statement, balance sheet and cash flow statement are not three reports. They are one model seen from three angles, wired together by two hinges. Get the wiring right and a change anywhere flows everywhere and still balances.

A balance sheet is a photograph of what a company owns and who has a claim on it, taken on one specific day. Everything on it hangs off a single identity that cannot break, assets equal liabilities plus equity.

Profit is an opinion; cash is a fact. The cash flow statement rebuilds the bank balance from reported earnings, splitting every movement into operating, investing and financing, and it is the hardest of the three statements to dress up.

Paying staff in shares is a real cost settled in a currency the company can print. The accounting freezes that cost at the grant date, which means the number on the income statement rarely matches what employees actually received or what shareholders actually gave up.

Give an analyst two consecutive balance sheets and one income statement and they can rebuild the entire cash flow statement from scratch. Every line of it is a balance sheet change wearing a different label, and one identity determines all the signs.

Two companies spend the same hundred million dollars. One reports it all as cost this year, the other spreads it over five. Both follow the rules, both end the period with identical cash, and their reported EBITDA differs by a factor of two.

A business is worth the cash it will hand you, shrunk for the fact that future money is worth less than money now. DCF turns that one sentence into a number, and most of the number usually comes from the part you can least defend.

The cash a business throws off after paying to keep itself running and growing. It is the number a valuation actually discounts, and unlike reported earnings there is no single official definition, which is exactly where the arguments start.

The same company, reporting the same year's results, can show a meaningfully different balance sheet and income statement depending on whether it follows IFRS or US GAAP, and an analyst comparing a US company to a European one has to know exactly where those gaps hide.

A bank's balance sheet is its inventory, loans and securities instead of widgets, and its income statement runs on a spread instead of a markup, which means the ratios that work for an ordinary company are close to useless for one.

Two firms report the same profit, but one collected the cash and the other booked receivables. Sloan showed the cash half of earnings persists better than the accrual half, and that share prices behave as if it did not.

Before any model tells you what an option is worth, pure logic tells you what it cannot be worth. A handful of inequalities pin every option price into a band, and a quote outside that band is free money you can collect without a forecast.

The whole option chain at one expiry contains the market's entire probability distribution for where the underlying will land, not just its average and its spread. Differentiate the call price twice with respect to strike and the distribution falls out.

An option's bent payoff can be manufactured out of shares and cash, provided you keep adjusting the mix as the price moves and never add or remove money along the way. The cost of setting that machine running on day one is the option's fair price.

A set of prices contains no free money if and only if you can find a set of made-up probabilities that reprices everything as a fair bet. That equivalence is the theorem the whole derivatives industry stands on, and a second half tells you when those probabilities are unique.