Every concept, endless questions
Each concept is a small financial model. Read how it works, then draw a real interview question from it: forwards, backwards or what-if, with fresh numbers and a worked solution every time.
Forward runs a concept the usual way. Inverse runs it backwards, which is what separates understanding from memorizing.
Probability
6 conceptsConditional probability and Bayes
A signal that is right 95% of the time can still be wrong most of the times it fires. What matters is not how accurate the signal is but how rare the thing it looks for is: if the event happens on one day in a hundred, then even a small false-alarm rate on the other ninety-nine days produces more false alarms than true ones. Bayes’ rule is the bookkeeping that weighs the true alarms against the false ones. Every question of the form "the test came back positive, how worried should I be" is this calculation, and the intuitive answer — the sensitivity — is almost always far too high.
Expected value of a game
The price of a bet is the average of what it pays, weighted by how often it pays it. That number — the expected value — is what you would earn per play if you played forever, and it is the only thing that decides whether a game is worth entering at a given fee. What it does not tell you is how bumpy the road is: two games with the same expected value can have wildly different standard deviations, and a small positive edge on a very volatile bet can bankrupt a player long before the average shows up. Repetition shrinks the noise relative to the edge, but only with the square root of the number of plays, which is why a casino wants millions of small bets and a gambler with one big bet is not "investing".
Dice sums and distributions
Two dice do not produce a flat distribution. Each face of each die is equally likely, but the totals are not, because a total in the middle can be made in many ways and a total at the edge in only one. Counting those ways — the ordered pairs — is the whole technique, and it generalises: every "probability of a sum" question, from dice to card totals to the number of defaults in a portfolio, is a question about how many equally likely outcomes land on each value. The counting also exposes the two habits interviewers test for: listing ordered pairs rather than unordered ones, and reaching for the complement when "at least one" appears.
Coin runs and sequences
Independent flips have no memory, but sequences of flips have structure. The number of heads in n flips is binomial, and its probabilities come from counting which flips were the heads. Waiting times are different: how long until a pattern first appears depends on the pattern, not just on its length. HT arrives faster than HH on a fair coin — four flips against six — because a failed attempt at HT (a second head) is already the start of the next attempt, while a failed attempt at HH (a tail) throws everything away. The two habits this rewards are reaching for the complement when the question says "at least", and setting up a first-step equation when it says "how long until".
Expected number of draws
If something happens with probability p on each try, you expect to wait 1/p tries for it. That single fact — the geometric mean — answers most "how long until" questions, and it has a property people find hard to accept: it does not shrink as you wait. After ten failures the expected remaining wait is still 1/p, because independent trials carry no memory. The same idea chains: the wait for the m-th success is m waits of 1/p, and the wait to collect every one of N types is a sum of geometric waits whose success probabilities fall as the set fills up, which is why the last few coupons take longer than all the others combined.
Combinatorics
Most probability questions are counting questions in disguise: how many outcomes are there, and how many of them are the ones you want. The single decision that matters is whether order counts. Picking a committee does not care who was named first; assigning titles does. Get that right and the rest is arithmetic — and a surprising number of "hard" questions collapse once you notice you can count the complement, or fix one person in place and count the rest, or notice that a randomly chosen committee contains any particular person with probability k/n by pure symmetry, without counting anything.
Brainteasers
2 conceptsFair value of a betting game
An option to try again is worth money, and the way to price it is backwards. Without the option a die is worth its average face. With one reroll you compare each face you might see against that average: keep it if it is higher, throw it back if it is lower, and the value of the game is the average of those decisions. A second reroll is priced the same way against the one-reroll value, and so on. Each extra reroll is worth less than the last, because the bar you are rerolling against keeps rising. This is the same logic that prices an American option: at every point, compare what you hold with what waiting is worth.
Clock angles and rate puzzles
Clock problems are relative-speed problems wearing a costume. The minute hand runs at 6° a minute, the hour hand at 0.5°, so the gap between them changes at exactly 5.5° a minute — and once you hold that number, every question is one division. The angle at any time is the hour hand’s position minus the minute hand’s; the next time they meet is the current gap divided by 5.5; the number of meetings in twelve hours is eleven, because the minute hand laps the hour hand once for every full lap it gains. Interviewers ask these to see whether you reach for the mechanism or try to remember an answer.
Options & Greeks
6 conceptsPut-call parity
Owning a call and selling a put at the same strike gives you exactly the payoff of owning the stock and owing the strike at expiry: above the strike the call pays, below it the put costs you, and either way you end up paying K for the share. Two positions with identical payoffs must cost the same today, or someone buys the cheap one, sells the dear one and pockets the difference with no risk. So C − P = S − PV(K), with no model and no volatility anywhere in it. That is why parity is the first thing an options desk checks: it holds whatever anyone thinks about the future, and when it breaks it is either a stale quote, a dividend or borrow cost nobody mentioned, or free money.
Delta hedging
An option’s delta is how many shares it behaves like for a small move in the stock. A dealer who sells calls is short that many shares of exposure, and buys exactly that many shares to cancel it. The hedge is only right for an instant: as the stock rises a call behaves more like stock, its delta climbs, and the dealer must buy more — and sell again when it falls. Buying high and selling low is the cost of being short an option, and it is exactly what the option premium pays for. Delta hedging does not remove risk; it swaps directional risk for the risk that the stock moves more than the premium assumed.
Gamma P&L
A delta-hedged long option makes money whenever the stock moves, in either direction, because the option’s delta changes in your favour: it gains more on the way up than the hedge loses, and loses less on the way down. That profit grows with the square of the move — half of gamma times the move squared. Nothing is free, though: the option loses a little value every day through theta. The trade reduces to one comparison. If the stock moves more than the market priced in, gamma beats theta; if it moves less, theta wins. That is what it means to be long volatility, and the break-even daily move is just the implied volatility restated in dollars.
Implied versus realized volatility
Implied volatility is a price: it is the volatility you have to plug into the model to get the option’s market price back out. Realized volatility is a measurement: how much the stock actually moved. An options trader’s whole job is the gap between the two. Buy a straddle when you think the stock will move more than the implied volatility is charging for, sell it when you think it will move less. The at-the-money straddle makes the link concrete — its price is roughly 0.8 × stock × vol × √time, which is also roughly the average size of the move the market expects by expiry.
Theta decay
An at-the-money option is worth roughly its expected move, and the expected move grows with the square root of time, not with time itself. So an option’s value falls like √T as expiry approaches — slowly at first and then very fast. Halve the time left and the option keeps about 71% of its value; quarter it and half the value is gone. That is why theta, the daily decay, accelerates into expiry for at-the-money options: the last month costs far more per day than the first. Option sellers love the final weeks; option buyers pay dearly for holding through them.
Spread payoffs at expiry
Buying a call and selling a higher-strike call gives up everything above the upper strike in exchange for a cheaper position. The sold call pays for part of the bought one, so the spread breaks even sooner than the outright call and loses less if the stock goes nowhere — but its profit is capped at the distance between the strikes. The payoff is three straight lines: flat at a loss of the premium below the lower strike, rising dollar for dollar between the strikes, flat again at the maximum above the upper one. Every vertical spread, collar and risk reversal is read the same way: add up the hockey sticks.
Statistics
5 conceptsAnnualizing volatility
If each day’s return is independent of the last, the variances of the days add up — so a year of 252 days has 252 times the variance of one day, and √252 ≈ 15.9 times the volatility. That square root is the whole trick, and it runs in every direction: from daily to annual, from annual to a ten-day risk horizon, from a daily standard deviation to a daily value at risk. It is also where the assumption bites. Returns that trend make long-horizon risk larger than √t says; returns that mean-revert make it smaller. The rule is a default, not a law.
Regression beta and R-squared
Regressing a stock on the market splits its risk in two. The slope, beta, says how many percent the stock moves for each percent the market moves, and it is correlation rescaled by the ratio of the two volatilities — so a volatile stock can have a high beta with a modest correlation, or a low beta with a high one. R-squared, the correlation squared, says what share of the stock’s variance the market explains; the rest is idiosyncratic, the part a hedge cannot touch. Alpha is whatever return is left after paying beta its due. Mixing up beta and correlation is the most common mistake, and the formula makes it easy to avoid.
Standard error and confidence intervals
A sample average is itself a random number: run the strategy over a different stretch of days and you would get a different mean. Its standard error — the volatility of daily returns divided by the square root of the number of days — says how much. A confidence interval is just the mean plus or minus about two standard errors. The square root is the painful part: to halve the uncertainty you need four times the data. That is why daily strategy returns, with a mean of a few basis points buried under a standard deviation of a hundred, take years to tell apart from zero.
Two-asset portfolio variance
The variance of a sum is the sum of the variances plus twice the covariance, and every portfolio calculation is that identity with weights attached. Covariance is correlation with units: the correlation scaled up by both volatilities. Because variance is a quadratic in the weight, it has a minimum, and the weight that reaches it depends only on the two variances and their covariance — not on expected returns. Flip the sign on one asset and the cross term changes sign too, which is why a long-short spread between two highly correlated assets can be far less volatile than either leg, and a spread between two uncorrelated ones is more volatile than both.
Is a Sharpe ratio significant?
A Sharpe ratio is a mean divided by a volatility, so testing whether the mean is really above zero is testing the Sharpe — and the t-statistic turns out to be just the annual Sharpe times the square root of the number of years. That one line explains why track records are so hard to judge. A genuinely good strategy with a Sharpe of one needs about four years before it clears the usual bar, and a backtest that tried fifty variations is almost guaranteed to find one that clears it by luck. Significance is a property of the evidence, not of the strategy.