Quant interview questions
One question a trading desk actually asks, worked through in full. You see the trap first, and the way almost everyone walks into it, then the thinking that gets you out.
∂Options, Greeks & marketsOptions, the Greeks, volatility, rates and portfolios — the questions a desk asks once it knows you can count.59 questions◇Trading brainteasersPuzzles that test how you reason out loud, not what you memorised. Most have a trap, and the trap is the point.46 questionsℙProbability & statisticsConditioning, expectation, distributions and the paradoxes that make strong candidates answer too fast.30 questions𝔼Expected value & market makingQuoting a price, sizing a bet, knowing when the edge is worth taking — and when winning is the bad news.13 questions×Mental math & estimationArithmetic under a clock and Fermi estimates from nothing. The filter that runs before anyone reads your CV.10 questions An at-the-money call with a zero interest rate is worth one over the root of two pi, which is 0.39894, times the absolute swing of the terminal price, so a 10 dollar standard deviation prices it at 3.9894 and the closest of the offered 1, 5 and 10 is 5. The two-step estimate is exact under a symmetric terminal law, where the option finishes above the strike exactly half the time and the average gain when it does is 7.979. Calibrate a lognormal to the same 10 dollar swing and those two factors become 0.4801 and 8.285, whose product is still 3.98, which is why the qualifier about half cannot be cut.
A plain t sitting next to the t squared in a Gaussian exponent looks like a new function and is only a shift. Completing the square turns the integral of e to the minus a t squared over two plus b t, from x to infinity, into e to the b squared over 2a times the root of 2 pi over a times the standard normal at a rescaled and shifted argument, never at x itself. The worked case comes out as exactly half a bell, e root pi over two or 2.40901455, but only because its lower limit happens to land on the centre b over a.
An estimated line of expected return against market sensitivity that sits entirely above the theoretical one is not a market on sale, because pricing errors scatter above and below instead of lifting everything by the same amount. In a two-factor world where every asset carries the same 0.75 exposure to the second risk, the fitted single-factor line comes out exactly parallel to the theoretical one and exactly 3 percentage points above it, with residuals of zero, while a mispricing world engineered to have the same average lift leaves errors of both signs as large as 6.5 points. Let the second exposure grow with sensitivity and the slope moves too, at which point the two lines can cross inside an ordinary sample.
Two stocks each swinging 20 percent a year with a correlation of one half give their product a volatility of 20 root 3, or 34.641 percent, rather than 40. Adding the two numbers is correct at exactly one correlation, namely 1, because the composite volatility is the law of cosines with the correlation as the cosine of the angle between two arrows. Pricing the call at 40 percent overstates it by 14.2 percent and ignoring the correlation understates it by 16.9 percent, and the other diagonal of the same parallelogram prices the ratio of the two stocks.
A call whose payoff is the square of the stock minus 100 does not start paying at 100, it starts paying at 10, because the square clears the strike exactly when the stock clears its square root. Placing the kink at the written strike prices the option at essentially zero on a stock at 12, when its real value is 53.8168. The closed form is ordinary Black-Scholes on the transformed asset with a growth term of 4 percent and a strike leg that still discounts at the riskless rate, and the value curve sits above intrinsic everywhere while being shallower than it at the spot in question.
An American call that only wakes up at 80 and dies for good at 125 has no closed form, and it cannot be simulated either, because a path runs forward while the exercise decision looks back. Every path that avoids the ceiling either visited the floor or never did, so the contract is one knock-out minus another and both come off a standard tree. The identity is exact to machine precision for European exercise at all seven grids tested, and for American exercise only in the continuous limit: the finite-tree residual falls from 0.532 percent at 45 steps to 0.043 percent at 3,394.
Buying two-year notes because you expect the curve to steepen is a position on the level of rates: the same correct view loses two dollars if the steepening arrives with everything rising. Matching the two legs on dollar duration removes the parallel part of the move as an algebraic identity, so half a point of widening pays four dollars whatever the level does. Matching market value as well is impossible with only two bonds and needs a third leg carrying no duration, and on a full cash-flow reprice the level survives at second order, worth 0.07 against a four-dollar profit at fifty basis points.
A share at 100 that jumps to either 80 or 130 gives a call an exact price of 12, from two equations in two unknowns and no probability at all. Let the jump size be random, so 110 is also reachable, and that same hedge pays 18 where the option pays 10 while no other portfolio does better. The arbitrage-free prices then fill the whole interval from 20/3 to 12, and the obstruction turns out to be the kink in the payoff rather than the number of states.
A riskless zero-coupon bond at 100 has a six-month forward of 102.531512, a premium. Give the same bond an 8% coupon and the forward drops to 98.511194, a discount, because the sign of the premium is the sign of the rate minus the coupon and nothing else. Quoting the forward at spot when the coupon is rich hands the other side a riskless 1.518802 per hundred, and the discrete-coupon version shows the answer also turns on whether a payment date falls before delivery.
An American put struck at 100 on a stock at 100, with no expiry date at all, is worth 23.21 when the rate is 5% and the volatility 30%. Removing the clock removes the time derivative from the pricing equation, which turns it into an ordinary differential equation solved by powers, and the exercise boundary collapses from a curve into the single level 1000/19 = 52.63. Its European twin, which cannot be exercised early, is worth exactly nothing, so every cent of the value is the right to stop.
A futures settles up every day, so its fair price is a plain expectation, while a forward settles once, so its fair price is a discounted expectation renormalised. The difference between the two is exactly the covariance of the discount factor with the contract price divided by the expected discount factor, and for a deposit contract quoted as 100 minus the rate both fall together, so the fair forward price is 95.019999 against the futures' 95.000000. Long the forward and short the futures is worth 0.0195 points at inception, the gap reaches 39.48 basis points at ten years, and on an asset whose price rises with rates the whole answer reverses.
In a rally you want positive convexity, and a mortgage pool has negative convexity, because the borrowers hold the right to prepay and you are short that option. Modelled as a ten-year 6 percent bond minus a three-year call struck at 105, the straight bond has convexity plus 68.8 and the pool minus 177.4, and doubling a rally from 100 to 200 basis points takes the bond from 7.79 percent to 16.35 while the pool goes only from 4.21 to 6.13. The pool still gains, so the reflex is right about the sign and wrong about the size, and the single parameter set in 108 with positive curvature is one whose prepayment option is far out of the money.