Derivatives Foundation interview preparation

Questions

100

Traced to a firm

29

Firms

19

Updated

September 2026

Asked at

All firmsMSMorgan Stanley4Nomura4Akuna Capital2Amundi2HSBC2PIMCO2Bank of America1Barclays1Citadel1DRW1Goldman Sachs1Jane Street1Millennium Management1Mizuho1Old Mission Capital1RCRBC Capital Markets1Scotiabank1UBS1Wells Fargo Securities1

Topic

All topicsForwards and futures10Options basics8Option pricing7The Greeks10Volatility7Option strategies9Swaps and rates7Credit derivatives4Market structure and clearing6Indian derivatives8Trading and markets9Brainteasers6Fit9

Level

AnyCoreIntermediateHard

Type

AnyTechnicalCaseMarket viewBrainteaserFit

1. 006You run a 50 million dollar equity portfolio with a beta of 1.2. Index futures are at 5,000 with a 50 dollar multiplier. Hedge it, and tell me what you are left with.Forwards and futuresHardtechnicalEquity derivativesAsset management

   Say this

   Short 240 contracts. One contract is 5,000 times 50, so 250,000 dollars of notional. You need beta times portfolio value of index exposure, which is 1.2 times 50 million, or 60 million, and 60 million divided by 250,000 is 240. What you are left with is the alpha, plus basis risk, plus the fact that beta is an estimate.

   Then walk it

   1. Contract notional first: 5,000 index points times the 50 dollar multiplier is 250,000 dollars per contract. Always state this before dividing, because it is where candidates drop a factor.
   2. Number of contracts equals beta times portfolio value over contract notional. 1.2 times 50 million is 60 million of index-equivalent exposure; divided by 250,000 that is 240 contracts, sold.
   3. Check the hedge does what you want. If the index falls 10 percent, your book falls about 6 million on a 1.2 beta, and the short 240 contracts gain 60 million times 10 percent, which is 6 million. Flat, by construction.
   4. The minimum-variance version is more honest than beta from a regression on the wrong window: h equals the correlation times the ratio of the standard deviations, which is the same thing as the slope of portfolio returns on futures returns. Estimate it on the horizon you actually intend to hedge.
   5. What remains: idiosyncratic return, which is the point if you think you can pick stocks. Plus basis risk between the futures and the cash index, dividend risk in the futures basis, and the cash drag of posting margin.
   6. And beta drifts. It is unstable across regimes and it rises in crashes, so the hedge that looks right in calm markets under-hedges in the event you bought it for. I would re-estimate and adjust rather than set it once.

   Where candidates lose it

   Forgetting the multiplier, or hedging notional rather than beta-adjusted notional. On a 1.2 beta portfolio, hedging 50 million instead of 60 leaves you a fifth under-hedged. Say the two-step — beta-adjust, then divide by contract notional — out loud so the interviewer can follow.

   Expect next

   • Would you use futures or buy puts, and how would you choose?
   • Your beta was estimated over three years. What if the market regime just changed?
   • What is left in the portfolio after the hedge, and is that what you wanted?
2. 007A refiner wants to hedge crude purchases for the next three years but only the front months are liquid. What do you do, and what could go wrong?Forwards and futuresHardcase studyCommodities tradingCorporate treasury

   Say this

   Stack the whole exposure in the liquid front contracts and roll it forward each month, or use a smaller strip out the curve and accept a partial hedge. Either way the thing that kills you is not price — it is the funding of variation margin on a position that is economically flat.

   Then walk it

   1. The stack-and-roll: put on the full three years of notional in the front two contracts, then roll month by month. You get liquidity and a tight bid-offer, but you take the roll basis twenty-plus times.
   2. The strip alternative: sell what you can in each maturity out to three years, accepting wide spreads and a smaller hedge ratio. Less basis risk, more transaction cost, and possibly no liquidity at all beyond eighteen months.
   3. The funding problem is the real answer. Your futures leg settles in cash daily. Your physical purchases happen over three years. If crude rallies, you fund margin calls today against a benefit that arrives in 2029.
   4. This is exactly what sank Metallgesellschaft in 1993. The hedges were economically sound, but the position was stacked in the front, the curve went from backwardation to contango, and the margin calls ran to over a billion dollars. They closed the hedges near the bottom.
   5. So the practical structure: size the stack to what you can fund under a stress scenario, arrange a committed credit line specifically for margin, and pre-agree with the board what a mark-to-market loss on a hedge means, so nobody panics at the wrong moment.
   6. I would also swap some of it into an OTC commodity swap with a bank. You give up the clearing-house credit protection and pay a wider spread, but the collateral terms are negotiable under a CSA, which is precisely the problem you are trying to solve.

   Where candidates lose it

   Answering only with the mechanics of stacking and rolling. The interviewer is fishing for the funding-liquidity failure — a perfect hedge that gets closed out because of a cash call. Name Metallgesellschaft or an equivalent, and say how you would size the position to survive it.

   Expect next

   • How would you size the position so a margin call cannot force you out?
   • Would you rather hedge with an OTC swap? What do you give up?
   • How do you explain a 200 million mark-to-market loss on a hedge to a CFO?
3. 010In April 2020 WTI settled at minus 37 dollars. How can a price be negative, and how does that break the models?Forwards and futuresHardsuperdayCommodities tradingClearing and risk

   Say this

   Because WTI is physically delivered at Cushing, and if every tank is full, taking delivery costs you money. A negative price is just storage scarcity expressed as a price: holders were paying to not receive barrels they had nowhere to put. It broke two things — models that assume lognormal prices, and margin systems built on percentage moves.

   Then walk it

   1. Mechanics first: the May contract required physical delivery at Cushing. Demand had collapsed, storage was effectively sold out, and long holders facing delivery with no tank had to pay someone to take the contract.
   2. So convenience yield went sharply negative. The carry identity still holds — it is the storage term that exploded, because the marginal unit of storage was unobtainable at any price.
   3. The modelling failure: lognormal price dynamics, which is what Black-Scholes and most commodity option models assume, put zero probability on a negative price. Every option model on the screen was undefined the moment the price crossed zero.
   4. The industry response was to shift crude options to the Bachelier model, where prices are normally distributed and can go negative, and to quote volatility in dollars rather than percent. Rates desks had already done this when European yields went negative in 2015.
   5. The clearing consequence was worse. Margin models scaled to percentage moves cannot size risk on a price near zero, and several brokers had systems that could not even represent a negative price. Retail products tracking the front contract — including a large Chinese bank's oil product — took catastrophic losses.
   6. The lesson I would draw is about the delivery mechanism rather than oil. A financially settled contract on the same underlying did not go negative in the same way. Physical delivery is what converts a full tank into a price, and any contract with physical settlement can do this.

   Where candidates lose it

   Treating it as a freak event with no lesson. The point is that the model assumption — prices cannot be negative — was an assumption, not a fact, and it was load-bearing in every option pricer and margin system. Name the switch from lognormal to Bachelier.

   Expect next

   • How do you price an option when the underlying can be negative?
   • Why did the financially settled contract behave differently?
   • What should a clearing house change after an event like that?
4. 017An option buyer has limited loss and unlimited gain. Why does anyone take the other side?Options basicsIntermediatetechnicalProp trading firmsMarket making

   Say this

   Because the seller gets paid, and on average the premium is more than the payout. Buyers are buying insurance and insurers earn a premium for bearing tail risk. The payoff shape is asymmetric but the expected value is not the same as the shape.

   Then walk it

   1. The empirical fact first: implied volatility exceeds subsequent realised volatility most of the time, in most markets. That wedge — the variance risk premium — is the seller's edge, and it is typically a couple of volatility points on index options.
   2. Why it exists: investors are structurally short the market and want protection, and protection pays off precisely when the rest of the portfolio is losing. That correlation makes it worth paying above fair value for, exactly like fire insurance.
   3. So the seller is not stupid, they are an insurer. The trade works the way insurance works: small steady income, occasional large loss, positive expectancy if priced right and sized right.
   4. The real risk is not the expectancy, it is the path. A short option book has negative skew, so it grinds up and then gives back years of premium in a week. February 2018 wiped out short-vol products in a single session on a move that was not even a large one by historical standards.
   5. Which is why sellers hedge. A market maker is not taking a directional view; they sell the option, delta hedge it, and try to earn the spread between implied and realised volatility. The naked seller and the hedged seller are completely different businesses.
   6. The honest framing: buyers pay for convexity and certainty of maximum loss, sellers earn a premium for supplying it. Neither is a free lunch, and the seller's version has a fatter left tail than a Sharpe ratio will show you.

   Where candidates lose it

   Answering 'because most options expire worthless'. That is a statistic about frequency, not about expected value, and an interviewer will immediately ask whether you would sell 1-in-1000 lottery tickets at any price. Name the variance risk premium and the negative skew of the seller's return.

   Expect next

   • So would you rather be systematically long or short volatility?
   • What happened to short-volatility products in February 2018?
   • How does a market maker sell options without taking a directional view?
5. 020Price a one-period call with a binomial tree. Stock at 100, up to 120 or down to 80, strike 100, rate zero.Option pricingIntermediatetechnicalProp trading firmsQuant trading

   Say this

   Ten. The risk-neutral up probability is 0.5 because the up and down moves are symmetric around 100 with a zero rate, so the option is worth 0.5 times 20 plus 0.5 times 0, undiscounted. And I can prove it with a hedge rather than a probability.

   Then walk it

   1. Risk-neutral probability: p equals one plus r minus d over u minus d. With u of 1.2, d of 0.8 and r of zero, p is 0.2 over 0.4, so 0.5. Payoffs are 20 up and 0 down, so the value is 10.
   2. Now the replication, which is the answer they actually want. Delta is the payoff spread over the price spread: 20 minus 0 over 120 minus 80, so 0.5. Hold half a share, which costs 50, and borrow 40. In the up state the half share is worth 60, repay 40, net 20. In the down state 40 minus 40 is zero. Both match, and the portfolio cost 10.
   3. So the hedge ratio and the price come out of the same arithmetic, and the 0.5 that appears twice is a coincidence of the symmetric tree — one is a delta, the other a probability.
   4. Note that p is not a forecast. If I told you the stock has a 90 percent chance of going up, the option is still 10, because I can hedge it. The real probability affects whether you want the trade, not what it costs.
   5. Add a rate and both pieces move: p shifts up because the risk-neutral drift is higher, and you discount the expectation. With r at 5 percent p becomes 0.625 and the value rises to about 11.9.
   6. The limitation worth saying: a one-period tree is a cartoon. It only works because two states and two instruments make the market complete. Add a third state and I can no longer hedge exactly, and the price becomes a range rather than a number — which is the real world with jumps in it.

   Where candidates lose it

   Using the real-world probability, or averaging 120 and 80 to get an expected stock price and working from there. Also, do the replication: the interviewer wants to see you derive delta as the ratio of payoff spread to price spread, not quote a formula.

   Expect next

   • Redo it with the rate at 5 percent.
   • What if I tell you the real probability of the up move is 90 percent?
   • Now make it two periods and tell me what changes about the hedge.
6. 024When would you use Monte Carlo rather than a closed form or a tree, and what goes wrong with it?Option pricingHardtechnicalQuant tradingStructured products

   Say this

   Monte Carlo when the payoff is path-dependent or there are several underlyings, because both break closed forms and blow up a tree. Its weaknesses are slow convergence, difficulty with early exercise, and the fact that it will happily give you a confident answer to a badly specified model.

   Then walk it

   1. Use it for Asian options where the payoff depends on an average, barriers where it depends on whether a level was touched, and baskets or worst-of structures where the dimensionality kills a tree.
   2. A tree is fine up to two or three factors and is the right tool when you need early exercise, because you can compare continuation against exercise at every node. Monte Carlo runs forward, so American features need something like Longstaff-Schwartz regression, which is doable but adds its own error.
   3. Convergence is the headline cost. The standard error falls as one over root N, so cutting your error in half needs four times the paths. Getting a Greek to three decimal places on a complex payoff is genuinely expensive in compute.
   4. The fixes: antithetic variates, control variates where you simulate a similar payoff with a known closed form and correct by the difference, and quasi-random low-discrepancy sequences. A good control variate is often worth more than a hundred times the paths.
   5. Discretisation bias is the subtle one. Barriers are systematically mispriced by daily time steps, because the simulated path can cross and return between observations. You either use a fine grid, a Brownian bridge correction, or you accept a bias you can measure.
   6. And the real danger, which is not numerical at all: the simulation is only as good as the process you assumed. A Monte Carlo on a geometric Brownian motion gives you a precise answer to a model that has no jumps in it. Precision is not accuracy, and a tight confidence interval around a wrong model is how structured products get mispriced.

   Where candidates lose it

   Treating this as a pure numerical-methods question. The answer that lands names the American-exercise difficulty, gives a variance reduction technique by name, and finishes on model risk — a narrow confidence interval around the wrong dynamics.

   Expect next

   • How would you handle an American feature in a simulation?
   • Name a variance reduction technique and say what it buys you.
   • How would you compute a Greek in a Monte Carlo without four times the runtime?
7. 025How would you price and risk-manage a down-and-in barrier put?Option pricingHardsuperdayStructured productsEquity derivatives

   Say this

   Price it off the vanilla surface using the in-out parity relationship, then adjust for the fact that a barrier is enormously sensitive to skew and to the dynamics near the barrier. The hard part is not the price, it is that delta and gamma become discontinuous at the barrier, so the hedge is unstable exactly where you need it.

   Then walk it

   1. Start with the identity: a down-and-in put plus a down-and-out put with the same strike and barrier equals a vanilla put. That gives you a sanity check and lets you price the harder one from the easier one.
   2. The naive route is a closed form under Black-Scholes with constant volatility. It is wrong in a specific direction, because a barrier payoff depends on the whole distribution near the barrier, which is precisely where the skew lives. You have to price it on a model calibrated to the smile — local volatility at minimum, stochastic volatility if the book is big.
   3. Then the risk. Just above the barrier the knock-in has almost no value; just below it is a live vanilla put. So delta jumps, and gamma is effectively infinite at the barrier. Hedging through it means trading a large amount of stock in a market that is already moving.
   4. That discontinuity is why desks apply a barrier shift — pricing as if the barrier were slightly further away — to build in the cost of the hedging error. It is a reserve dressed up as a model input, and it should be sized to the liquidity of the underlying.
   5. Pin risk near expiry compounds it. A barrier close to spot in the last days combines the discontinuity with almost no time to hedge, which is when the losses actually happen.
   6. The commercial context worth naming: these sit inside autocallable notes sold to retail and private banking clients, which is where most barrier risk in the world lives. The client is short a knock-in put and often does not know it, and in a sharp drawdown the whole book knocks in at once — the dealers end up with correlated, one-way risk, which is exactly what happened to Korean autocallable books in early 2020.

   Where candidates lose it

   Pricing it with a Black-Scholes closed form and stopping. The interviewer wants the two hard parts: skew dependence, because a barrier reads the wing of the distribution, and hedge instability at the barrier. Naming the barrier shift shows you have seen how a desk actually handles it.

   Expect next

   • Why is a barrier option so much more skew-sensitive than a vanilla?
   • How do you hedge through the barrier in an illiquid name?
   • Where does barrier risk actually sit in the market, and why does it correlate?
8. 031You have sold a call and you are delta hedging it. Walk me through what you actually do over the option's life.The GreeksIntermediatetechnicalMarket makingEquity derivatives

   Say this

   You buy delta shares against the short call and rebalance as spot moves. Because you are short gamma, every rebalance means buying higher and selling lower, so you lose money on the hedge and you are paid theta to compensate. Over the life, your profit is the premium you collected less what the hedging actually cost you.

   Then walk it

   1. Day one: sell the 100-strike call at, say, 4.00 with a 0.5 delta, so buy 50 shares per contract. You are locally flat.
   2. Stock rises to 105. Delta is now 0.65, so you buy 15 more shares at 105. Stock falls back to 100, delta is 0.5, so you sell 15 shares at 100. You have just bought at 105 and sold at 100. That loss is what being short gamma means, and it is unavoidable.
   3. Do that repeatedly and the total hedging loss is roughly proportional to the realised variance of the stock. You keep money only if the stock realises less volatility than the 20-ish implied you sold at.
   4. So the profit and loss decomposition is clean: premium received, minus the realised variance cost, plus or minus the error from hedging discretely rather than continuously, minus bid-offer and financing on the share position.
   5. How often to rebalance is a real decision, not a technicality. Hedge too often and transaction costs eat you; too rarely and you run naked gamma between hedges. Desks usually hedge on a delta band — rebalance when delta moves more than some threshold — rather than on a clock.
   6. The failure mode to state: a gap. If the stock jumps from 100 to 130 overnight on a takeover, no rebalancing schedule saves you, because you were hedged for the 0.65 delta and you needed 1.0. Delta hedging manages diffusion risk, not jump risk, and that is exactly the assumption Black-Scholes makes and reality does not.

   Where candidates lose it

   Describing the mechanics without ever stating that the hedge loses money. Short gamma means the rebalancing is systematically adverse, and the theta you collect is the payment for it. If your answer does not contain 'buy high, sell low', you have not understood the trade.

   Expect next

   • How often would you rebalance, and what decides it?
   • Decompose your final profit and loss into its pieces.
   • The stock gaps 30 percent overnight. What happens to you?
9. 032You are long a one-month at-the-money straddle at 20 volatility. Under what conditions do you make money?The GreeksHardtechnicalVolatility tradingProp trading firms

   Say this

   If you delta hedge it, you make money when realised volatility over the month exceeds 20. If you do not hedge it, you make money only if the stock finishes far enough from the strike to cover the combined premium, which is a much harder bar. Those are two completely different trades and the distinction is the answer.

   Then walk it

   1. Unhedged: you paid, say, 4.6 percent of spot for the straddle, so the stock must move more than 4.6 percent in either direction by expiry. Direction does not matter, distance does — and crucially, the path does not help you at all.
   2. Hedged: you rebalance delta daily and harvest the moves. Now the path is everything. A stock that oscillates 2 percent a day and finishes flat pays you handsomely while the unhedged straddle expires worthless.
   3. The break-even in the hedged case is the realised-versus-implied comparison. Twenty percent annualised is about 1.25 percent a day. If the stock is genuinely moving more than that, gamma harvesting beats the theta you bleed.
   4. Arithmetic to make it concrete: 20 volatility on a 30-day option costs roughly 20 times root of 30 over 365, about 5.7 percent of spot for the straddle at a 0.8 scaling — call it 4.5 to 5 percent. So you need about a 5 percent move unhedged, or sustained daily moves above 1.25 percent hedged.
   5. Two things can still beat you even if you are right about realised volatility. Implied volatility can fall, which hits your vega mark immediately, and the realised moves can arrive as one gap rather than as daily oscillation — a gap gives you the payoff once rather than repeatedly.
   6. The honest limitation: long gamma is not a free option on chaos. You pay theta every day, and in a market that grinds quietly for three weeks and then explodes on day 25, you may have been stopped out of the position before the payoff arrives. Sizing and horizon matter as much as the volatility view.

   Where candidates lose it

   Answering only the unhedged version — 'the stock has to move more than the premium'. Any interviewer on a volatility desk is testing whether you know the hedged straddle is a bet on realised variance and the unhedged one is a bet on the terminal price. Say both, and say which one you meant.

   Expect next

   • What if the stock moves 5 percent but in one overnight gap?
   • You were right about realised volatility and still lost money. How?
   • Would you rather own the straddle or a variance swap for this view?
10. 033What is pin risk, and how do you manage a large position into expiry?The GreeksHardsuperdayMarket makingEquity derivatives

    Say this

    Pin risk is the risk that the underlying closes almost exactly at your strike, so you do not know whether you will be assigned. You go into the weekend not knowing whether you are flat or hugely long or short stock, and by the time you find out, the market has moved. It is a settlement risk, not a pricing risk.

    Then walk it

    1. The mechanism: you are short 1,000 at-the-money calls and the stock settles at the strike. If they are exercised you are short 100,000 shares; if not you are flat. You cannot hedge a position you do not know you have.
    2. Gamma explodes into expiry for at-the-money options, so your delta swings between near zero and near one on tiny price moves. The hedge you put on at 3:29 can be completely wrong at 3:30.
    3. How desks manage it: reduce the at-the-money position before the last hour, close out rather than let it go to assignment, and avoid being short large size at a strike where open interest is concentrated.
    4. Cash settlement solves it. Indian index options — Nifty and Bank Nifty — are cash-settled on a weighted average of the last half hour, so there is no assignment ambiguity. But cash settlement creates a different problem: settlement-price manipulation risk, which is exactly why SEBI moved to a VWAP of the closing period rather than a closing print.
    5. That closing-period mechanic is why you see the volume spike into the last thirty minutes on expiry day in India. Large positions need to hedge against the same average that determines their settlement.
    6. The broader lesson to volunteer: pin risk is one of a family of expiry-day operational risks — assignment, settlement price, and the exercise cut-off being after the market closes so you can be assigned on news that broke post-close. These are the risks that lose money on well-hedged books, and they are the reason expiry-day process discipline exists.

    Where candidates lose it

    Explaining pin risk as a pricing phenomenon. It is an operational and settlement risk, and the answer should include what you do about it — reduce size, close rather than assign, and know whether your contract is cash or physically settled. Naming the Indian cash-settlement mechanic shows local knowledge.

    Expect next

    • How does cash settlement change the problem, and what new problem does it create?
    • Why does volume spike in the last half hour of an Indian expiry day?
    • You are assigned on news that broke after the close. What is your exposure?

Firm tags come from public, anonymous candidate reports on Wall Street Oasis: strong signal, not sworn testimony. Firms are named as the places a question was reported, not as partners of Fin Maverick. Answers are written for this page to show how to think out loud; they are not scripts to recite.