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. 008Is a futures price a forecast of where spot will be?Forwards and futuresHardtechnicalCommodities tradingProp trading firms

   Say this

   No. A futures price is the spot price plus carry, set by arbitrage. It only equals the expected future spot price if the asset carries no risk premium, which is rarely true. Confusing the two is how people talk themselves into thinking a contango curve is a bullish forecast.

   Then walk it

   1. For a storable financial asset, the futures price is pinned by replication. It contains no view at all: spot times the carry factor, and nothing else fits without an arbitrage.
   2. The expected spot price is a different object. It equals the futures price plus whatever risk premium hedgers are paying to lay off the risk. Keynes called the version where futures sit below expected spot normal backwardation.
   3. So the curve tells you about carry and inventory, not direction. A steep contango in oil says storage is full and financing is expensive; it is not the market predicting a rally.
   4. Empirically the futures curve is a poor forecaster, and that is exactly what makes carry strategies work. If futures were unbiased forecasts, there would be no systematic return to being long backwardated markets and short contango ones.
   5. Where the distinction bites in practice: a long commodity index investor in a 10 percent annualised contango loses roughly that much to roll before spot has moved at all. People buy the index expecting spot exposure and get spot minus carry.
   6. The caveat: for a non-storable like electricity, or for VIX futures where there is no arbitrage to hold the underlying, the curve does carry genuine expectational content, because replication is impossible and the price is set by supply and demand for the risk.

   Where candidates lose it

   Saying yes because the curve is upward-sloping and that must mean the market expects higher prices. It is the single most common misread of a futures curve. Lead with no, give the replication argument, then concede where the curve does contain a forecast.

   Expect next

   • So why do carry strategies earn a return?
   • Where does the curve genuinely contain expectations rather than carry?
   • What does a steep VIX contango tell you, and how do short-vol products harvest it?
4. 009When is a futures price different from a forward price on the same asset?Forwards and futuresHardtechnicalRates derivativesProp trading firms

   Say this

   When interest rates are random and correlated with the asset. The daily margin on a future means your gains get reinvested and your losses get funded at the prevailing short rate, so the correlation between the asset and rates has value. Positive correlation makes the future worth more than the forward; negative correlation the reverse.

   Then walk it

   1. With deterministic rates the two prices are identical. The proof is a replication where you scale the futures position by the discount factor each day, which is only possible if you know that factor in advance.
   2. Introduce stochastic rates and the asymmetry appears. If the asset tends to rise when rates rise, a long future receives variation margin exactly when it can be reinvested at a high rate, and pays it when funding is cheap. That is worth something, so the futures price sits above the forward.
   3. Negative correlation flips it, which is why this matters most in fixed income: bond prices fall when rates rise, so the correlation is strongly negative and the gap is real rather than theoretical.
   4. The size depends on the correlation, the volatility of rates and the maturity. On a three-month contract it is a rounding error. On a long-dated Eurodollar or SOFR strip it is material enough to have its own name: the convexity adjustment.
   5. That adjustment is why you cannot read a swap curve straight off futures. A futures strip needs a convexity correction before it gives you the forward rates a swap is priced off, and in the 1990s misunderstanding this was a genuine source of losses.
   6. Second-order effects also drive a wedge: the future is collateralised and the forward may not be, so the forward carries a credit and funding charge — CVA and FVA — that has nothing to do with rates at all.

   Where candidates lose it

   Saying the two are always equal because the textbook proof says so. The proof assumes deterministic rates and the interviewer knows it. Name the convexity adjustment and say where it is large enough to matter, which is long-dated rates.

   Expect next

   • Which way does the convexity adjustment go for a Eurodollar future, and why?
   • Why can you not read forward rates directly off a futures strip?
   • How do collateral and funding costs drive a separate wedge between the two?
5. 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?
6. 018Give me the bounds on a European call price without using any model.Options basicsHardtechnicalProp trading firmsMarket making

   Say this

   The price sits between max of zero and spot minus the discounted strike, and spot itself. Below that lower bound or above spot, there is a static arbitrage that does not depend on any model, any distribution or any volatility assumption.

   Then walk it

   1. Upper bound: a call can never be worth more than the stock, because the most it ever delivers is the stock, and only after you pay the strike. If a call traded above spot I would sell the call, buy the stock, and be guaranteed a profit.
   2. Lower bound: the call must be worth at least spot minus the present value of the strike. If it were cheaper, I buy the call, short the stock, invest the proceeds. At expiry I exercise or buy in the market, and I have locked in the gap risk-free.
   3. Note the discounting in the lower bound. Naive intrinsic, spot minus strike, is the wrong floor for a European option, and getting that right is the point of the question.
   4. Beyond bounds there are model-free shape constraints. Call prices must be decreasing in strike, and convex in strike — the butterfly constraint. A violation means a butterfly spread with a negative cost and a non-negative payoff.
   5. And calendar monotonicity: a longer-dated call cannot be cheaper than a shorter-dated one at the same strike, for European options on a non-dividend payer.
   6. These constraints are what production systems police. An arbitrage-free volatility surface is defined by exactly these inequalities, and a fitted surface that violates butterfly convexity will let a trader book a position the risk system prices as free money. That is why the checks run before the model does.

   Where candidates lose it

   Giving spot minus strike as the lower bound. The discounting is the whole test. And the strong follow-up is the convexity-in-strike condition — if you can name the butterfly argument, you have said something most candidates cannot.

   Expect next

   • Why must call prices be convex in strike?
   • Show me the arbitrage if they are not.
   • How do those constraints get used when fitting a volatility surface?
7. 022List the Black-Scholes assumptions and tell me where each one fails.Option pricingHardtechnicalQuant tradingRisk management

   Say this

   Six that matter: lognormal returns with constant volatility, continuous trading with no transaction costs, no jumps, a constant known risk-free rate, no dividends or a known continuous yield, and European exercise. Every one of them fails, and the market patches each failure in a different way — which is why the volatility surface has the shape it does.

   Then walk it

   1. Constant volatility fails first and worst. Volatility clusters and mean-reverts, so a single number cannot price every strike and maturity. The market's patch is to quote a different implied volatility per strike and expiry, which is the smile and the term structure.
   2. Lognormality fails in the tails. Real return distributions are fat-tailed and negatively skewed — a 5 standard deviation daily move should be a once-in-millennia event and happens every few years. The patch is the skew: out-of-the-money puts trade at higher implied volatility because the model underprices crashes.
   3. No jumps fails on any event date. A takeover or an earnings gap moves price discontinuously, so a delta hedge does not protect you through it. The patch is jump-diffusion and local-volatility models, and in practice a trader simply pays up for gamma into events.
   4. Continuous costless hedging fails always. You hedge at discrete intervals and pay bid-offer, so your realised profit and loss has a hedging error term whose size scales with your rebalance frequency. The patch is to widen the spread you quote.
   5. Constant rates and known dividends fail on long-dated equity options, where dividend uncertainty is a first-order risk and you need a stochastic rate model. For short-dated index options neither matters much.
   6. And European exercise fails for single-stock American options, where dividends make early exercise rational. That needs a tree or a numerical method, not a formula. The meta-point: Black-Scholes is used as a quoting convention rather than a belief, and the smile is the accumulated record of everywhere it is wrong.

   Where candidates lose it

   Listing assumptions without connecting each to an observable market feature. The strong version of this answer says 'fat tails plus no jumps equals the skew you see on the screen'. If you cannot link the failure to the smile, you are reciting a textbook page.

   Expect next

   • Which single assumption would you relax first if you were building a pricer?
   • So is the smile a model failure or market information?
   • How does discrete hedging show up in your profit and loss?
8. 023What do d1 and d2 actually mean in Black-Scholes?Option pricingHardsuperdayQuant tradingProp trading firms

   Say this

   N of d2 is the risk-neutral probability the option finishes in the money. N of d1 is the delta of the call, and it is also the probability of finishing in the money under a measure where the stock, rather than cash, is the numeraire. The gap between them, which is the volatility times root time, is why delta is not the same as the probability of exercise.

   Then walk it

   1. Read the formula as two pieces. The discounted strike times N of d2 is what you expect to pay, weighted by the chance you pay it. Spot times N of d1 is what you expect to receive, and it has to be weighted by the value of the stock conditional on exercise, not just the chance of it.
   2. That conditioning is the whole distinction. When the option finishes in the money, the stock is on average higher than its unconditional expectation, so the receipt leg needs a bigger weight. Hence d1 exceeds d2 by sigma root t.
   3. d2 equals the log of forward over strike, minus half the variance, all divided by sigma root t. It is a standardised distance to the strike measured in volatility units — a z-score.
   4. N of d1 being the delta is not a coincidence, it is the hedging argument: the number of shares you hold equals the probability-weighted claim on the stock.
   5. Useful desk consequence: a 25-delta option does not have a 25 percent chance of expiring in the money. For a call the probability is lower than the delta, and the gap widens with volatility and maturity. Retail traders systematically get this wrong when they size positions off delta.
   6. The caveat: all of this is under the risk-neutral measure, so N of d2 is not the real-world probability of exercise. Real-world probability needs the actual drift, which nobody knows. Quoting risk-neutral probabilities as real ones is the standard error.

   Where candidates lose it

   Calling N of d1 a probability without saying under which measure, or claiming delta is the chance of expiring in the money. The interviewer is specifically testing whether you know delta and probability of exercise are different numbers, and why.

   Expect next

   • So is a 25-delta call's probability of exercise higher or lower than 25 percent?
   • Why is d1 greater than d2 by exactly sigma root t?
   • Is N of d2 the real-world probability of exercise?
9. 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?
10. 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?

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.