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. 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?
2. 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?
3. 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?
4. 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?
5. 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?
6. 034Beyond the five standard Greeks, which second-order sensitivities actually get managed?The GreeksHardsuperdayExotics tradingVolatility trading

   Say this

   Vanna and volga, mainly. Vanna is how delta changes when volatility moves, and it is the same thing as how vega changes when spot moves. Volga is the convexity of vega in volatility. On any book with skew — which is every real book — those two determine whether your vega hedge holds up.

   Then walk it

   1. Vanna matters because implied volatility and spot are correlated. In equities the correlation is strongly negative: spot falls, volatility rises. So a position with vanna gets a second hit at exactly the moment the first one arrives, and the two are not independent risks.
   2. Volga is the reason a vega-neutral book is not volatility-neutral for large moves. Wing options have positive volga, so a long-wings, short-body position is vega-flat and still profits from a volatility spike. Every risk reversal and butterfly carries it.
   3. Charm is the decay of delta with time, and it matters near expiry and on barrier books, where your delta changes overnight without the market moving at all.
   4. The FX market prices in these terms directly. The vanna-volga approach prices an exotic by taking the Black-Scholes value and adding the cost of hedging its vanna and volga with the market-quoted risk reversal and butterfly. It is not elegant, but it reproduces market prices better than a naive smile interpolation.
   5. How desks actually manage it: a risk report showing profit and loss under a grid of spot and volatility shocks, rather than a list of Greeks. The grid captures vanna and volga implicitly and does not require you to trust a Taylor expansion.
   6. And the limitation: these are still local derivatives. A scenario grid with a minus 20 percent spot and plus 30 volatility shock tells you more than any second-order Greek, because in a real dislocation the correlations you assumed break and the higher-order terms are no longer small.

   Where candidates lose it

   Rattling off exotic Greek names without connecting them to spot-volatility correlation. Vanna matters in equities specifically because the skew is one-sided and spot and volatility are negatively correlated. If you cannot say that, the names are decoration.

   Expect next

   • Why is vanna so important in equities specifically?
   • How can a vega-neutral book still profit from a volatility spike?
   • Would you rather have a Greek report or a scenario grid, and why?
7. 038Why does the volatility smile exist?VolatilityHardtechnicalVolatility tradingEquity derivatives

   Say this

   Because the real distribution of returns is not lognormal, and Black-Scholes assumes it is. The market corrects the model by charging a different implied volatility per strike. Fat tails make both wings expensive relative to the body, and in equities negative skewness makes the downside wing much more expensive than the upside — that asymmetry is why the smile is really a smirk.

   Then walk it

   1. Mechanically, implied volatility is just the number you plug into a wrong formula to get the right price. If the true distribution has more mass in the tails than a lognormal, you need a higher volatility input to reproduce the market price of a wing option.
   2. Two distinct effects. Kurtosis — fat tails on both sides — lifts both wings and gives you a symmetric smile, which is roughly what you see in FX on a major pair. Negative skew lifts the left wing only, which is what you see in equity indices.
   3. Why equities are skewed: leverage means falling equity raises the debt-to-equity ratio and therefore the volatility of the remaining equity. Crashes are correlated across names while rallies are not, so index skew is steeper than single-stock skew. And there is a real demand effect — everyone wants to buy puts and sell calls.
   4. So the skew is not purely a distributional statement. Part of it is the price of insurance, which means part of the steepness is a risk premium rather than a forecast of the crash probability.
   5. The models that generate it endogenously are stochastic volatility, where negative spot-volatility correlation produces skew, and jump-diffusion, where a downward jump component produces exactly the left-wing fat tail. Local volatility fits the observed surface exactly but has unrealistic dynamics.
   6. One thing to be careful about: the skew steepened permanently after the 1987 crash. Before it, index implied volatility was roughly flat across strikes. So a large part of the smile is a learned behaviour about crash risk, not a timeless property of returns — which means it can reprice when beliefs change.

   Where candidates lose it

   Explaining the smile as fat tails only. In equities the dominant feature is skew, not kurtosis, and it has three causes — leverage, correlated crashes, and put demand. Also worth saying that the shape post-dates 1987, because it makes clear you understand it is a market convention as much as a statistical fact.

   Expect next

   • Why is index skew steeper than single-stock skew?
   • What did the 1987 crash change about the surface?
   • Which model would you use if you had to price a barrier off this surface?
8. 042If you want pure exposure to volatility, why use a variance swap rather than a straddle?VolatilityHardsuperdayVolatility tradingHedge funds

   Say this

   Because a straddle's exposure to volatility changes as spot moves away from the strike, and a variance swap's does not. The variance swap pays realised variance minus a fixed strike, with constant exposure regardless of where spot goes. It is the clean instrument; the straddle is a path-dependent approximation to it.

   Then walk it

   1. The straddle problem: gamma and vega are concentrated at the strike. Spot moves 10 percent and your straddle is now a directional position with little volatility exposure left, so you have to keep re-striking to maintain the view.
   2. A variance swap pays the notional times realised variance less the strike variance. No re-striking, no delta to manage in the same way, and the payoff is linear in variance by construction.
   3. How it exists at all: you can replicate it statically with a portfolio of options across all strikes weighted by one over strike squared, plus a dynamic futures hedge. That replication is why it can be quoted without a model.
   4. The catch, and it is a big one: variance is the square of volatility, so the payoff is convex in volatility. A short variance position loses quadratically. At a 20 strike, realised of 60 is nine times the variance, not three times — which is how short variance books were destroyed in 2008.
   5. Which is why the market largely moved to capped variance swaps after 2008, typically capped at 2.5 times the strike. A volatility swap — linear in volatility rather than variance — is the other answer, but it needs a model to price because it is not statically replicable.
   6. And one practical limitation: the replication needs a continuum of strikes. In reality you have a finite strike grid, so a genuine jump produces a payoff the replicating portfolio did not deliver. Single-stock variance swaps on names with takeover risk are notorious for this, which is why dealers price them wide or refuse them.

   Where candidates lose it

   Saying 'variance swaps give pure volatility exposure' without naming the convexity. Variance is the square, so short positions lose non-linearly, and the 2008 blowups plus the move to capped structures are the evidence. That detail is what makes the answer sound like it came from a desk.

   Expect next

   • So why did the market start capping them?
   • What is the difference between a variance swap and a volatility swap?
   • Why are single-stock variance swaps dangerous for a dealer?
9. 060A five-year CDS trades at 400 basis points. What does that tell you about the probability of default?Credit derivativesHardsuperdayCredit tradingRisk management

   Say this

   Roughly a 6 to 7 percent annual risk-neutral default probability, using the rule of thumb that spread equals default probability times loss given default. At a 40 percent recovery, 400 over 0.6 is about 667 basis points a year, so around 28 percent cumulative over five years. But that is a risk-neutral number, and it is meaningfully higher than the real-world probability.

   Then walk it

   1. The approximation: spread is approximately the hazard rate times one minus recovery. Invert it — hazard rate equals spread over loss given default. With 40 percent recovery assumed, 400 basis points implies about 6.7 percent a year.
   2. Cumulative over five years, compounding the survival probability: 0.933 to the fifth is about 0.71, so roughly a 29 percent chance of default over the life.
   3. The recovery assumption does a lot of work here. Assume 20 percent recovery instead and the implied hazard rate drops to 5 percent. So you cannot read a default probability out of a spread without a recovery view, and the two are jointly unidentified from the spread alone.
   4. The bigger point: this is a risk-neutral probability, which embeds a risk premium. Empirically risk-neutral default probabilities run perhaps two to three times realised default rates for investment grade credits, because investors demand compensation for default risk being correlated with bad times. Quoting 29 percent as the actual chance the company fails would be wrong.
   5. The spread also contains things that are not default risk: liquidity premium, the cost of dealer balance sheet, and for index CDS, the demand for macro hedges. In a stress event, spreads widen more than any credible reassessment of default odds justifies.
   6. So how I would use the number: as a market-implied ranking and a hedging cost, not as a forecast. If I wanted the real-world probability I would look at rating agency transition matrices or a structural model like Merton, and I would expect a considerably lower figure — and the gap between the two is itself the credit risk premium I might want to harvest.

   Where candidates lose it

   Quoting the risk-neutral number as the probability of default. The whole test is whether you know that the spread contains a risk premium, a liquidity component and a recovery assumption. Do the arithmetic, then correct it out loud.

   Expect next

   • How sensitive is your answer to the recovery assumption?
   • How would you get a real-world default probability instead?
   • Why do risk-neutral and real-world probabilities differ by so much?
10. 061What is the CDS-bond basis, and what does it mean when it goes negative?Credit derivativesHardsuperdayCredit tradingHedge funds

    Say this

    The basis is the CDS spread minus the bond's credit spread on the same issuer and maturity. In theory it should be near zero, because buying the bond and buying protection creates a near-riskless position. A negative basis — CDS cheaper than the cash bond spread — means the cash market is under stress and nobody has the balance sheet to arbitrage it.

    Then walk it

    1. The arbitrage in principle: buy the bond, buy CDS protection to the same maturity, and you have hedged default risk. The residual spread you earn should be roughly the risk-free rate, so a large positive residual is a negative basis trade.
    2. Why it does not get arbitraged away: the trade needs funding for the bond position and balance sheet at a dealer. In a crisis funding is expensive or unavailable, so the trade is theoretically profitable and practically impossible. That is why the basis went to hundreds of basis points negative in late 2008 and again in March 2020.
    3. So a deeply negative basis is a funding stress indicator, not a credit signal. It measures the scarcity of balance sheet rather than the probability of default.
    4. Structural reasons for a non-zero basis in normal times: the cheapest-to-deliver option in CDS makes protection worth slightly more than a specific bond's spread; CDS is unfunded so it attracts different investors; and CDS documentation covers restructuring events that a bond spread does not price identically.
    5. A positive basis usually reflects demand for protection that cannot easily be expressed in cash — for example when bonds are impossible to borrow, so the negative view has to be taken in CDS.
    6. The limitation for anyone thinking of the trade: it is not riskless. You carry the counterparty on the protection leg, you carry funding and margin risk that can force you out, and you carry basis risk on the exact maturity and deliverable. The 2008 version of this trade destroyed funds who were right about the convergence and could not survive the path.

    Where candidates lose it

    Calling a negative basis an arbitrage. The point is precisely that it is not — it persists because funding and balance sheet are the binding constraint, and that is why the size of the basis is a stress gauge. Say that, and name March 2020.

    Expect next

    • So why does the arbitrage not close?
    • What does a persistently positive basis tell you?
    • How would you fund and size a negative basis trade?

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.