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. 021Explain Black-Scholes.Option pricingIntermediatetechnicalGoldman SachsWealth Management · Zurich · 2025

   Say this

   It is a closed-form price for a European option, derived from the insight that an option can be perfectly hedged with the underlying. If you can hedge it, its price cannot depend on your view or on risk appetite — only on volatility, time, rates and the distance to the strike. The formula is the answer to that hedging argument, not a forecasting model.

   Then walk it

   1. The derivation in one line: build a portfolio that is long the option and short delta of the stock, and the random term cancels. What is left grows at the risk-free rate, and imposing that gives a differential equation whose solution is the formula.
   2. In words, the call price is the discounted expected payoff under a lognormal distribution: spot times N of d1, less the discounted strike times N of d2. N of d2 is roughly the risk-neutral probability of finishing in the money; spot times N of d1 is the expected value of the stock you receive if you do.
   3. So the inputs are spot, strike, time, rate, dividend and volatility. Five are observable. Volatility is not, which means in practice the formula is used inverted — you put the market price in and read the implied volatility out.
   4. That is its real job on a desk. Nobody believes the assumptions. It is a translation device that turns option prices in dollars into a single comparable number in volatility terms, so you can compare a one-month Nifty option with a two-year S&P option on the same axis.
   5. For a private client I would put it plainly: the formula prices the insurance. The further out of the money, the shorter the time and the calmer the market, the cheaper the insurance — and the fair price of that insurance is what the model gives you.
   6. The limitation, said before being asked: it assumes constant volatility and continuous, costless hedging, and it assumes prices do not jump. All three are false, which is why the market charges a different implied volatility for every strike. That pattern is the volatility smile, and it is the market's way of correcting the model.

   Where candidates lose it

   Writing out the formula and naming the terms without ever saying why the hedging argument removes the drift. And on a wealth management desk specifically: if you cannot restate it in one plain sentence about the price of insurance, you have failed the actual test, which was whether you can explain it to a client.

   Expect next

   • Which assumption fails worst in practice?
   • What does N of d2 actually represent?
   • If nobody believes the assumptions, why is it still on every screen?

   Reported by candidates at Goldman Sachs (Wealth Management, Zurich, 2025). Source: Wall Street Oasis.

2. 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?
3. 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?
4. 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?
5. 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?
6. 026What is delta, and give me three different ways to think about it.The GreeksCorephone / first roundMarket making

   Say this

   Delta is the change in the option's value for a one-unit change in the underlying. Three readings: it is the sensitivity, it is the hedge ratio — the number of shares you hold to be flat — and it is approximately the risk-neutral probability of finishing in the money, though only approximately.

   Then walk it

   1. As a sensitivity: a 0.4 delta call gains 40 paise for a 1 rupee move in the stock. Calls run from 0 to 1, puts from 0 to minus 1, and at the money sits near 0.5 in absolute terms.
   2. As a hedge ratio: short one 0.4 delta call and buy 40 shares and you are locally flat. This is the reading that matters on a desk, because it is the trade you actually put on.
   3. As a probability proxy: it is close to N of d2 but not equal to it, so treating a 25-delta option as a 25 percent chance of exercise is a real error that widens with volatility and time.
   4. As exposure: delta times the number of contracts times the contract multiplier gives you delta-equivalent notional, which is how the whole book gets aggregated. A book of 400 different options collapses to one number you can hedge with futures.
   5. The thing that makes delta interesting is that it is not constant. It moves with spot, which is gamma, and it moves with time and volatility, which are charm and vanna. A delta hedge is a snapshot that starts going stale the moment you put it on.
   6. The limitation to state: delta is a first-order local approximation. In a gap move it is nearly useless — a 0.4 delta call in a 15 percent overnight drop does not lose 0.4 times the move, it loses far less, because gamma works in the buyer's favour. Anyone who managed risk through March 2020 with delta alone learned this.

   Where candidates lose it

   Giving one definition and stopping. Junior interviews ask this to see whether you connect the maths sensitivity to the actual hedging trade. If you cannot say 'so I buy 40 shares', you have described a partial derivative rather than a job.

   Expect next

   • What is the delta of an at-the-money option, exactly?
   • How does delta change as expiry approaches?
   • Is delta the probability of expiring in the money?
7. 027What is gamma and why do traders care about it more than delta?The GreeksCoretechnicalMarket makingProp trading firms

   Say this

   Gamma is the rate of change of delta — the curvature of the option's value. Traders care more because delta can be hedged away in one trade, while gamma is what determines whether that hedge keeps working. Gamma is the risk you actually carry between rebalances.

   Then walk it

   1. Long gamma means your delta moves in your favour: you get longer as the market rises and shorter as it falls, so mechanical rebalancing sells high and buys low. Every rebalance banks a small profit.
   2. Short gamma is the reverse and it is vicious. You get shorter into a rally and longer into a selloff, so hedging forces you to buy high and sell low. The losses compound with the size of the move because the payoff is concave.
   3. Gamma is largest at the money and increases sharply as expiry approaches. A one-day ATM option has enormous gamma and almost no vega, which is why expiry-day books are managed completely differently from long-dated ones.
   4. You pay for gamma with theta. A long gamma position bleeds every day it does not move, and the break-even is roughly whether realised volatility exceeds the implied volatility you paid. That trade-off is the core of a market maker's daily profit and loss.
   5. Concrete version: long a 100-strike straddle at 20 implied on a stock that then realises 30 percent volatility. You lose theta every quiet day and make it back on the days that move, and the sum over the life is positive because realised beat implied.
   6. Market-wide, gamma positioning explains a lot of intraday behaviour. When dealers are short gamma they must hedge in the direction of the move, which amplifies it; when they are long they dampen it. That is the mechanism behind the gamma-squeeze stories, and it is real, though usually overstated in the press.

   Where candidates lose it

   Defining gamma as the second derivative and stopping. The interviewer wants the trading consequence: long gamma means your hedges make money, short gamma means they lose money, and you are paying or receiving theta for the privilege. Say what you do on the rebalance.

   Expect next

   • So how do you make money from being long gamma?
   • Where is gamma largest, and what does that do to your hedging on expiry day?
   • What does it mean for the market when dealers are collectively short gamma?
8. 028What is theta, and is collecting theta a strategy?The GreeksCoretechnicalIndian broking

   Say this

   Theta is the change in option value from one day passing, all else equal. For a long option it is negative — you lose a little every day. Collecting theta is not a strategy on its own; it is the premium you receive for being short gamma and short vega, and you only keep it if realised volatility comes in below implied.

   Then walk it

   1. Theta is largest in absolute terms for at-the-money options close to expiry, and it accelerates in the final week. A one-week ATM option can lose a fifth of its value a day near the end.
   2. The relationship to remember: theta and gamma are two sides of one trade. In Black-Scholes, theta is approximately minus half gamma times spot squared times variance. That identity says you are paid theta in exact proportion to the gamma you are short.
   3. So the break-even is not the passage of time, it is a volatility comparison. If you sell an option at 20 implied and the stock realises 15, you keep money. Realise 25 and you lose, no matter how much theta accrued on the way.
   4. This is why 'theta decay strategies' marketed to retail are misleading. Selling weekly index options collects theta reliably and looks like a bond until the week it does not, and one bad week removes many months of accrual. SEBI's own studies on Indian index option traders point at exactly this pattern of losses.
   5. The honest version of a short-theta business is the market maker's: sell options, delta hedge continuously, and earn the implied-minus-realised spread while keeping gamma small and diversified across names and expiries.
   6. Also note theta is not uniform through the day or week. Weekend decay is priced in on Friday, and on a low-realised-volatility day the effective bleed is higher than the model number suggests, because you also fail to earn the gamma.

   Where candidates lose it

   Presenting theta as free income. The interviewer's follow-up is always 'so you would just sell options every week?' Name the theta-gamma identity and say plainly that theta is compensation for short convexity, not a yield.

   Expect next

   • Write down the relationship between theta and gamma.
   • Why do weekly-expiry sellers in India lose money on average?
   • When is theta actually your friend?
9. 029What is vega, and where on the surface is it biggest?The GreeksIntermediatetechnicalEquity derivativesVolatility trading

   Say this

   Vega is the change in option value for a one-point move in implied volatility. It is biggest for long-dated at-the-money options, and it scales roughly with the square root of time. That is the key structural fact: gamma lives at the front of the curve, vega lives at the back.

   Then walk it

   1. Both calls and puts have positive vega, because higher volatility widens the distribution and both are convex payoffs. Being long options is being long volatility, full stop.
   2. Magnitude: vega is proportional to spot times root time times the standard normal density at d1. Root time means a two-year option has about five times the vega of a one-month option on the same notional.
   3. So the division of labour on a desk: if you want to trade the level of volatility you use long-dated options or variance swaps. If you want to trade the movement of the underlying you use short-dated options, where gamma dominates and vega is almost irrelevant.
   4. Vega is not one risk, it is a surface of risks. A book can be vega-flat in total and still be badly exposed if it is long front-month and short back-month volatility — that is vega term structure risk, and desks bucket vega by expiry rather than summing it.
   5. Numbers: an at-the-money one-year index option with 10 million of notional might carry 40,000 of vega, so a three-point volatility spike is 120,000. In March 2020 index implied volatility went from 15 to 80 in three weeks. That is the scale of the risk.
   6. The limitation: vega assumes a parallel shift in implied volatility, and real surfaces do not shift in parallel. Front-month volatility moves far more than back-month, and the skew steepens as the level rises. So you need vanna and volga — the cross-sensitivities to spot and to volatility itself — before your vega number means anything on a skewed book.

   Where candidates lose it

   Saying vega is biggest at the money and stopping, without the root-time scaling. The distinction that matters is that gamma is a short-dated risk and vega a long-dated one; if you cannot say which instrument to use for which view, you have not answered a trading question.

   Expect next

   • So which option would you buy to express a pure view on the level of volatility?
   • Why is a vega-neutral book still exposed to volatility?
   • What are vanna and volga, and when do they matter?
10. 030Rho gets ignored. When does it actually matter?The GreeksIntermediatetechnicalRates derivativesEquity derivatives

    Say this

    Rho is the sensitivity of an option's value to the interest rate. It is negligible on short-dated equity options, which is why nobody talks about it, and it is first-order on long-dated options, on FX, and on anything with a large strike relative to spot. It came back into focus when rates went from zero to five percent.

    Then walk it

    1. Sign: calls have positive rho, puts negative. A higher rate lowers the present value of the strike you will pay, which helps the call and hurts the put.
    2. Size: rho scales with time and with the discounted strike. On a one-month at-the-money option a 25 basis point rate move is a rounding error. On a five-year option it can be worth more than a volatility point.
    3. So where it bites: long-dated structured products, LEAPS, and the embedded options in insurance and pension liabilities. Anyone running a long-dated book in 2022 saw their option values move on rates as much as on volatility.
    4. In FX, rho is not one number but two, because you have the rate on each currency. The forward points are the rate differential, so an FX option's exposure to rates is really an exposure to the carry, and on a high-differential pair like USD/INR that dominates.
    5. Rates also change behaviour, not just value. Early exercise on American puts becomes optimal at higher rates, and the cost of carrying a delta hedge is a financing cost that grows with the rate. At five percent, financing a one-million-share hedge is a real line item in the profit and loss.
    6. The honest caveat: for the typical short-dated index option trade, rho is genuinely ignorable and pretending otherwise is false precision. The judgement being tested is whether you know which Greeks to care about for which instrument, rather than whether you can list five of them.

    Where candidates lose it

    Dismissing rho entirely, or over-claiming its importance. The right answer is a judgement about maturity and instrument: irrelevant for weekly index options, first-order for a five-year structured note and for FX carry. And name the financing cost of the hedge, which most candidates miss.

    Expect next

    • How does rho work differently in FX?
    • What did the 2022 rate cycle do to long-dated option books?
    • How does a higher rate change your delta hedging cost?

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.