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. 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?
2. 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?
3. 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?
4. 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.