Private Wealth Management puzzles, solved step by step
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- 30
037Explain Black-Scholes through one number. What is an at-the-money one-year call on a Rs 1,000 stock with 25% volatility worth, roughly, and what happens to that price if volatility doubles?Goldman SachsZurich · 2025
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Volatility doubles from 25% to 50%, everything else fixed. What happens to the call's price?
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About Rs 100, and doubling volatility roughly doubles it to about Rs 200. With no interest or dividends, Black-Scholes prices an at-the-money call at very nearly 0.4 x stock price x volatility x the square root of time: 0.4 x 1,000 x 0.25 x 1 = Rs 100. The exact formula gives Rs 99.5. At 50% volatility the exact price is Rs 197.4. The option is priced as the expected value of its payoff, and that expectation grows with how far the stock can move.
What is Black-Scholes actually doing?
Imagine a rain insurance policy for an outdoor wedding that pays only if it pours. Its fair price depends on how uncertain the weather is: in a steady dry season it is nearly worthless, in a volatile monsoon it is dear. Black-Scholes prices an option as the average payoff over every path the stock could take, where the spread of those paths is set by volatility. The owner of a call keeps the upside of big moves and loses at most the premium on the downside, so a wider spread of paths is worth more.
At the money, with rates set to zero, the formula collapses to a clean shortcut. The standard normal distributionThe bell curve with mean zero and standard deviation one, used to describe the spread of possible stock returns in the model. has height 1 over the square root of 2 pi at its centre, which is 0.399, and that is where the 0.4 comes from.
The relationshipS the stock price, Rs 1,000, equal to the strike \sigma the stock's annual volatility, 25% T time to expiry in years, 1 1/\sqrt{2\pi} about 0.399, the height of the bell curve at its centre What it says in wordsAn at-the-money call is worth about 40% of one standard deviation of the stock's move over the option's life.For an at-the-money one-year call on a Rs 1,000 stock, the exact Black-Scholes price is Rs 99.5 at 25% volatility and Rs 197.4 at 50%, almost a straight line. The 0.4 shortcut tracks it closely, so doubling volatility roughly doubles the price. Why would a wealth interviewer care about this?
Because structured products sold to private clients are bundles of options, and the client is often the seller of volatility without knowing it. A note that pays a high coupon while volatility is high is usually paying for an option the client has written, and the coupon rises with volatility for exactly the reason on this chart. The same shortcut shows time matters by its square root: a three-month option on the same stock costs about half the one-year option, Rs 49.8, not a quarter.
The limits: the shortcut is only good at the money and for short maturities with low rates. The model assumes constant volatility and smooth prices, which real markets break, and that is why traders quote different volatilities for different strikes.
Where candidates lose it
Candidates recite the formula, N of d1 and N of d2, and cannot say what any of it means or produce a number. A wealth interviewer wants the intuition and one sanity check, not the algebra.
The other slip is guessing that doubling volatility quadruples the price because variance quadruples. At the money the price moves with volatility, not its square. Give Rs 100, then Rs 200, then say why the client selling that volatility in a structured note should care.
What the interviewer asks next
- What is the same option worth with three months to expiry?
- How does the shortcut change for an at-the-money put?
- Why does a reverse convertible pay a higher coupon when volatility rises?
Asked at Goldman Sachs, 2026 | EMEA | Zurich | Wealth Management | Summer Analyst Interview, Zurich, 2025 (Wall Street Oasis):
Moreover, I was asked to explain Black and Scholes.
064How does a private bank make money on one client? Take a Rs 50 crore client with Rs 20 crore under a 1% advisory mandate, Rs 15 crore in funds and notes that pay the bank a 0.8% trail, and a Rs 10 crore loan against securities earning the bank a 2% spread.J.P. MorganCharlotte · 2026
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Which stream earns the bank the most from this client?
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About Rs 52 lakh a year, from three streams at once. The advisory fee earns 1% of Rs 20 crore, Rs 20 lakh. Product trails earn 0.8% of Rs 15 crore, Rs 12 lakh. The loan against securities earns a 2% spread on Rs 10 crore, Rs 20 lakh. Together that is about 1.04% of the client's Rs 50 crore, before the cost of the banker and the team who serve him.
What are the three ways a private bank earns on one client?
A family jeweller earns on making charges, on the margin in the gold he sells, and on the gold loan counter at the back of the shop, all from the same customer. A private bank earns a fee for advice, a commission for products it distributes, and a spread on money it lends, and a good relationship uses all three. Each is a balance times a rate, so the whole puzzle is three multiplications and an addition. The lending line is a loan against securitiesA loan secured by the client pledging shares, bonds or fund units he owns, so he can raise cash without selling them., and the bank keeps the difference between what it charges and what the money costs it.
From one Rs 50 crore client the bank earns Rs 20 lakh of advisory fees, Rs 12 lakh of product trail and Rs 20 lakh of lending spread, Rs 52 lakh a year or about 1.04% of the relationship. Why does the lending line matter so much?
Look at the figure: the Rs 10 crore loan earns as much as the Rs 20 crore advisory mandate. Lending earns on a balance the client does not have to move away from anyone else, which is why private banks work hard to offer credit to wealthy clients. The loan also keeps the pledged assets with the bank, which protects the other two streams. The spread is not free money; the bank carries the risk that the pledged securities fall faster than it can sell them.
What should a candidate add after the arithmetic?
Two things. First, cost: a relationship manager and the specialists behind him are paid out of this Rs 52 lakh, so the bank cares about revenue per banker, not just per client. Second, conflict: a product trail pays the bank more if the client holds certain products, which is why rules in many markets, including India, separate paid advice from commission-based distribution. Confirm the current rules; the point for the interview is that you see the incentive.
Where candidates lose it
The first trap is answering a generic bank's model, deposits and loans, and missing that a private bank earns mostly on fees and on the client's assets. The question is about the relationship, not the balance sheet.
The second is summing the balances, Rs 45 crore, and applying one rate. Each stream has its own base and its own rate; keep them apart and the lending line's weight becomes obvious.
What the interviewer asks next
- The client moves Rs 10 crore from products into the advisory mandate. What happens to revenue?
- Markets fall 20%. Which of the three streams falls, and which does not?
- Why might a bank accept a lower advisory fee to win the lending business?
Asked at J.P. Morgan, Private Banking, Charlotte, 2026 (Wall Street Oasis):
What is happening in the US economy right now? How does a bank make money?
076A private bank lists its risks and asks you to name the greatest. Two candidates: credit risk on a Rs 2,000 crore loan book with a 1% default rate and 45% loss given default, or a conduct fine of Rs 60 crore that it expects once every ten years. Which is the bigger risk?UBSNew York · 2026
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Which risk costs the bank more in an average year?
Show the worked solution
Credit is the bigger risk on expected loss, Rs 9 crore a year against Rs 6 crore, but conduct is the bigger risk in a bad year. Rs 2,000 crore x 1% x 45% is 9; Rs 60 crore x one in ten is 6. If a bad credit year triples defaults to 3%, credit loses 27, still well short of the 60 crore fine landing whole, and the fine also damages the franchise.
Why is the Rs 60 crore headline the wrong number to rank on?
Think of two risks to a household. A leaking tap wastes a little every month; a burglary costs a lot but comes rarely. You cannot compare them by the size of one burglary against one month of leaking, because one is certain and the other is not. Risks are ranked first by expected loss, the chance of the loss times its size, so a large but rare fine and a small but steady credit loss can be put on one scale.
The relationshipPD probability of default, the share of borrowers who stop paying in a year LGD loss given default, the share of the loan not recovered after default EAD exposure at default, the amount lent, here Rs 2,000 crore What it says in wordsCredit's expected loss is the default rate times the share lost times the book, Rs 9 crore a year.The conduct fine goes on the same scale the same way: Rs 60 crore with a one in ten chance each year is Rs 6 crore a year. On the average year credit costs half as much again as conduct. For a private bank the loan book is usually Lombard lendingLoans to wealthy clients secured against their investment portfolios, which the bank can sell if the loan is not repaid. against portfolios, which is why the loss given default here is well below what an unsecured book would suffer.
On expected loss credit ranks first, Rs 9 crore a year against Rs 6 crore for conduct; in a one in ten bad year conduct ranks first, the whole Rs 60 crore fine against Rs 27 crore of credit loss even with defaults tripled. So what do you say is the greatest risk?
Say both rankings, then pick with a reason. Expected loss tells you what to price and provision for; the bad year tells you what can hurt capital and the name, and for a bank that lives on client trust the bad year usually decides. A conduct fine rarely comes alone: clients leave, regulators restrict new business, and the cost runs well past the Rs 60 crore. That is why many private banking interviewers expect you to land on conduct and reputation as the greatest risk, and to show the arithmetic that says credit costs more on an ordinary year. The 3% bad-year default rate is an assumption; say it as one.
Where candidates lose it
Most candidates answer from the headline: Rs 60 crore sounds bigger than a 1% default rate, so conduct wins. They never weight the fine by its one in ten chance, and the interviewer hears a guess dressed as a judgement.
The opposite trap is stopping at expected loss and calling credit the winner. Give both numbers, 9 against 6 and 27 against 60, and say which one decides for a business that sells trust.
What the interviewer asks next
- Loss given default rises to 70% because collateral falls with the market. Does the ranking change?
- How would you set capital against each of these two risks?
- Name a risk on the wealth side of the bank that has neither a default rate nor a fine attached.
Asked at UBS, Private Wealth Management, New York, 2026 (Wall Street Oasis):
What is the broad range of risks a bank has, and what is the greatest risk?
