Private Wealth Management puzzles, solved step by step
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023An at-the-money option's value rises roughly with the square root of its time to expiry. If a one-year at-the-money call is worth Rs 100, what is a three-month at-the-money call on the same stock worth?Private banking
Try it first
Pick the three-month call's value.
Show the worked solution
About Rs 50. Three months is a quarter of a year, and the square root of one quarter is one half, so the option keeps half its one-year value. The flip side matters as much: the one-year option loses Rs 50 over its first nine months and the other Rs 50 in its last three, so time decay speeds up sharply as expiry approaches.
Why the square root of time?
Think of a person taking random steps left or right. After four steps they are typically about two steps from the start, not four, because steps partly cancel. After sixteen steps, about four. Price uncertainty spreads with the square root of time, and an at-the-money option's value is roughly proportional to that spread. The standard rule of thumb for an at-the-money call is 0.4 times volatility times price times the square root of time: at 25% volatility on a Rs 1,000 stock over one year that gives Rs 100.
An at-the-money option worth Rs 100 at one year is still worth Rs 50 at three months because value follows the square root of time, so it loses 13.4, 15.9 and 20.7 in the first three quarters and 50 in the last. What does the curve mean for a client who sells options?
It means time decayThe fall in an option value as expiry approaches with nothing else changing, often called theta. is not even. Half of a one-year option's value is lost in its final quarter, so a seller collects decay fastest close to expiry, which is also when a sudden move hurts most. For a client selling covered calls every month, this is why short-dated options are the usual choice, and why the income comes with gap risk.
The relationshipC_1y the one-year at-the-money call, Rs 100 3/12 three months as a fraction of the year C_3m the three-month call, same stock and volatility What it says in wordsScale an at-the-money option's value by the square root of the ratio of the times to expiry.State where the rule breaks. It holds for at-the-money options with low interest rates; deep in or out of the money options do not scale this way, and volatility for three months need not equal volatility for a year. It is a desk estimate, not a pricing model.
Where candidates lose it
The trap is Rs 25: scaling value in a straight line with time. It is the natural first answer and it is off by half, which tells the interviewer the candidate has not met the idea that uncertainty grows with the square root of time.
The second loss is stopping at Rs 50 without the decay point. The follow-up is almost always about when the option loses its value, and the curve answers it.
What the interviewer asks next
- What is a one-month at-the-money call worth on the same basis?
- If volatility doubles, what happens to the one-year call's value?
- Why do many option sellers prefer to sell one-month options rather than one-year options?
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
Try it first
Volatility doubles from 25% to 50%, everything else fixed. What happens to the call's price?
Show the worked solution
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.
087A client buys a put option 10% below the market every year to protect his portfolio, at a cost of 2% of the portfolio a year. If the market goes nowhere for ten years, how much wealth has the insurance cost him?Private banking
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After ten flat years, what share of his wealth has gone on premiums?
Show the worked solution
About 18.3% of his wealth. In a flat market the puts, struck 10% below, never pay out, so each year he simply loses 2%. Ten years leave 0.98 to the power 10, which is 0.817 of the starting wealth. That is slightly under 20% because each premium is taken from a pot the earlier premiums have already shrunk.
Why is the cost close to 18% and not 20%?
Paying 2% of your salary every year for a subscription feels like a small fixed charge, but if it were 2% of your savings each year, the savings would shrink and each charge would be a little smaller in rupees. A yearly premium set as a share of the portfolio compounds downwards, so ten years of 2% cost 1 minus 0.98 to the 10th, about 18.3%, not a flat 20%. In rupees, a Rs 10 crore portfolio would stand at about Rs 8.17 crore.
In a flat market the uninsured portfolio stays at 100 while the insured one pays 2% a year for puts that never pay and falls to 81.7, a gap that widens every year to 18.3% of wealth. The relationship0.02 the yearly premium as a share of the portfolio 10 years the insurance is bought W_10 wealth after ten years, as a share of the start What it says in wordsEach year keeps 98% of the pot, so ten years keep 0.98 to the tenth power.So is the insurance a bad idea?
Not by itself. A put optionA contract that gives the holder the right to sell an asset at a set price, so it pays out when the price falls below that level. pays out in a crash, which is when a client most needs cash and is most tempted to sell at the bottom. Permanent insurance is a permanent drag, so the question is whether the protection is worth about 2 points of return every year, not whether it pays out in any one year. A flat decade is the worst case for the buyer, because he pays every premium and collects nothing.
Say the limitation: in a rising market the drag is the same 2% a year taken from a growing pot, and in a crash the puts pay. The honest comparison sets the yearly cost against the losses avoided in the bad years, and the answer depends on how often and how deep those falls are.
Where candidates lose it
The fast answer is 20%, ten times 2%. It ignores that each year's premium is a share of a smaller pot, and the interviewer uses it to see whether you compound costs as naturally as returns.
The bigger loss is judging the insurance by the flat decade alone. Give the 18.3% and then say what the client gets for it, and when.
What the interviewer asks next
- Over the same ten years the market falls 30% once. What would the puts have paid?
- How could the client cut the premium, and what does he give up?
- What is a collar, and why do clients use one?
