Private Wealth Management puzzles, solved step by step

Puzzles
100
Traced to a firm
3
Topics
13
Hard
30
Topic
All topicsCompounding and doubling8Returns arithmetic9Fee and cost drag8Inflation and real return7Tax arithmetic7Probability and risk of loss9Retirement and withdrawal8Fixed income numeracy8Behavioural traps8Estimation and sizing8Options and structured products6Leverage and borrowing6Wealth business economics8
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 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?Options and structured productsCoreGoldman 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 relationship
    CATM≈12π S σT≈0.4×1000×0.25×1=100C_{\text{ATM}} \approx \frac{1}{\sqrt{2\pi}}\,S\,\sigma\sqrt{T} \approx 0.4 \times 1000 \times 0.25 \times 1 = 100
    Sthe stock price, Rs 1,000, equal to the strike
    \sigmathe stock's annual volatility, 25%
    Ttime 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.
    At the money, the call's price scales almost in step with volatilityRs 100Rs 200Rs 3000%20%40%60%80%Volatility, a yearshortcut 0.4 x S x volexact curve: Rs 311 at 80%25% vol: Rs 99.550% vol: Rs 197.4Double the volatility, roughlydouble the price of the option
    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.