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  1. 019Explain the Black-Scholes intuition by pricing, in your head, a one-year at-the-money call on a Rs 100 stock with 20% volatility and near-zero interest rates.Rates, risk and optionsHardGoldman SachsZurich · 2025

    Try it first

    Roughly what is the call worth?

    Show the worked solution

    About Rs 8. With near-zero rates, an at-the-money call is worth roughly 0.4 times the price times the volatility times the square root of time: 0.4 x 100 x 0.20 x 1 = 8.0. The full Black-Scholes formula gives Rs 7.97. The intuition: the option collects the upside of a bell-shaped spread of outcomes and nothing on the downside, and the average of that upside half is about 0.4 of one standard deviation.

    What is Black-Scholes actually doing?

    Picture a ticket that pays you whatever a stock gains over the next year and nothing if it falls. You would pay something for it, even though an at-the-moneyAn option whose strike price equals the current stock price, so exercising it today would pay nothing. ticket is worth nothing if cashed in today. Black-Scholes prices that ticket as the average payoff across all the ways the stock could move, with the spread of those moves set by volatility and time. Its inputs are the stock price, strike, time, volatility and interest rate, and with zero rates the formula reduces to S times N(d1) minus K times N(d2), where the two N terms come from the bell curve.

    At the money: worth nothing if exercised today, about Rs 8 with a year to run608010012014010203040Stock price, Rs (strike 100)Call value, RsRs 7.97 todaypayoff 0value today, 1 year leftpayoff at expirySpread of outcomes1 sd = 100 x 20% x 1 = Rs 20Call keeps the upper half;its average is sd x 0.40.4 x 20 = Rs 8.0Full formula100 x (N(0.1) - N(-0.1))= 100 x (0.5398 - 0.4602)= Rs 7.97
    At a stock price of 100 the call pays nothing if exercised today, yet with a year to run at 20% volatility it is worth Rs 7.97, almost exactly the rule of thumb of 0.4 times the Rs 20 one-year standard deviation.

    Where does the 0.4 come from?

    Over a year at 20% volatility, the stock's likely outcomes form a bell curve around Rs 100 with a standard deviation of about Rs 20. The call keeps the right half and treats the left half as zero. The average of the positive half of a bell curve, counting the negative half as zero, is the standard deviation divided by the square root of 2 pi, about 0.4 of it. So the call is worth about 0.4 x Rs 20, which is Rs 8. Halve the time to six months and the spread shrinks by the square root of 2, so the option is worth about Rs 5.6, not Rs 4.

    The relationship
    C≈12π S σT=0.399×100×0.20×1=7.98CBS=100 [N(0.1)−N(−0.1)]=7.97C \approx \frac{1}{\sqrt{2\pi}}\,S\,\sigma\sqrt{T} = 0.399 \times 100 \times 0.20 \times 1 = 7.98 \qquad C_{BS} = 100\,[N(0.1) - N(-0.1)] = 7.97
    Sthe stock price, Rs 100, equal to the strike
    sigmavolatility, 20% a year
    Ttime to expiry in years, here 1
    N(.)the cumulative bell-curve probability used in the full formula
    What it says in wordsAn at-the-money option is worth about 0.4 of one standard deviation of the stock's price over the life of the option.

    What does the shortcut leave out?

    It holds for at-the-money options with modest volatility and low rates. Away from the strike, the payoff is no longer the clean right half of the curve, and with meaningful rates the forward price drifts above the spot, which raises a call's value. The rule is a sanity check on a quoted price, not a replacement for the formula. Black-Scholes itself assumes constant volatility and smooth price moves, which is why traders quote options in implied volatility and adjust for the skew they see in the market. Doubling volatility to 40% roughly doubles this call's value, to about Rs 15.9.

    Where candidates lose it

    The common slip is to say an at-the-money option is worth nothing because exercising it today pays zero. That is intrinsic value; the whole price of this option is time value, the chance of a favourable move before expiry.

    The second slip is reciting the formula without any feel for the number. Give the 0.4 rule, reach Rs 8, and then say the exact formula agrees; that is what explaining Black-Scholes sounds like in an interview.

    What the interviewer asks next

    • What is the matching at-the-money put worth here, and why?
    • If volatility doubles to 40%, roughly what happens to the call's value?
    • What is the delta of this call, roughly, and what does it tell you?

    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.

  2. 052An equity index stands at 38,000 and its annual volatility is about 18%. The interviewer asks where it will close in four months. What range do you give, and how confident are you in it?Rates, risk and optionsHardMorgan StanleyTokyo · 2025

    Try it first

    How wide is a one standard deviation range over four months?

    Show the worked solution

    Roughly 34,050 to 41,950, with about two-thirds confidence. Four months is a third of a year, and volatility scales with the square root of time, so 18% a year becomes 18% times 0.577, about 10.4%, over four months. One standard deviation is about 3,949 points either side of 38,000. If returns are roughly normal, about 68% of outcomes land inside that band and about 95% inside 30,100 to 45,900. Give the range and the confidence, never a single number.

    Why does volatility shrink with the square root of time?

    Picture someone leaving a party and taking steps at random, forward or back. After four steps they are not four steps from the door; some steps cancelled, and on average they are about two away. Random moves partly cancel, so the spread of where you end up grows with the square root of the number of moves, not with the number itself. Four months is a third of a year, the square root of a third is 0.577, and 18% times 0.577 is 10.4%. Dividing 18% by three to get 6% would be right only if every month moved the same way, which is the opposite of random.

    Four months of an 18% annual volatility is 10.4%, not 6%: a two-in-three band30,10034,05038,00041,95045,9001 sd = 18% x sqrt(1/3) = 10.4% = 3,949 pointsabout 68%of outcomes2 sd: about 95%2 sd: about 95%-10.4%+10.4%Index level at the four-month close. Red bar: the wrong 6% band from 18% / 3One sd band: 34,050 to 41,950, two in three. Two sd: 30,100 to 45,900, nineteen in twenty.
    Over four months an 18% annual volatility becomes 10.4%, about 3,949 points, so one standard deviation runs from 34,050 to 41,950 and holds about two-thirds of outcomes, while dividing 18% by three gives a 6% band that is far too narrow.

    How do you turn 10.4% into a range?

    10.4% of 38,000 is about 3,949 points. One standard deviationA measure of how far outcomes typically land from the centre. Under a normal distribution about 68% of outcomes fall within one of the centre and 95% within two. either side is 34,050 to 41,950; two is 30,100 to 45,900. A one standard deviation band is a two-in-three bet, so the confidence attached to the width is the answer, and a range without one is just a guess with margins. Say the band, say the two in three, and offer the wider nineteen-in-twenty band if the interviewer wants more certainty.

    The relationship
    σ4m=18%×412≈10.4%38,000×10.4%≈3,949 points\sigma_{4m} = 18\% \times \sqrt{\tfrac{4}{12}} \approx 10.4\% \qquad 38{,}000 \times 10.4\% \approx 3,949\text{ points}
    18%the annual volatility, one standard deviation of the yearly return
    4/12the fraction of a year, four months
    3,949one standard deviation in index points
    What it says in wordsScale the annual volatility by the square root of the time fraction, then apply it to the index level.

    What do you add to sound like a trader rather than a textbook?

    Three refinements, each one sentence. Returns compound, so the band is slightly lopsided: 38,000 grown and shrunk by 10.4% continuously is about 34,250 to 42,150, a little more room on the upside. The expected drift over four months is a percent or two, small next to 10.4%, so ignoring it is fair. Real markets have fatter tails than the bell curve, so the two standard deviation band gets breached more often than one time in twenty. Say the range, say the confidence, then say what would make you wrong: the 18% itself is a historical figure, and the volatility the options market implies for the next four months may be higher or lower. This is a way of describing uncertainty, not a forecast, and the interviewer is testing whether you think in distributions.

    Where candidates lose it

    Dividing 18% by three and giving 6% either side is the common miss. It produces a band that gets breached far more often than the candidate expects, and a trader will catch it at once.

    The second loss is giving a single number. The question is a test of whether you think in distributions; a point estimate with no range and no confidence says that you do not.

    What the interviewer asks next

    • What is the one standard deviation range over one year, and over one week?
    • Why might the volatility implied by index options differ from the 18% historical figure?
    • If the index closes at 44,000, was the 18% volatility assumption wrong?

    Asked at Morgan Stanley, Sales and Trading, Tokyo, 2025 (Wall Street Oasis): what do you think x index will close at by end of year and why