Derivatives Foundation puzzles, solved step by step

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All topicsMental maths and estimation9Random walks and Markov chains7Conditional probability and Bayes7Volatility and correlation7Option pricing intuition7Expected value and optimal stopping10Market making11Option payoffs and no-arbitrage10Probability and counting11Distributions and statistics8Games and logic8Betting and sizing5
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  1. 088Rs 1 was invested in a broad equity index 30 years ago. If its log returns average 9% a year with 16% volatility, give a 95% range for what the Rs 1 is worth today, and explain why the range is lopsided.Distributions and statisticsHardOld Mission CapitalChicago · 2025

    Try it first

    Which quantity do you build the 95% range on first?

    Show the worked solution

    Roughly Rs 2.7 to Rs 82.9, around a median of Rs 14.9. Thirty years of log returns sum to a mean of 2.7 with a standard deviation of 0.16 x the square root of 30, about 0.876. The 95% band in logs is 2.7 plus or minus 1.72, from 0.98 to 4.42. Exponentiating gives Rs 2.7 to Rs 82.9. It is lopsided because e to the power stretches gains and compresses losses: 12.2 below the median, 68.0 above.

    Why work in logs and not in rupees?

    A savings balance grows by multiplying: up 10% then down 10% leaves you at 0.99, not 1. Multiplications are awkward to average, but their logarithms add, and sums of many independent pieces tend towards a bell curve. Log returns add across years, so their total over 30 years has a mean that grows with time and a spread that grows with the square root of time, and the 95% band is symmetric in logs. Here the mean is 30 x 0.09 = 2.7, the standard deviation is 0.16 x 5.477 = 0.876, and 1.96 standard deviations is 1.72. The log of the final value lies between 0.98 and 4.42 with 95% confidence.

    Symmetric in logs, lopsided in rupees0102030405060708090100what Rs 1 is worth after 30 years, in rupeeslikelihoodmost likely about Rs 6.9median Rs 14.9mean Rs 21.8Rs 2.7Rs 82.92.5% beyond95% of outcomes sit in the shaded bandIn logs: 2.7 plus or minus 1.72a symmetric bandIn rupees: 12.2 below the median,68.0 above it
    Thirty years of log returns averaging 9% with 16% volatility give a final value of Rs 1 that peaks near Rs 6.9, has a median of Rs 14.9 and a mean of Rs 21.8, and lies between Rs 2.7 and Rs 82.9 with 95% confidence, a band that is symmetric in logs but runs 12.2 below the median and 68.0 above it in rupees.

    Why does the rupee range lean so far to the right?

    Converting back means raising e to each end. e to the 0.98 is 2.67 and e to the 4.42 is 82.9, around a median of e to the 2.7, 14.9. The same 1.72 step in logs is a factor of 5.6 either way, and dividing by 5.6 moves you 12.2 rupees while multiplying by 5.6 moves you 68.0. That is why the distribution of wealth has a long right tail: the typical outcome, the median, sits well below the average, Rs 21.8, which is pulled up by the rare decades that compound very well.

    The relationship
    ln⁡W30∼N ⁣(30μ,  30σ2)=N(2.7,  0.8762),W30∈[e2.7−1.96×0.876,  e2.7+1.96×0.876]=[2.67,  82.9]\ln W_{30} \sim N\!\left(30\mu,\; 30\sigma^2\right) = N(2.7,\; 0.876^2), \qquad W_{30} \in \left[e^{2.7 - 1.96 \times 0.876},\; e^{2.7 + 1.96 \times 0.876}\right] = [2.67,\; 82.9]
    W_30the value after 30 years of Rs 1 invested at the start
    muthe average yearly log return, 9%
    sigmathe yearly volatility of log returns, 16%
    1.96the number of standard deviations that covers 95% of a normal
    What it says in wordsBuild a symmetric 95% band for the total log return, then raise e to each end to get a lopsided band in rupees.

    Two cautions the interviewer will reward. First, the 9% and 16% here are assumptions for the exercise; confirm the index's actual history before quoting any real figure, and note that a 30-year average return is itself estimated with a wide error. Second, the band assumes yearly log returns are independent and normal. Real markets have fat tails and runs of bad years, which widen the lower end in particular. The model gets the shape right, a long right tail with the median below the mean, even if the exact ends are only approximate.

    Where candidates lose it

    The common loss is building the band in rupees with a symmetric plus or minus, which gives a lower bound below zero or a band centred on the wrong number. Wealth cannot go below zero, and any method that says it can has skipped the logs.

    The second loss is scaling the volatility by 30 instead of the square root of 30. That gives a standard deviation of 4.8 in logs, a band from almost nothing to tens of thousands, and tells the interviewer you have not met the square-root-of-time rule.

    What the interviewer asks next

    • What is the probability the Rs 1 is worth less than Rs 1 today?
    • Why is the mean of the final value higher than the median?
    • If volatility were 25% instead of 16%, what happens to the median and to the mean?

    Asked at Old Mission Capital, Prop Trading, Chicago, 2025 (Wall Street Oasis): Confidence interval of portfolio value if you invested $1 in S&P 500 30 years ago

  2. 099Daily returns come from a normal with 1% volatility on 90% of days and a normal with 3% volatility on the other 10%, both with mean zero. What are the overall volatility and kurtosis, and what does this do to out-of-the-money option prices?Distributions and statisticsHardTwo SigmaNew York · 2025

    Try it first

    What is the overall daily volatility?

    Show the worked solution

    Volatility 1.34% a day and kurtosis 8.33, against 3 for a normal, which makes far out-of-the-money options worth much more than a single-volatility model says. Variance is 0.9 x 1 + 0.1 x 9 = 1.8. The fourth moment is 3 x (0.9 x 1 + 0.1 x 81) = 27, so kurtosis is 27/1.8^2 = 8.33. With the same volatility, a 4% move is 6.4 times as likely as the normal says, and a one-day put 4% out of the money is worth about 23 times as much.

    How do the moments of a mixture combine?

    Picture a road with quiet days and occasional storms. Most days the traffic varies a little; on storm days it varies a lot. Averaged over a year, the variability you see is dominated by the storm days, out of proportion to how rare they are. For a mixture with mean zero, each even moment is the weighted average of the components' moments: variances add up by weight, and fourth moments, which grow with the fourth power of volatility, are dominated by the rare wild days. Variance: 0.9 x 1 + 0.1 x 9 = 1.8, a volatility of 1.34%. Fourth moment: each normal's is 3 sigma^4, so 3 x (0.9 x 1 + 0.1 x 81) = 27. The 3% days are a tenth of the time but supply 90% of the fourth moment.

    Same volatility, taller peak, fatter tails-6%-4%-2%0%2%4%6%daily returnmixturenormal, 1.34% volzoomed at right3%4%5%6%right tail, height scale 13 times largercross 3.7%mixture above: fat tailkurtosis 8.33 against 3 for a normal; a move beyond 4% either way: 1.83% against 0.29%, 6.4 times as likelyillustration: the inputs are the ones given in the question, not market data
    Against a normal with the same 1.34% daily volatility, the mixture of 1% and 3% days is taller in the middle and lies above the normal in the tails beyond about 3.7%, so its kurtosis is 8.33 instead of 3 and a move beyond 4% either way is 6.4 times as likely.

    What does the fat tail do to option prices?

    Kurtosis is the fourth moment over the square of the variance: 27/3.24 = 8.33. A single normal fitted to the same data has the same volatility, 1.34%, but kurtosis 3, so it puts far too little weight on big moves. An option far out of the money pays only on a big move, so its value is driven by the tail, and a mixture that keeps the volatility but fattens the tail makes it worth many times what the matched normal says. A one-day put 4% out of the money is worth 0.0127% of spot under the mixture against 0.00055% under the normal, about 23 times as much. To match the mixture's price, a single-volatility model needs 1.91% for that put but only 1.20% at the money: the smile.

    The relationship
    σ2=0.9(1)2+0.1(3)2=1.8,κ=3 [ 0.9(1)4+0.1(3)4 ](1.8)2=273.24≈8.33\sigma^2 = 0.9(1)^2 + 0.1(3)^2 = 1.8, \qquad \kappa = \frac{3\,[\,0.9(1)^4 + 0.1(3)^4\,]}{(1.8)^2} = \frac{27}{3.24} \approx 8.33
    sigma^2the variance of the daily return, in square percentage points
    kappathe kurtosis, 3 for any single normal
    3 sigma^4the fourth moment of a normal with mean zero
    What it says in wordsVariances average by weight, fourth moments average by weight, and the ratio comes out near 8.3 because the rare 3% days dominate the fourth power.

    Say what it means on a desk. This mixture is the simplest model of regime-switching volatility, and it reproduces two things a single normal cannot: a peak that is too tall and tails that are too fat, the shape seen in daily returns of most liquid assets. It is also why at-the-money options can look rich and far wings cheap if you price both off one historical volatility. The limitation: the mixture here is symmetric and draws each day independently. Real markets cluster their wild days and fall harder than they rise, which adds skew to the smile, not just curvature.

    Where candidates lose it

    The common loss is averaging the volatilities, 0.9 x 1 + 0.1 x 3 = 1.2%, and then getting the kurtosis wrong as a result. Volatilities never average in a mixture; variances do.

    The second loss is computing the moments correctly and stopping, without connecting kurtosis to option prices. The question asks what the fat tail does to the wings, and the answer is a smile: more implied volatility the further out of the money you go.

    What the interviewer asks next

    • What is the kurtosis if the wild days are 5% volatility but only 4% of days?
    • Which is more mispriced by a single-volatility model, an at-the-money option or a far out-of-the-money one, and in which direction?
    • How would you add skew to this model?

    Asked at Two Sigma, Quantitative Research, New York, 2025 (Wall Street Oasis): They asked a couple questions involving Mixture Gaussians (e.g., probability density and moments).