Derivatives Foundation puzzles, solved step by step

25.98% at a correlation of 0.5, and 21.21% at zero. Basket variance is the sum of the two weighted variances plus twice the weighted covariance: 0.25 x 0.09 + 0.25 x 0.09 + 2 x 0.25 x 0.5 x 0.09 = 0.0675, whose square root is 25.98%. At zero correlation the cross term vanishes, leaving 0.045, whose root is 21.21%. The basket is less volatile than either stock because they do not move in step.

Why is the basket calmer than the stocks inside it?

Two commuters who each arrive late by a random ten minutes rarely arrive late together; the average of their lateness swings less than either does alone. Volatilities do not add; variances do, and the cross term that joins them is scaled by the correlation, so anything below a correlation of 1 cuts the basket's swing below its parts. At a correlation of 1 the two stocks are one stock and you get 30% back. At minus 1 they cancel exactly and the basket is flat.

How do you do it in your head?

Use the shortcut in the formula: with two equal stocks the basket volatility is 30% times the square root of (1 plus rho) over 2. At rho 0.5 that is 30% times the root of 0.75, about 0.866, giving 26.0%; at rho 0 it is 30% times the root of 0.5, about 0.707, giving 21.2%. Say the structure first, then the number, so a slip in the arithmetic does not look like a slip in the thinking.

What is the limitation you should name?

The formula treats correlation as a fixed number, and it is not. Correlations between stocks tend to rise in a sell-off, which is exactly when a basket holder wants the diversification, so the 25.98% is a fair-weather figure. On a derivatives desk that is why basket options and dispersion trades are priced with a correlation assumption that is marked, stressed and hedged rather than looked up once. Say that the answer depends on the correlation you assume, and that the assumption is the risk.

The stock ends at about 28% of its start; the three-times fund at roughly 8 millionths of its start, effectively zero. Each up-and-down pair multiplies the stock by 1.1 x 0.9 = 0.99, and 125 pairs give 0.99^125 = 0.285. The leveraged fund moves 30% each way, so each pair is 1.3 x 0.7 = 0.91, and 0.91^125 is about 7.6e-06. The arithmetic average return is zero in both cases; the compounded return is not.

Why does a zero average return lose money?

Take a 100 rupee note to a shop that marks everything up 10% in the morning and discounts 10% in the afternoon. The afternoon discount is taken off a bigger number, so the price ends at 99, not 100. A gain and a loss of the same percentage do not cancel, because the loss acts on the larger base; the pair costs the square of the move, 1% for a 10% swing. Repeat that 125 times and the 1% losses compound to a 72% fall. This is volatility drag: the gap between the average return and the compounded return.

Why is three times the move so much worse than three times the loss?

The drag per pair is the square of the move. Tripling the move multiplies the drag by nine, not three: the stock loses 1% per pair, the fund loses 9%. That is why a daily-rebalanced leveraged fund in a choppy, sideways market bleeds even when the underlying ends flat. The general rule you can quote: over many periods the compounded growth rate is roughly the average return minus half the variance, and leverage multiplies the variance by the square of the leverage.

What would you say to a client who holds the three-times fund?

That the product tracks three times the daily move, exactly as promised, and that this is not the same as three times the return over a year. A leveraged fund is a tool for a view on the next day or week; held through a sideways year it loses to its own rebalancing. The limitation to state: the alternating path is the worst case for drag, and a strongly trending market can make a leveraged fund return more than three times the underlying. The drag is about path, not just direction.

About 27.2%. Variance is volatility squared times time, and variances add across periods. The three months carry 0.25 squared x 3 = 0.1875 of variance; the first month carries 0.20 squared x 1 = 0.04; so months two and three carry 0.1475 between them, 0.0737 per month on an annualised basis. The square root is 0.272. The forward volatility must sit above 25% because the quarter's average has to be pulled up from the 20% start.

Why do you add variances and not volatilities?

If you walk a random distance each hour, your spread after three hours is not three times the hourly spread, because some hours cancel others. What grows in a straight line is the variance, the spread squared. Implied volatility is quoted per year, so the variance an option carries is volatility squared times its life, and a longer option's variance is the sum of the variances of the pieces of time inside it. That is the whole mechanism. Twenty per cent for one month is 0.04 of variance; twenty-five per cent for three months is 0.1875; the difference belongs to the two months in between, and dividing by two months and taking the square root turns it back into a volatility.

What does the number tell a trader?

The 27.2% is the volatility you would lock in for months two and three by selling the one-month option and buying the three-month one in variance-weighted sizes, a calendar spread. If you believe realised volatility in those two months will be well below 27.2%, the three-month option is rich relative to the one-month, and the calendar is the trade that expresses it. Say the rounding: an upward sloping term structure, 20% then 25%, hides a forward that is steeper than either quote, 27.2%, and the trap of reading 25% as the forward is what the question is built to catch. The calculation uses calendar time in months; a desk would use trading days or variance-weighted business days, which shifts the number by a few tenths.

What is the limit of the calculation?

The subtraction has to leave something positive. If the three-month volatility were below 11.5% with the one-month still at 20%, the quarter would carry less variance than its first month alone, which is impossible without arbitrage: you could sell the one-month, buy the three-month and hold a position with negative forward variance. Forward variance can be small but never negative, so a term structure that inverts too sharply is a mispricing, not a forecast. The second limit is that both quotes must refer to the same strike in forward terms; mixing an at-the-money one-month with a three-month that has rolled away from the money adds skew to the comparison and the forward you compute is no longer clean.

The eigenvalues are 1 + 2 rho, once, and 1 - rho, twice; rho can go no lower than -1/2. The vector (1, 1, 1) is an eigenvector with eigenvalue 1 + 2 rho, and any vector whose entries sum to zero is an eigenvector with eigenvalue 1 - rho, which gives a two-dimensional space and hence a double root. A correlation matrix must have no negative eigenvalue, since each eigenvalue is a portfolio variance, so 1 + 2 rho is at least 0 and rho is at least -1/2.

Why can three assets not all be strongly negatively correlated?

Three friends cannot all sit opposite each other at a table: if A faces B and B faces C, then A and C are on the same side. Correlation has the same constraint. If asset A moves against B and B moves against C, A and C are pushed towards moving together, so there is a floor on how negative a common correlation can be, and for three assets that floor is -1/2. You can see the floor without any algebra by holding an equal-weight portfolio of the three, each with unit variance: its variance is (3 + 6 rho) / 9, which is (1 + 2 rho) / 3, and a variance cannot be negative. At rho = -1/2 the portfolio has zero variance; it is perfectly hedged, and nothing below that is possible.

How do you find the eigenvalues without expanding a determinant?

Write the matrix as (1 - rho) times the identity plus rho times the all-ones matrix J. The identity leaves every vector alone, so you only need the eigenvalues of J, and J is easy: it maps (1, 1, 1) to (3, 3, 3), eigenvalue 3, and it maps any vector whose entries sum to zero to the zero vector, eigenvalue 0, with a two-dimensional space of such vectors. Shifting and scaling by (1 - rho) turns those into 1 - rho + 3 rho = 1 + 2 rho for the market direction and 1 - rho for the two spread directions. Check with the trace: the eigenvalues add to 1 + 2 rho + 2(1 - rho) = 3, the sum of the diagonal, as they must. At rho = 0.3 they are 1.6, 0.7 and 0.7.

What does the structure tell a risk or trading desk?

The eigenvectors are the principal components. The (1, 1, 1) direction is the market factor, and its eigenvalue over the trace, (1 + 2 rho) / 3, is the share of total variance it explains: 53% at rho = 0.3 and 80% at rho = 0.7. The two spread directions carry the rest, equally. A long-short book that nets to zero across the three assets lives entirely in the 1 - rho directions, which is why pairs trades get calmer as correlation rises and why a correlation of 1 collapses them to nothing. For n assets the same argument gives eigenvalues 1 + (n - 1) rho and 1 - rho, so the floor is -1 / (n - 1): -1/3 for four assets and -1/9 for ten. Say the limitation as well: a historical correlation matrix estimated from more assets than observations is only barely positive semi-definite, and a hand-edited one, where a trader overrides a few pairs, can fail the test entirely, which is exactly the fault a risk system is built to catch.

Between about -0.75 and 0.95. The three correlations must form a positive semi-definite matrix, and for three variables that means its determinant 1 - a^2 - b^2 - c^2 + 2abc cannot be negative. Solving for c with a = 0.2 and b = 0.5 gives c = ab plus or minus sqrt((1 - a^2)(1 - b^2)) = 0.1 plus or minus sqrt(0.96 x 0.75) = 0.1 plus or minus 0.849.

Why can the third correlation not be anything it likes?

If Amit is tall when Bela is tall, and Bela is tall when Chitra is tall, then Amit and Chitra cannot be perfectly opposite; the middle person ties them together. Three pairwise correlations describe one joint set of three variables, and a joint set has a variance for every combination of them that cannot be negative, which is the positive semi-definite condition on the correlation matrix. With weak links, 0.2 and 0.5, the tie through Y is loose and the band is wide; with links of 0.9 and 0.9, X and Z would be forced to be strongly positively correlated.

How do you get the band without writing out eigenvalues?

Use the determinant. For a 3 x 3 correlation matrix with off-diagonal entries a, b and c, the determinant is 1 - a^2 - b^2 - c^2 + 2abc, and the matrix is valid exactly when that is at least zero. Treat it as a quadratic in the unknown c: c^2 - 2ab c + (a^2 + b^2 - 1) must be at most zero, so c lies between the two roots ab plus or minus sqrt((1 - a^2)(1 - b^2)). With a = 0.2 and b = 0.5 the centre is 0.1 and the half-width is sqrt(0.96 x 0.75) = sqrt(0.72), about 0.849. The geometric reading is the same thing: think of each variable as a unit vector, correlations as cosines of the angles between them, and the third angle is bounded by the sum and difference of the first two.

Where does this show up on a desk?

In any correlation matrix that was assembled by hand or from mismatched histories. A desk that marks the stock-to-index correlation at 0.9, the index-to-sector correlation at 0.9, and the stock-to-sector correlation at 0.3 has written down a matrix that no world can produce, and a basket option priced from it will give nonsense, often a negative variance for some combination. The fix is to project the matrix back to the nearest valid one before pricing. The limitation of the puzzle is that it covers only three variables; with more, every principal sub-matrix must pass, and the feasible set has no simple closed form.

About 0.17, exactly one sixth. For n equally weighted stocks with the same volatility and the same pairwise correlation rho, index variance is stock variance times (1/n + (1 - 1/n) rho). Index variance over stock variance is 0.15 squared over 0.30 squared = 0.25, so 0.25 = 0.1 + 0.9 rho and rho = 0.15/0.9 = 1/6. The large-n shortcut of 0.25 overstates it because it ignores the 1/n term.

Why does the index have less volatility than its members?

Ten shopkeepers in one market each have noisy daily takings, but the market's total takings are steadier, because one shop's bad day is often another's good day. Only the part of the noise they share, the weather or a festival, survives the averaging. Index variance has two parts: the stocks' own noise, which averages away as 1/n, and the common movement, which survives in proportion to the correlation. With ten stocks at 30% and no correlation at all the index would still have 9.5% volatility, and that floor is why the implied correlation is below the naive 0.25.

How do you derive the formula on the spot?

Write the index as the average of n returns and expand its variance. There are n variance terms, each sigma squared over n squared, and n(n - 1) covariance terms, each rho sigma squared over n squared, so index variance is sigma squared times (1/n + (n - 1) rho / n). Set that equal to 0.15 squared with sigma = 0.30 and n = 10: 0.0225 = 0.09 x (0.1 + 0.9 rho), so 0.25 = 0.1 + 0.9 rho and rho = 1/6. The derivation takes four lines, and the interviewer wants to hear the count of covariance terms, n(n - 1), said out loud.

What does a desk do with implied correlation?

It compares it with realised correlation and trades the gap. Selling index options and buying single-stock options is a short correlation position, called a dispersion trade, and it pays when the stocks move more independently than the 1/6 the market has priced in. The limitation of the puzzle is its symmetry: real indices have unequal weights and unequal volatilities, so the implied correlation is a weighted average that can differ from the simple formula, and the implied figure also moves with the skew of the index options, which the single number cannot show.

Give both assets a shared monthly drift and opposing daily noise. Let A's daily return be m/21 + e + u and B's be m/21 - e + v, where m is a monthly drift common to both, e is noise that hits them in opposite directions, and u and v are their own noise. Inside a month m is fixed, so daily returns co-move through -e alone: correlation -0.50 with equal variances. Over a month e sums to a modest total while m is large, so monthly returns correlate at +0.66.

Why can one pair of series carry two correlations?

Two shops on the same street both do better in a festival month and worse in a quiet one, so their monthly takings rise and fall together. But on any given day, the customer who buys lunch at one is not buying it at the other, so their daily takings move against each other. Nothing is paradoxical: the two effects live at different speeds. Correlation is not a property of two assets, it is a property of two assets at a horizon, because different components of the return dominate at different horizons. The slow shared component barely moves within a day; the fast opposing component barely survives a month.

How do you put numbers on the construction?

Take a drift m with a standard deviation of 12 points a month, opposing noise e of 1 point a day, and own noise u and v of 1 point a day. Within a month m is a constant, so it drops out of the daily covariance: cov = -var(e) = -1, and each variance is var(e) + var(u) = 2, so the daily correlation is -1/2. Over 21 days the sums of e, u and v each have variance 21, and the monthly covariance is var(m) - 21 var(e) = 144 - 21 = 123 against a variance of 144 + 21 + 21 = 186. The monthly correlation is 123/186 = 0.66, positive, because the shared drift's variance grows with the square of the month while the noise variance only grows in proportion to it.

Say where this shows up on a desk, because that is why the question is asked. Two stocks in one sector share the sector's slow repricing but trade against each other intraday as market makers hedge one with the other. A hedge ratio estimated from daily data will be wrong for a position held for months, and the reverse. The limitation of the example: a flat drift inside a month is a simplification, and in real data the slow component is itself noisy, so the sign flip is usually a fade from negative towards positive rather than a clean reversal.