Derivatives Foundation puzzles, solved step by step

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  1. 069An index of ten equally weighted stocks has 15% implied volatility, and each of the ten stocks has 30% implied volatility. What average correlation is the market implying?Volatility and correlationHardVolatility tradingExotics trading

    Try it first

    Index vol is half the single-stock vol. Rough instinct for the implied correlation?

    Show the worked solution

    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.

    Index volatility rises with correlation; read 15% off the curve at about 0.1700.20.40.60.810%10%20%30%average pairwise correlationindex volatility, 10 stocks each at 30%15% index vol at rho = 1/6 = 0.167rho 0: 9.5%rho 1: 30%no correlation can push index vol below 9.5%Solve itindex var = stock var x(1/n + (1 - 1/n) rho)0.15^2 / 0.30^2 = 0.250.25 = 0.1 + 0.9 rhorho = 0.15 / 0.9 = 1/6Shortcut for large n:rho = 0.25, which overstatesby ignoring the 1/n term
    For ten equally weighted stocks at 30% volatility, index volatility rises from 9.5% at zero correlation to 30% at a correlation of one, and a 15% index volatility is reached at an average correlation of one sixth, about 0.17.

    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.

    The relationship
    σI2=σ2(1n+n−1nρ)  ⇒  ρ=σI2/σ2−1/n1−1/n=0.25−0.10.9=16\sigma_I^2 = \sigma^2\left(\frac{1}{n} + \frac{n-1}{n}\rho\right) \;\Rightarrow\; \rho = \frac{\sigma_I^2/\sigma^2 - 1/n}{1 - 1/n} = \frac{0.25 - 0.1}{0.9} = \frac{1}{6}
    sigma_Iindex volatility, 15%
    sigmasingle-stock volatility, 30% for every stock
    nthe number of equally weighted stocks, ten
    rhothe average pairwise correlation implied by the two volatilities
    What it says in wordsIndex variance is stock variance times one over n plus the correlation times the rest, which solves to a correlation of one sixth.

    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.

    Where candidates lose it

    The common wrong answers are 0.5 and 0.25: the first divides volatilities and the second divides variances but forgets the 1/n floor from the stocks' own noise. With only ten stocks the floor is a tenth of the variance, which is a big correction.

    The second loss is knowing the formula but failing to say what trade it supports. Say dispersion, and say which side is short correlation.

    What the interviewer asks next

    • Same numbers with 50 stocks instead of 10. What is the implied correlation?
    • What is the lowest possible index volatility for ten stocks at 30%, and at what correlation?
    • Implied correlation is 1/6 and realised correlation turns out to be 0.4. Which side of a dispersion trade made money?
    • The stocks have different weights and volatilities. How does the formula change?