Derivatives Foundation puzzles, solved step by step

Puzzles
100
Traced to a firm
66
Topics
12
Hard
29
Topic
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
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 004Two stocks each have 30% annual volatility and a correlation of 0.5. What is the volatility of a basket that holds half of each? What if the correlation were zero?Volatility and correlationWarm upEquity derivativesRisk management

    Try it first

    Before the formula: can a 50/50 basket of two 30% stocks ever be more volatile than 30%?

    Show the worked solution

    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.

    Basket volatility against correlation: it reaches the parts' 30% only at rho = 1-1.0-0.50+0.5+1.010%20%30%correlation between the two stockseach stock alone: 30%rho 0: 21.21%rho 0.5: 25.98%rho 1: 30.00%rho -1: 0%, a perfect hedgeThe gap below 30% is the diversificationit exists only because rho is below 1
    Basket volatility rises with correlation from 0% at minus 1 through 21.21% at zero and 25.98% at 0.5 to the parts' 30% only at a correlation of 1, so the gap below 30% is the diversification and it exists only because the stocks are imperfectly correlated.
    The relationship
    σB2=w2σ2+w2σ2+2w2ρ σ2=2×0.25×0.09 (1+ρ)⇒σB=0.301+ρ2\sigma_B^2 = w^2\sigma^2 + w^2\sigma^2 + 2w^2\rho\,\sigma^2 = 2 \times 0.25 \times 0.09\,(1+\rho) \quad\Rightarrow\quad \sigma_B = 0.30\sqrt{\tfrac{1+\rho}{2}}
    wthe weight of each stock, 0.5
    sigmaeach stock's volatility, 0.30
    rhothe correlation between the two stocks
    sigma_Bthe basket's volatility
    What it says in wordsWith equal weights and equal volatilities, the basket's volatility is the single-stock volatility times the square root of (1 plus rho) over 2.

    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.

    Where candidates lose it

    The fast wrong answer is 30%, from averaging the two volatilities. Volatility is a square root, and square roots do not average. Add the variances and the covariance, then take the root.

    The second loss is forgetting the factor of 2 on the cross term. With it, the correlation 0.5 answer is 25.98%; without it, you get 23.72% and an interviewer who knows the number immediately.

    What the interviewer asks next

    • Three stocks at 30% volatility, all pairwise correlations 0.5, equal weights. What is the basket volatility?
    • As the number of equally correlated stocks grows large, where does the basket volatility settle, and why?
    • The basket option is quoted at 24% implied volatility. What correlation is the market pricing?