Two stocks each swinging 20 percent a year with a correlation of one half give their product a volatility of 20 root 3, or 34.641 percent, rather than 40. Adding the two numbers is correct at exactly one correlation, namely 1, because the composite volatility is the law of cosines with the correlation as the cosine of the angle between two arrows. Pricing the call at 40 percent overstates it by 14.2 percent and ignoring the correlation understates it by 16.9 percent, and the other diagonal of the same parallelogram prices the ratio of the two stocks.
A futures settles up every day, so its fair price is a plain expectation, while a forward settles once, so its fair price is a discounted expectation renormalised. The difference between the two is exactly the covariance of the discount factor with the contract price divided by the expected discount factor, and for a deposit contract quoted as 100 minus the rate both fall together, so the fair forward price is 95.019999 against the futures' 95.000000. Long the forward and short the futures is worth 0.0195 points at inception, the gap reaches 39.48 basis points at ten years, and on an asset whose price rises with rates the whole answer reverses.
Two stocks with equal expected returns, variances 0.10 and 0.40, and correlation 0.5: the reflex differentiates the portfolio variance and reports an interior weight. The minimum sits at 100% in the calmer stock, and the usual explanation for that, which blames the no-shorting rule, is wrong. The vertex of the variance parabola lands exactly on w = 1, so the constraint does no work at all and the answer survives dropping it.
Six sevenths in the quieter share beats holding it alone, because what the jumpy share adds at the margin is its correlation times its own swing, fifteen against twenty. The derivative of the variance at a full allocation is exactly +1/50, so the informal test and the first-order condition are one statement. The article carries the exact optimum sqrt(27/700), the convexity making it a minimum, and the correlation threshold of two thirds above which the dip disappears entirely.
A shift leaves every deviation from the mean untouched, and a stretch multiplies the covariance and one standard deviation by the same factor, so both cancel out of the ratio. The tempting answer of five times rho is worse than wrong: at rho = 0.40 it names 2.0, which Cauchy-Schwarz forbids any correlation from reaching. The article carries the general affine rule, the sign flip a negative factor produces, and the curved maps the invariance does not survive.