In a rally you want positive convexity, and a mortgage pool has negative convexity, because the borrowers hold the right to prepay and you are short that option. Modelled as a ten-year 6 percent bond minus a three-year call struck at 105, the straight bond has convexity plus 68.8 and the pool minus 177.4, and doubling a rally from 100 to 200 basis points takes the bond from 7.79 percent to 16.35 while the pool goes only from 4.21 to 6.13. The pool still gains, so the reflex is right about the sign and wrong about the size, and the single parameter set in 108 with positive curvature is one whose prepayment option is far out of the money.
A bond paying 100 in ten years costs 67.5564 at a four percent yield. The first two points of yield cost 11.7169 and the next two only 9.5201, so the curve bends. The usual explanation blames duration falling as yields rise, and this bond refutes it: with a single cash flow its Macaulay duration is exactly ten at every yield. The slope is minus duration times price over one plus the yield, and the general statement needs no duration at all, only that every discount factor is convex.
Let a fair coin decide at the start of the year whether the share runs at 15 or 35 percent, price calls in that world, then read the volatilities back out with the constant-volatility formula: 28.43, 25.91, 24.97, 25.68 and 27.16 percent across five strikes. The floor sits at the money and below the 25 percent average of the two regimes, which one second derivative settles without any numerics. The usual explanation for the wings is refuted here, because the coin-flip world is less likely to clear 130 than a flat 25 percent and its option is still worth 27 percent more.
Moving the average inside the exponential returns 1, and 1 happens to be the exact median and the exact geometric mean of e^X, which is why the mistake survives every re-check of the arithmetic. Completing the square in the exponent gives the true value e^(sigma squared over two), or 1.6487 at unit spread, because multiplying a Gaussian density by e^x slides its centre and scales its mass. Convexity settles the direction before any integral is set up, and on a heavy-tailed variable the quantity stops being finite at all.
Raise your right hand at a mirror and the reflected hand stays on the same side of the room, which means the usual question has a false premise. A plane mirror is the matrix diag(1, 1, -1): it fixes both axes lying in the glass and reverses only the direction you look along. Its determinant is -1, so no rotation reproduces it, and the sideways flip everyone reports belongs to the half turn you perform in your head.
The hedge on a long call is the slope of its value, and a convex curve flattens as you slide left, so a falling share forces a smaller short and a smaller short is a purchase. No volatility, maturity or distribution enters that argument. The rebalance buys twenty shares, and the position gains 0.9824 per share on the fall.
Strip the outer shell off a ten-by-ten-by-ten block of unit cubes and count what falls. The reflex answer, 271, is arithmetic done correctly on the wrong picture: a shell leaves from both opposing faces, so every axis loses two units and not one. Two independent counts land on 488, and the general shell turns out to grow like a surface rather than a volume.