A plain t sitting next to the t squared in a Gaussian exponent looks like a new function and is only a shift. Completing the square turns the integral of e to the minus a t squared over two plus b t, from x to infinity, into e to the b squared over 2a times the root of 2 pi over a times the standard normal at a rescaled and shifted argument, never at x itself. The worked case comes out as exactly half a bell, e root pi over two or 2.40901455, but only because its lower limit happens to land on the centre b over a.
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.
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.
A pays the floating rate L and receives 24 per cent minus 2L, which nets to 24 minus 3L and factors as three times 8 minus L: three vanilla swaps at eight per cent, so the fixed rate was never the twenty-four printed on the deal. Reading it as twenty-four is a 48-point error at a floating rate of twenty-four. The article also carries the version that does need a model, where a floor on the inverse leg adds two caplets struck at twelve and the factorisation fails.
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.
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.
The long hand turns 6 degrees a minute and the short one half a degree, so a 90 degree gap closes at 5.5 degrees a minute and the hands coincide 180/11 minutes past three, at 3:16:21.8181. At 3:15 the long hand has reached the 3 and the short hand is 7.5 degrees ahead of it, which is the whole content of the wrong answer. Consecutive coincidences are 720/11 minutes apart, so there are eleven per twelve hours and twenty-two per day, and the common phrasing "eleven times a day" is wrong by a factor of two.
With no air, v squared equals 2gD gives 89.4 metres per second and t equals root of 2D/g gives 8.94 seconds, both stable under g = 10 or g = 9.81. Dividing the height by the impact speed returns 4.47 seconds, wrong by exactly a factor of two at every drop height, because a body released from rest averages half its final speed. Real air reverses the picture: a coin-sized disc reaches terminal velocity near 11.9 metres per second and takes about 34.6 seconds, so nine seconds is a floor and 200 miles an hour a ceiling.
Running straight out from the centre loses, because a radius costs you 1 while half the fence costs the dog pi over 4. Inside a quarter of the radius your angular speed beats his, so you can orbit until he is diametrically opposite and then sprint three quarters of a radius against his pi over 4, and the whole escape reduces to 3 being less than pi with a margin of 0.0354 R. That two-phase plan works only up to a speed ratio of pi + 1, while the best known strategy for the problem reaches 4.60334.
Servings follow area and area follows the square of the width, so feeding eight instead of six multiplies the diameter by the square root of four thirds: exactly 8 root 3, or 13.8564 inches, about 15.5 percent wider. Sixteen inches carries 16/9 of the area and would feed 10.67 people, so the reflex over-orders by nearly three servings. Allowing a one inch bare crust moves the answer down to 13.55, because a wider pizza spends proportionally less of itself on edge.
Differentiating y = L tan(wt) gives a spot speed of w R squared over L, so the footprint accelerates with the square of its distance from the lamp: a tenth of pi directly opposite, and exactly pi miles per second nine miles along. The 9 is the along-shore leg, which makes 90 the squared hypotenuse rather than the square of nine, and that misreading is the usual failure. Nothing physical moves at that speed, and a straight coast running 2310 miles would carry a nominally faster-than-light spot carrying no information at all.
The two correlations you are handed do not pin the third one down, but they fence it into exactly [−1/50, 1], and that interval dips below zero. The fence falls out of a 3×3 determinant read as a quadratic in the unknown, and out of a picture: 0.7 is an angle of 45.573°, both stocks live on a cone of that half-angle around the index, and putting them on opposite sides opens 91.146° between them. Also here: why the real tipping point is ab ≥ 0 together with a² + b² ≥ 1 rather than "both above 0.707", why 0.9 and 0.5 force a positive answer while 0.9 and −0.9 allow −1, why standing on the floor costs a rank, and why three Bernoulli(0.5) indicators with the same two correlations are confined to [0.40, 1] instead.
Take the centre, then mirror every move through it, and you place the last coin. The proof has three requirements and only one of them needs that opening move, which is the step a one-line answer skips. Central symmetry alone is not the condition: an annulus is centrally symmetric and the first player loses on it.
Nine slots hold the digits 1 to 9 and four overlapping windows each add to 20. One subtraction pins the centre digit to 5 with no search at all, and the same structure classifies every arrangement: there are 96 of them, falling into 48 mirror pairs. Change the target and the centre is forced to an odd digit that sometimes admits nothing.
One radius drawn to the rectangle's far corner turns the whole problem into a right triangle with legs R - 10 and R - 5, and the quadratic that follows has roots 5 and 25. Both satisfy the equation exactly, so rejecting 5 takes geometry rather than arithmetic: at that radius the corner really does touch the circle while the rectangle has already swallowed half the disk. The general a by b rectangle shows the discarded root is a permanent feature of squaring.
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.
Factoring into p minus one times p plus one stops the question being about p: the two neighbours are consecutive even numbers so their product carries eight, and one of the three consecutive integers around p is a multiple of three which cannot be p itself. Since eight and three are coprime, 24 divides, and 24 is exactly maximal.
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.
Two riders twenty-five miles apart close at fifty miles an hour, so a forty mile an hour fly shuttling between them flies exactly twenty miles. The series of shuttle legs gives the same twenty, with a first leg of 100/7 and a round-trip ratio of 1/21, but no finite number of legs ever reaches it: after eighty legs the exact total is twenty minus about 2.6e-52. The general law is wD/(a+b), and it holds identically in the rider speeds rather than by luck at 20 and 30.
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.
One condition, a two-line recurrence, and the fifth power falls out as a clean 123 with no radicals left. Climb the same ladder far enough and the golden ratio and the Lucas numbers are hiding underneath.
A line with rational coefficients maps ℚ onto ℚ. Nothing curved ever does. Three filters — interpolation, shape, denominators — leave the full classification.