Reading the RND: PDF, CDF, Expected Moves

In the last chapter you extracted a risk-neutral density from a strip of option prices. That was the hard part. This chapter is about something more valuable: actually reading it. Because a density sitting on a screen is not yet an answer to any question you care about. The questions you care about sound like "What is the chance this finishes above four hundred and fifty by Friday?" and "Where, honestly, does the market think this thing lands, and how wide is the doubt?" Those are not questions the density answers by being looked at. They are questions you answer by integrating it.

So let us build the intuition first, and the calculus will follow without a fight.

Picture the forecast density as a pile of probability spread along the price axis at expiry. Tall where the market thinks the price is likely to settle, low and trailing off in the tails where it thinks settlement is unlikely but not impossible. This shape is the probability density function, the PDF. Here is the single most important thing to internalize about a PDF, and it is the thing beginners get wrong: the height of the curve at a given price is not a probability. A continuous price can land at infinitely many exact values, so the chance of hitting any one of them exactly is zero. The height is a probability density — a rate. Probability only appears when you take an area. The area under the curve between two prices is the probability the underlying finishes between those two prices. The total area, from zero to infinity, is exactly one, because the price has to finish somewhere.

That move — from height to area — is the whole game. And the tool that performs it is the cumulative distribution function, the CDF.

The CDF answers one question and answers it everywhere at once: "What is the probability the underlying finishes at or below this price?" You build it by sweeping a running total of area from the far left tail rightward. Formally, the CDF evaluated at a price K is the integral of the density from zero up to K. So if you call the density f and the CDF F, then F of K equals the integral, from zero to K, of f of s with respect to s. The CDF starts at zero on the far left, climbs monotonically — it can never go down, because you are only ever adding non-negative area — and flattens out at one on the far right. Its slope at any price is exactly the density there: steep where the density is tall, flat where the density is thin. The PDF is the derivative of the CDF; the CDF is the integral of the PDF. They are two readings of the same object.

Once you have the CDF, the questions you actually care about become subtraction. The probability of finishing above a strike K is one minus F of K — the whole pile minus everything at or below K. The probability of finishing inside a band, between a lower price a and an upper price b, is F of b minus F of a. Every "what are the odds" question collapses to evaluating F at one or two prices.

Now run the CDF backwards and you get the other thing you want: quantiles. Pick a probability, say zero-point-five, and ask which price has that much area to its left. That price is the median of the risk-neutral forecast — the level the market puts even odds on the price being below. Pick zero-point-one-six and zero-point-eight-four and you have carved out the central sixty-eight percent of the distribution, the risk-neutral analogue of a one-standard-deviation band. Pick zero-point-zero-five and zero-point-nine-five and you have a ninety percent confidence interval. These quantiles, plotted across a range of expiry dates, are exactly what stack up to form the probability cone: a fan of confidence bands widening with time, each edge a quantile of the density for that horizon.

A word on the expected move, because it is easy to quote the wrong number. The naive expected move is the at-the-money straddle price, which approximates the expected absolute move — and in a symmetric lognormal world that expected absolute move is only about eight-tenths of a standard deviation, sigma times the square root of two over pi, not a full one. So if you want the actual one-standard-deviation move, you have to scale the straddle up: the one-sigma move is roughly the straddle divided by zero-point-eight, about one-and-a-quarter times the straddle. The honest expected move, read from the RND, is taken from its own quantiles — and when the density is skewed, as equity-index densities almost always are, the downside quantile sits farther from the spot than the upside one. The cone is not symmetric, and that asymmetry is information. It is the market pricing crash risk into the left tail.

This is where the chapter ties back to the throughline of this institute: judging when a number is genuine evidence an asset will move. A wide cone is not a forecast of a big move — it is a confession of uncertainty. A narrow cone that suddenly widens, or a left tail that fattens while the right stays put, is the market revising its odds. That revision is the signal. The level is noise; the change in the shape is evidence.