Probability-Targeted Strike Selection

In the last chapter you did something most traders never do: you pulled a full probability distribution out of the options market. You took a calibrated, arbitrage-free smile, ran the Breeden–Litzenberger machinery, and out came a density — a curve over every price the asset could land on at expiry, telling you how much risk-neutral probability the market is assigning to each outcome. That density is a beautiful object. But a density, on its own, is not yet a decision. Today you turn it into one.

Here is the question a strike picker actually has to answer. You do not want to know the probability of every price at once. You want to go the other way. You want to say: find me the strike that the asset has, say, a twenty percent chance of closing above. That is the question behind sizing a covered call, setting a target, or placing a stop where the market itself thinks the odds sit. And answering it is an exercise in inverting a cumulative distribution function — what this chapter is really about.

Let us build the intuition first. The density you extracted, call it little f, tells you probability per unit price at each point. To get the probability of finishing below some level, you sweep from the far left and add up — you integrate the density from minus infinity up to that level. The result is the cumulative distribution function, big F. So F of K is the risk-neutral probability that the asset finishes at or below K. It starts at zero far to the left, climbs as you walk right, and approaches one far to the right. It never goes back down, because you are only ever adding non-negative probability as you sweep. That single fact — that F only ever rises — is the hinge the whole method swings on.

Now notice what you have. The CDF is a function that takes a price and returns a probability. The strike picker wants the opposite: hand it a probability, get back a price. That reverse map is the inverse CDF, written F-inverse, and it is also called the quantile function. If you want the strike with probability p of finishing below it, the answer is exactly K equals F-inverse of p. And if you instead think in terms of finishing above — the more natural way to talk about a call — you simply target one minus your probability. A strike with a twenty percent chance of closing above it is the strike with an eighty percent chance of closing below it, so you compute F-inverse of zero-point-eight. The whole feature is that one formula: K equals F-inverse of p.

The reason this is well-posed — the reason there is exactly one strike for each probability and not zero and not several — is monotonicity. Because F is strictly increasing across the support of the density, every probability level between zero and one corresponds to one and only one price. There is no ambiguity to resolve and no choice to make. The inverse exists and is unique. This is the mathematical gift the density hands you, and it is why the picker can be trusted to give a single, definite strike rather than a fuzzy region.

So how do you actually compute F-inverse, given that you almost never have a clean formula for F? You invert it numerically, and you exploit monotonicity to do it cheaply. The most robust way is bisection. You want the K where F of K equals your target p. Bracket it: pick a low price where F is clearly below p and a high price where F is clearly above p. Look at the midpoint. If F of the midpoint is too small, your answer lies to the right, so the midpoint becomes your new low bound; if it is too large, the answer lies to the left, and the midpoint becomes your new high bound. Each step halves the interval, and because F is monotone you can never bracket the wrong root. A dozen or so halvings pin the strike to a precision finer than any tick you would ever trade. In practice, since the density was extracted on a discrete price grid, you already hold F as a cumulative sum along that grid; then inversion is even simpler — you find the two grid points whose cumulative probabilities straddle p and linearly interpolate between them for the strike. Same idea, same guarantee, less work.

One discipline before you act on the number. The probability you are inverting is the risk-neutral probability, not the real-world one. It is the market's risk-adjusted measure, the one consistent with no-arbitrage pricing, and it is systematically not the same as the physical probability of the move actually happening — the equity wing, in particular, carries a risk premium that fattens the downside. So when the picker tells you a strike has a twenty percent risk-neutral chance of finishing in-the-money, read that precisely: it is the probability priced into the options, which is the right input for hedging and relative value, but it is the market's belief plus a premium, not a crystal ball. Knowing which measure you are standing in is, once again, half of not making an error — and it is exactly the discipline that separates a number that is genuine evidence an asset will move from a number that merely looks like one.