Estimated price-outcome distribution at expiry

An estimated probability distribution, centered on the live spot and built from AAPL's own realized volatility (how much it has actually moved) — where it could sit at expiry given that, not a forecast. The volatility level is a realized-vol estimate and the skew shape is modeled — neither is market-implied. With the mean, mode and median, the probability above versus below spot, an arbitrage-free check, and notes on each figure.

AAPLspot $333.08realized vol 31.9%arbitrage-free ✓realized-vol estimate · skew modeled

Likely range at expiry$310.60 – $370.55 — the 16th–84th percentile (68%) band, read off the estimated distribution; spot $333.08.Two ways of measuring the same spread: half the band above is $29.97, while the distribution's own typical swing is $30.40 — a quantile band and a moment need not match on a curve leaning -0.33; both are read off the same served grid.

estimated density68% rangespotmedian

Mean (realized-vol est.)$334.18

Mode (most likely)$340.57

Median$335.76

P(below spot) at expiry46.4%

P(above spot) at expiry53.6%

Horizon30d

1-σ swing (±)$30.40

Skew-0.33

Probability mass1.000

realized vol 31.9%63d close-to-closeannualized ×√2521-σ $30.4030d horizonmoments of the served density grid (trapezoid)skew -0.3330d horizonthird moment of the served density grid (trapezoid)mean $334.1830d horizonBreeden-Litzenberger density · modeled skew shape

AAPL's realized volatility puts the most-likely landing near $340.57 with a $30.40 one-sigma swing, a left-skewed shape, and is arbitrage-free. The agent below explains how a vol level and a strike grid become a probability distribution, and how that curve weights a strategy's payoff. The vol level is a realized-vol estimate from daily closes and the skew shape is modeled — neither is read off the chain's quoted implied vols, so neither is market-implied; the Breeden-Litzenberger math is real. Your decisions, your risk.

Each card walks one piece of the distribution — what it shows, what the math means, and what it suggests — so a beginner can read this realized-vol probability curve.

What is this probability distribution?

Built from AAPL's realized volatility, the most-likely landing is near $340.57 (the peak), with a mean of $334.18 and a one-sigma spread of about $30.40.

A volatility level + a strike grid, run through Breeden & Litzenberger (1978), becomes an entire probability distribution: line up call prices across strikes and the curvature of that line literally IS the density. The peak is the most-likely landing spot; the width is how uncertain the move is. Here the vol level is a realized-vol estimate from daily closes — not the quoted implied vols on the options chain, so it is not market-implied.

→ Most-likely close ~ $340.57; a realized-vol estimate of a $30.40 one-sigma swing.

How the skew bends the distribution

The distribution is left-skewed (fatter downside tail — more weight on a downside surprise) and close to a normal bell curve in the tails. Skewness -0.333, kurtosis 3.2467.

A flat-IV (one number for every strike) world gives a clean lognormal bell. Real markets charge MORE implied vol for downside puts — that "skew" fattens the left tail of this curve, so the distribution leans toward downside surprises. The shape of the IV smile is the shape of the fear.

→ Downside protection is bid — the left tail is fat.

Is the curve arbitrage-free?

Yes — the call-price curve is convex everywhere, so every recovered probability is non-negative.

Probabilities can never be negative. Mathematically that requires the call-price curve to be CONVEX (bend upward) across strikes. If a middle call is too cheap relative to its neighbors, the second difference goes negative — a free-lunch "butterfly" arbitrage. The lab checks this automatically.

→ No-arbitrage: the recovered density is valid.