Options-Implied Probability: How the Calculator Works

Options-implied probability is the chance of a price move that is consistent with option prices under a standard pricing model. Vyreon reports, for each security, the probabilities of a move of at least 5% or 10% up or down over 30 days, and the one-standard-deviation price range, all from its 30-day implied volatility. These are what option prices imply, not forecasts.

How Options Imply A Probability

The Black–Scholes model assumes that log prices follow a normal distribution, with spread set by the volatility. Once you choose a volatility, the model gives the probability of ending above or below any price. Using implied volatility as that input turns option prices into probabilities.

Options Probability Formula

For a current price S, a target S·(1 + x), volatility σ, time t in years and interest rate r:

d2 = [ ln(1 / (1 + x)) + (r − σ²/2) · t ] / (σ · √t)
P(end at or above S·(1 + x)) = N(d2)

N is the standard normal cumulative distribution. "Down at least 5%" is one minus the probability of ending above S·(1 − 0.05). The one-standard-deviation range is S·e^(−σ√t) to S·e^(+σ√t).

Illustrative example: price $100, σ = 20%, t = 30/365, r = 4%. The model gives about 21% for up 5% or more, 18% for down 5% or more, 5% for up 10% or more, 3% for down 10% or more, and a one-standard-deviation range of about $94.43 to $105.90. This is the same arithmetic an options probability calculator performs.

How Vyreon Measures It

Volatility: Vyreon's 30-day at-the-money implied volatility (IV30). If IV30 is withheld, so are the probabilities.
Horizon: 30 calendar days.
Rate: the federal funds rate. Dividends and other carry are not included.
Outputs: P(up ≥ 5%), P(down ≥ 5%), P(up ≥ 10%), P(down ≥ 10%), and the one-standard-deviation low and high, measured from the day's closing price.

Risk-Neutral Vs Real-World Probability

These are risk-neutral probabilities: the odds under which option prices would be fair if nobody required payment for bearing risk. Real-world odds differ, for three reasons:

Risk premiums. Option buyers pay for protection, and studies of index and stock options have found implied volatility, on average, above later realized volatility (see Sources). Where that holds, priced odds of large moves are higher than the volatility that followed would imply.
Model shape. Real returns have fatter tails and skew. Extreme moves have happened more often than a lognormal model says (see Sources), and a single at-the-money volatility ignores the IV skew.
Drift. The model grows prices at the interest rate, not at any expected return.

So "21% chance of up 5%" means "option prices are consistent with 21% under this model". It does not mean "this will happen 21% of the time".

What It Does Not Tell You

It is not a forecast and not the probability that a trade is profitable.
It does not use the whole options chain. Probabilities derived from many strikes would reflect skew and could differ.
It does not say anything about the path the price takes before day 30.

Sources

Fischer Black and Myron Scholes, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy (1973): the model and the N(d2) probability.
Douglas T. Breeden and Robert H. Litzenberger, "Prices of State-Contingent Claims Implicit in Option Prices", Journal of Business (1978): probabilities implied by option prices across many strikes.
Peter Carr and Liuren Wu, "Variance Risk Premiums", Review of Financial Studies (2009): found option-implied variance on average above later realized variance for all five stock indexes and most of the 35 individual stocks studied.
Benoit Mandelbrot, "The Variation of Certain Speculative Prices", Journal of Business (1963), and Eugene F. Fama, "The Behavior of Stock-Market Prices", Journal of Business (1965): returns with fatter tails than a normal distribution.
John C. Hull, Options, Futures, and Other Derivatives (Pearson, many editions): risk-neutral valuation.