Unlocking Market Expectations: Option-Implied Probability Distributions

What Options Tell Us About the Future
When traders buy or sell options, they implicitly reveal their expectations about where a stock's price might go. By analyzing option prices, we can extract the market's implied probability distribution—a mathematical snapshot of how likely different price levels are at future dates. This article explains the theory behind these probabilities and introduces a practical tool to visualize them.

The Core Idea: From Prices to Probabilities

Options are contracts that give the right (but not obligation) to buy or sell a stock at a set price (strike) by a specific date (expiration). Their prices reflect:

Probability assessments (the chance the stock will be above/below a strike by expiration).

Expected volatility (how much the stock might swing).

Key Insight

The market's implied probability that a stock (e.g., AAPL) will exceed price K at expiration T can be derived from option prices. Two elegant approaches exist:

Approach 1: The Intuitive Butterfly Spread

A butterfly spread combines three options at nearby strikes:

Buy 1 call at strike K+ΔKK+ΔK

Buy 1 call at strike K−ΔKK−ΔK

Sell 2 calls at strike KK

The payoff resembles a "spike" centered at K. Its cost approximates:Butterfly Price≈e−rτ⋅f(K)⋅(ΔK)2Butterfly Price≈e−rτ⋅f(K)⋅(ΔK)2

where f(K)f(K) is the probability density at K, rr is the interest rate, and ττ is time to expiry. Rearranging gives:f(K)≈Butterfly Price⋅erτ(ΔK)2f(K)≈(ΔK)2Butterfly Price⋅erτ​

This method is simple but limited to discrete strikes.

Approach 2: The Rigorous Breeden-Litzenberger Formula

For a continuous probability distribution, we start with a call option’s fair value:C(K,τ)=e−rτ∫K∞(x−K)f(x)dxC(K,τ)=e−rτ∫K∞​(x−K)f(x)dx

where f(x)f(x) is the probability density function (PDF). Taking derivatives with respect to K:

1. First derivative:

∂C∂K=−e−rτ∫K∞f(x)dx=−e−rτ⋅P(ST>K)∂K∂C​=−e−rτ∫K∞​f(x)dx=−e−rτ⋅P(ST​>K)

2. Second derivative:

∂2C∂K2=e−rτf(K)∂K2∂2C​=e−rτf(K)

Solving for the PDF:f(K)=erτ∂2C∂K2f(K)=erτ∂K2∂2C​​

This is the Breeden-Litzenberger formula—it converts call prices into a probability distribution.

The Interpolation Challenge

In practice, strikes are discrete (e.g., AAPL options at 155, $160). To compute the second derivative, we need continuous call prices. Naively interpolating prices can create arbitrage opportunities.

Solution: Interpolate in Volatility Space

1. Convert prices to implied volatilities (IVs) using the Black-Scholes inverse function.
2. Interpolate IVs smoothly across strikes (e.g., using splines).
3. Convert interpolated IVs back to prices.
   Why this works: IVs vary more smoothly than prices, avoiding arbitrage and producing stable PDFs.

Introducing the Options Implied Probability Terminal

The live tool operationalizes these concepts:

1. Inputs:
   • Stock symbol (e.g., AAPL).
   • Threshold price (defaults to the last price).
2. Outputs:
   • Probability Curve: Chance the stock exceeds the threshold at future expirations.
   • Heatmap: Implied probabilities across strikes and dates (see below).
   • Backtest: Historical accuracy of the 75% probability threshold.

How It Works

1. Fetches option data (from Alpha Vantage).
2. Calculates probabilities using Breeden-Litzenberger and IV interpolation.
3. Visualizes results:
   • Line chart: Probability trend across expirations.
   • Heatmap: Probability "landscape" (darker = more likely).
   • Backtest: Validates predictions against historical prices.

https://plot.ly/~demo/heatmap.png
Implied probability heatmap for AAPL. X-axis = price, Y-axis = expiration dates.

Why This Matters

• Traders: Identify overpriced options or misaligned market expectations.
• Risk Managers: Quantify tail risks (e.g., crash probabilities).
• Researchers: Test market efficiency ("Do implied probabilities predict reality?").

The Backtest Module answers this last question by:

1. Selecting 20 random dates (past 3 years).
2. Calculating the 75% probability threshold for 5 future expirations.
3. Comparing thresholds to actual prices.
4. Reporting success rate and Brier score (lower = better calibration).

Conclusion

Options encode the market’s probabilistic forecasts. Through the Breeden-Litzenberger formula and smart interpolation, we can extract these insights—transforming prices into probabilities. The Options Implied Probability Terminal makes this accessible, letting you visualize expectations and test their accuracy. As options markets grow, tools like this will become indispensable for data-driven investors.

"The future is uncertain, but options force us to quantify it."