Probability Distributions

Overview

Accurate probability modeling is the foundation of expected value calculations. FlashAlpha uses sophisticated probability distributions that account for real-world market behavior.

The Problem with Black-Scholes

The classic Black-Scholes model assumes:

• Log-normal distribution - Symmetric price movements
• Constant volatility - Same vol at all strikes
• No jumps - Continuous price movements

Reality is different:

• Markets have fat tails - Extreme moves happen more often
• Volatility smiles and skews - Different strikes imply different volatility
• Gap risk exists - Prices can jump overnight

FlashAlpha's Approach

Implied Distribution Extraction

We extract the market's implied probability distribution directly from option prices:

1. Collect the volatility surface - IV at all strikes and expirations
2. Interpolate and smooth - Fill gaps, remove arbitrage
3. Apply Breeden-Litzenberger - Convert option prices to probabilities
4. Result: Risk-neutral distribution - What the market is pricing

Distribution Adjustments

The risk-neutral distribution isn't the real-world distribution. We apply adjustments:

• Volatility risk premium - Markets overprice volatility
• Historical calibration - Blend with realized outcomes
• Regime detection - Adjust for current market conditions

Distribution Types

Standard (Market-Implied)

Uses the raw market-implied distribution. Best for:

• Liquid underlyings
• Normal market conditions
• Short-dated options

Historical-Adjusted

Blends market-implied with historical outcomes. Best for:

• Less liquid names
• When IV seems mispriced
• Longer-dated options

Fat-Tail Enhanced

Increases probability of extreme moves. Best for:

• Earnings plays
• Binary events
• High-uncertainty environments

Custom

Define your own distribution parameters:

• Skewness adjustment
• Kurtosis (tail fatness)
• Mean shift

Viewing Distributions

In the Analysis module:

1. Select any position or spread
2. Click Probability tab
3. View the probability density function (PDF)
4. See cumulative distribution function (CDF)

Key Metrics

• Expected Move - 1 standard deviation range
• Probability ITM - Chance of finishing in-the-money
• Probability of Touch - Chance of hitting strike before expiration
• Tail Probabilities - Chances of extreme outcomes

Practical Applications

Identifying Mispriced Options

Compare:

• Market-implied probability of strike X finishing ITM
• Your model's probability

If they differ significantly, there may be an opportunity.

Stress Testing

Use fat-tail distributions to:

• Model worst-case scenarios
• Size positions appropriately
• Set realistic stop levels

Event Trading

Around earnings or events:

• Use higher kurtosis (fatter tails)
• Consider bimodal distributions
• Account for gap risk