How the numbers are made

Every number on screen is computed on your device from the stated assumptions by the same pricing engine. Nothing is generated or fetched, and no result is typed in by hand. This page lists the model, its assumptions and what it leaves out.

Model

• Option prices and Greeks: the Black-Scholes model with a continuous dividend yield.
• Delta is used as a rough stand-in for the probability of finishing in the money. It is an approximation, not a forecast.
• Expected value is shown two ways: under the lognormal (Black-Scholes) assumption, and under a fat-tail version that reuses the shape of real 21-trading-day log returns from the preceding history window (at least 60 samples, otherwise the row is not shown).
• The chance of reaching the 50% take-profit point before expiration is a Monte Carlo estimate with 2,000 paths and a fixed seed, so the same inputs always give the same number.
• Tail risk (CVaR) is the average result of the worst 5% of those fat-tail outcomes.

Assumptions

• Implied volatility = historical volatility × 1.15 in the archived scenarios and the wheel simulation (historical volatility over the previous 20 trading days, annualised). Option sellers are paid for implied volatility usually sitting above realised volatility; the factor makes that gap explicit instead of hiding it.
• No VIX: it is a separate licensed index that exists for very few underlyings. Deriving volatility from the same price history keeps every number reproducible from one bundled data set.
• Interest rate: 4% in the wheel simulation; between 0.1% and 5% by year in the archived scenarios. Dividend yield: 1.3% for SPY in the archived scenarios; 0 in the wheel simulation, because its prices are dividend-adjusted.
• Every trade fills at the theoretical (Black-Scholes) price.
• In the calculator, every input is yours; the app does not fill in market values.

What is not included

• Bid-ask spreads and slippage.
• Commissions and fees.
• Taxes.
• Margin requirements.
• A full treatment of early assignment and ex-dividend risk.
• Any promise of returns. Results under these assumptions are teaching illustrations, not predictions.

Data

• SPY daily closing prices, dividend-adjusted, from 1993-01-29 to 2026-09-11 (8,462 trading days), bundled with the app. 216 archived daily scenarios are built from historical closes. No live or delayed market data is used.
• Source and licence: to be confirmed.

Checks

• The pricing, expected-value and rule engines are checked number by number against an independent Python implementation.
• Key statistics quoted in lesson text are recomputed from the data by a script, and the release checks fail if they drift.

Thetify is for educational purposes only and does not provide financial, investment, or trading advice.