Documentation / Pricing Model & Assumptions

Pricing Model & Assumptions

How option prices and Greeks are calculated

Black-Scholes-Merton Model

All theoretical option prices and Greeks are calculated using the Black-Scholes-Merton (BSM) model. This is the industry-standard closed-form formula for pricing European-style options. The model takes five inputs:

  • Underlying price: The current (or simulated) price of the stock/index
  • Strike price: The option's strike
  • Time to expiration: Precisely calculated from the current (or virtual) time to the expiration date
  • Implied volatility (IV): Sourced from market data or calculated from OHLC prices
  • Risk-free interest rate: Real Treasury yield curve rates matched to the option's time to expiration

Two-phase IV solver: When solving IV from a market price, the system first uses Newton-Raphson for fast convergence, then falls back to put-call parity plus bisection for robustness — particularly for deep in-the-money options where Newton-Raphson can struggle.

Model Assumptions

The BSM model relies on several simplifying assumptions. Understanding these helps explain any differences between theoretical prices and actual market prices:

  • Log-normal price distribution: Stock prices follow geometric Brownian motion, meaning returns are assumed to be normally distributed. In reality, markets exhibit "fat tails" (extreme moves happen more often than the model predicts).
  • No dividends: The model does not account for dividend payments. This can cause slight mispricing on dividend-paying stocks, especially near ex-dividend dates.
  • Constant volatility: IV is assumed to remain the same over the life of the option. In practice, volatility changes continuously.
  • European-style exercise: The formula prices European options (exercisable only at expiration). This is technically incorrect for American-style options (like most equity and ETF options), though the difference is usually small except for deep in-the-money puts.
  • Continuous trading: Markets are assumed to be open and liquid at all times, with no gaps or halts.
  • No transaction costs: Commissions and bid/ask spreads are not factored into theoretical prices (though the platform tracks commissions separately for P&L).

Risk-Free Interest Rates

Rather than using a single flat interest rate, the platform fetches real U.S. Treasury yield curve rates and matches them to each option's days to expiration (DTE):

  • Under 30 days: 1-month T-bill rate
  • 30 to 90 days: 3-month T-bill rate
  • 90 to 180 days: 6-month T-bill rate
  • 180 days to 1 year: 1-year Treasury rate
  • Over 1 year: 2-year Treasury rate

Rates are refreshed from the U.S. Treasury every 6 hours. If the Treasury data is temporarily unavailable, the platform falls back to a default rate of 5%. This DTE-matched approach provides more accurate pricing than using a single flat rate for all options.

The current Treasury rates are visible in the Settings page so you can verify the rates being used.

Greeks Calculations

Greeks are computed analytically from the closed-form BSM partial derivatives (not approximated numerically):

  • Delta: Rate of change of option price per $1 move in the underlying
  • Gamma: Rate of change of delta per $1 move in the underlying
  • Theta: Daily time decay (how much value the option loses per day)
  • Vega: Sensitivity to a 1% change in implied volatility
  • Rho: Sensitivity to a 1% change in the risk-free interest rate

You can choose between two Greeks sources in Settings:

  • Market Data: Uses Greeks provided directly by the market data feed
  • Calculated (BSM): Recalculates all Greeks using the BSM model with the current inputs

Practical Limitations

The two most significant real-world impacts of the model's assumptions are:

  • No dividend adjustment: For stocks that pay dividends, the model may overvalue calls and undervalue puts slightly, especially for options spanning an ex-dividend date. Index options (SPX, NDX, RUT) are less affected since indices are typically quoted on a total-return or adjusted basis.
  • European vs. American exercise: American-style options (most equity and ETF options) can be exercised early, which gives them slightly more value than the European model predicts. This difference is most noticeable for deep in-the-money puts with significant time remaining. Index options like SPX are actually European-style, so the BSM model is a better fit for those.

Despite these limitations, BSM remains the standard pricing framework across the industry and provides reliable theoretical prices for most trading scenarios.