Documentation / Pricing Model & Assumptions

Pricing Model & Assumptions

How option prices, implied volatility, and Greeks are calculated

The Black-Scholes-Merton Model

Theoretical prices and Greeks come from the Black-Scholes-Merton (BSM) model, the industry-standard closed-form formula, extended to account for dividends. Stratsigma uses it everywhere it needs a theoretical value: the options chain, position modeling, payoff and T+ lines, and historical/backtesting valuation.

It takes six inputs:

  • Underlying price: the current or simulated price
  • Strike price
  • Time to expiration: computed precisely from the current (or virtual) time to expiry
  • Implied volatility: solved per strike (see below)
  • Risk-free interest rate
  • Dividend yield: see assumptions below

Price and all Greeks are derived analytically in a single pass (no numerical approximation of the derivatives), using a high-accuracy approximation of the normal distribution. At and after expiration, an option is valued at its intrinsic value.

Implied Volatility (Per-Strike)

Implied volatility is solved independently for each strike from that contract's market price, producing a full volatility surface. The skew across strikes is preserved, which matters for getting spread Greeks right.

The solver uses Newton-Raphson for fast convergence, with a put-call-parity bisection fallback for deep in-the-money options, where Newton-Raphson can stall. If a quoted price implies no valid volatility (for example, a price at or below intrinsic value), so that strike simply has no IV rather than a misleading number.

Dividend Assumptions

Dividends are modeled as a continuous yield:

  • Index options: 0%. Recognized index roots (SPX/SPXW, NDX/NDXP, RUT/RUTW, VIX, XSP, DJX, MNX) carry no dividend yield.
  • All other underlyings: a flat 2%. Stocks and ETFs are priced with an assumed 2% annual dividend yield.

Because the 2% figure is a single assumption rather than each name's actual yield, theoretical prices for high-yield ETFs or non-dividend payers can differ slightly from a model tuned to their real yield. For index options, which are genuinely dividend-free, the 0% assumption is exact.

A risk-free interest rate is also applied. See Risk-Free Interest Rates below.

Risk-Free Interest Rates

Rather than a single flat rate, Stratsigma pulls the real U.S. Treasury yield curve and matches a tenor to each option's days to expiration (DTE):

  • Under 30 days: 1-month rate
  • 30 to 90 days: 3-month rate
  • 90 to 180 days: 6-month rate
  • 180 days to 1 year: 1-year rate
  • 1 year or more: 2-year rate

The curve is refreshed from the U.S. Treasury about every 6 hours and saved locally, so pricing always has a rate available, even right after a restart.

If a tenor is briefly missing from the Treasury feed, it's filled from the nearest available real tenor, and if the feed is temporarily unreachable the most recent real rates are used until the next refresh succeeds, so under normal operation the rate is always a genuine Treasury rate, not a fixed guess.

The one exception is a safety net: if an in-browser calculation such as the payoff diagram can't reach the latest rates at all, it temporarily uses a 5% placeholder so charts keep rendering until the real rates load.

Greek Conventions

All Greeks are dividend-aware and computed from the closed-form BSM derivatives. The scaling matters for reading them correctly:

  • Delta: price change per $1 move in the underlying
  • Gamma: delta change per $1 move in the underlying
  • Vega: per 1 percentage point (one vol point) change in implied volatility
  • Theta: decay per calendar day (not per trading day)
  • Rho: per 1 percentage point change in the risk-free rate

Assumptions & Limitations

BSM rests on simplifying assumptions; knowing them explains any gap between theoretical and market prices:

  • Log-normal returns: real markets have fatter tails. Extreme moves happen more often than the model implies.
  • Volatility: each strike is priced with its own solved IV, so skew across strikes is represented, but BSM still treats that IV as constant over the option's remaining life, which real volatility isn't.
  • European exercise: the formula prices European-style options. That's an exact fit for cash-settled index options (SPX, NDX, RUT), which are European. Equity and ETF options are American and can be exercised early, giving them a small premium over the model, most noticeable in deep in-the-money puts with time remaining.
  • No transaction costs: theoretical prices ignore commissions and spreads; commissions are tracked separately in your P&L.