Methodology

Exactly how every number is computed

Every price and Greek on OptionWit comes from the Black-Scholes model for European options. This page states the exact formulas, conventions, and limitations so you can reproduce any figure yourself. The engine is a single small file of JavaScript that runs in your browser; the same file produces the worked examples in every guide.

Pricing formulas

For a spot price S, strike K, time to expiry T in years, volatility σ, and risk-free rate r:

d1 = [ ln(S / K) + (r + σ² / 2) · T ] / (σ · √T) d2 = d1 - σ · √T Call = S · N(d1) - K · e^(-rT) · N(d2) Put = K · e^(-rT) · N(-d2) - S · N(-d1)

N(x) is the standard normal cumulative distribution function. We evaluate it with the Abramowitz & Stegun 26.2.17 rational approximation, whose maximum absolute error is about 7.5 × 10^-8, far finer than any price is displayed to.

The Greeks

Delta, gamma, theta, and vega are the standard Black-Scholes sensitivities, with two display conventions worth stating explicitly:

Delta: N(d1) for a call, N(d1) - 1 for a put Gamma: N'(d1) / (S · σ · √T) Vega: S · N'(d1) · √T shown per 1 vol point (value / 100) Theta: the annual BS theta, shown per calendar day (value / 365)

N'(x) is the standard normal density. Theta is reported per calendar day, not per year, because that is the number traders actually watch. Vega is reported per one volatility point (a move from 45% to 46%), not per unit of decimal volatility.

Conventions

Day count

Time to expiry is measured in calendar days divided by 365. A 30-day option uses T = 30 / 365. This counts weekends and holidays as elapsed time, which is why an option can lose value over a weekend even though no session traded. A trading-day count (252 per year) is a defensible alternative; it front-loads theta into fewer, larger daily steps and lands in a similar place. The calculator's advanced inputs let you switch between a calendar (365) and trading (252) basis.

Day count on expiry day (0DTE)

A position expiring today has no days left to count, so the hour-by-hour board measures what is left of the session instead. The same two conventions carry over, with hours in place of days:

Trading: T = remaining_session_hours / 6.5 / 252 Calendar: T = remaining_hours / 24 / 365

A regular US equity session runs 9:30am to 4:00pm ET, which is 6.5 hours, so the trading basis treats one full session as one of its 252 days. The calendar basis keeps counting wall-clock hours against a 365-day year, the same way it counts overnight and weekend hours the market is shut.

The two disagree by a lot intraday, far more than they do at 30 days out. With a whole session left, the trading basis gives T = 6.5 / 6.5 / 252 = 1/252 and the calendar basis gives T = 6.5 / 24 / 365, which is roughly 5.3 times smaller. Option value scales with the square root of time, so the trading basis carries a little over twice as much time value. On a $743.29 stock, a $743 put at 15% IV and a 4.5% rate is worth $2.59 with a full session left on the trading basis, but only $1.06 on the calendar basis.

The hourly board defaults to the trading basis, because that is closer to how intraday options are actually quoted: the market prices the session it can trade, not the hours it is closed. The calendar basis is still one click away in the advanced inputs. Either way, both are approximations. Real intraday volatility is not spread evenly across the session, so treat the final hour before the close as the least reliable part of the board.

Risk-free rate

The rate is treated as continuously compounded, entering only through the discount factor e^(-rT). It is a minor input at typical levels; the default of 4.5% is a reasonable stand-in for short-term Treasury yields and can be changed in the position panel.

Breakeven at expiry

The panel's breakeven is the stock price where the option's intrinsic value at expiry equals the premium you paid: strike + premium for a call, strike - premium for a put. Before expiry your true breakeven sits closer to the current price, because the option still carries time value, which is exactly what the projection board shows.

A worked check

With S = 100, K = 105, σ = 45%, r = 4.5%, and 30 days to expiry (T = 30/365), the model returns a call value of $3.27, delta 0.388, theta -$0.087 per day, and vega $0.110 per vol point. These are the default inputs on the calculator, so you can confirm them in a few seconds, and they are the same numbers our test suite pins the engine to.

Known limitations

European exercise. The model assumes exercise only at expiry. Most listed US equity options are American and can be exercised early, which adds a small premium the model does not capture, mostly relevant deep in the money or around dividends.
Dividends via continuous yield. The engine takes an optional dividend yield q (default 0) in the advanced inputs, applied as a continuous yield rather than discrete ex-dividend dates. For most single stocks 0 is fine; set it for ETFs or high-yield names. Discrete dividends and the early-exercise premium they can create are not modeled.
Constant volatility. Each projection holds implied volatility fixed at your input (plus any IV change you set). Real IV moves with price and time and has a skew across strikes. The IV change control lets you stress this manually.
Model, not market. Outputs are theoretical values. Real quotes differ with bid-ask spread, liquidity, and live IV. Treat the board as a map of the terrain, not a fill price.

If any of these matters for your position, the honest move is to treat the numbers as a well-calibrated estimate and confirm against your broker's live chain before acting.