Options calculator methodology

Applies to Build, Chain, Scan and Positions

PayoffPlot separates observed market inputs from modelled estimates. This page documents the data, the pricing model and the rules behind every figure in the calculator and finder, so results can be interpreted and reproduced.

Assumptions at a glance

Input	What the app uses
Quotes	Alpaca. Option quotes are indicative (adjusted rather than exact exchange quotes); the quote time is on screen.
Fill price	Natural by default: buy at the ask, sell at the bid. Midpoint or last trade on request.
Pricing model	European Black–Scholes–Merton with a continuous dividend yield, floored at intrinsic value.
Risk-free rate	US Treasury daily par yield curve, interpolated to each leg's time to expiry.
Dividends	Continuous yield from announced and trailing cash dividends.
Volatility	Each leg's IV calibrated to its own bid/ask midpoint under the same model.
Time	Calendar time to the 16:00 New York close on expiry day (13:00 on early closes).
Contract size	100 shares per option contract.
Probability	Risk-neutral, at the first expiration, with a skew correction at each breakeven.

Market data and pricing

Stock quotes, option contracts, bid/ask prices, last trades, implied volatility (IV), Greeks, volume and open interest come from Alpaca. Option quotes are indicative: adjusted rather than exact exchange quotes, with last trades about 15 minutes delayed. Before the market opens, option quotes are those of the previous session, so they can lag a stock that has moved overnight. Where Alpaca has no IV for a contract, it is solved from the contract's own quote; where no IV reproduces the quote (a stale price below intrinsic value, for example), it is interpolated from neighbouring strikes.

The calculator shows the market session and how long ago prices were fetched beside the stock price, and refreshes them every minute while the regular session is open and the page is visible.

Quote basis

Natural: Buy legs use the ask and sell legs use the bid. This is the conservative default.
Midpoint: Uses (bid + ask) / 2 for a valid, non-zero two-sided market, then falls back to last.
Last trade: Uses the latest positive trade, then falls back to midpoint when necessary.

A manually edited premium, including a premium restored from a legacy shared URL, is not overwritten when the global quote basis changes.

Where you'll see itSPY options chain All options chains

Payoff and commissions

At expiration, a call's intrinsic value is max(stock price − strike, 0); a put's is max(strike − stock price, 0). Long-leg profit is intrinsic value minus premium, while a short leg reverses that cash flow. Each result is multiplied by contract quantity and the standard 100-share option multiplier. Stock legs use quantity × (future price − entry price), with the sign reversed for short shares.

P/L(S_T) = Σ ±q · 100 · (intrinsic(S_T) − premium) + shares · (S_T − S₀)

Commissions

Estimated commission = option contracts × commission per contract per side × charged sides. Opening-only calculations use one side; enabling the closing estimate uses two. Commission is subtracted once from every position outcome and from expected value. It does not alter Greeks.

Positions with several expirations

Calendars, diagonals and other multi-expiry positions are summarised at the first expiration, when the nearest leg expires. Max profit, max loss, breakevens, probability of profit and expected value all refer to that date, with the later legs valued by the pricing model below at that moment.

Where you'll see itBreakeven calculator Covered call payoff Iron condor payoff

Rates, dividends and time

Risk-free rate: From the US Treasury daily par yield curve, interpolated to each leg's own time to expiry. Refreshed daily, with a recent snapshot as a fallback.
Dividends: A continuous dividend yield derived from announced and trailing cash dividends.
Time to expiry: Calendar time to the 16:00 New York close on expiration day (13:00 on early-close days), with daylight saving time handled.

Where you'll see itBlack-Scholes calculator

Option pricing model, IV and Greeks

Before expiration, option values come from the European Black–Scholes–Merton model with a continuous dividend yield. Listed equity options are American-style, so each value is floored at its intrinsic value — an option is never worth less than exercising it now.

C = S·e^−qT·N(d₁) − K·e^−rT·N(d₂)
P = K·e^−rT·N(−d₂) − S·e^−qT·N(−d₁)
V = max(V_BSM, intrinsic)

Each leg's IV is calibrated to its bid/ask midpoint under this same model, rate and clock. A provider's IV comes from the provider's own model; re-pricing with it would show a profit or loss on a position opened a second ago that is nothing but model mismatch. After calibration, a position's P/L at today's price is just the difference between the midpoint and what was paid. The IV control scales every calibrated IV.

Delta, gamma, theta, vega and rho are calculated per leg and signed by buy/sell direction and quantity. The Greeks panel shows them for the whole position, as a broker would:

Greek	Shown as
Delta Δ	Share equivalents
Gamma Γ	Change in that delta per $1 move in the stock
Theta Θ	Dollars per calendar day
Vega ν	Dollars per one-point change in IV
Rho ρ	Dollars per one-percentage-point change in rates

Where you'll see itBlack-Scholes calculator Greeks calculator Implied volatility calculator IV and Greeks on the SPY chain

Probability of profit and expected value

Probability of profit is risk-neutral: it is the probability implied by option prices, at the first expiration, of the stock finishing inside the position's profitable ranges. Each breakeven K is evaluated at the IV of the smile at that strike, plus a correction for the smile's slope there (taken from the position's own legs at that expiration):

P(S_T > K) = N(d₂) − e^rT · S · e^−qT · φ(d₁) · √T · ∂σ/∂K

S is the stock price, T the time to expiry in years, r the risk-free rate, q the dividend yield, σ the IV at K, N and φ the standard normal distribution and density, and d₁ = [ln(S/K) + (r − q + σ²/2)T] / (σ√T), d₂ = d₁ − σ√T. Without the correction, an equity put skew understates the chance of finishing above a short put's breakeven by several points.

Expected value is market-consistent. Under the same risk-neutral measure, every instrument's expected value at the horizon is its value today grown at the risk-free rate, so the expectation reduces to Σ(model value now × e^rT − premium paid) across the legs, plus financing on any shares, less commissions. For a fairly priced trade that is close to the bid/ask spread and fees given up on entry; it is positive only when the entry price was better than the model's fair value.

Both are model estimates, not forecasts. Risk-neutral probabilities include the market's risk premia and are not the real-world chance of any outcome.

Where you'll see itProbability of profit in Build Breakeven calculator Iron condor guide

Income metrics

For positions opened for a credit, cash-secured puts and covered calls, the summary adds:

Capital at risk: Cash-secured put: strike × 100 × contracts − credit. Covered call: share cost − credit. Other defined-risk credit trades: the max loss. None when the max loss is unlimited.
Return on risk and annualised: Max profit ÷ capital at risk, and that return × 365 ÷ calendar days to the first expiration (simple, not compounded).
Theta per day: The position's theta in dollars: the change in value over one calendar day with price and IV unchanged.
Expected move: One standard deviation by the first expiration, S · σ · √T, with σ the at-the-money IV (the average of the call and put IV at the strike nearest the stock price).
50% profit target: The buy-back cost that keeps half of the premium received, before fees.

Debit positions show theta per day and the expected move only.

Where you'll see itCovered call guide Cash-secured put guide SPY expected move

Liquidity score

The 100-point score combines relative bid–ask spread (50 points), open interest (30), and volume (20). Grades are A at 80+, B at 65+, C at 45+, D at 25+, and F below 25. Crossed or invalid markets and zero-bid contracts score zero. Warnings identify wide spreads, low open interest, and absent volume. Multi-leg finder grades are quantity-weighted averages; liquidity breaks ties after the selected optimization score.

Grade	Score
A	80 and above
B	65–79
C	45–64
D	25–44
F	Below 25

Where you'll see itLiquidity grades in Scan

Limitations

Option quotes are indicative, with last trades about 15 minutes delayed; outside the regular session they are from the last session. Displayed prices do not guarantee an executable fill.
Early exercise is modelled only through the intrinsic-value floor. There is no further early-exercise premium, so deep in-the-money American puts and calls ahead of a dividend can be slightly undervalued, and assignment risk is not modelled.
Dividends are a continuous yield, not discrete payments on their ex-dates.
Earnings and other scheduled events are not modelled: each leg keeps its current IV, so a post-earnings drop in volatility is not anticipated.
Taxes, exercise and assignment fees, slippage beyond the selected quote basis, and margin requirements are not modelled.
Probability, expected value and the expected move are sensitive to IV, rates, time and distribution assumptions.

Use the results for education and scenario analysis, not as financial advice. Return to the options calculator to inspect the active assumptions beside your trade.