Methodology: How CalcOptions Calculates Every Number

Bjerksund and Stensland 2002 (default model)

The default model is the Bjerksund and Stensland (2002) closed form approximation for American options. It values a call with a two period flat exercise boundary using normal and bivariate normal probabilities (Hart 1968 and Genz 2004 algorithms), and it never returns less than the Black Scholes Merton value or intrinsic value. Puts use the put call transformation: a call with S and K swapped and r and q swapped. When q is zero or below, early exercise of a call is never optimal, so the European value is used.

CRR binomial tree

The Cox Ross Rubinstein tree splits T into equal steps. Each step moves the stock up by e raised to volatility × the square root of the step length, or down by the inverse, with a risk neutral probability based on r minus q. Each node takes the greater of its discounted expected value and its exercise value. To damp the oscillation between odd and even step counts, CalcOptions averages trees of 150 and 151 steps.

Barone Adesi and Whaley 1987

The Barone Adesi and Whaley (1987) quadratic approximation adds an early exercise premium to the Black Scholes Merton price. Newton iteration finds the critical stock price where immediate exercise becomes optimal; short of that boundary the option is worth the European price plus the premium, and beyond it, intrinsic value. Calls with q of zero or below, and puts when r is zero or negative, use the European price.

Black Scholes Merton for index options

Black Scholes Merton is the European model, with a cost of carry of r minus q. The calculator switches to it automatically for SPX, NDX, RUT, VIX and XSP, which are European style and cash settled, and you can select any model manually.

Time to expiration

T is the number of calendar days from now to 4 pm New York time on the expiration date, divided by 365. The close is set at 21:00 UTC, within an hour of 4 pm New York all year. Weekends and holidays count, so theta is per calendar day.

Risk free rate

r defaults to the latest US Treasury 13 week bill rate, converted to a continuously compounded rate. If the rate cannot be loaded, 4 percent is used. You can type any rate.

Dividends

Dividends enter every model as a continuous yield q that you set as a percent. A higher yield lowers call values, raises put values and makes early exercise of deep in the money calls more likely. An entered next dividend and ex dividend date do not change prices; they only drive the dividend check in the assignment risk panel.

Implied volatility

For each market priced contract, implied volatility is solved with the selected model from the bid and ask midpoint (or the provider's mid or last price). The model then reproduces the market price, so today's P/L starts near zero. Typing your own entry price solves the volatility again. The solver is a safeguarded Newton method with a bisection fallback over 0.1 to 500 percent, stopping within a millionth of a dollar; if nothing matches, the provider's IV is used. Manual mode uses the volatility you enter.

P/L at any date

P/L at a stock price and date is the sum over legs of value minus entry price, times the quantity and 100 shares per contract, with long legs positive and short legs negative. Options are valued with the selected model at each leg's IV plus the IV slider shift (floored at 0.5 percent), r and q. Expired legs are worth intrinsic value and stock legs the stock price. The chart and the P/L table apply this at every price and date they show.

Max profit and max loss

Max profit and max loss are the highest and lowest P/L at the first expiration across 600 evenly spaced prices from near zero to 3 times the higher of S and the highest strike, plus every strike and S. A slope test compares P/L at 10 and 20 times that level: still rising means unlimited profit, and still falling means unlimited loss. With several expirations, open legs are valued with the model, so the figures are estimates.

Breakevens

Breakevens are the stock prices where P/L at the first expiration crosses zero. Each sign change on the same price grid is refined by bisection until P/L is within a ten millionth of a dollar of zero.

Chance of profit

Chance of profit is the probability that the position is profitable at the first expiration, with the stock price modeled as lognormal: mean log return (r minus q minus half of sigma squared) × T and standard deviation sigma × the square root of T. Sigma is the position's implied volatility, an average of leg volatilities (including the IV shift) with each leg weighted by its contracts × 1 / (0.02 + |ln(K / S)|), so far out of the money wings do not dominate. The price axis is split at the breakevens, and the probabilities of the profitable intervals are added.

Greeks

Greeks are finite differences of the selected model, so they include early exercise. Delta and gamma use central differences with a stock price bump of 0.5 percent, theta is the value change over one calendar day, and vega and rho are central differences per 1 point change in implied volatility and in r. Position Greeks sum each leg's Greeks times its signed quantity and 100 shares per contract, and each share of stock adds one delta.

Capital at risk and margin estimate

When max loss is limited, capital at risk is the larger of the net debit and the max loss. When max loss is unlimited, an estimated margin is shown instead: any net debit plus, for each short option leg, 20 percent of S minus the out of the money amount (minimum 10 percent of S for calls or of K for puts) plus the premium, times 100 per contract. This follows the spirit of Reg T rules, but your broker's requirement may differ.

Early exercise and assignment flags

The assignment risk panel rates each short option leg and lists long legs for information. Time value is model value minus intrinsic value, and the first matching rule applies:

• High: an in the money call, a dividend you entered goes ex before expiration, and the dividend exceeds the time value.
• High: in the money with time value below the larger of 5 cents and 0.1 percent of K.
• Medium: an in the money put whose time value is below K × r × T, the interest the holder could earn.
• Medium: in the money with less than 25 cents of time value.
• Medium: out of the money within 1 percent of K with under 3 days left (pin risk).
• Low: none of the above.

Data sources and delays

Ticker search covers about 7,400 US stocks, ETFs and indexes. Quotes and option chains, with implied volatility and Greeks, come from a market data provider and are delayed; outside market hours they show the previous session's close. Chains cover most US stocks; for ETFs and indexes you type the price and the calculator prices every contract with its model. Prebuilt strategies pick expirations near 30 days, about 30 days later and about one year for front month, back month and LEAPS legs.

Limitations

Every figure is a model estimate, with these limits:

• Each leg keeps one implied volatility, while real volatility moves with price, time, skew and events such as earnings.
• A continuous yield approximates discrete dividends, so prices near a large ex dividend date can differ from the market.
• Chance of profit uses a risk neutral drift, so it is not a forecast.
• Midpoint entry prices may differ from your fills, and taxes and fees are not modeled.
• The three American models are approximations and can differ slightly from one another.
• Margin is an estimate, and assignment flags are heuristics: exercise is always the holder's decision.

Frequently asked questions