How Options Are Priced

An option’s premium is the price a buyer pays and a seller receives for the option. Its value depends on the possible payoff, time remaining and contract terms. A pricing model estimates that value from assumptions about the underlying, volatility, interest rates and dividends. An actual trade, however, takes place at a price available in the market—not necessarily the model’s estimate. 1

Principal valuation inputs

Underlying price and strike

A call gives its holder the right to buy at the strike; a put gives the right to sell. With other inputs fixed, a higher stock price makes a call more valuable and a put less valuable. With the stock price and other inputs fixed, a lower strike makes a call more valuable because its holder would pay less for each share bought on exercise. A higher strike makes a put more valuable because its holder would receive more for each share sold on exercise.

The response depends on how far the stock is above or below the strike—its moneyness. A deeply in-the-money call generally tracks the stock more closely than an out-of-the-money call, whose expiration payoff is still zero unless the stock crosses the strike. Delta and gamma measure this changing price sensitivity.

Time and volatility

Time sets the horizon over which the stock can move; volatility describes the scale of that variability. Higher volatility generally increases standard call and put values because it expands the potential for favorable payoffs while leaving the holder free not to exercise after an unfavorable outcome. An option can therefore have value even when it is out of the money. 2

Time value usually declines toward expiration, but more time is not always worth more. An otherwise identical later-expiring American option includes the earlier option’s exercise opportunities. A European option instead moves its only exercise date later. At positive interest rates, that delay can reduce a deeply in-the-money European put’s value by postponing receipt of the strike proceeds.

Because these effects combine, a call can lose value despite a modest stock rise if falling volatility or time decay has a larger effect. A put can likewise lose value despite a modest stock decline.

Interest rates and dividends

Higher rates reduce the present value of a future strike payment. That generally benefits a call holder who can pay later, but reduces the value of the proceeds a put holder may receive. This comparison holds the stock price fixed; it does not predict the stock’s reaction to changing rates.

Shareholders, not option holders, receive dividends. Expected distributions reduce the future ex-dividend stock price relative to an otherwise identical path without them, generally lowering call values and raising put values. Distribution dates and amounts also matter for early exercise, particularly for an in-the-money call with little time value left. 3

Why is the calculator different from my broker?

Often, the two numbers measure different things. A model value, a broker’s mark and the price of an executed trade are not interchangeable. Before looking for a calculation error, check the price type, contract, units, valuation time and opening premium. As OIC’s pricing-model explanation notes, formulas estimate theoretical values; markets determine actual premiums.

Compare prices that mean the same thing

The bid is an offered buying price and the ask an offered selling price. Their midpoint is simply an average, not a completed transaction. The last trade records an earlier execution that may no longer reflect the market. Quote timing and available quantity matter too; the option-quote guide explains these fields.

Suppose you bought one option at $8.70 per share. A later hypothetical quote shows an $8.30 bid, an $8.70 ask and a $9.10 last trade. The midpoint is $8.50. With a 100-share multiplier:

Comparison with the $8.70 purchase	Calculation before costs	Meaning
Closing sale actually fills at $8.30	($8.30 − $8.70) × 100 = −$40	Realized loss for that hypothetical fill
Broker values the open position at $8.50	($8.50 − $8.70) × 100 = −$20	Unrealized estimate, if that broker uses the midpoint
Display shows last trade $9.10	No new sale has been established	An earlier trade does not establish your exit price

These are illustrative figures, not live quotes. Brokers can use different marks, and fees or other costs change the result. Even a currently displayed bid does not guarantee a later fill at that price.

Match the contract and units

Check the underlying, call or put, strike, expiration, exercise style, quantity and multiplier. A per-share premium is not a whole-position value: $8.50 per share is $850 for one 100-share contract, before costs. Other products and adjusted contracts can have different multipliers or settlement terms.

Align the assumptions and valuation time

Compare the same underlying price, remaining time, IV, interest rate and dividends. A stale stock price, different dividend assumption or different treatment of early exercise can explain a discrepancy without either calculation being arithmetically wrong.

This calculator uses your assumptions, not broker data. The ticker labels a scenario; it does not load quotes. All legs share one expiration, volatility, interest rate and dividend yield, whereas market IV can vary by strike and expiration. Valuation uses whole days: expiry today has zero remaining model time and intrinsic value, not intraday decay. See methodology for the models and units.

Check what the opening premium represents

Entry price is the opening underlying price, not the option premium you paid. The calculator models opening premiums at that price and today’s date. Editing Entry price, IV, rates, dividends or contract details recalculates those premiums and the position summary. The time-view price slider edits the same opening price.

Analysis date changes the stock-price graph’s valuation date, not the opening premium or today’s position summary. The stock-price view varies the underlying on that date; the time view holds Entry price fixed and varies the date. Reading a point on a curve is different from editing an opening assumption.

There is no actual-fill input or broker import. Adjusting assumptions until a premium matches your purchase price does not turn the graph into a record of that trade. Reconcile an actual trade using entry and exit premiums, quantity, multiplier and costs; for an open position, identify the mark used to value it.

Read IV in its price and model context

Implied volatility is the volatility that makes a selected model match a selected premium, with other inputs fixed. The bid, midpoint and ask can therefore imply different IVs. Time, dividend and exercise assumptions also affect the answer. An IV figure is best read alongside those inputs, as explained in implied volatility.

European and American pricing models

Black–Scholes–Merton values European options, which permit exercise only at expiration. It assumes continuous stock-price changes, constant volatility and interest rates, and frictionless trading. Its dividend-adjusted form uses a continuous yield rather than individual dividend payments. 4

The model can be derived by combining a changing shareholding with borrowing or lending to reproduce the option’s payoff. Under its assumptions, this replicating portfolio and the option must have the same value to prevent arbitrage. The calculation does not require a forecast of the stock’s expected investment return.

American options also permit early exercise. A binomial tree splits the remaining term into steps with possible up and down stock moves. Working backward from expiration, it compares the immediate exercise value with the discounted value of continuing to hold the option at each exercise opportunity, choosing the greater amount. 1

For a non-dividend-paying call at nonnegative interest rates, early exercise adds no value under frictionless assumptions: holding the option preserves its remaining choice and delays the strike payment. Its American and European model values are therefore equal. A dividend-paying call or an in-the-money put can benefit from early exercise. The calculator respects this distinction rather than applying the European formula to every American option.

European pricing formulas

Let $S$ be the current underlying price, $K$ the strike and $T$ the remaining time in years. Annual volatility is $\sigma$ , the continuously compounded interest rate is $r$ and the continuous dividend yield is $y$ . Use decimals for volatility and rates: 20% volatility is $0.20$ . For positive $S$ , $K$ , $T$ and $\sigma$ , per-share call value $C$ and put value $P$ are:

C=Se^{-yT}N(d_1)-Ke^{-rT}N(d_2)

P=Ke^{-rT}N(-d_2)-Se^{-yT}N(-d_1)

Here $N(z)$ is the cumulative standard normal distribution—the probability that a standard normal variable is no greater than $z$ —and $\ln$ is the natural logarithm. The arguments are:

d_1=\frac{\ln(S/K)+(r-y+\sigma^2/2)T}{\sigma\sqrt{T}}

d_2=d_1-\sigma\sqrt{T}

The factor $e^{-rT}$ discounts the strike payment, while $e^{-yT}$ adjusts the underlying-price term for the continuous dividend yield. The $d_1$ and $d_2$ terms combine the stock-to-strike relationship, financing, dividends and volatility over the remaining term.

For a European option with a $750 underlying price, a $775 strike, 45 days remaining, 20% volatility, a 4% interest rate and a 1% continuous dividend yield, the model gives approximately $12.03 for the call and $34.14 for the put per share, using $T=45/365.25$ .

At expiration, value is intrinsic value. Zero time or zero volatility needs a limiting case rather than direct substitution into $d_1$ and $d_2$ . With zero volatility, the model follows a deterministic stock-price path and discounts the resulting payoff.

Model limitations

Markets can jump, volatility changes and actual dividends arrive on specific dates. Constant volatility and a continuous yield simplify those features; a finite tree also limits exercise decisions to its time steps. A model value is therefore conditional on its assumptions, not an executable quote. 4

Exercise and assignment can create share positions and funding obligations outside the calculator’s option-only P/L. It does not calculate broker margin or account returns, and separate option contracts retain their own exercise and assignment treatment.