Options Calculator
✦ Finance & MoneyUse this free options calculator to see the payoff, break-even price and max profit or loss for long and short calls and puts — plus a European Black-Scholes theoretical price and option Greeks, with a payoff chart and worked examples.
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The formula
Options payoff and Black-Scholes theoretical price
Payoff at expiry:
Long call: max(S_T - K, 0) - premium
Short call: premium - max(S_T - K, 0)
Long put: max(K - S_T, 0) - premium
Short put: premium - max(K - S_T, 0)
x contracts x multiplier - commission
Break-even: K + premium (call) | K - premium (put)
Black-Scholes-Merton (European):
d₁ = [ln(S/K) + (r - q + σ²/2)T] / (σ√T)
d₂ = d₁ - σ√T
Call: C = S·e^(-qT)·N(d₁) - K·e^(-rT)·N(d₂)
Put: P = K·e^(-rT)·N(-d₂) - S·e^(-qT)·N(-d₁)The payoff formula is deterministic — it requires no model and holds for any option at expiry. The Black-Scholes-Merton formula models European options only; it assumes constant volatility and a continuous risk-free rate and is not valid for American-style early exercise. Dividend yield q uses a continuous approximation (Merton 1973); discrete dividends are only approximated. N(·) is the standard normal CDF. T is days ÷ 365; theta is shown per calendar day. All inputs are manual — no live data is retrieved.
Calculating it by hand
- Select the option type (Call or Put) and your position (Long or Short).
- Enter the strike price and the premium per share you paid (or received).
- Set the number of contracts and the contract multiplier (default 100 for U.S. equity options).
- Add a commission per contract in the Advanced section if your broker charges one.
- Enter a spot price at expiry in the Spot price field to see the P&L at that level.
- The Black-Scholes block in the results shows the theoretical price and Greeks from its own inputs (underlying price, days to expiry, risk-free rate, implied volatility, dividend yield); open the Black-Scholes section of the form to change them.
- Read the break-even price, max profit and max loss from the result panel; check the payoff chart and table below for the full range of outcomes.
Worked examples
Long call: K = $100, premium = $5, spot at expiry = $110
Intrinsic value at expiry: max(110 − 100, 0) = $10.P&L per share = $10 − $5 premium = $5 profit.Total P&L = $5 × 1 contract × 100 multiplier = $500.Break-even = $100 + $5 = $105.Max loss (if the option expires worthless) = $5 × 100 = $500.
P&L at $110: +$500. Break-even: $105. Max loss: −$500.
Long put: K = $100, premium = $3, spot at expiry = $90
Intrinsic value at expiry: max(100 − 90, 0) = $10.P&L per share = $10 − $3 premium = $7 profit.Total P&L = $7 × 1 contract × 100 multiplier = $700.Break-even = $100 − $3 = $97.Max profit (if the underlying goes to $0) = ($100 − $3) × 100 = $9,700.
P&L at $90: +$700. Break-even: $97. Max profit (at $0): $9,700.
Black-Scholes call: S = $100, K = $100, 90 days, r = 5%, σ = 20%, no dividend
T = 90 / 365 = 0.246575 years; r = 0.05; σ = 0.20; q = 0.d₁ = [ln(1) + (0.05 + 0.02) × 0.246575] / (0.20 × √0.246575) = 0.017260 / 0.099313 = 0.1738.d₂ = 0.1738 − 0.0993 = 0.0745.N(0.1738) = 0.5690; N(0.0745) = 0.5297.C = 100 × 0.5690 − 100 × e^(−0.012329) × 0.5297 = 56.90 − 52.32 = $4.58.Delta = 0.5690; theta = −$0.0288 a day (−$10.53 a year ÷ 365); vega = $0.195 per 1-point change in σ.
Theoretical price: $4.58. Delta: 0.569. The option has a 53% risk-neutral probability of expiring in the money (N(d₂)).
Methodology
How options pricing and payoff work
Options have two different questions: what will this be worth at expiry? (deterministic — only the strike, premium and spot price at expiry matter) and what is it worth right now, before expiry? (probabilistic — volatility, time and interest rates all feed the model). Most confusion starts from mixing the two.
Read sources and limitsWhat does an option payoff diagram show?
A payoff diagram plots profit or loss against the underlying price at expiry. For a long call it is the familiar hockey stick: flat at the maximum loss, the premium paid, for every price below the strike, then rising one-for-one above the break-even. Flat means the option expires worthless; rising means it has intrinsic value.
The payoff at expiry is entirely deterministic. Given a strike, a premium and a price at expiry, the profit or loss is fixed arithmetic — no model, no probability, no assumption about volatility or interest rates. That is why the payoff diagram is the right first step for any option trade.
All four basic positions follow the same arithmetic with different signs. A short call mirrors a long call: its maximum profit is the premium received and its loss is unlimited above the break-even. A long put profits when the underlying falls below the strike; a short put loses then. The calculator adds a risk warning when the selected position carries unlimited loss.
How do you calculate break-even, max profit and max loss on an option?
For a long call, break-even at expiry is the strike plus the premium per share; for a long put, the strike minus the premium. At break-even the intrinsic value exactly repays the premium. A per-contract commission moves it slightly further away: break-even = strike ± (premium + total commission ÷ (contracts × multiplier)).
A buyer's maximum loss is the premium plus commissions, however far the underlying moves — a known number before you trade. A short call's maximum loss is unlimited, because the underlying can rise without limit; a short put's is the strike minus the premium, reached if the underlying falls to zero.
A seller's maximum profit is the premium received, less commissions. A long put's maximum profit is the strike minus the premium, since the underlying cannot fall below zero: a $100 put bought for $3 can make at most $97 a share, $9,700 a contract. A long call's maximum profit is unlimited.
How does the Black-Scholes model price an option before expiry?
Before expiry an option has time value too. The Black-Scholes model (Black and Scholes, 1973) estimates it from five inputs — underlying price S, strike K, time to expiry T in years, risk-free rate r and implied volatility σ — and Merton (1973) added a dividend yield q. The result is a European option's theoretical price.
The formula uses two intermediate values, d₁ and d₂. N(d₂) is the risk-neutral probability that the option expires in the money; the call price is S × e^(−qT) × N(d₁) minus K × e^(−rT) × N(d₂). The results block shows d₁, d₂, N(d₁) and N(d₂) beside the price so you can check the arithmetic.
Implied volatility is the input that moves the price most after the strike and the underlying. It is the market's estimate of how much the underlying will move, as an annualised standard deviation: 20% implies a typical daily move of about 1.26% (20% ÷ √252 trading days). Higher implied volatility makes every option more expensive; that sensitivity is vega.
What do the option Greeks delta, gamma, theta, vega and rho measure?
Delta (Δ) is how much the option price changes for a $1 move in the underlying: 0.57 means about $0.57. Call deltas run from 0 to +1 and put deltas from −1 to 0; an at-the-money option sits near ±0.5. Gamma (Γ) is how fast delta itself changes as the underlying moves, so high gamma means delta swings quickly.
Theta (θ) is time decay — what the option loses in a day with everything else unchanged; it is negative for a long option, and this calculator shows it per calendar day. Vega (ν) is the change in price for a 1-point move in implied volatility, and rho (ρ) for a 1-point move in the risk-free rate, which matters most for long-dated options.
What are the limitations of Black-Scholes?
Black-Scholes assumes constant volatility, which markets contradict: implied volatility changes over time and differs by strike (the smile or skew). It assumes prices move without jumps, so it underprices options that can gap around earnings. And it assumes European exercise, while most U.S. equity and ETF options are American and can be worth more.
Practitioners use Black-Scholes mainly as a quoting convention — converting market prices to implied volatility and back — and use binomial trees or stochastic-volatility models for American and exotic options. For education and rough European pricing it is a good reference; for a trade, use your broker's option chain and its bid-ask spread.
This calculator provides educational estimates based on the assumptions you enter. It is not investment, tax, legal or financial advice. Options involve significant risk, including loss of the entire premium paid and, for short uncovered positions, potentially unlimited loss; read the OCC's Characteristics and Risks of Standardized Options before trading.
Options questions
What is the break-even price for a call option?
The strike price plus the premium per share. A $100-strike call bought for $5 breaks even when the underlying reaches $105 at expiry; below that the option is worth less than the premium paid, and above it every extra dollar is a dollar of profit per share. Commission moves it slightly higher: strike + premium + total commission ÷ shares controlled.
How do I calculate options profit before expiry?
Before expiry an option's price is its intrinsic value plus time value, which the Black-Scholes model estimates from the underlying price, strike, time to expiry, risk-free rate and implied volatility. This calculator shows that theoretical price and the Greeks in the results; change the inputs in the Black-Scholes section of the form. A 90-day at-the-money call at $100 with 20% volatility and a 5% rate is worth $4.58.
What is the maximum loss on a long call or put?
The premium paid plus commissions. If the underlying stays below the strike (call) or above it (put) at expiry, the option expires worthless and the premium is the whole loss — $500 on one contract of a $5 option. That defined, known worst case is the main feature of buying options rather than selling them.
Can you lose more than you invest in options?
As a buyer, no — the loss is limited to the premium. As a seller, yes: a short call has unlimited theoretical loss because the underlying can rise without limit and you must deliver at the strike, and a short put can lose the strike minus the premium if the underlying falls to zero. The calculator shows a warning for unlimited-loss positions; the OCC's Characteristics and Risks of Standardized Options sets out these risks in full.
What do option Greeks measure?
How the option price responds to each input. Delta: change per $1 move in the underlying (about 0.5 at the money). Gamma: how fast delta changes. Theta: value lost per day, negative for long options. Vega: change per 1-point move in implied volatility. Rho: change per 1-point move in the risk-free rate. For the 90-day $100 call at 20% volatility, delta is 0.569, theta −$0.0288 a day and vega $0.195.
What is implied volatility and why does it matter for options pricing?
Implied volatility (IV) is the volatility that, put into the Black-Scholes formula, reproduces the option's market price — the market's estimate of how much the underlying will move, as an annualised percentage. Higher IV makes every option more expensive. IV often rises before an earnings announcement and falls straight after, so an option bought before the announcement can lose value even when the stock moves the right way.
What is a contract multiplier in options?
The number of shares one contract covers, which converts the per-share premium into a contract price. Standard U.S. equity and ETF options use 100, so a $5 premium on one contract costs $5 × 100 = $500. Index and other non-standard contracts can differ, and adjusted contracts after a split or merger often do; the calculator defaults to 100 and lets you change it under Advanced.
How do I calculate the break-even price on a put option?
For a long put, the strike minus the premium paid per share. A $100-strike put bought for $3 breaks even at $97; below that every dollar of further decline is a dollar of profit per share, and its maximum profit is $97 a share if the stock goes to zero. A short put uses the same number from the other side. Commission moves a buyer's break-even slightly lower.
What happens to an option at expiration?
If an equity option is at least $0.01 in the money at expiration, the Options Clearing Corporation exercises it automatically unless the holder instructs otherwise: a call buyer receives 100 shares per contract at the strike, a put buyer sells 100 shares at the strike. Out-of-the-money options expire worthless. Cash-settled index options pay the difference instead of delivering shares. Brokers may close positions an account cannot fund, so check their policy.
What is assignment risk when you sell an option?
When you sell an American-style option, the buyer can exercise at any time before expiration and you are assigned: a short call must deliver 100 shares per contract at the strike, a short put must buy them. Early assignment is uncommon because exercising gives up remaining time value, but it becomes likely deep in the money with little time value left, and for short calls just before an ex-dividend date. Buying the contract back removes the risk.
What does probability of profit mean for an option trade?
Probability of profit (POP) estimates how likely the underlying is to finish past your break-even at expiration, usually from the same lognormal assumption Black-Scholes uses. N(d₂) is the risk-neutral chance a call expires in the money — 53% for the 90-day example here — and delta is a common rough proxy. POP says nothing about size: a 70% chance of keeping $50 can still lose money against a 30% chance of losing $500.
What is the difference between intrinsic and extrinsic value?
Intrinsic value is what an option is worth if exercised now: for a call, the stock price minus the strike (or zero); for a put, the strike minus the stock price (or zero). Extrinsic, or time, value is the rest of the premium. With the stock at $105, a $100 call trading at $7 has $5 of intrinsic and $2 of extrinsic value. Extrinsic value decays to zero at expiration.
Source notes
Method, sources & limitations
Every figure on this page comes from the formula above applied to the values you enter. Nothing is estimated from market data, and no result is a forecast. These are the sources behind the method and any fixed figures it uses:
- Options Clearing Corporation — Characteristics and Risks of Standardized Options
- Black, F. & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81(3), 637–654
- Merton, R. C. (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science 4(1), 141–183
- Cboe — Options education
Educational information only. Results are estimates produced from the assumptions you enter, and are not financial, tax, legal or medical advice. Check anything that matters against a qualified professional and the cited sources.