Inputs
Loads the price, the ~30-day expiry and the at-the-money strike, with live r and q.
Results
- Delta
- 53.3 sh
- 0.5326 per share
- Gamma
- 5.548
- 0.0555 per share
- Theta
- −$5.30/day
- −0.0530 per share
- Vega
- $11.40/pt
- 0.1140 per share
- Rho
- $4.13/1%
- 0.0413 per share
- Model value
- $302.11
- $3.02 per share
Position figures are per share × 100 × 1 contract. Black-Scholes-Merton with a 0.00% dividend yield and 0.0822 years to expiration.
Understanding the Greeks
The Greeks measure how an option's price responds to each input of the Black-Scholes-Merton model: the stock price (delta and gamma), time (theta), volatility (vega) and the interest rate (rho). This calculator works them out per share and for a whole position, on the same model and calendar-day basis as the calculator.
Delta (Δ)
Calls 0 to 1, puts −1 to 0
How much the option's price changes for a $1 move in the stock. A delta of 0.50 means the option gains about $0.50 when the stock rises $1. Delta is often used as a rough proxy for the chance of finishing in the money; the model's risk-neutral probability is N(d₂).
Gamma (Γ)
Positive for long options
How fast delta changes as the stock moves. High gamma means delta swings quickly. At-the-money options close to expiration have the highest gamma.
Theta (Θ)
Usually negative for long options, positive for short
Time decay: how much value the option loses per calendar day if nothing else changes. It speeds up as expiration nears, most for at-the-money options.
Vega (ν)
Positive for long options
How much the price changes for a 1-point change in implied volatility. Longer-dated, at-the-money options have the most vega.
Rho (ρ)
Positive for calls, negative for puts
How much the price changes for a 1-point move in the risk-free rate. Higher rates raise call values and lower put values; the effect grows with time to expiration.
Greek formulas
Delta (Δ)
Calle−qT·N(d1)
Put−e−qT·N(−d1)
Price change per $1 move in the stock.
Gamma (Γ)
Calle−qT·φ(d1) / (S·σ·√T)
Put: same as the call.
Change in delta per $1 move.
Theta (Θ)
Call−S·e−qT·φ(d1)·σ / (2√T) − r·K·e−rT·N(d2) + q·S·e−qT·N(d1)
Put−S·e−qT·φ(d1)·σ / (2√T) + r·K·e−rT·N(−d2) − q·S·e−qT·N(−d1)
Per year in the formula; shown here per calendar day (÷ 365).
Vega (ν)
CallS·e−qT·φ(d1)·√T
Put: same as the call.
Per 1.00 of volatility in the formula; shown here per vol point (÷ 100).
Rho (ρ)
CallK·T·e−rT·N(d2)
Put−K·T·e−rT·N(−d2)
Per 1.00 of rate in the formula; shown here per 1% (÷ 100).
φ(d1) = e−d₁²/2 / √(2π). Positions scale these per-share figures by 100 shares per contract, with the sign flipped for a short option.
Worked example
Hull's Greeks example: a 20-week call on a stock at $49 with a $50 strike, 20% volatility and a 5% risk-free rate. The textbook gives delta 0.522, gamma 0.066, theta −4.31 a year, vega 12.1 and rho 8.91 (per 1.00 change in σ and r). The calculator's figures match once the units are lined up: theta per day is the yearly figure ÷ 365, and vega and rho per point are ÷ 100.
| Option | Buy 1 call |
|---|---|
| Stock price (S) | $49.00 |
| Strike (K) | $50.00 |
| Time (T) | 0.3846 years (20 weeks) |
| Volatility (σ) | 20% |
| Risk-free rate (r) | 5% |
| Call price | $2.40 |
|---|---|
| Delta | 0.5216 |
| Gamma | 0.0655 |
| Theta per day | −0.0118 (−4.31 a year) |
| Vega per vol point | 0.1211 (12.1 per 1.00) |
| Rho per 1% | 0.0891 (8.91 per 1.00) |
Assumptions & limitations
- European exercise. For American options (most US stock options) the Greeks of deep in-the-money puts, and of calls before an ex-dividend date, differ.
- Constant volatility and rate. Each Greek is the sensitivity to one input with the others held fixed; in markets they move together.
- Greeks are local: delta and gamma describe small moves, and a large move changes them.
- Dividends as a continuous yield q.
- Time in calendar days ÷ 365; theta is per calendar day, including weekends.
For a multi-leg position, add the legs in the options calculator, which nets the Greeks across the trade. See also the methodology.
Frequently asked questions
Why don't my Greeks match my textbook?
Check the time input and the units. Textbooks give T in years (Hull's 20 weeks is 0.3846); this calculator starts in days, so switch the time field to Years. Typed into the days field, 0.3846 is nine hours. The units differ too: textbooks quote theta per year (−4.31) where this shows it per calendar day (−4.31 ÷ 365 = −0.0118), and vega and rho per 1.00 change (12.1 and 8.91) where this shows them per point (0.121 and 0.0891). If your course divides theta by 252 trading days rather than 365 calendar days, its daily theta is about 45% larger.
Is delta the probability of expiring in the money?
Delta is often used as a rough proxy for the chance of finishing in the money; the model's risk-neutral probability is N(d₂) for a call and N(−d₂) for a put. The two are close for short-dated options near the money and drift apart as volatility and time to expiration grow.
Why is theta negative?
A long option loses time value each day if nothing else changes, so its theta is negative: it's the cost of waiting. A short option earns that decay, so its position theta is positive. Deep in-the-money European puts can have positive theta.
What do the position figures mean?
They scale the per-share Greeks by 100 shares per contract and the number of contracts, with the sign flipped when you sell. A position delta of 52 shares gains and loses like 52 shares of stock for small moves; a theta of −$1.18 a day is the dollars the position loses per calendar day if nothing else changes.
What does rho measure?
How much the option's price changes when the risk-free rate moves by one percentage point. Calls gain when rates rise and puts lose. Rho is small for short-dated options and matters most for long-dated ones.
References
- 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.
- Hull, J. C. Options, Futures, and Other Derivatives. Pearson (11th ed., 2021).
Related tools
- Black-Scholes CalculatorTheoretical call and put prices from the Black-Scholes-Merton model, with dividends and Greeks.
- Implied Volatility CalculatorSolve the volatility an option's price implies, with the expected move it prices in.
- Options Breakeven CalculatorBreakevens, max profit and max loss at expiration for up to four legs, with the payoff chart.
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Or open the options calculator to build any strategy on a live chain, read how we price options, or browse the strategy guides.