Options Greeks Calculator
Calculate Delta, Gamma, Theta, Vega, and Rho for options using the Black-Scholes model.
Formula
S = spot price, K = strike, r = risk-free rate, σ = implied volatility, T = time to expiry in years.
Divided by 365 to express as daily decay.
Change in option price per 1 percentage-point increase in IV. Displayed per 1% move.
Examples
- S = $60,000, K = $60,000, T = 30/365 = 0.0822 years
- σ = 0.65, r = 5%
- d₁ = [ln(1) + (0.05 + 0.65²/2) × 0.0822] / (0.65 × √0.0822) = 0.1098
- d₂ = 0.1098 − 0.1863 = −0.0765
- Call price ≈ $4,526
- Delta ≈ 0.544, Gamma ≈ 0.0000146, Theta ≈ −$86.40/day
- S = $3,200, K = $3,000, T = 14/365 = 0.0384
- σ = 0.70, r = 5%
- d₁ = [ln(3200/3000) + (0.05 + 0.70²/2) × 0.0384] / (0.70 × √0.0384) = 0.529
- d₂ = 0.529 − 0.137 = 0.392
- Put price ≈ $65, Delta ≈ −0.298
- S = $150, K = $160, T = 7/365 = 0.0192
- σ = 0.90, r = 5%
- d₁ = [ln(150/160) + (0.05 + 0.90²/2) × 0.0192] / (0.90 × √0.0192) ≈ −0.448
- d₂ ≈ −0.573
- Call price ≈ $3.50, Delta ≈ 0.327, Theta ≈ −$5.20/day
Key Concepts
What is Delta?
Delta measures how much an option's price changes for a $1 move in the underlying. A delta of 0.5 means the option gains ~$0.50 for every $1 increase in spot. Calls have positive delta (0 to 1), puts have negative delta (−1 to 0). Delta also approximates the probability of expiring in-the-money.
What is Gamma?
Gamma is the rate of change of delta per $1 move in the underlying. High gamma means delta shifts rapidly — common for at-the-money options near expiry. Gamma is highest for ATM options and decreases for deep ITM/OTM. It's always positive for long options.
What is Theta?
Theta measures time decay — how much value an option loses per day, all else equal. Theta is always negative for long options: time works against buyers. Decay accelerates as expiry approaches, especially for ATM options. Theta is the cost of holding an option position.
What is Vega?
Vega measures sensitivity to implied volatility changes. A vega of $10 means the option price increases by $10 for each 1 percentage-point rise in IV. Vega is highest for ATM options with more time to expiry. In crypto, vega exposure is significant because IV swings are large.
What is Rho?
Rho measures sensitivity to interest rate changes. It's typically the least significant Greek for short-dated crypto options but matters for longer-dated ones. Calls have positive rho (higher rates increase call value), puts have negative rho.
Black-Scholes Assumptions
The Black-Scholes model assumes log-normal price distribution, constant volatility, no dividends, and continuous trading. Crypto markets violate several of these (jumps, changing volatility), so the model is an approximation. Traders use it as a baseline and adjust via the volatility smile.
Understanding Options Greeks
The Greeks are partial derivatives of the option pricing model, each measuring sensitivity to a different variable. Together they give a complete picture of how an option's price will change as market conditions shift — underlying price, time, volatility, and interest rates.
For crypto options traders, Delta and Theta are the most actively monitored Greeks. Delta determines directional exposure and is used for hedging, while Theta quantifies the daily cost of carrying an options position. In crypto's high-volatility environment, Vega exposure is also critical.
This calculator uses the Black-Scholes model, the standard framework for European-style options pricing. While crypto options have features that deviate from Black-Scholes assumptions (price jumps, varying vol), the model remains the industry standard baseline for calculating Greeks and theoretical value.
Frequently Asked Questions
Why does the calculator show a different price than the market?
The Black-Scholes model produces a theoretical price based on your inputs. Market prices reflect supply/demand, skew, and factors the model doesn't capture. The difference is often due to the volatility smile — the model uses a single IV, but markets price different strikes at different IVs.
What risk-free rate should I use for crypto?
Use the US Treasury rate for the matching duration (e.g., 3-month T-bill rate for 90-day options). For short-dated options (under 30 days), the rate has minimal impact. Common values are 4-5% as of recent years. Some traders use stablecoin lending rates as an alternative.
How do I interpret Theta for crypto options?
Theta represents the dollar amount your option loses per day from time decay alone. If Theta is −$50, your option loses roughly $50 of value each day, all else equal. Crypto options with high IV have higher Theta because more premium means more to decay.
Can I use this for perpetual options?
Perpetual options (everlasting options) don't have a fixed expiry and use different pricing models. This calculator is for standard European-style options with a defined expiry date, which is what most crypto exchanges (Deribit, OKX, Bybit) offer.