The five greeks each answer one question: if one input of an option changes a little, how much does the price move. The Options Greeks Calculator ships with a default scenario, a call on a 100 underlying with a 100 strike, 60 days to expiry, 25 percent volatility and a 5 percent risk free rate, and returns 0.5524 of delta, 0.0390 of gamma, -0.0404 of theta per day, 0.1603 of vega per volatility point and 0.0835 of rho per rate point.
d1, d2 and the model behind the five numbers
Every greek is a derivative of the same Black-Scholes price, so all five reduce to two intermediates, d1 and d2. In the default case d1 is 0.1318 and d2 is 0.0304: delta reads the cumulative bell curve at d1, gamma is the bell curve height at d1 divided by the underlying, the volatility and the square root of time, theta and vega both carry that height term, and rho is the discounted strike. The Black-Scholes Option Pricing prints the price these sensitivities differentiate, 4.4479 for the call and 3.6294 for the put on the same inputs.
Delta: a hedge ratio that also behaves like a probability
Delta doubles as a hedge ratio and as a rough probability. As a hedge, 0.5524 means each call is worth 0.5524 of a share, so a book of 100 calls is hedged with 55.24 shares sold. As a probability it sits close to the chance of finishing in the money, which is 0.5121 here, but the two differ because delta reads d1 while the probability reads d2, and the gap widens with volatility and time. Push the strike to 110 and the call delta falls to 0.2094 while the in the money probability is 0.1814; drop it to 90 and delta climbs to 0.8792. The Implied Probability Calculator makes that d2 probability the whole output.
Gamma: the speed of delta, highest at the money
Gamma is the speed of delta: 0.0390 says a one point move in the underlying shifts delta by about 0.039, which the model confirms, 0.5909 at 101 and 0.5130 at 99. It peaks at the money, 0.0198 for the 90 strike against 0.0284 for the 110 strike, and it grows as expiry shortens, 0.0554 at 30 days against 0.0317 at 90. Volatility works the other way: gamma is 0.0773 at 12.5 percent and only 0.0195 at 50, because the height term divides by volatility. Market volatility usually comes from data rather than a guess; the Implied Volatility Calculator backs it out of option prices before it feeds any greek.
Theta: the daily carry the position pays
Theta is the carry the position pays for time, and the tool prints it per day after dividing by 365: the call bleeds 0.0404 a day and the put 0.0268, the put slower because the discounted strike term enters with a plus sign. Decay accelerates as expiry approaches, 0.0544 per day at 30 days against 0.0341 at 90, closer to an inverse square root of time than to a linear decline. Set the rate to zero and the call theta eases to 0.0337, the financing part of the cost disappearing. For positions that roll on a schedule, the Funding Rate Calculator prices the explicit carry of the futures side of the hedge.
Vega and rho: the vol and rate levers
Vega is the response to volatility in points: 0.1603 means one point of extra vol is worth about 0.1603 on this call, and the put carries exactly the same 0.1603. At the money vega barely depends on the vol level, 0.1604 at 35 percent and 0.1601 at 50, but it grows with the square root of time, 0.1139 at 30 days and 0.1955 at 90. Rho is the quietest of the five, 0.0835 per rate point for the call and -0.0795 for the put, which is why long dated books get more attention from it. The IV Percentile & IV Rank shows where the current vol sits in its own range before you decide that vega exposure is rich or cheap.
Where the greeks sit in a hedged book
In a live workflow the greeks are the rebalancing schedule of a hedged book. The delta leg keeps the book market neutral, the gamma leg tells you how often that delta leg has to be rebuilt, and the theta leg is the daily bill for holding volatility. Each rebalance trade then carries its own entry discipline, the Risk/Reward Ratio Calculator checks that the next move pays more than it costs. When the book as a whole gets sized against the account, the Kelly Criterion guide converts the edge per trade into a fraction of equity to risk.