Time to expiry is this divided by 365.
Annualized. The model has no value at zero.
Continuously compounded, a year.
Continuous, a year. Merton's extension. Zero for a non-payer.
| Measure | Unit | Call | Put | Call, x100 |
|---|---|---|---|---|
| Value | per share | 3.5911 | 3.2629 | 359.11 |
| Delta | per $1.00 of spot | 0.5324 | -0.4676 | 53.24 |
| Gamma | delta per $1.00 of spot | 0.04623 | 0.04623 | 4.62 |
| Vega | per 1 volatility point | 0.1140 | 0.1140 | 11.40 |
| Theta | per calendar day | -0.0624 | -0.0515 | -6.24 |
| Rho | per 1 point of rate | 0.0408 | -0.0411 | 4.08 |
Theta is one calendar day of decay, not one trading day. Vega is a one-point move in volatility, from 30% to 31%. Rho is a one-point move in the rate. Gamma and vega are identical on the call and the put.
Hypothetical illustration, computed only from the figures entered. Every input is typed; nothing here reads a live quote or a chain.
European exercise only. The model does not price the early-exercise right of an American option, so it is a lower bound for an American put and for an American call on a dividend-paying underlying. It assumes one constant volatility across all strikes, which a real volatility surface is not, along with a constant rate, a continuous dividend yield rather than discrete payments, no transaction costs and continuous trading. Time to expiry is calendar days over 365.
Black-Scholes puts a value on a European option from six inputs: spot, strike, time, volatility, the risk-free rate and the dividend yield. The Greeks beside it are the partial derivatives of that value, which say how much the price moves when one input moves and the rest hold still.
A $100 call struck at $100 with 30 calendar days left, 30% volatility and a 4% rate prices at $3.5911 a share, $359.11 for one 100-share contract. Its delta is close to 0.53, so a $1 rise in the stock lifts the option about 53 cents, and its theta is roughly -$0.06, the value one calendar day removes.
Reading vega or theta in the wrong unit. Theta here is per calendar day, not per year, and vega is per one point of volatility, not per 1.00 of it. Getting either wrong moves the figure by a factor of 365 or 100.