Options Greeks Explorer: Worked Examples
The price is not the point; the Greeks are. These scenarios hold the model constant, Black-Scholes at 5% risk-free rate and no dividends, then vary moneyness and tenor so you can see how delta, gamma, theta, and vega change across the surface. Theta is per calendar day, vega per one percentage point of volatility, rho per one percentage point of rate. Compare the same option across moneyness to see how each Greek moves rather than reading any single row in isolation.
Worked Examples
See the inputs and outcome together
Each scenario keeps the starting point, the outcome, and the actual lesson in one place so the page reads like a decision notebook, not a data dump.
- 1
At-the-money call
The classic textbook position. The strike sits exactly at the spot, 30 days out, so the option is pure time value with maximum sensitivity to small moves.
Price 2.49, delta 0.54, gamma 0.069, theta minus 0.045/day, vega 0.114, rho 0.042.
Type
Call
Spot
100
Strike
100
Volatility
20%
Days to expiry
30
Risk-free rate
5%
Delta sits just above 0.50 because the drift and time value tilt the call slightly in the money. Gamma and vega peak near the money, so this is where small price and volatility moves hit your P&L hardest. The whole 2.49 premium is time value that theta bleeds away.
- 2
At-the-money put, same contract
Identical inputs but a put. Put-call parity links the two, so this row is a check on the call above rather than a new bet.
Price 2.08, delta minus 0.46, gamma 0.069, theta minus 0.031/day, vega 0.114, rho minus 0.040.
Type
Put
Spot
100
Strike
100
Volatility
20%
Days to expiry
30
Risk-free rate
5%
The put delta is the call delta minus one, so 0.54 minus 1 gives minus 0.46. Gamma and vega are identical to the call because they do not depend on type. The put is cheaper than the call here purely because the positive carry from the rate favors the call.
- 3
Out-of-the-money call, more time and vol
A 110 strike on a 100 underlying, 90 days out, with volatility lifted to 30 percent. This is the typical lottery-ticket call traders buy for cheap upside.
Price 2.80, delta 0.31, gamma 0.024, theta minus 0.033/day, vega 0.176, rho 0.071.
Type
Call
Spot
100
Strike
110
Volatility
30%
Days to expiry
90
Risk-free rate
5%
Delta drops to 0.31 because the option needs a 10 percent move just to reach the strike, but vega jumps to 0.176, the highest in this set. An out-of-the-money call with time on it is primarily a long-volatility position, not a directional one.
- 4
Deep in-the-money call
A 90 strike on a 100 underlying, back to 30 days and 20 percent vol. The option is mostly intrinsic value and behaves almost like the stock.
Price 10.43, delta 0.97, gamma 0.011, theta minus 0.018/day, vega 0.017, rho 0.071.
Type
Call
Spot
100
Strike
90
Volatility
20%
Days to expiry
30
Risk-free rate
5%
Delta near 0.97 means the option tracks the stock almost one for one, while gamma and vega have nearly vanished. A deep in-the-money call is a leveraged stock proxy with little optionality left to pay for, which is why long-dated deep calls are a common stock-replacement trade.
Patterns
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Sources & References
- The Pricing of Options and Corporate Liabilities — Black, F. and Scholes, M., Journal of Political Economy (1973)
- Theory of Rational Option Pricing — Merton, R. C., Bell Journal of Economics (1973)
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