Options Greeks explained: what delta, gamma, vega, theta and rho each measure, where the Black-Scholes model comes from, and what it leaves out

An option's price is a single number. The Greeks are the five directions it can move in, and most losses that surprise people come from a Greek they were not watching. This page says what each one measures, where the model that produces them came from, and what that model assumes, on one sample contract.

Where the model comes from

Fischer Black and Myron Scholes published "The Pricing of Options and Corporate Liabilities" in the Journal of Political Economy, volume 81, number 3, May to June 1973, pages 637 to 654 (source 1). Robert C. Merton published "Theory of Rational Option Pricing" in the Bell Journal of Economics and Management Science, volume 4, number 1, 1973, beginning on page 141 (source 2). The pricing model used in the Bindler workbook is the one that grew out of those two papers and is usually called Black-Scholes-Merton; the workbook's version includes a continuous dividend yield, so it prices options on dividend-paying stocks and on indices as well as on assets that pay nothing.

The model gives a European call and put price from six inputs: the underlying price, the strike, the time to expiry, a continuous risk-free rate, a continuous dividend yield and a volatility. The Greeks are the model's sensitivities to five of those inputs.

The five Greeks in plain words

Delta is how much the option price moves for a one-unit move in the underlying. A call's delta sits between 0 and 1, a put's between minus 1 and 0. It is also read as a rough hedge ratio: the number of units of the underlying that offset the option. Gamma is how much delta itself moves for a one-unit move in the underlying. It is the same for the call and the put at the same strike and expiry, largest near the money, and it is why a hedged position needs re-hedging as the price moves. Vega is how much the option price moves for a one-point change in volatility. Also the same for the call and the put. Long options have positive vega: they gain when the market's expectation of movement rises, whatever the direction. Theta is how much the option price changes as one day passes with everything else fixed. For a long option it is usually negative: time is running out. The Bindler workbook quotes it per calendar day, and its specification notes that theta's sign is one of the two things most options spreadsheets get wrong. Rho is how much the option price moves for a one-point change in the risk-free rate. Calls have positive rho, puts negative.

The sample contract

The workbook's Pricer ships with one illustrative case: underlying 100, strike 105, six months to expiry, a 3% continuous rate, no dividend, 25% volatility. The prices, every Greek and the parity check were recalculated with a formula engine and matched to an independent Python replica before listing (source 3).

• Call price 5.576, put price 9.013. Put-call parity holds to zero, which is the workbook's built-in check that the two prices agree.
• Call delta 0.459, put delta minus 0.541. The two differ by exactly one, as they must for a non-dividend stock.
• Gamma 0.0224 for both.
• Vega 0.281 per volatility point for both: a move from 25% to 26% volatility adds about 0.28 to each price.
• Theta minus 0.0225 a day for the call, minus 0.0140 a day for the put.
• Rho 0.202 for the call and minus 0.316 for the put, per point of rate.

One finding: the out-of-the-money call at 105 loses about 2.3 cents a day to time with the underlying standing still, roughly 0.4% of its price each day, and that rate accelerates as expiry approaches. Theta is a cost that compounds against a long option holder who is right about direction and slow about timing.

From one option to a position

The same workbook's Strategy sheet takes up to four legs on the same underlying and expiry, prices each leg by the model at its own strike and by the fill you enter, and lays out the payoff at expiry across forty-one underlying prices. Its sample is a two-leg bull call spread: long one 100 call at a 6.20 fill, short one 110 call at a 2.35 fill. Net premium 385 for one contract of 100 units, model value 386.17, position delta 21.2 units. At expiry the grid shows a P&L of minus 385 at any price up to 100, plus 615 at any price from 110 upward, so max profit 615 and max loss 385 (source 3).

That grid is the part of options that is exact: the payoff at expiry depends only on the strikes and what you paid. The Greeks and the model price are estimates that hold under the model's assumptions.

What the model assumes, and what a real position does not

The workbook's Guide states its assumptions: European exercise, constant volatility and rates, lognormal prices, no transaction costs (source 3). Real listed equity options are mostly American and can be exercised early; volatility moves every day, so vega is a live exposure rather than a fixed input; fills differ from model prices; and the P&L at expiry "is exact for the payoff and only as good as your premium inputs for the cost". Volatility surfaces, American exercise and live data are outside the workbook and the listing says so.

Where the workbook fits

The Options P&L and Greeks Workbook is a Pricer sheet (six inputs, call and put price, the five Greeks, put-call parity), a Strategy sheet (up to four legs priced leg by leg, net premium, model value, position delta, and the forty-one-price payoff grid with max profit and max loss) and a Guide. Live formulas, no macros, no locked cells, Excel and Google Sheets. $19, one price.

A model, not a market. No signals, no forecasts, not investment advice.

Sources

1. Fischer Black and Myron Scholes, "The Pricing of Options and Corporate Liabilities", Journal of Political Economy 81(3), May to June 1973, pp. 637 to 654: https://doi.org/10.1086/260062

2. Robert C. Merton, "Theory of Rational Option Pricing", The Bell Journal of Economics and Management Science 4(1), 1973, p. 141 onward: https://doi.org/10.2307/3003143

3. Bindler, Options P&L and Greeks Workbook specification, Guide sheet and verification log (products/options-pl/spec.json, build.py and verify.py), figures as listed 27 September 2026.

Last checked against the sources on 28 September 2026.