Every Greek in this module answers one question: if one input changes and nothing else does, how much does the option's price change? To see why there are five of them, and why they behave the way they do, it helps to stop thinking of an option price as a number and start thinking of it as a curve.
The price is a curve
Fix the strike, the time left, the volatility and the rate, and ask what a call is worth at every possible stock price. Far below the strike the call is worth almost nothing. Far above, it is worth about the stock minus the strike. In between, the curve bends smoothly from one to the other. The dashed line in the chart is the value at expiry, the hockey stick from Module 1. The solid lines are the same call with time still left: always above the hockey stick, because time value is never negative, and closest to it far from the strike, where the outcome is nearly settled.
Call value against the stock price with 90, 30 and 7 days left, and the value at expiry (dashed). Hover to read any point. Strike $100, 30% volatility and a 4% rate until you move the sliders.
The lines with less time left sit lower and hug the hockey stick more closely. That sinking, day by day, is theta. Move the volatility slider up and every line lifts. That lift is vega.
Slope, bend and shifts
- Delta is the slope of the curve at today's stock price: how steeply value rises per $1. Near zero on the left, near one on the right.
- Gamma is how fast that slope changes, the bend. The curve bends most near the strike, so gamma is largest there.
- Theta is how far the curve drops when one day passes.
- Vega is how far it moves when volatility rises one point.
- Rho is how far it moves when the interest rate rises one point.
Delta and gamma describe the shape of one curve. Theta, vega and rho describe how the whole curve shifts when an input other than the stock changes. That is the entire first-order family. The second-order Greeks later in this module ask how the slope itself shifts with time and volatility.
Using them together
Because delta is a slope and gamma is a bend, you can estimate an option's new price after a move without running a model. The change is roughly delta times the move, plus half of gamma times the move squared. Say a call is worth $5.00 with a delta of 0.50 and a gamma of 0.03. After a $4 rise, 0.50 times 4 is $2.00, and half of 0.03 times 16 is $0.24, so the call is worth about $7.24. Delta alone would have said $7.00; the bend adds the rest. After a $4 fall, delta says $3.00 and the bend adds the same $0.24 back, so about $3.24. The bend helps the owner in both directions, which is the whole appeal of owning gamma.
The maths: the pricing formula behind the curve
The curves in these charts come from the Black-Scholes formula for a European call: C = S e^(-qT) N(d1) - K e^(-rT) N(d2), where d1 = [ln(S/K) + (r - q + v²/2) T] / (v √T) and d2 = d1 - v √T. S is the stock price, K the strike, T the years to expiry, v the volatility, r the interest rate, q the dividend yield, and N the cumulative normal distribution.
Each Greek is a partial derivative of C: delta is ∂C/∂S = e^(-qT) N(d1), gamma is ∂²C/∂S², theta is ∂C/∂t, vega ∂C/∂v and rho ∂C/∂r. The estimate in the last section is a second-order Taylor expansion, ΔC ≈ Δ·dS + ½·Γ·dS². It is accurate for moves that are small next to the stock's typical range and drifts for large ones, because gamma itself changes along the way. The Terminal's chain carries the data vendor's model values, which start from the same idea with refinements for dividends, early exercise and skew.
Your turn
Open the SPY chain on the monthly expiration. Read the mid price, delta and gamma of the at-the-money call, then use them to estimate what the call would be worth after a $5 rise.