The Greeks from zero
1/6What is a Greek?
An option price depends on several inputs: spot price, time, volatility, rates. A Greek tells you how much the option price moves when one of those inputs changes by a small amount.
If you remember slopes from calculus, a Greek is a partial derivative. If you don't, think of it this way: a Greek is the answer to "if I nudge this one input, how much does my option price react?"
That is all. Each Greek corresponds to a different input being nudged. Delta nudges spot. Theta nudges time. Vega nudges volatility. Same idea, different knob.
The interactive widget below shows a call price curve as a function of spot. The tangent line at each point has a slope. That slope is delta. Every Greek works the same way, just along a different axis.
Drag the spot slider. Watch the tangent line rotate. Deep in the money, the slope approaches 1. Far out of the money, it approaches 0. At the money, it sits near 0.5. That tangent slope is delta.
Delta
Delta is the first Greek everyone learns, and the one you use the most. For a call, delta ranges from 0 to 1. It answers: "how many dollars does my option move per $1 in the underlying?"
In Black-Scholes, call delta is simply N(d₁) — the cumulative normal distribution evaluated at d₁. The deeper in the money, the closer delta is to 1. The further out of the money, the closer to 0.
Practical interpretation: delta also tells you the approximate probability that the option expires in the money. A 25 delta call has roughly a 25% chance of finishing ITM. Not exact, but close enough for intuition.
Hedge ratio: if you sold one call, you need to buy delta shares to be delta-neutral. If delta is 0.50, buy 50 shares per option. As spot moves, delta changes, and you adjust.
Gamma
Gamma is the rate of change of delta. If delta tells you where you are, gamma tells you how fast delta is changing as spot moves.
Mathematically, gamma is the second derivative of option price with respect to spot. In practice, it matters because delta-hedging isn't a one-shot deal. As spot moves, delta shifts, and you have to rehedge. Gamma measures how much.
Drag the slider and watch gamma (blue) peak right at the strike. Far from the strike, delta barely changes — the option either moves dollar-for-dollar with spot (deep ITM) or hardly at all (deep OTM). Near the strike, delta is changing rapidly, so gamma is high.
Why gamma matters for PnL: gamma creates the curvature in the price curve. For a $2 spot move, delta contributes Δ × $2, but gamma contributes an additional ½ Γ × $2². That extra term is the gamma PnL — it's why long options outperform their delta hedge on large moves.
Theta
Theta is time decay. Every day that passes, an option loses some value — even if nothing else changes. Theta tells you how much.
For long options, theta is negative: you're bleeding value daily. For short options, theta is positive: you're collecting rent. This is the core trade-off in options — you pay theta for the right to earn gamma on big moves.
Key pattern: theta accelerates near expiry. An ATM option loses more value per day in its last week than in any prior week. The curve steepens dramatically — this is why short-dated options are a theta-collection favorite and a blow-up risk.
Gamma and theta are two sides of the same coin. If you're long gamma (benefiting from big moves), you're paying theta. If you're collecting theta, you're short gamma (getting hurt by big moves). There is no free lunch.
Vega
Vega measures how much the option price changes when implied volatility moves by 1 percentage point. It is always positive for both calls and puts — higher vol means higher option prices.
Vega is not actually a Greek letter (there is no letter "vega" in the Greek alphabet). The convention stuck anyway. Some people use nu (ν) instead.
Where vega matters most: ATM options have the highest vega. Deep ITM or OTM options barely react to vol changes — they're already dominated by intrinsic value or worthlessness.
Practical use: if you're trading a vol event (earnings, FOMC), you want to know your vega exposure. A $0.15 vega on 10 contracts means a 1% IV crush costs you $150.
Putting it together
In real trading, everything moves at once: spot, time, and vol. The Greeks let you decompose your PnL into pieces — what came from delta, what came from gamma, what you lost to theta, and what vol gave or took away.
The Taylor expansion of an option's price change is:
Move the sliders below. Watch each Greek's contribution. The "residual" row shows what the first-order approximation misses — it's small for tiny moves and grows for large ones.
What to notice: for small spot moves, delta dominates. For large spot moves, gamma kicks in. Theta is steady and predictable. Vega is the wild card — it depends entirely on how vol moves, which you can't predict.
This decomposition is how professional desks think about PnL every day. The question is never just "did I make or lose money?" It's "where did the PnL come from?"