Pricing the impossible
In 1973 Black, Scholes and Merton showed that an option can be perfectly replicated by continuously trading the stock and cash. No-arbitrage then forces a unique price, governed by the . Because the hedge is riskless, the stock's real drift drops out — only the and the remain.
Solving it with the option's payoff as a boundary condition gives the famous closed form. A European is worth:
Read as the risk-neutral probability the option finishes in the money, and as the option's . The whole formula is "expected payoff if exercised, discounted" — nothing more exotic than that.
One option, live
Set the five inputs. Everything downstream — price, the intermediate /, and all the Greeks below — recomputes instantly. The chart shows the option's value today against the jagged it collapses to at expiry.
Four ways the price moves
Each Greek is a partial derivative — the price's sensitivity to one input, holding the rest fixed. to spot, to volatility, to the passage of time, to rates. Pick one and watch its whole profile across spot.
Delta is the bridge to the Desk. An option's delta is exactly the number of shares that replicate it right now — the option's , the same idea as a position's to the market. Sell that many shares and you're delta-neutral: immune to small moves, left holding .
The Greeks of the Greeks
First-order Greeks drift as the market moves, so desks track their derivatives too. is how fast delta changes; , , , and are the next layer — the ones that bite when markets move fast or vol is unstable.
Why gamma is the one that matters. Delta-hedging cancels the straight-line risk, but the option's value curves — that curvature is gamma. When you're long gamma, every move helps you (you buy low, sell high re-hedging); the cost is , the rent you pay each day for that convexity. That trade-off is the entire life of an options market-maker.
The rent and the payoff
A option's daily P&L is a tug-of-war: gamma pays you when the stock moves, theta charges you for holding. They balance at one specific size of move — the . Below it, theta wins; above it, gamma does.
This is why implied vol matters. Set and the break-even move and the vol-implied move line up — the option is fairly priced against its own volatility. That's not a coincidence: is precisely the σ that makes the theta you pay equal the gamma you expect to earn.
Which Greek is which?
The word "Greeks" gets stretched. are portfolio ideas from the Desk; delta, gamma and the rest are option sensitivities. They rhyme — beta is to a portfolio what delta is to an option — but they're not the same animal. Here's the map.
| Symbol | What it measures | Lives in |
|---|---|---|
| Skill / excess return above the benchmark | Portfolio | |
| Sensitivity to the market's move | Portfolio | |
| Sensitivity to the underlying's move | Option | |
| How fast that sensitivity itself changes (convexity) | Option | |
| Sensitivity to volatility | Option | |
| Θ — Theta | Sensitivity to time passing | Option |
The one-line version: alpha is the edge you're paid for, beta is the market risk you hedge away, and gamma is the curvature that a straight beta-hedge can't catch. The Desk lives in alpha/beta; this page lives in delta/gamma; delta-hedging is the same reflex as beta-hedging, one level more precise.