The greeks, operationally

The greeks are partial derivatives of the option price. That description is accurate and tells you nothing about why anyone computes them. Each one exists because it answers a question somebody has to answer before the end of the day.

   greek   derivative     the operational question
   ─────────────────────────────────────────────────────────────────────
   Δ delta   ∂V/∂S        how many shares do I hold to be hedged?
   Γ gamma   ∂²V/∂S²      how fast does that hedge go stale?
   Θ theta   ∂V/∂t        what does holding this cost me per day?
   ν vega    ∂V/∂σ        what happens if the market repriced volatility?
   ρ rho     ∂V/∂r        (usually nothing. Long-dated options only.)

Only vega is not a Greek letter, which tells you something about who named these.

Delta — the hedge ratio you already derived

M2.2 built delta from scratch: it is the number of shares in the replicating portfolio, N(d₁) in Black–Scholes. Calls run from 0 (deep out) to 1 (deep in); puts from −1 to 0; at the money it sits near ±0.5.

One correction worth internalising, because the sloppy version is everywhere: delta is not the probability of finishing in the money. That is N(d₂). They are close for short-dated near-the-money options and diverge as maturity and moneyness grow. Using delta as a probability is a decent field approximation and a bad thing to say to someone who prices options.

Gamma — the curvature that makes options interesting

Gamma is the rate at which delta changes as the stock moves. A hedge set this morning is wrong by this afternoon, and gamma says how wrong.

   long gamma                        short gamma
   ──────────────────────────────────────────────────────────────
   stock rises → delta rises →       stock rises → delta falls →
   you must SELL to re-hedge         you must BUY to re-hedge
   stock falls → delta falls →       stock falls → delta rises →
   you must BUY to re-hedge          you must SELL to re-hedge
   ──────────────────────────────────────────────────────────────
   rehedging buys low, sells high    rehedging buys high, sells low
   every move pays you a little      every move costs you a little

That table is the entire economics of the thing. Long gamma means your own hedging mechanically buys low and sells high. Nothing about the stock's direction is required — only that it moves. Short gamma is the mirror: you are forced to chase, and a violently moving market bleeds you.

Gamma is largest for at-the-money options near expiry, which is exactly when delta is most unstable — a stock hovering at the strike on expiry day has a delta lurching between 0 and 1.

Theta — the rent

Nothing that good is free. Long gamma is always short theta: the option decays, every day, and the decay is what you pay for the privilege of being long curvature. The relationship is not a tendency, it is an identity in the Black–Scholes world, which the next section makes exact.

M0.8 called an option a wasting asset. Theta is the waste, and gamma is what you bought with it.

The result that ties them together

Take a long option, delta-hedged, and Taylor-expand its value over a short interval:

   dV  =  Δ·dS  +  ½Γ·(dS)²  +  Θ·dt   +  …

   you are delta-hedged, so the Δ·dS term is cancelled by the hedge:

   hedged P&L  =  ½Γ·(dS)²  +  Θ·dt

Now substitute Black–Scholes theta. With r = 0,

   Θ  =  −½ Γ S² σᵢ²        (σᵢ = the IMPLIED vol you paid)

   hedged P&L  =  ½Γ·(dS)²  −  ½ Γ S² σᵢ² dt
               =  ½Γ [ (dS)²  −  S² σᵢ² dt ]

   and over a short step the stock actually delivers  E[(dS)²] = S²σ_r² dt
   ──────────────────────────────────────────────────────────────────────
        hedged P&L  ≈  ½ Γ S² ( σ_realized²  −  σ_implied² ) · dt

Read what has happened. The direction of the stock has vanished. What remains compares the volatility the stock actually delivered against the volatility you paid for. Gamma is the exchange rate between them.

Which gives the sentence this lesson exists for:

A delta-hedged option is not a bet on the stock. It is a bet that realized volatility will exceed implied volatility.

Checking it actually works

Worth verifying rather than believing. Buy a 3-month at-the-money call on a £100 stock at 20% implied, delta-hedge it daily to expiry, and let the stock genuinely realize various volatilities:

   at S = K = 100, T = 0.25, σᵢ = 20%:
      price = £3.99      delta = 0.520      gamma = 0.0398
      ½ΓS² = 199.2   ← the exchange rate between vol² and money

   realized vol    simulated hedged P&L    ½ΓS²(σr²−σi²)T
   ────────────────────────────────────────────────────────
       15%               −1.00                 −0.87
       20%               +0.00                  0.00     ◀ breaks even
       25%               +0.99                 +1.12
       30%               +1.98                 +2.49

The middle row is the point: when realized equals implied, the hedged position earns nothing — the gamma you harvest exactly pays the theta you owe. You bought volatility at 20% and it delivered 20%, so you broke even, which is what a fairly priced asset should do.

Note also that the closed form drifts high at 30%. That is honest and instructive: ½ΓS² uses today's gamma, but gamma changes as the stock moves away from the strike and as time runs out. The formula is a local statement, excellent for a day and rough over a quarter.

Vega — the exposure that actually gets traded

Since the whole position is a view on volatility, the sensitivity to volatility is the one desks manage most closely. Vega is ∂V/∂σ — what you make if the market simply reprices volatility, with the stock unmoved.

Distinguish the two ways a vol view pays, because they are different trades:

• Gamma pays off when volatility is realized — the stock actually moves, and you harvest it by rehedging.
• Vega pays off when volatility is repriced — the market's implied vol rises and your option is marked up, whether or not the stock ever moves.

Vega is largest for long-dated options (more time for vol to matter); gamma is largest for short-dated ones. So "long vol" is ambiguous until you say which — and a desk long gamma and short vega has a coherent, common, and quite specific view.

All of which leaves one question: implied volatility is the price, so what does the market actually charge, and why isn't it one number? That is M2.4.

Source: Hull ch.19 for the greeks in full. For the gamma-theta relation as practitioners actually use it, Euan Sinclair, Volatility Trading, is the best plain-spoken treatment — it is written by someone who has hedged the positions rather than only differentiated them.