Delta, gamma, vega and theta

The Greeks are the partial derivatives of the option price. Knowing the definitions is table stakes; what desks actually test is the *shape* — where each one peaks, what happens to it near expiry, and why being long gamma always means being short theta.

Greek	Measures sensitivity to	Long call	Long put
Delta $\Delta$	The underlying	Positive, $0$ to $1$	Negative, $-1$ to $0$
Gamma $\Gamma$	How delta changes	Positive	Positive
Vega $\nu$	Implied volatility	Positive	Positive
Theta $\Theta$	The passage of time	Negative	Negative

Table 1.1 · The four that matter. Gamma and vega are positive for any long option, call or put — you are long optionality either way. Only delta distinguishes them.

Proposition 1.2

Three readings of delta

Delta is the hedge ratio — how many shares to hold against the option. It is the sensitivity — how much the option moves for a one-point move in the underlying. And it is roughly the risk-neutral probability of finishing in the money, which is $N(d_2)$ exactly and $N(d_1)$ near enough for an at-the-money option.

Holds when

The probability reading is an approximation, and it is $N(d_2)$ that is the true risk-neutral probability, not delta.
An at-the-money call has delta slightly above $0.5$ , because the forward sits above the spot.

Greek	Largest when	As expiry approaches
Delta	Deep in the money (approaches 1)	Becomes a step function at the strike
Gamma	At the money	Explodes at the money, vanishes elsewhere
Vega	At the money, long-dated	Falls towards zero everywhere
Theta	At the money	Accelerates — decay is fastest at the end

Table 1.3 · Where each Greek is largest. Gamma and theta both peak at the money and both blow up near expiry, in opposite directions. That is not a coincidence.

Figure 1.4 · Gamma peaks where the option is most uncertain. A schematic, normalised to show the shape rather than quote actual Greeks. Near the strike, a stock move changes the exercise outcome most sharply, so gamma is largest there; the option’s time decay is also most costly there. Exact heights depend on expiry and volatility.

Equation 1.5

The P&L equation

A second-order Taylor expansion of the option price. Every day on an options desk is an argument about which of these four terms explained the day.

\mathrm{d}\Pi \approx \Delta\,\mathrm{d}S + \tfrac{1}{2}\Gamma\,(\mathrm{d}S)^2 + \Theta\,\mathrm{d}t + \nu\,\mathrm{d}\sigma

$\Delta\,\mathrm{d}S$: Directional exposure — removed by delta hedging.
$\tfrac{1}{2}\Gamma (\mathrm{d}S)^2$: Always positive when long gamma, whichever way the move goes.
$\Theta\,\mathrm{d}t$: The rent you pay for that convexity.
$\nu\,\mathrm{d}\sigma$: What you make if the market reprices volatility.

Delta, gamma, vega and theta

After this lesson you should be able to

Three readings of delta

The P&L equation

The rest of this lesson is in Premium