Section 26.3

Greeks and Their Relationships

Intuition for delta, gamma, vega and theta; parity relations between the Greeks of calls, puts and straddles; delta as a probability proxy.

Toolkit

Call and put on the same strike/maturity are linked by put-call parity, $C-P=S-Ke^{-rT}$ ; differentiating in the relevant variable transfers identities directly to the Greeks. Differentiating w.r.t. $S$ gives

\Delta_C-\Delta_P=1,\qquad \Gamma_C=\Gamma_P,\qquad \text{Vega}_C=\text{Vega}_P,

while differentiating w.r.t. $t$ shifts theta by the cost of carry. Greeks are linear in position size and add across legs of a portfolio (e.g. a straddle’s delta is $\Delta_C+\Delta_P$ ).

Under Black–Scholes, $\Delta_C=N(d_1)$ is treated as the (risk-neutral) probability the call finishes in the money, with $\Delta_P=N(d_1)-1=-\mathbb{P}(\text{put ITM})$ .

Gamma and vega are largest near the money and decay away from the strike; at the money $d_1\approx(r+\sigma^2/2)\sqrt T/\sigma$ is independent of $S$ , so

\Gamma=\frac{\phi(d_1)}{S\sigma\sqrt T}\propto \frac1S,\qquad \text{Vega}=S\phi(d_1)\sqrt T\propto S.

A short position negates the sign of every Greek.

Problems