Finance & Options

Lesson 9 of 12

Delta and Gamma Hedging

Delta says how much an option moves when the stock moves one dollar, and gamma says how fast delta itself changes, which is why a hedge has to be redone as the stock moves.

Delta: how much the option moves

When the stock moves by $1, the option moves by about delta, the slope of the option price against the stock price:

Δ=∂C∂S.\Delta = \frac{\partial C}{\partial S}.

Take the running example from the Black-Scholes lesson: S=K=100S = K = 100, σ=20%\sigma = 20\%, T=1T = 1 year, r=0r = 0. The call is worth 7.97 and its delta is N(d1)=N(0.1)=0.540N(d_1) = N(0.1) = 0.540. If the stock ticks up to 101, delta predicts the call gains 0.54 and ends near 8.51. Repricing with the formula gives 8.515, so the slope is good for a one-point move.

Two readings of delta are worth keeping. It is the hedge ratio: for small moves one call behaves like 0.54 shares, so a trader who is short the call buys 0.54 shares to cancel the exposure. It is also a gauge of moneyness, running from 0 to 1 as the call goes from deep out of the money to deep in the money. At S=80S = 80 the delta is 0.15; at S=120S = 120 it is 0.84.

A put has delta N(d1)−1N(d_1) - 1, between −1-1 and 00. That follows from put-call parity: C−P=S−Ke−rTC - P = S - Ke^{-rT}, and the right side has delta 1. The at-the-money put in the example has delta −0.46-0.46.