Gamma Hedging With Options

Delta hedging with stock cancels the slope of an option position at today's price, but does nothing about the fact that the option's value curve is bent while the stock's value line is dead straight. A share of stock always has delta exactly 1 and gamma exactly 0 — its payoff line never curves. So no amount of stock can offset an option position's gamma; stock zeroes out delta at one instant, but the instant the stock moves, the option's delta has already changed while the hedge's delta (still exactly 1) has not. Only something with curvature — another option — can neutralise that.

A ruler can straighten a line, never a curve

Lay a straight ruler against a bent wire and you can match the two at exactly one point, tangent to the curve. Move an inch either side and the wire has already diverged, because the ruler cannot bend to follow it. Stock is the ruler: it matches an option position's slope at the current price perfectly, with zero capacity to follow its curvature. To flatten the curve, lay a second curved wire against the first, chosen so the two bends cancel — that second wire is another option.

Solving for the gamma hedge

To flatten a position's gamma, add $n_h$ contracts of a hedging option with gamma $\Gamma_h$ per contract, chosen so the total gamma is zero:

$$n_h \;=\; -\frac{\Gamma_{\text{position}}}{\Gamma_h}.$$

In plain English: divide the gamma you need to cancel by the gamma of one contract of your hedging instrument, with a sign flip, to get how many contracts to trade. Adding options changes delta too, so after solving for $n_h$ you then re-hedge delta with stock as a final step — gamma first (because only options can fix it), delta last (because stock is free to fix without disturbing gamma again, since stock's own gamma is zero).

Worked example 1: flattening gamma with a different strike

A market maker is short 20 contracts (100 shares/contract) of a near-the-money call with gamma $0.06$ per share. Total position gamma: $-20 \times 0.06 \times 100 = -120$ (in shares-of-delta-per-$1-move units). An available hedging option, a slightly out-of-the-money call, has gamma $0.04$ per share, i.e., $4$ per contract (100 shares).

$$n_h = -\frac{-120}{4} = 30 \text{ contracts long}.$$

Buying 30 contracts of the hedging call flattens the book's gamma to zero: $-120 + 30 \times 4 = -120 + 120 = 0$. Suppose the original short calls had delta $-0.50$ each (short, so $-20 \times 0.50 \times 100 = -1{,}000$ shares of delta) and the hedging calls have delta $0.30$ each ($30 \times 0.30 \times 100 = 900$ shares of delta). Combined delta before any stock trade: $-1{,}000 + 900 = -100$ shares. The desk buys 100 shares of stock to flatten delta too, without touching gamma at all, since stock's gamma is zero by construction.

Worked example 2: what a gamma hedge is actually worth

Before the hedge, the desk's dollar gamma of $-120$ meant a 1% move in a $100 stock ($1) would change its delta by roughly $-120 \times 1 = -120$ shares, forcing a scramble to trade 120 shares just to stay hedged. After flattening gamma to zero, a 1% move changes delta by essentially nothing. If the stock makes five such wiggles a day, the unhedged book trades roughly $5 \times 120 = 600$ shares a day purely chasing gamma; at $0.02 per share of spread cost, that's about $12 a day, or roughly $3,000 a year, avoided by buying the 30 hedging contracts once.

The strategy below combines several option legs into one payoff, the same trick a gamma hedge uses: watch how the combined curve's bend changes as you compare it to a single-leg payoff, and note that no amount of straight-line stock could ever reproduce that bend.

What this means in practice

Any market maker running a book of any size gamma-hedges with options first and delta-hedges with stock last, because stock rebalancing under unhedged gamma is expensive and gets worse with position size — the direct fix for the cost quantified in Discrete Hedging Error. Desks routinely carry small, deliberately curved option positions purely as gamma hedges for a much larger book, with no directional view of their own.