Delta hedging

Position Greeks first. An option's Greeks are per unit. A position's are (contracts) × (multiplier) × (per-unit Greek) × (+1 long / −1 short). Do this explicitly, every time. Short 200 contracts of a 100-multiplier call with delta 0.5477 gives position delta −10,954 shares: for a small move this book behaves exactly like being short 10,954 shares.

Delta hedging holds the offsetting underlying so position delta is zero. For a small move the book no longer cares about direction. The word doing all the work is small.

Why it fails, in one equation. Expand the position value to second order:

ΔV ≈ Δ·ΔS + ½·Γ·(ΔS)² + Θ·Δt + ν·Δσ

Delta-hedging zeroes the first term and does nothing to the second. Gamma is the curvature the delta hedge cannot see, and it enters as the square of the move, negligible for small moves, dominant for large ones. A short-option book has negative gamma, so ½Γ(ΔS)² is negative whichever way the price moves: the hedge loses on up-moves and down-moves alike. That is not a flaw; it is what being short an option is.

What are the risks of delta hedging?

The compensation is theta. The same book earns Θ·Δt daily. Set the two against each other and you get the identity that runs the volatility-trading industry: the daily P&L of a delta-hedged book is approximately

½ · Γ_position · S² · (σ_implied² − σ_realized²) · Δt

Sell at 30% implied and the stock realizes 20% and you make money; realize 40% and you lose. Worked example 4 computes this book's P&L at four realized-volatility levels and shows it crossing zero exactly at 30%. A delta hedge does not make you flat, it converts a directional bet into a bet on realized versus implied volatility. A smaller and better-understood risk, but a bet. That sentence is why this module exists.

Discrete rebalancing is a third cost. Theory assumes continuous re-hedging; you rebalance discretely and delta drifts in between. The residual error has mean near zero and a standard deviation falling roughly as 1/√n. A Monte Carlo over 20,000 paths of the Worked-example-4 book (short one at-the-money three-month call, premium $9.47) gives hedging-error standard deviations of $6.80 at one rebalance, $1.66 at 21, $0.50 at 252 and $0.25 at 1,008, each quadrupling of frequency roughly halving the error. Note what daily hedging leaves: ±$0.50 against a $9.47 premium, 5.3% of the option's value, irreducible. Transaction costs run the other way, which is why desks rebalance on delta bands, never on the clock.

A units rule, once, because it is the most expensive small mistake in this section. Closed-form vega is the derivative with respect to σ itself, so it is quoted per 1.00 of volatility, meaning a move from 20% to 120%. Traders talk in volatility points, so the per-point figure is the closed form divided by 100. Every vega in this module is stated per 1.00 with the per-point figure alongside; DV1.02's Greeks section states the same convention for the closed forms. Confusing the two scales a risk number by a hundred in either direction, and the direction that flatters the book is the one nobody questions.

Delta-gamma hedging: solving the two-instrument system

If gamma is the problem, the fix is arithmetic: you cannot hedge curvature with a linear instrument. The underlying has delta 1 and gamma 0 by definition, so no quantity of stock changes your gamma. You need a second option, then you re-neutralize delta with stock, a two-equation system solved in a fixed order:

1. Gamma first. With book gamma Γ_P and hedging-option gamma Γ₂, choose n₂ units (share-equivalents) so Γ_P + n₂Γ₂ = 0: n₂ = −Γ_P / Γ₂.
2. Delta second. The hedging option brought n₂Δ₂. Take n_S = −(Δ_P + n₂Δ₂) shares. Stock has zero gamma, so this cannot undo step 1, which is why the order is gamma-then-delta and never the reverse.

Worked example 5 runs this and the results are dramatic: on a one-day +10% move the delta-only book loses $34,585 while the delta-gamma book gains $3,057; on −10% the losses are $39,207 versus $3,366. An order of magnitude, from one extra instrument.

What does a delta-gamma hedge cost?

But look at what bought it. Neutralizing gamma required buying 20,791 share-equivalents of a 160-strike call, and that purchase carries vega, which here cancels the book's vega almost exactly. That is not luck: for two options on the same underlying and expiry, vega/gamma = S²T, identical at every strike, so gamma-neutralizing with a same-expiry option is automatically vega-neutralizing. Go to a different expiry and it is not: vega scales with √T while gamma scales with 1/√T, so a long-dated option delivers a lot of vega per unit of gamma. Choose the hedging expiry and you are choosing your residual vega, whether or not you meant to.

And theta does not disappear. The delta-only book earned $1,084/day; the delta-gamma book earns $29/day, you gave up 97% of the theta income to buy the gamma protection. That is exactly the trade: theta is the rent you collect for being short gamma, and if you stop being short gamma the rent stops. Any presentation of gamma hedging that does not show the theta you gave up is selling you something.