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Explain how a delta-neutral and gamma-neutral position is put in place and then kept in place.

FRM® Part I · Valuation and Risk Models · Option Sensitivity Measures: The “Greeks” · FRM1.VR.16.f

The idea

Delta-neutral means total delta is zero, and gamma-neutral means total gamma is zero, so delta stays near zero as the stock moves. The underlying has zero gamma, so gamma can only be changed with a traded option, and the procedure has two steps. First, buy or sell the number of traded options that cancels the portfolio gamma, given by Options = −Γp ÷ ΓT. Here Γp is the portfolio gamma and ΓT is the gamma of one traded option, while Δp is the portfolio delta and ΔT is the delta of one traded option. Second, trade the underlying to cancel the delta that the options have added, given by Shares = −(Δp + Options × ΔT). The order matters because the option trade changes delta, and the usual error is to size the share trade before adding the options. The position is neutral only locally, so both Greeks drift with price and time and must be rebalanced.

Formulas

Number of traded options for gamma neutrality

nO = −ΓP ÷ ΓO

ΓP
the portfolio's gamma
ΓO
the gamma of one traded option

Size this first, because the options also add delta. Dividing by the option's delta instead of its gamma is the trap.

Shares to trade after the gamma hedge

nS = −(ΔP + nO × ΔO)

ΔP
the portfolio's delta before the option trade
nO
the number of traded options bought (negative if sold)
ΔO
the delta of one traded option

Positive means buy shares. Leaving out the delta the options add is the usual error.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why can a trader not make a portfolio gamma neutral by trading only the underlying stock?

  1. The stock has a gamma of one, so trading it would change delta by too much.Why not: A share has a delta of one but a gamma of zero, because its delta never changes.
  2. The stock has a gamma of zero, so trading it does not change the portfolio's gamma. - correct
  3. Gamma can be adjusted with futures on the stock, but not with the stock itself.Why not: Futures are also linear and have zero gamma; only options, which are non-linear, can offset gamma.
  4. Trading the stock changes gamma but only after the options expire.Why not: Gamma does not depend on expiry in this way; a linear position simply has no gamma to add.

Gamma measures curvature (change in delta). A share is linear: its delta is always 1, so its gamma is zero. A traded option is needed to alter portfolio gamma.

A harder one

A portfolio has delta 0 and gamma -4,500. A traded call has delta 0.55 and gamma 1.80. Which trades make the portfolio delta neutral and gamma neutral?

  1. Buy 2,500 calls and sell 1,375 shares - correct
  2. Buy 2,500 calls and buy 1,375 sharesWhy not: Right option trade but the wrong sign on the stock: the calls add +1,375 delta, so shares must be sold.
  3. Sell 2,500 calls and buy 1,375 sharesWhy not: Sells options, which would make gamma more negative (-9,000); gamma is offset by buying.
  4. Buy 8,182 calls and sell 4,500 sharesWhy not: Divides by the delta (0.55) instead of the option's gamma when sizing the option position.

Calls needed = 4,500 / 1.80 = 2,500 (bought, to add positive gamma). Their delta = 2,500 x 0.55 = +1,375, so sell 1,375 shares to bring delta back to zero.

The hardest tier

A dealer is short 20,000 calls, each with delta 0.50 and gamma 0.040. The only traded option available for gamma hedging is a put with delta -0.35 and gamma 0.025. Which trades make the position delta and gamma neutral?

  1. Buy 32,000 puts and sell 21,200 sharesWhy not: Right option trade but the wrong sign for the shares: the net delta is -21,200, so shares must be bought.
  2. Buy 32,000 puts and sell 1,200 sharesWhy not: Treats the put delta as +0.35; puts have negative delta, so buying puts adds negative delta.
  3. Buy 32,000 puts and buy 21,200 shares - correct
  4. Buy 20,000 puts and buy 17,000 sharesWhy not: Matches the number of puts to the number of short calls instead of matching gammas (800 / 0.025).

Short calls: gamma -800, delta -10,000. Buy 800 / 0.025 = 32,000 puts: delta 32,000 x (-0.35) = -11,200, total delta -21,200. Buy 21,200 shares to neutralise.

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