Running a book: aggregating Greeks and neutralising them

GRK · Chapter 412 min readAsked at Optiver, SIG, IMC, Akuna

Assumes Delta hedging in practice: how often, and what it costs.

After this lesson you should be able to

  • Aggregate Greeks across a book and say which ones add.
  • Construct a delta- and gamma-neutral position with two instruments.
  • Read a risk report and say where the book actually hurts.

A single option is a teaching example; a book is the job. Greeks add linearly across positions on the same underlying, which makes the aggregation easy — and creates the real problem, which is that a book can look flat on every number and still lose badly to a move the report does not show.

Proposition 4.1

What adds and what does not

Delta, gamma, vega, theta and rho all add across positions on the same underlying, weighted by position size. What does not add is risk *across* underlyings: two hundred vega in one name and minus two hundred in another is not flat vega, because the two volatilities do not move together.

Holds when

  • Aggregate in dollar terms, not in raw Greeks, so different spot levels are comparable.
  • Vega should be bucketed by expiry — a one-month vega and a two-year vega are different risks.
  • Across names, aggregate through a factor model or not at all.

Derivation 4.2

Delta- and gamma-neutral with two instruments

Gamma cannot be hedged with stock, because stock has none. You need another option, and then stock to clean up the delta.

  1. n=−ΓbookΓhedgen = -\frac{\Gamma_{\text{book}}}{\Gamma_{\text{hedge}}}

    Choose the option quantity to zero the gamma.

  2. Δnew=Δbook+nΔhedge\Delta_{\text{new}} = \Delta_{\text{book}} + n\Delta_{\text{hedge}}

    The option hedge has changed the delta.

  3. shares=−Δnew\text{shares} = -\Delta_{\text{new}}

    Stock finishes the job without disturbing the gamma.

gamma first, then delta — the order matters\text{gamma first, then delta — the order matters}

Example 4.3

Your book has delta −2,000-2{,}000 and gamma −500-500. A traded option has delta 0.60.6 and gamma 0.050.05 per contract, on 100 shares. Neutralise both.

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Worked solution

  1. Formula
    n=−ΓbookΓhedge,shares=−Δnewn = -\frac{\Gamma_{\text{book}}}{\Gamma_{\text{hedge}}}, \qquad \text{shares} = -\Delta_{\text{new}}
  2. Substitute
    Γhedge per contract=0.05×100=5\Gamma_{\text{hedge per contract}} = 0.05 \times 100 = 5
  3. Solve
    n=−−5005=+100 contractsn = -\frac{-500}{5} = +100 \text{ contracts}
  4. Δnew=−2000+100×0.6×100=−2000+6000=+4000\Delta_{\text{new}} = -2000 + 100 \times 0.6 \times 100 = -2000 + 6000 = +4000
  5. shares=−4000\text{shares} = -4000
  6. Answer
    buy 100 contracts, sell 4,000 shares\text{buy } 100 \text{ contracts, sell } 4{,}000 \text{ shares}

Sanity check. Note the option hedge moved the delta by more than the original exposure, which is normal — you chose the quantity for gamma, so the delta it drags along is whatever it is. Doing it in the other order would have meant hedging the delta twice.

Flat on every number, still exposed. A book can be delta, gamma and vega neutral and still lose heavily, because those numbers are local derivatives and a real move is not local. Long the wings and short the body gives zero net gamma at the current spot and strongly positive gamma five per cent away; reverse it and a gap destroys you while the report shows zero. This is why desks run scenario grids — reprice the book at spot down ten to up ten, with volatility up and down — rather than trusting the Greeks alone. The Greeks tell you about today; the grid tells you about the day you are worried about.

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