Case 064Greeks and managing an options bookCore

A book has delta of plus 2,000 shares and gamma of minus 500 shares per rupee. Using the stock and one listed option with delta 0.6 and gamma 0.05, how many options and shares make it delta and gamma neutral?

The situation

Arkavati Derivatives makes markets in options on Ghodbunder Pharma. After a day of client trades its book has a delta of plus 2,000 shares, meaning it gains Rs 2,000 for each rupee the stock rises, and a gamma of minus 500 shares per rupee, meaning its delta falls by 500 shares for every rupee the stock rises.

The desk can trade the stock itself and one listed option whose delta is 0.6 and whose gamma is 0.05 per share per rupee. The head trader wants the book flat on both delta and gamma before the weekend.

Your task

Find the number of options and shares to trade, show why the order of the two steps matters, and say what the hedge leaves behind.

Quick check

Before working it: how many of the listed options does the desk need?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

Buy 10,000 options and sell 8,000 shares. The options add plus 500 of gamma, cancelling the book's minus 500, and bring 6,000 of delta with them, so delta rises to plus 8,000; selling 8,000 shares, which have no gamma, takes delta to zero. Doing it the other way round, delta first, leaves the book with plus 6,000 of delta after the gamma trade.

Step 1Which instrument fixes which Greek?

A shower with a hot tap and a mixer is a good picture. The mixer changes the temperature without changing the flow; the hot tap changes both. You set the hot tap first for the flow, and then use the mixer to fix the temperature it disturbed. Stock has delta but no gamma, so it can fix delta without touching gamma; the option has both, so it must be used for gamma first, and stock then cleans up whatever delta the option brought in. That ordering is the whole case.

The relationship
n=−ΓbookΓopt=5000.05=10,000shares=−(Δbook+n Δopt)=−(2,000+6,000)=−8,000n = \frac{-\Gamma_{\text{book}}}{\Gamma_{\text{opt}}} = \frac{500}{0.05} = 10{,}000 \qquad \text{shares} = -(\Delta_{\text{book}} + n\,\Delta_{\text{opt}}) = -(2{,}000 + 6{,}000) = -8{,}000
noptions to buy, positive for a purchase
Gamma book, Gamma optthe book's gamma, minus 500, and one option's gamma, 0.05
Delta book, Delta optthe book's delta, plus 2,000, and one option's delta, 0.6
What it says in wordsThe option count is chosen to cancel gamma; the share trade then cancels the delta the book had plus the delta the options added.
Fix gamma first, with the option; then clean up the delta, with stockBook nowDelta+2,000Gamma-500After the optionsDelta+8,000Gamma0HedgedDelta0Gamma0Buy 10,000 options+6,000 delta+500 gammaSell 8,000 shares-8,000 deltano gammaThe wrong orderSell 2,000 shares first: delta 0, gamma still -500.Then buy 10,000 options for the gamma: they bring 6,000 delta with them.Result: gamma 0 but delta +6,000, three times the delta you started with.
Buying 10,000 options cancels the minus 500 of gamma but lifts delta from plus 2,000 to plus 8,000; selling 8,000 shares then flattens delta, while hedging delta first and gamma second leaves plus 6,000 of delta behind.
Step 2What does each step actually remove from the profit and loss?

To second order, the book's profit for a stock move of x rupees is delta times x plus half of gamma times x squared. Unhedged, that is 2,000x minus 250x squared: a tilt that favours a rise and a bend that hurts any large move. Selling 2,000 shares removes the tilt and leaves the bend, minus Rs 9,000 for a Rs 6 move either way; the option trade removes the bend as well, and the book is flat to second order. The curve below makes the difference visible: the unhedged line loses Rs 21,000 on a Rs 6 fall, the delta-hedged line Rs 9,000, and the neutral line nothing.

What each step removes: the tilt, then the bend0-10k-20k-6-30+3+6Stock move, Rs-21,000-9,000unhedgeddelta hedged onlydelta and gammaneutralThe bend at a Rs 6 move:0.5 x gamma x move squared= 0.5 x -500 x 36 = -9,000
To second order, the unhedged book loses Rs 21,000 on a Rs 6 fall, the delta-hedged book loses Rs 9,000 on a Rs 6 move either way because of its minus 500 gamma, and the delta and gamma neutral book is flat.
Step 3What does the neutral book still carry?

Neutral on two Greeks is not neutral on everything. Buying 10,000 options adds long vega and pays time decay, so the book now loses a little every day the stock sits still and gains if implied volatility rises; whether that suits the desk depends on what the rest of the book holds. The hedge is also local: the listed option's own delta and gamma change as the stock moves and as expiry approaches, so the book drifts away from neutral and must be re-hedged. And if the listed option has a different expiry from the options in the book, the two gammas will decay at different speeds, so a book flat on Friday can be long or short gamma by Wednesday. A trader states the hedge with these limits attached, rather than calling the book riskless.

Where candidates lose it

Candidates sell 2,000 shares first because delta is the Greek they know, then buy 10,000 options and announce the book is neutral. It is gamma neutral with plus 6,000 of delta, three times worse than where it started. Fix gamma with the option first; the stock goes last.

The second miss is sizing the options on delta, 2,000 divided by 0.6. Delta can be fixed with stock at no cost to gamma, so the option count must come from gamma.

What the interviewer asks next

  • The book also has vega of minus Rs 30,000 per vol point, and a second listed option is available. How would you neutralise delta, gamma and vega together?
  • The stock rises Rs 5 after the hedge. Roughly what is the book's delta now, and why?
  • Why might a market maker prefer to stay a little short gamma rather than pay for 10,000 options?

Company names and figures are illustrative.