Flatten a Two-Leg Book

Question

Your book on one stock holds two positions: long 50 call contracts with delta 0.30, and long 80 put contracts with delta -0.25. Each contract covers 100 shares. What single stock trade makes the book delta-neutral right now?

Accepted Answer

Key insight: delta aggregates linearly in share-equivalents. Convert each leg, keeping signs. - Calls: 50 times 0.30 times 100 = +1500 share-equivalents. - Puts: 80 times (-0.25) times 100 = -2000 share-equivalents. Net book delta = 1500 - 2000 = -500 shares. You are synthetically short 500 shares, so the flattening trade is to buy 500 shares. The trap is dropping either the contract multiplier (answering 5 shares) or the put's sign (answering sell 3,500 share-equivalents). Both mistakes show up constantly in fast screens, so interviewers pick numbers where the sign flip changes the direction of the hedge, not just its size. Why it matters on a desk: every market maker's first risk line is net delta in share-equivalents, because that is the number the auto-hedger trades against. Greeks for a whole book are just this sum run over thousands of lines; if you cannot do the two-line version instantly, nothing downstream works. Note the hedge is only locally flat: the book has gamma from both legs, so after a move the delta will no longer be zero and the hedge must be refreshed.

Flatten a Two-Leg Book

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