Your convex P&L when the stock gaps overnight

Question

You hold a delta-hedged long call, gamma $\Gamma = 0.08$ per share, on a stock at $\$ 100 $. Overnight the stock gaps to$ $106 $, you had no chance to re-hedge. Each contract covers$ 100 $shares and you hold$ 10$ contracts.

Approximately what did the gap earn you (ignore theta), and why does the sign not depend on which way it gapped?

Accepted Answer

Because you were delta-hedged, the first-order (delta) exposure is cancelled. What survives a move is the second-order gamma term: 	ext{P\&L per share} \approx 	frac12\,\Gamma\,(\Delta S)^2 = 	frac12 	imes 0.08 	imes (6)^2 = 	frac12 	imes 0.08 	imes 36 = \1.44. Scaling to your size, 10 contracts 	imes 100 shares = 1{,}000 shares of option exposure: 1{,}000 	imes \1.44 = +\1{,}440. Why the sign is always positive (for long gamma): the gamma term depends on (\Delta S)^2, which is positive whether the stock gapped up or down. Being long gamma means you profit from movement itself, dire…

Your convex P&L when the stock gaps overnight

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