Short a call, the stock gaps up, what did it cost you?

Question

You are short 1 call contract (100 shares) with delta 0.40 and gamma 0.06 per share. You had hedged by buying 40 shares. Overnight the stock gaps from \100 to \110, and the call's delta jumps to 0.75. Estimate your P&L from the gap, and what you must do to re-hedge.

Accepted Answer

You were short gamma, so a gap hurts. Two ways to see the loss: Greek approximation. A short-gamma position loses 	frac12\Gamma(\Delta S)^2 on a move: 	ext{loss per share} \approx 	frac12 	imes 0.06 	imes (10)^2 = 	frac12 	imes 0.06 	imes 100 = \3.00. Over 100 shares of option exposure that is about -\300 from the gap (before theta, which was small overnight). Hedge-tracking view. You were long 40 shares as a hedge but the option behaved, over the move, like being short roughly the average delta, closer to 	frac{0.40+0.75}{2} = 0.575, i.e. \approx 57.5 shares of exposure. Your 40-sha…

Short a call, the stock gaps up, what did it cost you?

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