Rehedging a book after a move

You are long 100 identical calls and start perfectly delta-hedged. Each call currently has delta 0.450.45 and gamma 0.080.08 (per $1 move, per share). The stock then jumps up $3. To restore delta-neutrality, how many shares must you now trade — and in which direction? Give the number of shares (a positive number; state the direction in your reasoning).

Show hints (2)+
  1. Gamma updates delta: Δnew=Δold+ΓdS\Delta_{\text{new}}=\Delta_{\text{old}}+\Gamma\,dS.
  2. Trade the difference between the new hedge and the one you already hold, not the whole new hedge.

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Asked at: Citadel, DRW

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