How much movement pays for a day of gamma rent?

Question

You are long an option position that gains $500$ deltas for every $\$ 1 $the stock rises (position gamma$ = 500 $shares per$ $1 $). It costs you$ $50$ per day in theta. You re-hedge back to delta-neutral after each move.

Each re-hedge captures $\tfrac12\,\Gamma\,(\Delta S)^2$ . How much stock movement do you need in a day to cover the theta?

Accepted Answer

Every time you re-hedge, you lock in a gamma profit of 	frac12\,\Gamma\,(\Delta S)^2 = 	frac12 	imes 500 	imes (\Delta S)^2 = 250\,(\Delta S)^2. You want the day's total scalping profit to cover the \50 theta, so you need the sum of squared moves between re-hedges to reach \sum 	frac12\,\Gamma\,(\Delta S)^2 \ge 50 \;\Longrightarrow\; \sum (\Delta S)^2 \ge \frac{2 	imes 50}{500} = 0.2. Concretely, suppose the stock moves in \0.20 steps and you re-hedge each time. One step earns 250 	imes (0.20)^2 = 250 	imes 0.04 = \10. So five clean \0.20 swings (5 	imes \10 = \50) exactly pay the re…

How much movement pays for a day of gamma rent?

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