The gamma–theta trade-off of a hedged option

Question

You are long a call option and you delta-hedge it continuously (holding $-\Delta$ shares so the position is directionally flat). The stock is at $S = 100$ , your option's gamma is $\Gamma = 0.05$ , and it was priced at an implied volatility of $20\%$ .

Where does your daily P&L come from, and under what condition do you make money?

Accepted Answer

Two competing forces: gamma and theta. A delta-hedged option's P&L over a short interval, after the delta term cancels, is d\Pi \approx 	frac{1}{2}\,\Gamma\,(\Delta S)^2 \;+\; \Theta\,dt. - The gamma term 	frac12\Gamma(\Delta S)^2 is your friend when long: it is positive for any move (up or down), because (\Delta S)^2 \ge 0 and \Gamma 0. Being long gamma means you profit from movement, your hedge is always a step behind, so you end up buying low and selling high as you rebalance ("gamma scalping"). - The theta term \Theta\,dt is the rent you pay. A long option decays with time; \Theta < 0. …

The gamma–theta trade-off of a hedged option

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