Adding up a gamma-scalping week

Question

You are long an option, delta-hedged, with gamma roughly constant at $\Gamma = 0.05$ , $S \approx 100$ , and implied volatility $\sigma_{\text{impl}} = 20\%$ . Over three trading days the stock's absolute moves are $\$ 1 $,$ $3 $, and$ $0.50$.

What is your total P&L over the three days, and is realized volatility above or below implied?

Accepted Answer

Each day contributes a gamma gain 	frac12\Gamma(\Delta S)^2 and a theta rent 	frac12\sigma{	ext{impl}}^2 S^2\Gamma\,dt. Gamma gains (	frac12\Gamma(\Delta S)^2 with \Gamma = 0.05): 	ext{Day 1: } 	frac12(0.05)(1)^2 = 0.025, \quad 	ext{Day 2: } 	frac12(0.05)(3)^2 = 0.225, \quad 	ext{Day 3: } 	frac12(0.05)(0.5)^2 = 0.00625. Total gamma gain = 0.025 + 0.225 + 0.00625 = \0.25625. The convexity is the whole story: the single \3 day dominates, because gamma pays on the square of the move. Theta rent per day, with \sigma{	ext{impl}} = 20\%, S = 100, dt = 1/252: 	frac12(0.04)(10{,}000)(0.05…

Adding up a gamma-scalping week

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