Turning theta into a daily rent bill

Question

Your long-option book has a position theta of $-\$ 400 $per day and a position gamma of$ 600 $deltas per$ $1 $move, on a stock at$ $50$.

If the stock does nothing for $5$ trading days, what do you lose? Roughly what daily move would offset one day's theta?

Accepted Answer

The rent. Theta is a daily charge. If nothing moves for 5 days, 5 	imes (-\400) = -\2{,}000. That is the price of holding your convexity through a dead week. The offset. Theta is the rent you pay for gamma; a move earns gamma P&L of 	frac12\Gamma(\Delta S)^2. Set that equal to one day's theta and solve for the break-even move: 	frac12 	imes 600 	imes (\Delta S)^2 = 400 \;\Longrightarrow\; (\Delta S)^2 = \frac{800}{600} = 1.333 \;\Longrightarrow\; \Delta S \approx \1.15. So a stock move of about \1.15 (roughly 2.3\% on a \50 stock) in a day pays for that day's theta; anything larger is pr…

Turning theta into a daily rent bill

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