How big a move do you need to break even?

Question

You are long an option and delta-hedge it continuously. The stock is at $S = 100$ and the option was priced at an implied volatility of $16\%$ . Some gamma $\Gamma > 0$ , but its exact value is not given.

What size of daily move makes you break even for the day, and why does the answer not depend on gamma?

Accepted Answer

A delta-hedged long option earns the gamma term and pays the theta term: d\Pi \approx 	frac{1}{2}\,\Gamma\,(\Delta S)^2 + \Theta\,dt, \qquad \Theta \approx -	frac{1}{2}\sigma{	ext{impl}}^2 S^2 \Gamma. Break-even is where the gamma gain exactly offsets the theta rent: 	frac{1}{2}\Gamma(\Delta S)^2 = 	frac{1}{2}\sigma{	ext{impl}}^2 S^2 \Gamma\,dt. The 	frac12\Gamma cancels from both sides, leaving (\Delta S)^2 = \sigma{	ext{impl}}^2 S^2\,dt \quad\Longrightarrow\quad \Delta S = \sigma{	ext{impl}}\,S\,\sqrt{dt}. The break-even move is just the one-standard-deviation daily move implied by…

How big a move do you need to break even?

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