Your options book has theta of -$450 per day and gamma of 200 shares per point with the underlying at 80. What annualised realised volatility do you need to break even, assuming 252 trading days?

Your options book has theta of -$450 per day and gamma of 200 shares per point with the underlying at 80. What annualised realised volatility do you need to break even, assuming 252 trading days?

Approach: Set the daily convexity gain equal to the daily decay to get the break-even move in points, convert it to a return, then scale by the square root of the number of trading days.

42.1%. Break-even requires 0.5*Gamma*(dS)^2 = theta, so dS = sqrt(2*450/200) = sqrt(4.5) = 2.121 points. As a return that is 2.121/80 = 2.652% per day, and annualising by the square root of time gives 0.02652*sqrt(252) = 0.02652*15.875 = 0.421. So realised volatility above 42.1% makes the long gamma position pay and anything below it loses to theta. That number is the implied volatility the book is effectively long at, because the ratio of theta to gamma in Black-Scholes is fixed at theta = -0.5*Gamma*sigma^2*S^2, and inverting it recovers the implied volatility used to mark the position.

Follow-up: Can you derive the identity theta = -0.5*Gamma*sigma^2*S^2 for a zero-rate European option and say why it makes the break-even independent of strike?

Key concepts: break-even volatility, gamma, theta, square root of time.