You are long a book with gamma of 400 shares per point and theta of -$600 per day, spot 100. The stock closes 2 points higher after a smooth path and you hedged continuously. What is the day's profit and loss, and which term dominates on a quiet day?

You are long a book with gamma of 400 shares per point and theta of -$600 per day, spot 100. The stock closes 2 points higher after a smooth path and you hedged continuously. What is the day's profit and loss, and which term dominates on a quiet day?

Approach: Value the hedged position with the second-order term one half gamma times the squared move, then subtract the day's time decay.

$200. A delta-hedged position earns 0.5*Gamma*(dS)^2 from convexity, which is 0.5*400*2^2 = $800 of gamma scalping profit, and pays theta of $600, so the day nets 800 - 600 = $200. The convexity term is quadratic in the realised move while theta is linear in time, so a day whose move falls short of sqrt(2*600/400), or 1.73 points, loses money and every day above it gains. On a quiet day theta dominates and the book bleeds, which is why a long gamma position is a bet on realised movement rather than on direction. The path matters only through the sum of squared moves, so the same 2-point close reached by a single jump or by chopping back and forth pays differently once hedging is discrete.

Follow-up: The stock instead round-trips, closing unchanged after moving 2 points up then 2 points down intraday. What is the P&L if you rehedged at the high?

Key concepts: gamma scalping, theta, delta hedging, realised move.