Your delta hedged book has gamma of 500 shares per $1 move. The stock moves $3. What is the gamma profit and loss, and what daily theta would make that a breakeven day?

Your delta hedged book has gamma of 500 shares per $1 move. The stock moves $3. What is the gamma profit and loss, and what daily theta would make that a breakeven day?

Approach: Apply the second order term of the Taylor expansion of option value in the underlying, then set it equal to the daily time decay to find the breakeven move.

$2250. Gamma profit and loss on a delta hedged book is 0.5 * Gamma * (dS)^2, which is the second order Taylor term of option value in the underlying. Here 0.5 * 500 * 9 gives $2250. The square is what matters, since a $6 move pays $9000, four times as much, so gamma pays for range and pays nothing for direction. Against that sits theta as the daily rent. The book breaks even on the day when 0.5 * Gamma * (dS)^2 equals the theta bill, so with $250 of theta the breakeven move is sqrt(2 * 250/500), which is $1 exactly, and any move larger than a dollar in either direction leaves the day profitable.

Follow-up: The stock makes six separate $1 moves in the day and ends unchanged. What does the book earn if you rehedge after each move?

Key concepts: gamma, second order Taylor term, theta breakeven.