# Path dependency for Delta hedge value

This is actually a follow-up questions for the two threads below - value of a delta hedged option:

Delta hedge value formula

Continuous delta hedge formula

My question is that how the drift (mu) impact this hedged portfolio. In Paul Wilmott's book, he comments that “if we start off at the money ad the drift is very large (positive or negative), we will find ourselves quickly moving into territory where gamma, and hence the hedged portfolio value, is small. The best that could happen would be for the stock to end up close to the strike at expiration, this would maximize the total profit."

Can someone help to explain why? Thank you so much in advance.

• As discussed in the linked posts, the profit to a deltaheged position depends on the Dollar Gamma ($S^2 \Gamma$) integrated over the path. We know that Gamma is highest near ATM and near Expiration. ( mathworks.com/help/examples/finance/win64/… ) Conversely a path where Gamma dwindles to a low value will be a low profit path. Nov 8 '20 at 19:50
• @noob2 Thank you for the comments. If the drift is a large positive value, then the S^2 will get bigger fast. Does it mean that the Gamma decrease faster the increase of S^2, such that the Dollar Gamma as a whole gets smaller? Nov 8 '20 at 23:58