Delta hedged options, return estimate

So if we have a crude plan of delta hedging a long/short option position once a day, we'd like to know how far the underlying could go for us to make/lose money, as accurately as possible. Since gamma is associated with variance, and is a derivative of delta like acceleration off velocity; the return on a delta hedged options position is then like that of the distance formula from mechanical physics.

Distance formula: 
distance = (initial velocity)*(time) + (1/2)*(acceleration)*(time)^2

To translate this into a delta hedged options position, 

initial velocity -> delta
acceleration -> gamma
time -> change in the underlying

P&L of delta-hedged options

 









With a volatility forecast, the above gives us a pretty good idea of the position's EV (expected value).