Setting: Black-Scholes with deterministic volatility.
Vanilla options are priced with volatility equal to remaining variance/volatility [$] \int_t^T \sigma^2(u) du [$].
What kind of option, that is delta-hedgeable, has as input at current time [$] t [$] not the remaining variance, but the total variance [$] \int_0^T \sigma^2(u) du [$] from [$] u=0 [$] to [$] u = T[$]?
See also my question here.
Is it a timer option that I should be looking at?