# Remaining variance and historical variance in Black-Scholes with term structure

When pricing an European vanilla option in a Black-Scholes world with deterministic volatility term structure, what matters is the remaining variance between today $$t$$ and maturity $$T$$, i.e. the volatility parameter to use in the BS formula is $$\sqrt{\int_t^T \sigma^2(u) du}$$.

Is there an option where not only the future volatility, but also the realized volatility from some time in the past, say time $$0$$ is part of the input parameter? So for instance an option where the implied volatility parameter at time $$t$$ would be $$\sqrt{ \int_0^T \sigma^2(u) du }$$ ?

• What kind of kindergarten nonsense is this +1/ -1 voting possibility for a question asked? – ilovevolatility Jan 9 at 0:52