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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 $\sqrt{ \int_0^T \sigma^2(u) du }$ ?

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

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