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Let $v(t)$ be the instantaneous variance of an underlying stock or index at time $t\in[0,1]$ between the open at $t=0$ and close at $t=1$ of an exchange. Usually $v(t)$ achieves local maxima at $t\in\{0,1\}$. For example, $$v(t)\propto f(t)= a+bt+c\cos(2\pi t), \, c>0,$$ where $\int_0^1 f(t)dt=1$.

I can see $\int_{T_1}^{T_2}v(t)dt$ can be plugged into a model, such as the Black-Scholes formula. But I was told that the functional form is also used to compute some kind of effective variance-time in options pricing/trading?. I am not sure exactly how. Could someone elucidate?

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  • $\begingroup$ What matters for pricing an option at time $T_1$ that expires at time $T_2$ is the "integrated variance" $\int_{T_1}^{T_2} \sigma^2(t) dt$. Once you compute this parameter (via your variance schedule or any other) then the option pricing proceeds as normal. $\endgroup$
    – nbbo2
    Sep 7 at 5:52
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    $\begingroup$ @nbbo2: I understand that part. I just edited my question. Do you know of the other usage I am asking? $\endgroup$
    – Hans
    Sep 7 at 6:30
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    $\begingroup$ What is exactly meant here by variance-time and variamce-schedule? $\endgroup$
    – Frido
    Sep 7 at 14:28
  • $\begingroup$ @FridoRolloos: That is precisely what I want to know. :-D $\endgroup$
    – Hans
    Sep 7 at 16:43
  • $\begingroup$ Where does the proportionality relationship in your question come from (is this simply an ad hoc assumption?), and what "functional form" are you referring to - the former? I also don't understand what is the relation between $v(t)$ reaching a maximum and your question about "variance-time". $\endgroup$ Sep 7 at 19:16

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