# How is the variance schedule used?

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?

• 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. Sep 7 at 5:52
• @nbbo2: I understand that part. I just edited my question. Do you know of the other usage I am asking?
– Hans
Sep 7 at 6:30
• What is exactly meant here by variance-time and variamce-schedule? Sep 7 at 14:28
• @FridoRolloos: That is precisely what I want to know. :-D
– Hans
Sep 7 at 16:43
• 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". Sep 7 at 19:16