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so Implied Volatility is computed by equalizing the value of the call option given by the black and scholes model with the one observed.

Then, by inversing $C_{BS}$, one gets "$\sigma_{IMP}$". My question would be, is $\sigma_{IMP}$ a function of the time to maturity ? Or, is it as I understood, it is "annualized". In other words, it represents

I am baffled by the definition of "Total Implied Variance" which is : $ T \sigma_{IMP}^2$ (T time to maturity). I don't see the interest one can have in multiplying $\sigma$ by the time T. The IV is a random variable that would vary with T, and assuming it is constant for all Ts is weird to me. Perhaps I have understood something wrong.

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Implied vol is per unit time, unit being a year here, so yes it is per annum. So for a T maturity, the variance will have diffusive impact of $\sigma^2 T$. I can think of two ways that might convince you that it is the variance that should scale with time.

  1. Remember the coefficient of the diffusion equation is $\sigma^2$, so the square version is important from the diffusion point of view.
  2. Sum of T independent and identically distributed normal variable will have variance $\sigma^2 T$

And regarding $\sigma$ varying with time or being a random variable or a stochastic process, in the cases where the total implied variance would be of interest, one is assuming constant volatility (i.e., Black Scholes) so that's the reason. In the non-constant case, you will have integral and so on.

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Implied vol on the market depends on time, that's why we talk about vol surface as a function of maturity and strike. Here's an example from Wikipedia:

enter image description here

Notice, that this nothing to do with time scaling, this is quoted on annual terms.

In a simple model that Black and Scholes used there is no implied vol. There's only geometric Brownian motion (GBM) with $\sigma$, which is a volatility of underlying.

The concept of implied volatility is born from the market observation though. For instance, in a simple model all options of the same underlying should have the same volatility. However, in reality they are different. Hence, the implied vol of an option.

In theory you just use one volatility and nothing else is necessary. Obviously $Var[S_T]=T\sigma^2$ for GBM, where time $T$ is measured in years. It is a convention to quote volatility in annual terms, i.e. T=1.

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  • $\begingroup$ please excuse my ignorance, what GBM stands for? $\endgroup$ – Marine Galantin Jun 17 at 22:00
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    $\begingroup$ geometric brownian motion $\endgroup$ – Aksakal almost surely binary Jun 17 at 22:01
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There is a very popular parametrisation of the Implied Volatility (IV) Smile in terms of Total Implied Variance (see Gatheral & Jaquier). A condition for an Implied Volatility Surface (IVS) to be free of calendar spread arbitrage is also derived in terms of Total Implied Variance. Defining the latter by $w(k, t):=\sigma_{\mathrm{BS}}^{2}(k, t) t$, it must hold: $$\partial_{t} w(k, t) \geq 0, \; \forall \;k \in \mathbb{R} \text { and } t>0,$$ where $k$ is the log-moneyness defined as $k:=\frac{K}{F_t}$ and with $F_t$ being the forward price. This means that Total Implied Variance must be non-decreasing in $k$ to preclude calendar spread arbitrage (see Gatheral & Jacquier or Fengler).

The above property allows to visually check for arbitrage violations in your options data or IVS parametrisation. Indeed, the Total Implied Variance Smiles plotted in $k$-space need not to cross in order for the IVS to be free of calendar spread arbitrage.

enter image description here

Having an IVS which is free of static arbitrage (also butterfly arbitrage, which corresponds to having non-negative probability densities (see Breeden & Litzenberger), needs to be precluded) avoids the existence of negative Local Variances, and thus serves as a basis to build a robust Local Volatility pricing engine.

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