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As proven in Gatheral notes (or in this discussion)

The equations of the local volatility as a function of the vanilla calls can be written as $$ \sigma^2(T,K) = \frac{\frac{\partial C}{\partial T} + (r - q)K \frac{\partial C}{\partial K} + qC}{ \frac{1}{2} K^2 \frac{\partial^2C}{\partial K^2}} $$

And or as a function of implied volatility surface as $$ \sigma_{\mathrm{Dup}}(T,K)^2 = \frac{ \frac{\partial w}{\partial T} }{1 - \frac{y}{w} \frac{\partial w}{\partial y}+ \frac{1}{4}\left( - \frac{1}{4} + \frac{1}{w} + \frac{y^2}{w^2} \right) \left(\frac{\partial w}{\partial y}\right)^2 + \frac{1}{2}\frac{\partial^2 w}{\partial y^2} } $$ with $y = \ln(K/F_0^T)$ and $w(T,y) = T\Sigma^2(T,y)$

However, I would require a precision on the time-dimension variables meaning (as I feel a bit confused with all these variables, especially the $w$ and $y$, that in their writings show a dependance to a $T$ variable.

In short version:

  1. What is $T$ in these writings ( $y = \ln(K/F_0^T)$ and $w(T,y) = T\Sigma^2(T,y)$ ) ?
  2. What points are evaluated the variables and derivatives ?

In confused version (this may help provide more elements of my misunderstanding):

  1. What is the $T$ in Dupire local volatility ?
  2. is it a $ T= t $ ?
  3. For instance a simulation step between such that $t>t_0 $ ? where $t_0$ is the spot calibration date ? But in such case, at what grid point are evaluated $w, y,$ and all the partial derivatives ?
  4. For instance, $w$ is defined as $w(T,y) = T\Sigma^2(T,y)$, in such case, if $T=t$, taking $t\Sigma^2(t,y)$ for $w$ does not make sense as it should be defined as a total implied variance from a maturity $T$ point ? Or should it be $T-t$ instead ?
  5. I have the same questions for the variables and derivatives in the formula of local volatility from vanilla calls. Should the calls be evaluated at $T$, $t$ or $T-t$ ?
  6. I don't think $T$ in Dupire formula represent a time to maturity as the local volatility is supposed to be specific maturity independent. But, I would require some explanations for a better understanding I guess...

Note: I help writing my question with use of different pdf and some quant.stackexchange pages such as this one

user25844
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    $T$ is the option expiry in the Dupire local volatility formula, because you build a local volatility surface $\sigma(T,K)$ as viewed from $t=0$. Once you've done that (often you only precompute $\sigma(T,K)$ on a discrete lattice of $T \times K$ and then you interpolate, for faster calculations), you can do the MC simulation of $dS_t/S_t= (r-q) dt + \sigma(t, S_t) dW_t$ using for instance an Euler scheme. – Antoine Conze Mar 26 '21 at 12:08
  • Thank you for your comment. My understanding thank to you: at $t=0$, I can get the implied vol surface that has different quotations for $K_i,T_j$, for each point, calculate an equivalent $w,y$ grid. And on each point of this grid, have the $\sigma_Dup(T,k)$. This, would be the local volatility as viewed from $t=0$. If i need intermediate points, i could use some interpolations. Now, in the MC at time $t$, you use the local vol at point $t$, while $t$ is the elapsed time sicnce $t=0$. How is this consistent while we said first variable represent an option-expiry? – user25844 Mar 26 '21 at 13:23
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    the model is $dS_t/S_t = (r-q)dt + \sigma(t, S_t) dW_t$ so yes $t$ is elapsed time since pricing time $0$. How you obtain the Dupire Formula for $\sigma(.,.)$ however, is from an application of Tanaka's formula (a generalization of Itô's Lemma for non $C^2$ functions) to the vanilla call payoff $(S_T - K)^+$, hence the notation capital $T$ in the Dupire formula. It might be easier to understand if you consider the case where the implied $\Sigma(T)$ does not depend on strike, in which case you have $\Sigma(T)^2 T = \int_0^T \sigma^2(t) dt$. – Antoine Conze Mar 26 '21 at 14:05

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