# Initial forward variance curve calibration

Let $$V_t^{T_1, T_2}$$ be the forward variance swap rate for the period $$[T_1, T_2]$$, seen from $$t$$ (see for instance Lorenzo Bergomi's Smile Dynamics II) and let $$\xi_t^T = V_t^{T,T} = \frac{\partial}{\partial T} V_t^T$$ be the instantaneous forward variance swap rate at $$T$$ seen from $$t$$, where $$V_t^T = P_{t,T} \mathbf{E}^{\mathbf{Q}^T} \left[\left. \frac{252}{N} \sum_{i=1}^N \left(\ln\left(\frac{S_{T_{i+1}}}{S_{T_i}}\right)\right)^2 \right| \mathscr{F}_t\right]$$ for the forward measure $$\mathbf{Q}^T$$ associated to some risk-neutral measure $$\mathbf{Q}$$. (I assume that the underlying pays no dividend until $$T$$.)

The initial instantaneous forward variance curve $$(\xi_0^T)_T = \left(\frac{\partial}{\partial T} V_0^T\right)_T$$ is the starting point of calibration of variance curve models (for instance Bergomi's P1, P2 (and PN) models).

As the $$V_0^T$$'s are given by the market quotations of variance swaps (more precisely, it's the $$\widehat{\sigma}_0^T = \sqrt{\frac{V_0^T}{T}}$$'s that are quoted) what is common practice (numerical differentiation (but how ...) + interpolation (which one ?) or fitting of a parametrical form (which one ?) to derive $$\xi_0^T$$ ?

Remark : the question assumes implicitely that there's a variance swap market on the considered underlying, yes, but what could be done would we not have a VS market at all, or would we not have access to such market quotations (but would we at least have an options market) ?

(First, we could calibrate the local volatility and price variance swaps with it to obtain synthetic variance swap quotations, but that would be bad because of the obvious shortcomings of the local volatility model regarding forward implied volatility which variance swap are sensitive to. In fact, same would prevail for the Heston model -- this is classical, see Lorenzo Bergomi's Smile Dynamics I for instance. The point is that a good model kind for such a synthetization would be the very kind of models I'm trying to calibrate in the first place ...)

After investigation, practioners usually interpolate linearly the quoted $$T \left(\widehat{\sigma}_0^T\right)^2$$'s to obtain a piecewise linear full curve and then differentiale wrt $$T$$ to obtain the curve $$T\mapsto\xi_0^T$$.