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Let say I have a Variance Swap contract which is based on daily closing prices (not the continuous variance calculation) and will last between the day interval $T_1$ and $T_2$ against a strike with $K^2$.

Standing at time $T_0$, I need to value this contract.

Is there any analytical Valuation formula to achieve this?

Any pointer will be highly appreciated.

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  • $\begingroup$ No. I am looking to value (at time $T_0$) above Variance Swap where payoff is based on discretely sampled return and with strike $K$ i.e. payoff = $N \left(\frac{1}{T_2 - T_1} \sum_{t=T_1}^{T_2} {R_t}^2 - K^2 \right)$. I am basically looking for some Analytical formula for this valuation. Let me know if more information is required. $\endgroup$ – Daniel Sep 6 '20 at 18:02
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This resource surveys the main available replication-based approximations of discrete variance swap pricing:

  • continuous method
  • Derman's method
  • Trapezoidal/Simpson methods
  • Optimal Quadratic Hedge (Leung and Lorig)

Edit:

We have:

$$ A_{m,n}:=(t_n-t_m)^{-1}\sum_{i=m+1}^n R^2_i = (t_n-t_m)^{-1}\left(\sum_{i=1}^n R^2_i -\sum_{i=1}^m R^2_i\right) $$

$$ = w_1 (t_n-t_0)^{-1} \sum_{i=1}^n R^2_i - w_2 (t_m-t_0)^{-1} \sum_{i=1}^m R^2_i $$ $$ = w_1A_{0,n} - w_2A_{0,m}$$

with $w_1-w_2 =1$, $w_1 = (t_n-t_m)^{-1}(t_n-t_0) $, $w_2 = (t_n-t_m)^{-1}(t_m-t_0) $.

Forward-starting variance swap payoff is then a calendar spread of two spot-starting variance swap payoffs:

$$ A_{m,n} - K^2 = w_1 (A_{0,n} - K^2) - w_2 (A_{0,m}- K^2). $$

Edit 2:

Bossu et al. paper 'Everything you need to know about variance swaps' has, well, everything, including a term sheet sample.

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  • $\begingroup$ Thanks. But am not sure if that talks if the swap $(K)$ starts in some future date. $\endgroup$ – Daniel Sep 6 '20 at 20:01
  • $\begingroup$ Fixing date schedule is always in the future at trade time. Once one gets inside the schedule, one needs to keep track of the realized fixings in the pricing. Discrete replication-based methods should be able to take care of any discrete schedule. $\endgroup$ – ir7 Sep 6 '20 at 20:06
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    $\begingroup$ I added an edit. Forward-starting variance swap payoff is a calendar spread of two spot-starting variance swaps. $\endgroup$ – ir7 Sep 6 '20 at 20:58
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    $\begingroup$ Variance swaps are completely additive, so the price of a swap running from $t_1$ to $t_2$ and one running from $t_2$ to $t_3$ should exactly equal the price of one running from $t_1$ to $t_3$... nice property! $\endgroup$ – StackG Sep 6 '20 at 23:07
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    $\begingroup$ @noob2 From what I know, one can’t really escape SV or SLV models (in FX and even equity). MC method can easily price any variation of var or vol swap. Where there is liquidity, one might want to use standard variance swap as calibration instruments for dynamics. $\endgroup$ – ir7 Sep 6 '20 at 23:57

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