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.