VarSwap PnL formula

I came across this formula for the varswap PNL: let $r_i$ be the log return over $[t_i,t_{i+1}]$ and suppose we risk manage the VS at a fixed implied volatility sigma, the PnL of (the payoff) over time interval $[t_i,t_{i+1}]$ is: $$r_i^2-\sigma^2*\Delta T\$$ Do you know how the author gets this formula?

Thanks!

• If you "risk manage" your variance swap then hopefully your pnl is zero. The formula you refer to is the contribution to the VS payoff over time interval $[t_i, t_{i+1}[$. – Antoine Conze May 22 '18 at 10:07
• In any "swap" you receive one thing and you pay another. In this case each day you receive the squared daily return $r_i^2$ and you pay the agreed fixed variance at the annual rate $\sigma^2$, on a daily basis this translates to $\sigma^2 \Delta T$ where $\Delta T$ is one day i.e. $\frac{1}{252}$ years. Hence the "per day net" is the expression you quoted $r_i^2-\sigma^2 \Delta T$ – Alex C May 23 '18 at 0:15
• The issue is that the variance swap is not a swap per se, it's actually a forward and the only "leg" is at maturity. – JiLight May 23 '18 at 7:32
• It is still a swap in the sense that you receive/pay a floating leg (the realized variance, accruing over time as the sum of daily square log returns "fixings") and pay/receive a fixed leg (the strike). In a forward you usually receive/pay the difference between a single fixing and a fixed quantity. – Antoine Conze May 23 '18 at 8:37