Variance Swap Revisited (Redux)

It's amazing how much I learn every time I re-read the variance swap literatures.

1. The (model-independent) formula for the fair strike of a variance swap depends on the choice of S* (the truncating spot price). Most generally it is

$ rT - \frac {S_0 e^{rT} - S^*} {S^*} - ln \frac {S^*}{S_0} + e^{rT} \int_0^{S^*} \frac {P(K)}{K^2} dK + e^{rT} \int_{S^*}^{\infty} \frac {C(K)}{K^2} dK$

If S* is chosen to be the future stock price then the above expression can be simplified.

To hedge a variance swap, a trader statically holds the continuum of OTM options, plus rolling a future position of (1/F_t - 1/F_0). In a B-S universe the future position would hedge away the delta exposure, except that...

2. Delta of variance swap (one that comes from practicality):
In principal the terminal variance is measured as the INTEGRAL of INSTANTANEOUS variance; in practice, the contract is such that variance is calculated on a daily basis using daily prices. This leads to the need for intra-day Delta hedging.

3. Delta of variance swap (one that comes from smile)
The Coulombe paper show that variance swap Delta is a function of implied volatility skew. This implies that the Delta can be negative.

Reference:
Derman GS paper
de Weert "Exotic Options Trading" Ch. 23
Carr and Madan 2002 "Towards a Theory of Volatility Trading"
Coulombe et al 2008 "An Analytic Formula for the Delta of Variance Swap"