Returns on an asset are negatively correlated with own variance, and I would like to set up a hedge with a variance swap (no options are traded). I need to decide on the notional of the swap: any ideas how I could calculate it?
EDIT I do not want to trade variance, I want to (imperfectly) hedge the part of the return that is by my gut feeling low when the return variance is high.
My attempt: I will try to set up a \$1 portfolio of the asset and variance swap that has return $r^p$:
$$r^p_{t+1} = r_{t+1} +s_{t+1}, $$
where $r$ is the asset return and $s$ is the payoff of the variance swap:
$$ s_{t+1} = N_t ( rv_{t+1} - iv_{t} ), $$ where $N_t$ is the notional, $rv_{t+1}$ is the realized variance in month $t+1$, $iv_t$ is the swap price. I think I ultimately need the following to hold:
$$ E \frac{\partial r^p_{t+1}}{\partial \sigma^2_{t+1}} = E \frac{\partial r_{t+1}}{\partial \sigma^2_{t+1}} + N_t = 0. $$
I thought of modelling the dependence between $r$ and $\sigma^2$ as a GARCH-in-mean process:
$$ r_{t+1} = \alpha + \beta \color{red}{\sigma^2_{t+1}} + \varepsilon_{t+1} $$ $$ \varepsilon_{t+1} \sim N(0, \color{red}{\sigma^2_{t+1}}) $$ $$ \color{red}{\sigma^2_{t+1}} = \omega + \theta_1 \varepsilon_{t}^2 + \theta_2 \sigma^2_{t}, $$
from where it would follow that:
$$ E \frac{\partial r_{t+1}}{\partial \sigma^2_{t+1}} = \beta = -N_t. $$ What would you say? Thanks.
