Converting Variance Swap Notional into Vega Exposure

# Hedging with variance swaps: how to calculate the notional


# Hedging with variance swaps: how to calculate the notional












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.

## Answer by Richi Wa (score 2)

https://quant.stackexchange.com/a/39503

Variance swaps pay the difference in variance. However, people tend to think in volatility.

You usually want some effect on your portfolio if vol goes up by $x\%$. Changing implied volatility is called vega.

Thus people use the vega notional $N_{vega}$: $$ N_{vega} = 2 K N_{variance}, $$ where $K$ is the current strike in vol terms.

If you think that the pay-off of the variance swap is:

$$ N_{variance} (\sigma^2 - K^2), $$ where $\sigma^2$ is the current vola squared, then the position changes in the following way if $\sigma$ changes by $\epsilon$:

$$ N_{variance} ( (\sigma+\epsilon)^2 - K^2) = N_{variance} ( (\sigma^2 +2\sigma\epsilon +\epsilon^2 - K^2) $$ and if $\epsilon$ is small this is roughly the same as $$ N_{variance} ( (\sigma+\epsilon)^2 - K^2) = N_{variance} ( (\sigma^2 +2\sigma\epsilon - K^2). $$ Thus the increase of the contract is roughly $$ N_{variance} 2\sigma\epsilon $$ thus a change in vol of $\epsilon$ gives you $N_{variance} 2\sigma = N_{vol}$ Dollars.