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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.

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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.

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  • $\begingroup$ This makes sense, but how would you go about calculating $N_{variance}$ for the purpose of hedging the return that is known to pay off poorly when its variance is high? $\endgroup$ – Igor Pozdeev Apr 29 '18 at 23:27
  • $\begingroup$ As I have seen by portfolio manager, they form a view how much they want to participate in terms of vega as explained above. The basis risk is always high so I would not define a strict hedge ratio but rather think of mitigating losses with the variance swap. It depends on the shock that you want to hedge. If vol is low at the moment then think of scenarios of vol rising by 50% or 100% (doubling) and set the notional such that the gain fits your needs in Dollar terms. $\endgroup$ – Richard Apr 30 '18 at 16:16

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