# Capped Variance Swap // Fair volatility using replication portfolio

I know that the Heston volatility model should be the best approach for computing fair volatility on capped variance swap but is there a way to estimate it from replication portfolio?

What I call capped variance swap is a VS with a cap on the realized volatility in order to bound the maximum payoff for the buyer. Replication portfolio is the equivalent portfolio of vanilla options priced using BS model.

Thank you

A capped variance swap has terminal payoff $$N \left[ min(C,RV)- K^2_{var} \right]$$ where $$N$$ is the notional, $$C$$ is the cap, $$RV$$ is the realized variance over the life of the contract, and $$K^2_{var}$$ is the fair strike of an uncapped variance swap, i.e. a plain vanilla varswap.

When you say "fair volatility" I am assuming you mean how to compute $$K^2_{var}$$. For this you don't need Heston or any model (unless there are jumps). Googling "replicating variance swaps" will give you the answer, and this forum has some threads on it a well.

If you mean the price of the capped varswap with "fair volatility" then I am not sure fair volatility is the right name to call it as the payoff is not convex.

Also, there is clearly optionality in the term $$min(C,RV)$$ so there is no model-free way to price this if there are no variance/volatility options.

It is not immediately clear what the replicating portfolio is for a capped varswap as it's price will depend on the model used. To hedge/replicate a capped varswap you'll probably need to dynamically trade variance swaps or variance options.

Not sure Heston is the best choice of model for volatility derivatives even though it is 'nice' because it has some semi-analytical solution. For volatility derivatives you might want to look at pricing under e.g. the 3/2 model.

• Thank you, indeed, I'm talking about the $K_{var}^2$. My point is a VS struck at 20% with a cap at 40% should have a lower strike than the same VS with a cap at 60% as the payoff on the up side is more limited. I'll explore the 3/2 model and a model using variance call option struck at the cap. – Mamath Aug 12 '20 at 19:01