Pricing Capped Variance Swaps Beyond Vanilla Variance Replication
Summary
The document distinguishes the fair strike of a plain variance swap from the price of a variance swap whose realized variance is capped. A strip of vanilla options can be used to replicate the uncapped variance exposure under standard assumptions, so estimating that strike does not generally require a stochastic volatility model. The cap changes the problem: the payoff depends on the minimum of realized variance and a fixed ceiling, introducing optionality that the vanilla option strip alone does not price.
The answer explains that without traded variance or volatility options, pricing the capped contract requires a model, and any hedge or replication would likely involve dynamic trading in variance swaps or variance options. It cautions against calling the capped contract’s price a fair volatility because its payoff is not convex. Heston is presented as one possible model rather than a preferred choice; the response mentions the 3/2 model as another candidate. It provides no calibration, numerical comparison, or detailed pricing procedure, and the model-free comments assume the absence of jumps for the uncapped strike.
Key ideas
- A vanilla option strip can replicate the fair strike of an uncapped variance swap under standard assumptions.
- Capping realized variance adds optionality that prevents direct model-free pricing from the vanilla replication portfolio alone.
- Without traded variance or volatility options, valuing the capped payoff requires a model.
- A hedge for a capped variance swap may require dynamic trading in variance swaps or variance options.
- Heston is not the only model choice; the answer also points to the 3/2 volatility model.
Tags
Full text
# Capped Variance Swap // Fair volatility using replication portfolio
# 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
## Answer by user34971 (score 0, accepted)
https://quant.stackexchange.com/a/57290
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.