Relating Volatility Swap and Variance Swap Notionals

Summary

The document explains the approximate conversion between volatility swap notional and variance swap notional. A volatility swap payoff changes with volatility, while a variance swap payoff changes with squared volatility. Applying a local linear approximation to the relationship between volatility and variance gives the factor of twice the volatility level that appears in the conversion; expressed using volatility swap vega, the variance notional is approximately vega divided by twice the reference volatility.

The explanation uses the differential of squared volatility, whose first-order term is twice volatility times the change in volatility. For stochastic volatility, the squared-change term is nonzero and represents an additional convexity contribution. Therefore, the conversion is a local approximation and does not capture that higher-order effect exactly. The document provides the conceptual derivation but no numerical example, calibration procedure, or discussion of how the appropriate volatility reference level is selected.

Key ideas

• A variance swap payoff depends on squared volatility, while a volatility swap payoff depends on volatility itself.
• The local derivative of squared volatility with respect to volatility is twice the volatility level.
• This first-order relationship gives an approximate conversion between variance notional and volatility vega.
• The squared change in volatility contributes a convexity term for stochastic processes.
• The notional conversion is therefore approximate rather than an exact equivalence.

Tags

# relationship between notional amounts of volatility swaps and variance swaps


# relationship between notional amounts of volatility swaps and variance swaps












Taking volatility swap payoff as $$( \sigma_F - \sigma_S ) * volatility~notional $$

and Taking variance swap payoff as $$( \sigma_F^2 - \sigma_S^2 ) * variance~notional $$

I am trying to understand the origin of the relationship;

$$variance~notional = \frac{vega}{(2\sigma_s)}$$

I understand that vega is volatility notional as $\frac{\delta f} {\delta \sigma_F}$ is the change of payoff with respect to volatility point

I understand that variance notional is $\frac{\delta f} {\delta \sigma_F^2}$ as this is the change of payoff with respect to variance point

and $2\sigma_s$ is obviously the derivative of $\sigma_s^2$

## Answer by user34971 (score 3)

https://quant.stackexchange.com/a/45991

Look at the infinitesimal version of the change in variance: $$ d\sigma^2 = 2\sigma d\sigma + (d \sigma)^2 $$ The Ito term $(d\sigma)^2$ is non-zero for stochastic processes, and is of order $dt$, but if we ignore that then we get the approximate relation $$ d\sigma^2 \approx 2 \sigma d\sigma $$ which is where the factor $2 \sigma$ comes from in the translation between variance and vega notional.

As AlexC wrote it is based on a linearization of the P/L (the Ito term is a "convex" term if you will)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.