Vega Notional / Variance Notional

The notional for a variance swap can be expressed either as a variance notional or a vega notional.

The variance notional represents the P&L per point difference between the strike squared (implied variance) and the subsequent realised variance.

 

Since most market participants are used to thinking in terms of volatility, trade size is typically expressed in vega notional. 

The vega notional represents the average P&L for a 1% change in volatility. 

The vega notional = variance notional * 2K 

 

The P&L of a long variance swap can be calculated as:  Vs formula

When RV is close to the strike, the P&L is close to the difference between IV and RV multiplied by the vega notional. 

The variance swap payout, expressed in vega notional, is locally linear around the strike. 

 

Vega and variance notionals

 

The P&L of a variance swap is often expressed in terms of vega notional.

For a vega notional of €100k, a gain of €500k is expressed as a profit of 5 vegas (i.e. 5 times the vega notional).  

Comments (2)

Campuzan
  • 1. Campuzan (link) | 29/05/2022
Hi,

I understand the need for adjustment to go from Vega notional to Variance notional but I don’t understand why we multiply Variance notional by 2K to get the vega notional.

Thank you.
maxxman100
  • maxxman100 | 11/06/2022
Bonjour Florian, C'est simplement la dérivée première de la fonction x^2 (passage de la volatilité à la variance). Cette formule de P&L est une approximation qui sera cohérente lorsque la volatilité réalisée sera relativement proche du strike (puisqu'on approxime une fonction convexe par une droite). A l'initiation du swap, expected(RV) = K --> RV = K dans la formule et le swap vaut 0 quand l'investisseur rentre dedans. Ensuite, tu peux partir de la formule du P&L sur base du vega notionnel. Imagine un mouvement de RV vers (RV + dRV). Comment sera impacté RV^2? Une bonne approximation est de que RV^2 passera à RV^2 + 2*RV. Prenons un exemple chiffré très simpliste (je grossis un peu le trait en ne parlant pas directement en % pour faciliter la compréhension). Swap en T0, RV = K = 22. Vega notionnel = 100K. Grosso modo pour un passage de la vol de 22 à 23, le VS gagne 100K. Ce passage de RV de 22 à 23 entraine un passage de RV^2 de 484 à 529. Cette modification pouvait être approximée initialement par 22^2 + 2*22 (2x étant la dérivée de X^2) = 484 + 44 = 528. Alors que RV a augmenté de 1, RV^2 approximé a augmenté de 44 (2*K puisque RV = K à la conception du produit). Le P&L du VS étant évidemment le même, l'ajustement est fait au niveau du Notionnel qui est divisé par ce 44 (2K).

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