Introduction

For a given European option maturing at time T with a strike K, the forward volatility representation is the volatility input   that is necessary to input in a Black formula in order to match the market price.
To be more precise, if we write the forward as follows , then the non discounted price of the vanilla option is equal to:

This representation is the common market convention. It is so principally because it only assumes that the forward is a lognormal variable with a given volatility.
Other financial operators use a different approach to keep track of their volatility surfaces. The assumption behind is that only the “capifactor “ part of the forward is volatile but not the “capidiv” part. Under these assumptions the pricing formula is also simple and the non discounted price of the call option is given by:

We note  the spot volatility as the input in the previous formula in order to match the market price.

Transformation

 To derive the first guess solution  spot volatility as a function of the forward volatility  we use the fact that in the spot volatility representation the forward is the difference of a volatile lognormal term which mean value is the “capifactor” times the initial spot and a non volatile term which is the “capidiv”.

Therefore, the implied volatility for a strike K is given by the following formula, using the Jerome Busca approximation: