Piterbarg in Funding beyond discounting: collateral agreements and derivatives pricing using Black Scholes derives the value of an option that is not perfectly collateralised as an FVA adjustment to the value of a perfectly collateralised option:
$$ V_t = E_t \left[ e^{-\int_t^Tr_C(u)du}V_T\right]-E_t \left[\int_t^Te^{-\int_t^ur_C(v)dv}\left( r_F(u)-r_C(u)\right) \left(V_u-C_u\right)du \right]$$
Is this equation applicable to all derivatives in general and not just options? I.E: would we be able to use the same formula for applying an FVA to an interest rate swap, commodity forward, asian option (we would assume that cleared markets for these derivatives exist and so $V_T$ is observable)? My understanding is that the idea of applying a FVA, which reflects the cost of hedging any general derivative on a cleared market, is applicable to any derivative that is not perfectly collateralised.
The source of this question is that I've seen this similar (if not equivalent) equation to Piterbarg's used to calculate FVA for swaps:
$$FVA = \int_{h=0}^{h=t}DF(h)ENE(h)*(FundingRate(h)- RiskFreeRrate(h))dh$$
(written here) which to me is equivalent to the second term in Piterbarg's equation above. The referenced post uses this (second) equation to calculate the FVA of a swap.
