When Piterbarg’s FVA Framework Applies Beyond Options

Summary

The document asks whether Piterbarg’s funding valuation adjustment (FVA) equation, derived for a partially collateralized option, can also price other derivatives such as swaps, commodity forwards, and Asian options. It presents the equation’s funding spread multiplied by the difference between derivative value and collateral, and compares this with an exposure-based integral used to estimate swap FVA.

The discussion frames the key issue as whether the adjustment depends on the instrument being an option or on the general cost of funding an uncollateralized exposure. It gives no derivation or worked example establishing the formula’s scope, and does not resolve the question. Its useful contribution is identifying a generalization question and a similarity between two FVA expressions. Applying either equation requires attention to its modeling assumptions, including collateral, funding rates, exposure dynamics, and valuation conventions.

Key ideas

• The question is whether an FVA expression derived for an option can extend to other derivative types.
• The equation’s adjustment integrates the funding spread against value in excess of collateral.
• An exposure-based integral is presented as a similar way to estimate swap FVA.
• The document raises the generalization but supplies no proof or definitive answer.

Tags

# Is Piterbarg's FVA equation generally applicable


# Is Piterbarg's FVA equation generally applicable












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.

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