Deriving Funding-Rate Discounting in Collateralized Derivatives Pricing

Summary

The document asks how to follow a step in Piterbarg’s treatment of collateral agreements and derivatives pricing. The sticking point is a change in discount factor between two equations, and the questioner suspects integration by parts and notes that the collateral-adjusted value may grow at the funding rate, but cannot reconstruct the derivation.

The reply suggests starting from the pricing partial differential equation preceding the disputed equation. It adds and subtracts the collateral rate multiplied by the derivative value, then rearranges the funding and collateral terms into a collateral-rate component plus a funding-versus-collateral spread applied to the uncollateralized portion. Applying the Feynman–Kac formula then yields the valuation representation. The exchange gives a high-level route, not the full algebra or assumptions, so readers need the paper’s equations and model setup to verify each step.

Key ideas

• Begin with the pricing partial differential equation that precedes the disputed equation.
• Add and subtract the collateral rate multiplied by the derivative value.
• Rearrange the terms to isolate the funding spread applied to value net of collateral.
• Use the Feynman–Kac formula to obtain the pricing representation.
• The reply sketches the method but does not show the complete derivation or assumptions.

Tags

# About a step in Piterbarg's paper "Funding Beyond Discounting"


# About a step in Piterbarg's paper "Funding Beyond Discounting"












I have a question about Piterbarg's 2010 paper "Funding Beyond Discounting: Collateral Agreements and Derivatives Pricing".

The author mentions that the equation (3.4) follows the rearranging of the terms in the equation (3.2) as shown below:

However, the change in the discount factor is a bit surprising. I appreciate that there is a suspicion of integration by parts here and also that $d((V(u)-C(u))e^{-\int_t^u r_F(v)dv})=0$ because $V-C$ grows at rate $r_F$, but I could not get the full path from (3.2) to (3.4). Any idea ?

## Answer by Quantuple (score 2)

https://quant.stackexchange.com/a/85479

The most straightforward way to see this IMO is to start from the pricing PDE just above (3.2).

You then add and remove $r_c(t) V(t)$ to the right hand side and re-arrange terms to obtain that it equivalently evaluates to $r_c(t) V(t) + (r_f(t) - r_c(t))(V(t)-C(t))$.

You then apply Feynman-Kac formula to conclude.

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