self-financing condition for funding and collateral

The self-financing condition for derivative portfolios with funding and collateral must properly distinguish between price processes, dividend processes, and gain processes (Duffie 2001). The gain process of an asset $A_i$ is $G_i^A:=P_i^A+D_i^A$ (cumulative price + cumulative dividends). A trading strategy $({\theta }_1,\dots ,{\theta }_n)$ is self-financing if its portfolio dividend is zero, i.e., $d{G}^{\Pi }=d{V}^{\Pi }$, which gives:

$d\left(\sum {\theta }_i{P}_i^A\right)=\sum {\theta }_{i}d{G}_i^A$

This is not the same as $d(\sum {\theta }_i{P}_i^A)=\sum {\theta }_{i}d{P}_i^A$, which would only hold when all individual asset dividend processes are null. Confusing these two conditions is a common error arising from misapplication of the stochastic Leibniz rule.

Brigo et al. (2012) showed that both Piterbarg (2010) and Burgard & Kjaer (2011) state the incorrect form of the self-financing condition, implicitly assuming that subportfolios (the equity position or the risky asset group) are self-financing on their own. However, the final pricing PDEs in both papers are correct because the error cancels in the derivation.

Key Details

• The error: writing $d(\Delta (t)S(t))=\Delta (t)dS(t)$ implies zero rebalancing — clearly wrong
• The correction: use gain processes $d{G}^A=d{P}^A+d{D}^A$ and require $d{G}^{\Pi }=\sum {\theta }_{i}d{G}_i^A$
• For collateralised derivatives: the collateral position has price $P=C$ and gain $dG=r_{C}Cdt$; the funding position has price $P=\alpha$ and gain $dG=r_F\alpha dt$
• The corrected self-financing condition recovers Piterbarg’s PDE: $\Delta (r_D-r_R)S+r_{C}C+r_F(V-C)=V_t+\frac{1}{2}{\sigma }^2{S}^2{V}_{SS}$

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