I am rereading [1] and there is something I cannot get my head around this time.

In Section 3, page 6 of the paper, they derive the growth of the cash account when the hedging portfolio includes the underlying asset $S$, a zero-recovery bond from the seller $P_B$ and a zero-recovery bond from the counterparty $P_C$. When explaining the growth of the cash account $d\bar{\beta}_F(t)$, they write (my emphasis):

From the above analysis, any surplus cash held by the seller after the own bonds have been purchased must earn the risk-free rate $r$ [...]

They then derive the following differential equation for the cash account, Equation (8):


The other cash accounts growths, $d\bar{\beta}_S$ and $d\bar{\beta}_C$, account for the financing costs of the underlying asset $S$ and the counterparty bond $P_C$.

Why do the authors consider only the remaining cash after purchase of own bonds $P_B$ in Equation (8), instead of also accounting for the purchase/sale of $S$ and $P_C$?

It seems to me that the hedging portfolio incurs funding costs/benefits for $S$, $P_C$ and $P_B$ so that we should be looking at the residual funding cost/benefit including all purchases (given the cash account is the adjustment variable in the hedging portfolio which allows to equalise it to the derivative contract value at any time $t$).

So Equation (8) should instead be replaced by something along the lines of:

$$\begin{align} &d\bar{\beta}_B(t)=-\alpha_BrP_Bdt \\[2pt] &d\bar{\beta}_F(t)=\{r(-\hat{V}+\delta S+\alpha_CP_C+\alpha_BP_B)^++r_F(-\hat{V}+\delta S+\alpha_CP_C+\alpha_BP_B)^-\}dt \end{align}$$


[1] Burgard, Christoph and Kjaer, Martin (2011). "Partial Differential Equation Representations of Derivatives with Bilateral Counterparty Risk and Funding Costs", The Journal of Credit Risk, Vol. 7, No. 3, 1-19.


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