# Cash account growth in Burgard & Kjaer (2011)

I am rereading  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):

$$d\bar{\beta}_F(t)=\{r(-\hat{V}-\alpha_BP_B)^++r_F(-\hat{V}-\alpha_BP_B)^-\}dt$$

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}

References

 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.