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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):

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

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

[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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Following further analysis, the results derived by Burgard and Kjaer rely on the assumption that the funding of the asset $S$ and the counterparty bond $P_C$ is fully achieved through the repo market, whereas funding for one's own bonds is unsecured.

To make the derivation more rigorous, let us formally introduce into their model the following well-defined financing assets:

  1. A repurchase agreement on underlying $S$ with dynamics $\text{d}R_S(t)=q_SR_S(t)\text{d}t$;
  2. Another repo on the counterparty bond with dynamics $\text{d}R_C(t)=rR_C(t)\text{d}t$;
  3. A funding account $F>0$ with dynamics $\text{d}F(t)=r_F(t)F(t)\text{d}t$ for borrowing;
  4. A deposit account $D>0$ with dynamics $\text{d}D(t)=r(t)D(t)\text{d}t$ for lending.

We model the repos the same way as a bank account or a collateral account. This is consistent given a repo is another type of financing asset, and hence works much in the same way as e.g. a Treasury loan. However we should expect $q_S<r_F$ given a repo is to be interpreted as a secured loan, which is the assumption made by Piterbarg in [2]. On the other hand, Burgard and Kjaer assume the financing cost of $P_C$ is the risk-free rate, which seems odd given the bond is risky.

Their hedging portfolio, that is Equation (6), can then be rewritten: $$\begin{align} -\hat{V}&= \delta S+\alpha_BP_B+\alpha_CP_C+\beta \\ &= \delta S+\alpha_BP_B+\alpha_CP_C+(\beta_SR_S+\beta_CR_C+\beta_FF+\beta_DD) \end{align}$$ where their "units of cash" $\beta(t)$ are to be understood as the sum of all financing assets, each one held in units $\beta_X$. If we fully fund the asset $S$ and the bond $P_C$ through repos, then their values must cancel each other, that is: $$\beta_S=-\delta\frac{S}{R_S}, \qquad \beta_C=-\alpha_C\frac{P_C}{R_C}$$ We are then left with: $$-\hat{V}-\alpha_BP_B=\beta_FF+\beta_DD$$ We can set the units of the funding and deposit accounts to ensure the portfolio hedges $\hat{V}$. The former is drawn when cash needs to be borrowed, and vice versa for the latter. Hence: $$\beta_F=\left(\frac{-\hat{V}-\alpha_BP_B}{F}\right)^-, \qquad \beta_D=\left(\frac{-\hat{V}-\alpha_BP_B}{D}\right)^+$$ It then comes by positivity of $F$ and $D$: $$\begin{align} \beta_F\text{d}F+\beta_D\text{d}D &=\beta_Fr_FF\text{d}t+\beta_DrD\text{d}t \\ &=r_F(-\hat{V}-\alpha_BP_B)^-\text{d}t+r(-\hat{V}-\alpha_BP_B)^+\text{d}t \end{align}$$ which corresponds to their Equation (8).

References

[2] Piterbarg, Vladimir (2012). "Funding beyond discounting: collateral agreements and derivatives pricing", Risk.

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