Separating Price and Gain Processes for Repo, Collateral, and Funding
Summary
The document explains the distinction between an asset’s price process and its gain process in a funding and collateral framework. For an asset, the gain process combines price changes with dividends or other cash flows. The question concerns how this definition applies to a repo contract, collateral, and a funding account, and why the stated gain increments do not always include a separate dividend term.
The answer resolves the issue by describing how each position is represented. The repo is treated as a pure dividend process: its price is zero, while the gain reflects the underlying asset’s price change adjusted for the repo funding cost. Collateral and funding are represented as price processes that accrue interest, with no separate dividend process. Their price increments therefore equal their gain increments. This framework clarifies the bookkeeping conventions used in the cited paper, but the document does not derive the financing setup or explain how to apply it to other contracts.
Key ideas
- A gain process combines an asset’s price process and dividend process.
- The repo is modeled as a zero-price position whose gain is represented through cash flows.
- Collateral and funding are modeled as price processes that accrue interest without separate dividends.
- The process definitions determine whether interest appears in the price increment or dividend increment.
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# Dividend and Gain process in paper by Brigo, Buescu, Pallavicini and Liu
# Dividend and Gain process in paper by Brigo, Buescu, Pallavicini and Liu
In the paper Illustrating a problem in the self-financing condition in two 2010-2011 papers on funding, collateral and discounting (2012) (link) from Brigo, Buescu, Pallavicini and Liu, the above authors give a detailed description on how to work with the gain process, and how it differs from the price process.
Namely, given an asset $A$, we identify two processes with it, $P^{A}$ (price process) and $D^{A}$ (dividend process). The Gain process is then defined by $G^{A}=P^{A} + D^{A}$. Further ahead, they apply this definition to particular assets (I modified slightly their notation): $A_{1}$ a repo contract (with risky underlying asset having price $S_{t}$), $A_{2}$ a collateral position (with price being $C$) and $A_{3}$ a funding position, whose price is denoted by $\alpha$.
Now, the authors identify $P^{A_{1}} = 0$ (I guess here the repo rate chosen is such that the price of the contract is zero), $P^{A_{2}}=C$ and $P^{A_{3}}=\alpha$.
Then they claim: and the gain processes $dG^{A_{1}} = dS + (r_{D}-r_{R})Sdt$, $dG^{A_{2}}=r_{C}Cdt$, $dG^{A_{3}}=r_{F}\alpha dt$, where $r_{*}$ is some rate (not relevant for the question).
The question is, how do they get this? I thought, for example for $A_{3}$, that $dG^{A_{3}} = dP^{A_{3}}+dD^{A_{3}} = d\alpha + r_{F}\alpha dt$.
I hope my question is clear.
## Answer by Daneel Olivaw (score 1)
https://quant.stackexchange.com/a/69919
By definition of the authors, repos are defined as pure dividend processes where the price performance of $S$ is continuously exchanged against a funding cost $r_RS$, whereas collateral and the funding account are pure price processes which continuously earn and compound interest: \begin{align} &A_1:\begin{cases} dP^{A_1}=0, \quad P^{A_1}_0=0 \\ dD^{A_1}=dS+(r_D-r_R)Sdt,\quad D^{A_1}_0=0 \\ dG^{A_1}=dP^{A_1}+dD^{A_1}=dS+(r_D-r_R)Sdt,\quad G^{A_1}_0=P^{A_1}_0+D^{A_1}_0=0 \end{cases}\tag{1} \\[10pt] &A_2:\begin{cases} dP^{A_2}=r_CCdt, \quad P^{A_2}_0=C_0 \\ dD^{A_2}=0,\quad D^{A_2}_0=0 \\ dG^{A_2}=r_CCdt,\quad G^{A_2}_0=P_0^{A_2} \end{cases}\tag{2} \\[10pt] &A_3:\begin{cases} dP^{A_3}=r_F\alpha dt, \quad P^{A_3}_0=\alpha_0 \\ dD^{A_3}=0,\quad D^{A_3}_0=0 \\ dG^{A_3}=r_F\alpha dt,\quad G^{A_3}_0=P_0^{A_3} \end{cases}\tag{3} \end{align}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.