How Collateral, Stock Financing, and Funding Rates Affect Derivative Cash
Summary
The document asks how to interpret the cash-account changes in Piterbarg’s framework for pricing derivatives under collateral agreements. It distinguishes dividend income on a hedging stock position from the cost of financing that position, and asks why secured borrowing may be represented by a repo rate rather than an unsecured funding rate. It also raises whether collateral interest is truly a profit when the holder must pass that interest back to the counterparty.
The final question concerns funding the derivative value net of collateral: collateral received can offset the amount that must be financed. The document offers questions rather than a derivation or settled explanation, so it does not establish a single cash-account formula or resolve how the rates apply in practice. Its value is as a guide to the components that must be understood when modeling collateralized derivative funding; actual treatment depends on the agreement and financing setup.
Key ideas
- Dividend income on a hedging stock position is distinct from the cost of financing that position.
- Secured stock financing may be associated with a repo rate, while unsecured borrowing uses a different funding rate.
- Collateral may generate interest, but the holder may owe that interest to the counterparty.
- Collateral received can reduce the amount of derivative value that must be funded.
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# "Funding Beyond Discounting": understanding the rates impacting the cash account # "Funding Beyond Discounting": understanding the rates impacting the cash account I have a few questions about the various rates impacting the evolution of the cash account in Piterbarg's 2010 paper "Funding Beyond Discounting: Collateral Agreements and Derivatives Pricing". I guess the issue is related to understand exactly what happens in practice when we sell a product and hedge it. Let's look at the change between $t$ and $t+dt$ - Because we hold $\Delta(t)$ shares of $S$, I understand of course that we earn $r_D(t)\Delta(t)S(t)dt$ if $r_D$ is the dividend yield of the stock. - I am not sure I understand why the cost of holding the stock is $r_R$. Is it because we assume that we use those very same shares as guarantee when we borrow the money to buy these shares? That is, we borrow money to buy the stock, and instead of paying $r_F$ (unsecured), we directly agreee to give the shares we will buy as guarantee to secure this borrowing (thereby getting a cheaper rate $r_R$ associated with secured borrowing instead of unsecured rate $r_F$)? - Part of the cash comes from the collateral received from the counterparty ($C_t$), so it earns $r_C(t)dt$. I understand the principle (we get this collateral, we put it on a secured account because it is not "our money", so it earns risk-free rate), but usually, the interests earned on collateral need to be transferred to the counterparty, am I right? So, should we really consider this as a profit on our cash account? - Finally, I am not sure why we need to "fund $V-C$". I understand this at time $t=0$ if the bank buys the derivatives (pays $V$). Because in this case, should the bank receive $C$ from the collateral, it only needs to find the difference, ie, fund $V-C$ indeed. Any clarification about why this expression of $d\gamma$ is the correct one would be much appreciated.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.