Funding Account Dynamics with Asymmetric Rates and Stock Borrowing Fees
Summary
The document addresses how to express the funding account in a replication setup when borrowing and lending rates differ and short stock positions incur a borrowing fee. It decomposes the account into positive and negative funding balances, then assigns the applicable rate to each component. A separate adjustment captures the funding treatment of a short stock position, using the lending rate corrected by the funding rate.
The answer shows how this piecewise formulation matches a rate function that selects the lending or borrowing rate according to the sign of the balance, with separate cases for long and short stock positions. It also responds to whether a risk-neutral measure exists by citing work that extends the arbitrage-free framework to funding costs and collateralization. The post offers algebraic clarification, not a complete derivation of the pricing PDE or the conditions needed for a risk-neutral measure; it assumes positive stock prices and funding processes accruing continuously at their stated rates.
Key ideas
- Positive and negative funding balances accrue at different lending and borrowing rates.
- The funding account can be split into signed cash components and a short-stock funding adjustment.
- For a nonnegative stock position, the account dynamics reduce to a rate applied to the residual balance.
- For a short stock position, the stock borrowing fee adjustment enters separately.
- The cited framework treats arbitrage-free pricing with funding costs through an extended market model.
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# Question on derivation step in portfolio replication under different borrowing and lending rates
# Question on derivation step in portfolio replication under different borrowing and lending rates
I'm currently trying to understand the derivation of a pricing PDE on a european claim that considers stock lending fees: https://cs.uwaterloo.ca/~paforsyt/hjb.pdf
In Appendix A.2, the author talks about stock lending income being a complicated transaction (and I agree, I was not really able to connect the cited paper to what happens there) and mentions that the holder of the short position "will not receive the proceeds of the short sale, but rather effectively receives $r_l−r_f$". I can accept this conceptually.
However, I can't understand the evolution of the bank account in A.7). Could someone please explain to me how that one can be derived?
Also (somewhat unrelated): Is it possible that there is no longer a risk-neutral measure for this market?
## Answer by ir7 (score 3, accepted)
https://quant.stackexchange.com/a/63169
Noting that $$ B= V -\alpha S = V - (\alpha S)^+ + (\alpha S)^- $$ $$ = (V - (\alpha S)^+)^+ - (V - (\alpha S)^+)^- + (\alpha S)^-,$$ a clearer way to write the dynamics of the funding costs (funding account by funding account, the borrowing one, the lending one, and the one that funds at lending rate corrected by stock borrowing fees) is:
$$dB = r^l (V - (\alpha S)^+)^+ dt - r^b (V-(\alpha S)^+)^- dt + (r^l - r^f) (\alpha S)^-dt , $$
where $$ x^+ = x\cdot 1_{x\geq 0}, \: \: \: x^- = -x\cdot 1_{x< 0}, \: \: \: x=x^+ - x^-.$$
($S$ is assumed positive everywhere. All funding processes $F$ are assumed of the form $dF = rF dt$.)
We can then show that this is consistent with the paper's findings, expressed in terms of the function $\rho$, $$\rho(x) = r^l\cdot 1_{x\geq 0} + r^b \cdot 1_{x<0}.$$
Indeed, if $\alpha \geq 0$, then
$$ dB = r^l (V-\alpha S)^+ dt - r^b(V- \alpha S)^- dt $$ $$= \rho(V - \alpha S)(V - \alpha S) dt.$$
If $\alpha < 0$, then
$$ dB = r^l V^+dt - r^b V^- dt - (r^l-r^f)\alpha S dt $$ $$= (\rho(V)V - (r^l-r^f)\alpha S)dt.$$
Regarding your second question (that is related to the first), Bielecki and Rutkowski set up an "arbitrage-free model, by proposing an essential extension of the classic definition, and" and they "provide sufficient conditions for the no-arbitrage property of a market model under alternative assumptions about trading and netting" in their paper Valuation and Hedging of Contracts with Funding Costs and Collateralization.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.