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Self-Financing Hedging Portfolios with Different Borrowing and Lending Rates

Article Quant Q&A · Author: iefjkfdhfure
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Summary

The document derives the value dynamics of a self-financing portfolio that holds an underlying asset and a money-market account. With a single risk-free rate, the cash position earns that rate, while the risky holding contributes the asset’s drift and stochastic movement. This provides a way to express the change in portfolio value from its holdings rather than treating it as an independent process.

It then allows borrowing and lending rates to differ. The cash balance is split into positive and negative parts using maximum and minimum functions, applying the lending rate to surplus cash and the borrowing rate to a deficit. This piecewise formulation clarifies how funding costs enter a hedging equation. The response states that proving the required self-financing condition involves a cumbersome differential calculation, but does not provide that derivation. The model also assumes the stated rates and portfolio setup; broader market frictions are not analyzed.

Key ideas

  • A self-financing portfolio’s value changes through its money-market and risky-asset holdings.
  • With one funding rate, the cash balance earns that rate regardless of whether it represents borrowing or lending.
  • Separate rates can be modeled by applying the lending rate to positive cash and the borrowing rate to negative cash.
  • The document does not show the full proof that the stated portfolio condition is self-financing.

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OptionsDerivatives pricingRisk management
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# Justify a backward differential equation


# Justify a backward differential equation












Regards of 4.5.1, how we get 4.5.5?

## Answer by Daneel Olivaw (score 2, accepted)

https://quant.stackexchange.com/a/49793

Note that for the replicating portfolio to be self-financing it suffices that (1): $$\lambda_t=\frac{V_t-h_tS_t}{B_t}$$ where I have changed the notation by designating by $B_t$ the money market account: $$B_t=B_0e^{rt}$$ Hence, because the portfolio is self-financing, its dynamics are: $$\begin{align} dV_t&=\left(\frac{V_t-h_tS_t}{B_t}\right)dB_t+h_tdS_t \\[5pt] &=r(V_t-h_tS_t)dt+(\mu h_tS_tdt+\sigma h_tS_tdW_t) \end{align}$$ Now, you can either be in a position to lend ($V_t>h_tS_t$) or borrow money. If the rates to lend $r_1>0$ and borrow $r_2<0$ are different then the equation above changes to: $$dV_t=r_1\max(V_t-h_tS_t,0)dt-r_2\min(V_t-h_tS_t,0)dt+h_tdS_t$$ You lend at $r_1$ if you have excess cash in your hedging account, namely the value of your hedge $h_tS_t$ is lower than the value of the derivative $V_t$, otherwise you borrow at $r_2$. Note that: $$\begin{align} V_t>h_tS_t \Leftrightarrow &\ r_1\max(V_t-h_tS_t,0)-r_2\min(V_t-h_tS_t,0) \\ &= r_1(V_t-h_tS_t)>0 \\[3pt] V_t<h_tS_t \Leftrightarrow &\ r_1\max(V_t-h_tS_t,0)-r_2\min(V_t-h_tS_t,0) \\ &= -r_2(V_t-h_tS_t)<0 \end{align}$$ Namely, the $\max$ and $\min$ functions allow to separate the excess/deficit of cash cases.

(1) Proving this is extremely cumbersome. Basically you need to prove that: $$B_td\lambda_t+d\lambda_tdB_t+S_tdh_t+dS_tdh_t=0$$ If I find the time (and energy) I'll try to post a derivation, otherwise you can check this answer to see how to do this.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.