Risk-Neutral Measures Under Collateral and Repo Funding
Summary
The document examines a modeling question raised by Piterbarg’s collateralized Black–Scholes framework. It distinguishes the OIS rate used for a collateral account from the repo rate used for financing a stock, and asks how the associated risk-neutral measures should be understood when the stock and bond are driven by Brownian motions.
The central issue is whether the stock’s martingale measure, associated with a money-market account growing at the repo rate, can be identified with the measure introduced using the OIS bond. The excerpt poses this as an interpretation question but provides no resolution or derivation. It therefore highlights a subtlety in measure changes and hedging under different funding rates, while leaving the mathematical conditions and answer unspecified.
Key ideas
- The framework distinguishes collateral remuneration at the OIS rate from stock financing at the repo rate.
- A stock’s martingale measure can be associated with a different funding account than the collateral bond’s measure.
- The excerpt asks whether both measures can be treated as the same on a sufficiently rich probability space.
- It poses the measure question but does not provide a proof or definitive answer.
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Full text
# Risk-neutral measure(s) under collateralization and funding costs
# Risk-neutral measure(s) under collateralization and funding costs
In Piterbarg (2010) the author presents a modified Black-Scholes model with an economy with a CSA-collateral (OIS) rate $r_C(t)$, a repo rate $r_R(t)$ and considers a derivative $V(t)$ written on a stock share $S(t)$ and collateralized by $C(t)$. The market has two assets, a zero-coupon bond yielding the OIS rate $P_C(t)$ and the stock, both following under their respective risk-neutral measures a Geometric Brownian Motion with drifts the OIS rate and the repo rate respectively (assuming no dividends) $-$ numbering as in paper:
$$ \begin{align} dP_C(t)/P_C(t) &= r_C(t)dt + \sigma_C(t)dW_C(t) \qquad \text{(1)} \\[6pt] dS(t)/S(t) &= r_R(t)dt+\sigma_S(t)dW_S(t) \qquad \text{(4)} \end{align}$$
The author defines one risk-neutral measure $P$ from the zero-coupon bond's Brownian Motion $W_C(t)$, however note that Piterbarg's hedging argument shows that the stock price is a martingale under the risk-neutral measure associated to the money market account with rate $r_R(t)$. Then the author states just below equation $\text{(4)}$:
> Note that if our probability space [generated by the stock's Brownian Motion $W_S(t)$] is rich enough, we can take it to be the same risk-neutral measure $P$ as used in $\text{(1)}$.
What might the author mean by that sentence?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.