Deriving Expected Derivative Value Changes Under Multicurve Funding
Summary
The document explains how a derivative’s expected value change follows from its self-financing replication in a framework with different rates for unsecured funding and collateral. The portfolio holds a delta position in the underlying and a cash account; the cash earns the collateral rate on posted collateral and the funding rate on the remaining balance. Combining the cash-account dynamics with the underlying’s price dynamics cancels the underlying’s drift exposure, leaving a drift term based on the derivative value and collateral, plus a stochastic market-risk term. Taking the conditional expectation removes that stochastic term and gives the stated expected change.
A second answer gives a compact financing decomposition and invokes risk-neutral pricing. The explanation relies on the paper’s setup, including its rate conventions, self-financing assumption, and treatment of collateral. It is a derivation of the model relation, not empirical evidence or a general prescription for choosing funding rates or collateral amounts.
Key ideas
- A replicating portfolio combines a delta position in the underlying with a cash account.
- Collateral and unsecured balances accrue at different rates in the multicurve setup.
- The underlying’s drift term cancels against the corresponding cash-account term.
- The remaining stochastic exposure has zero conditional expectation under the stated framework.
- The resulting expected value change depends on funding, collateral, and the derivative’s current value.
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# expected change in value of a derivative in a multicurve framework
# expected change in value of a derivative in a multicurve framework
I'm reading Piterbarg paper, "Funding beyond discounting: collateral agreements and derivatives pricing." and have a question about equation $(6)$. There he says that for a derivative we have
$$E_t[dV_t]=(r_F(t)V(t)-(r_F(t)-r_C(t))C(t))dt = (r_F(t)V(t)-s_F(t)C(t))dt$$
where $C(t)$ amount in collateral, $r_F$ short rate for unsecured funding, $r_C$ the short rate for risk free rate which corresponds to the safest available collateral, cash and $s_F(t)$ is the funding spread $r_F-r_C$. Why is the above first formula for the expected change in the derivative true?
## Answer by Gordon (score 1, accepted)
https://quant.stackexchange.com/a/25951
From $(2)$ of Piterbarg, \begin{align*} V(t) = \Delta (t) S(t) + \gamma(t), \end{align*} where $\Delta (t)= \frac{\partial V(t)}{\partial S}$, and $\gamma(t)$ is the cash account that satisfies \begin{align*} d\gamma(t) &= \big[r_C(t) C(t) + r_F(t)(V(t)-C(t))-(r_R(t)-r_D(t))\Delta(t)S(t) \big]dt\\ &=\big[r_F(t)V(t) + (r_C(t)-r_F(t)) C(t)-(r_R(t)-r_D(t))\Delta(t)S(t) \big]dt. \end{align*} Moreover, based on Equation $(4)$ in the paper, \begin{align*} dS(t)/S(t) = (r_R(t)-r_D(t))dt + \sigma_S(t) dW_S(t). \end{align*} Then, from the self-financing condition, \begin{align*} dV(t) &= \Delta (t) dS(t) + d\gamma(t)\\ &=\big[r_F(t)V(t) + (r_C(t)-r_F(t)) C(t)\big]dt + \Delta (t)S(t)\sigma_S(t) dW_S(t). \tag{E1} \end{align*} It is obvious now that \begin{align*} E_t(dV(t)) = \big[r_F(t)V(t) + (r_C(t)-r_F(t)) C(t)\big]dt. \end{align*}
Note that Formulas $(3)$ and $(5)$ in Piterbarg can also be derived directly from Equation $({\rm E}1)$ above.
## Answer by M. Jeunesse (score 2)
https://quant.stackexchange.com/a/25950
self financed portfolio will give you :
$$ dV_t = r_F(t) \underbrace{(V(t)-C(t) - \Delta S_t )}_{\text{cash position}} dt + r_C(t) \underbrace{C(t)}_{\text{posted collateral}} dt + \underbrace{\Delta dS_t}_{\text{market move}} $$
then you retrieve his equation using that under risk-neutral measure : $$\mathbb{E}[dS_t|\mathcal{F}_t]=r_F(t)S_tdt$$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.