expected change in value of a derivative in a multicurve framework

Question

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 1

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 2

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 : $$\mathbb{E}[dS_t|\mathcal{F}_t]=r_F(t)S_tdt$$