Rewriting Funding and Collateral Pricing Under Alternative Discounting
Summary
The post asks how to transform one risk-neutral pricing expression for a derivative with funding and collateral rates into another expression using collateral-rate discounting and an adjustment based on the difference between funding and collateral rates. The accepted response derives both representations from the same self-financing valuation dynamics. It first writes the derivative value’s drift in terms of funding costs and collateral, then applies an integrating discount factor and takes conditional expectations.
For the second form, the derivation rearranges the drift to separate collateral-rate growth from the funding-minus-collateral spread applied to the unsecured value, meaning value net of collateral. Discounting at the collateral rate and taking conditional expectations yields the target expression. This illustrates why the two formulas can be equivalent under the stated model setup. The result relies on the paper’s assumptions and notation, including its dynamics and trading strategy; it is not a general pricing rule for every collateral agreement.
Key ideas
- The pricing expressions can be related through the derivative value’s self-financing dynamics.
- Rearranging the drift isolates collateral-rate growth and a funding spread on value net of collateral.
- Applying an integrating discount factor converts the dynamics into a conditional-expectation equation.
- Discounting at the collateral rate produces an adjustment involving the funding-collateral rate difference.
- The derivation depends on the assumptions and dynamics of the referenced model.
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Full text
# Risk-neutral expectation equation with collateral and funding costs
# Risk-neutral expectation equation with collateral and funding costs
I am looking at a paper by V. Piterbarg, Funding beyond discounting: collateral agreements and derivatives pricing, that you can download on the following link, in which the author adapts the Black-Scholes pricing framework to introduce collateral and funding at a non-risk-free rate.
Letting $V_t \equiv V(t)$ and $C_t \equiv C(t)$, I am having trouble to go from equation $(3)$...
$$ V_t = E_t \left[ e^{-\int_t^Tr_F(u)du}V_T+\int_t^Te^{-\int_t^ur_F(v)dv}\left( r_F(u)-r_C(u)\right) C_u \ du \right]$$
... to equation $(5)$:
$$ V_t = E_t \left[ e^{-\int_t^Tr_C(u)du}V_T\right]-E_t \left[\int_t^Te^{-\int_t^ur_C(v)dv}\left( r_F(u)-r_C(u)\right) \left(V_u-C_u\right)du \right]$$
According to the author, to go from $(3)$ to $(5)$ we only need to "rearrange terms".
Can anybody show how to go from one to the other?
## Answer by Gordon (score 8, accepted)
https://quant.stackexchange.com/a/30436
One derivation is to replace $V_u$ in Equation $(5)$ using the expression given by Equation $(3)$ and then work out to reach $(5)$; see Appendix A in this paper for more details. Here, we provide another derivation. See also this question.
We recall that, 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{*} \end{align*}
From $(*)$, \begin{align*} d\left(e^{-\int_0^t r_F(v)dv}V_t \right) &=-r_F(t)e^{-\int_0^t r_F(v)dv}V_tdt + e^{-\int_0^t r_F(v)dv}dV_t\\ &=e^{-\int_0^t r_F(v)dv}\big[(r_C(t)-r_F(t)) C(t)dt + \Delta (t)S(t)\sigma_S(t) dW_S(t)\big]. \end{align*} Therefore, \begin{align*} e^{-\int_0^T r_F(v)dv}V_T-e^{-\int_0^t r_F(v)dv}V_t &=\int_t^Te^{-\int_0^u r_F(v)dv}\big[(r_C(u)-r_F(u)) C(u)du\\ &\qquad + \int_t^T\Delta (u)S(u)\sigma_S(u) dW_S(u). \end{align*} Taking conditional expectation with respect to $\mathscr{F}_t$ on both sides, we obtain that \begin{align*} E_t\left(e^{-\int_0^T r_F(v)dv}V_T \right)-e^{-\int_0^t r_F(v)dv}V_t &=E_t\left(\int_t^Te^{-\int_0^u r_F(v)dv}\big[(r_C(u)-r_F(u)) C(u)du\right), \end{align*} which leads to Equation $(3)$ in Piterbarg, that is, \begin{align*} V_t &= E_t\left(e^{-\int_t^T r_F(v)dv}V_T + \int_t^Te^{-\int_t^u r_F(v)dv}\big[(r_F(u)-r_C(u)) C(u)du\right)\tag{3} \end{align*}
$$$$ To derive Equation $(5)$, we note that, from $(*)$ above, by rearranging terms, \begin{align*} dV_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)\\ &=\big[r_C(t)V_t + (r_F(t)-r_C(t))(V_t -C(t))\big]dt + \Delta (t)S(t)\sigma_S(t) dW_S(t).\tag{**} \end{align*} As above, \begin{align*} d\left(e^{-\int_0^t r_C(v)dv}V_t \right) &=-r_C(t)e^{-\int_0^t r_C(v)dv}V_tdt + e^{-\int_0^t r_C(v)dv}dV_t\\ &=e^{-\int_0^t r_C(v)dv}\big[(r_F(t)-r_C(t))(V_t -C(t))dt + \Delta (t)S(t)\sigma_S(t) dW_S(t) \big], \end{align*} and, consequently, \begin{align*} E_t\left(e^{-\int_0^T r_C(v)dv}V_T \right) -e^{-\int_0^t r_C(v)dv}V_t&=E_t\left( \int_t^Te^{-\int_0^u r_C(v)dv}\big[(r_F(u)-r_C(u))(V_u -C(u))du\right), \end{align*} which leads to Equation $(5)$ in Piterbarg immediately, that is, \begin{align*} V_t =E_t\left(e^{-\int_t^T r_C(v)dv}V_T \right) - E_t\left(\int_t^Te^{-\int_t^u r_C(v)dv}\big[(r_F(u)-r_C(u))(V_u -C(u))du \right). \tag{5} \end{align*}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.