Collateralized Discounting

Introduction

In the past, standard derivatives pricing theory assumed the existence of a risk-free rate for derivatives discounting. Until the global financial crisis(GFC), this assumption worked well, but has since been replaced by Collateral adjusted valuation(CAV). Collateralized discounting is standard practice on derivatives trading desks.

A risk-neutral measure can still be defined and much of the pricing technology developed in the traditional setting can be reused.

The theoretical foundations of collateralized discounting are the papers Cooking with collateral and Funding beyond Discounting by Piterbarg. I summarize the main arguments here.

Pricing under collateral

We replicate the derivative worth $V(t)$, by an amount $\theta_1$ of the underlying $X$, an amount $\theta_2$ of funding account $B_f(t)$ and an amount $\theta_3$ of collateral account $B_c(t)$. The value of the portfolio at time $t$ is:

\[ \begin{align*} V(t) = \theta_1(t) X(t) + \theta_2(t) B_f(t) + \theta_3(t) B_c(t) \end{align*} \tag{1}\]

The self-financing assumption implies that:

\[ dV(t) = \theta_1 dX_t + \theta_2 dB_f(t) + \theta_3 dB_c(t) \tag{2}\]

Assume that the dynamics of the three assets is as follows:

\[ \begin{align*} dX(t) &= \mu^{\mathbb{P}}(t) X(t) dt + \sigma(t)X(t)dW^\mathbb{P}(t)\\ dB_f(t) &= r_f(t)B_f(t) dt\\ dB_c(t) &= r_c(t)B_c(t) dt \end{align*} \tag{3}\]

The derivative’s price dynamics $dV(t,X_t)$ is obtained by the Ito’s lemma as:

\[ \begin{aligned} dV( t,X) & =\frac{\partial V}{\partial t} dt+\frac{\partial V}{\partial X} dX_{t} +\frac{1}{2}\frac{\partial ^{2} V}{\partial X^{2}}( dX_{t})^{2}\\ & =\frac{\partial V}{\partial t} dt+\frac{\partial V}{\partial X}\left( \mu ^{\mathbb{P}} X_{t} dt+\sigma _{t} X_{t} dW_{t}^{\mathbb{P}}\right) +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2} dt\\ & =\left(\frac{\partial V}{\partial t} +\mu ^{\mathbb{P}} X_{t}\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt+\sigma _{t} X_{t}\frac{\partial V}{\partial X} dW_{t}^{\mathbb{P}} \end{aligned} \tag{4}\]

Substituting Equation 3 and Equation 4 in Equation 2, we have:

\[ \begin{aligned} \left(\frac{\partial V}{\partial t} +\mu ^{\mathbb{P}} X_{t}\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt+\sigma _{t} X_{t}\frac{\partial V}{\partial X} dW_{t}^{\mathbb{P}} & =\theta _{1}\left( \mu ^{\mathbb{P}} X_{t} dt+\sigma _{t} X_{t} dW_{t}^{\mathbb{P}}\right)\\ & +\theta _{2}( r_{f}( t) B_{f}( t) dt) +\theta _{3}( r_{c}( t) B_{c}( t) dt) \end{aligned} \]

The perfect collateral condition implies that the collateral held at any time equals the mark-to-market(MtM) value of the derivative. So, $B_c(t) = V(t)$. So, we have:

Setting $\theta_3(t) = 1$ in Equation 1, we get :

\[ \begin{align*} \theta_2(t)B_f(t) = \theta_1(t)X(t) \end{align*} \tag{6}\]

Substituting Equation 6 in Equation 5, we get:

\[ \begin{align*} \left(\frac{\partial V}{\partial t} +\mu ^{\mathbb{P}} X_{t}\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt+\sigma _{t} X_{t}\frac{\partial V}{\partial X} dW_{t}^{\mathbb{P}} & =\theta _{1}\left( \mu ^{\mathbb{P}} X_{t} dt+\sigma _{t} X_{t} dW_{t}^{\mathbb{P}}\right)\\ & -\theta _{1}( r_{f}( t) X( t) dt) +( r_{c}( t) V( t) dt) \end{align*} \]

Re-arranging the terms, we get:

\[ \begin{align*} \left(\frac{\partial V}{\partial t} -\mu ^{\mathbb{P}} X_{t}\left( \theta _{1} -\frac{\partial V}{\partial X}\right) +r_{f}( t) \theta _{1}( t) X( t) +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt & =\sigma _{t} X_{t}\left( \theta _{1}( t) -\frac{\partial V}{\partial X}\right) dW_{t}^{\mathbb{P}}\\ & +( r_{c}( t) V( t) dt) \end{align*} \]

Setting $\theta_(t) = \frac{\partial V(t)}{\partial X}$, we get:

\[ \begin{align*} \left(\frac{\partial V}{\partial t} +r_{f}( t) X( t)\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt & =r_{c}( t) V( t) dt \end{align*} \]

or equivalently:

This is the pricing PDE. Applying Feynman-Kac, the solution to this PDE for the boundary condition:

\[ V(T,x) = g(x) \]

has the stochastic representation:

\[ V(t,x) = \mathbb{E}^{\mathbb{Q}^f}[e^{-\int_t^T r_c(t) dt } g(X_T)|\mathcal{F}_t] \tag{8}\]

where $\mathbb{Q}^f$ is the measure associated with the funding account numeraire $B_f(t)$ and the underlying risky asset has the dynamics:

\[ dX_t = r_f(t)X(t)dt+ \sigma(t)X(t)dW^{\mathbb{Q}^f}(t) \]

References

Cooking with collateral, Vladimir Piterbarg
Funding beyond Discounting, Vladimir Piterbarg

--- title: "Collateralized Discounting" author: "Quasar" date: "2025-01-27" categories: [Back to the basics] image: "image.jpg" toc: true toc-depth: 3 --- # Collateralized Discounting ## Introduction In the past, standard derivatives pricing theory assumed the existence of a **risk-free rate** for derivatives discounting. Until the global financial crisis(GFC), this assumption worked well, but has since been replaced by *Collateral adjusted valuation*(CAV). **Collateralized discounting** is standard practice on derivatives trading desks. A risk-neutral measure can still be defined and much of the pricing technology developed in the traditional setting can be reused. The theoretical foundations of collateralized discounting are the papers [Cooking with collateral](http://janroman.dhis.org/finance/Kreditrisk/Risk_Cooking%20with%20collateralpdf.pdf) and [Funding beyond Discounting](https://www.researchgate.net/profile/Vladimir-Piterbarg-2/publication/284078682_Funding_beyond_Discounting_Collateral_Agreements_and_Derivatives_Pricing/links/614c85a9a595d06017e55b0e/Funding-beyond-Discounting-Collateral-Agreements-and-Derivatives-Pricing.pdf) by Piterbarg. I summarize the main arguments here. ## Pricing under collateral We replicate the derivative worth $V(t)$, by an amount $\theta_1$ of the underlying $X$, an amount $\theta_2$ of funding account $B_f(t)$ and an amount $\theta_3$ of collateral account $B_c(t)$. The value of the portfolio at time $t$ is: $$ \begin{align*} V(t) = \theta_1(t) X(t) + \theta_2(t) B_f(t) + \theta_3(t) B_c(t) \end{align*} $${#eq-replicating-portfolio} The self-financing assumption implies that: $$ dV(t) = \theta_1 dX_t + \theta_2 dB_f(t) + \theta_3 dB_c(t) $${#eq-derivative-of-portfolio-price} Assume that the dynamics of the three assets is as follows: $$ \begin{align*} dX(t) &= \mu^{\mathbb{P}}(t) X(t) dt + \sigma(t)X(t)dW^\mathbb{P}(t)\\ dB_f(t) &= r_f(t)B_f(t) dt\\ dB_c(t) &= r_c(t)B_c(t) dt \end{align*} $${#eq-dynamics-of-asset-prices} The derivative's price dynamics $dV(t,X_t)$ is obtained by the Ito's lemma as: $$ \begin{aligned} dV( t,X) & =\frac{\partial V}{\partial t} dt+\frac{\partial V}{\partial X} dX_{t} +\frac{1}{2}\frac{\partial ^{2} V}{\partial X^{2}}( dX_{t})^{2}\\ & =\frac{\partial V}{\partial t} dt+\frac{\partial V}{\partial X}\left( \mu ^{\mathbb{P}} X_{t} dt+\sigma _{t} X_{t} dW_{t}^{\mathbb{P}}\right) +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2} dt\\ & =\left(\frac{\partial V}{\partial t} +\mu ^{\mathbb{P}} X_{t}\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt+\sigma _{t} X_{t}\frac{\partial V}{\partial X} dW_{t}^{\mathbb{P}} \end{aligned} $${#eq-dynamics-of-the-derivative-price} Substituting @eq-dynamics-of-asset-prices and @eq-dynamics-of-the-derivative-price in @eq-derivative-of-portfolio-price, we have: $$ \begin{aligned} \left(\frac{\partial V}{\partial t} +\mu ^{\mathbb{P}} X_{t}\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt+\sigma _{t} X_{t}\frac{\partial V}{\partial X} dW_{t}^{\mathbb{P}} & =\theta _{1}\left( \mu ^{\mathbb{P}} X_{t} dt+\sigma _{t} X_{t} dW_{t}^{\mathbb{P}}\right)\\ & +\theta _{2}( r_{f}( t) B_{f}( t) dt) +\theta _{3}( r_{c}( t) B_{c}( t) dt) \end{aligned} $$ The perfect collateral condition implies that the collateral held at any time equals the mark-to-market(MtM) value of the derivative. So, $B_c(t) = V(t)$. So, we have: $$ \begin{aligned} \left(\frac{\partial V}{\partial t} +\mu ^{\mathbb{P}} X_{t}\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt+\sigma _{t} X_{t}\frac{\partial V}{\partial X} dW_{t}^{\mathbb{P}} & =\theta _{1}\left( \mu ^{\mathbb{P}} X_{t} dt+\sigma _{t} X_{t} dW_{t}^{\mathbb{P}}\right)\\ & +\theta _{2}( r_{f}( t) B_{f}( t) dt) +\theta _{3}( r_{c}( t) V( t) dt) \end{aligned} $${#eq-price-pde} Setting $\theta_3(t) = 1$ in @eq-replicating-portfolio, we get : $$ \begin{align*} \theta_2(t)B_f(t) = \theta_1(t)X(t) \end{align*} $${#eq-rel-between-theta1-and-theta2} Substituting @eq-rel-between-theta1-and-theta2 in @eq-price-pde, we get: $$ \begin{align*} \left(\frac{\partial V}{\partial t} +\mu ^{\mathbb{P}} X_{t}\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt+\sigma _{t} X_{t}\frac{\partial V}{\partial X} dW_{t}^{\mathbb{P}} & =\theta _{1}\left( \mu ^{\mathbb{P}} X_{t} dt+\sigma _{t} X_{t} dW_{t}^{\mathbb{P}}\right)\\ & -\theta _{1}( r_{f}( t) X( t) dt) +( r_{c}( t) V( t) dt) \end{align*} $$ Re-arranging the terms, we get: $$ \begin{align*} \left(\frac{\partial V}{\partial t} -\mu ^{\mathbb{P}} X_{t}\left( \theta _{1} -\frac{\partial V}{\partial X}\right) +r_{f}( t) \theta _{1}( t) X( t) +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt & =\sigma _{t} X_{t}\left( \theta _{1}( t) -\frac{\partial V}{\partial X}\right) dW_{t}^{\mathbb{P}}\\ & +( r_{c}( t) V( t) dt) \end{align*} $$ Setting $\theta_(t) = \frac{\partial V(t)}{\partial X}$, we get: $$ \begin{align*} \left(\frac{\partial V}{\partial t} +r_{f}( t) X( t)\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) dt & =r_{c}( t) V( t) dt \end{align*} $$ or equivalently: $$ \begin{align*} \left(\frac{\partial V}{\partial t} +r_{f}( t) X( t)\frac{\partial V}{\partial X} +\frac{1}{2} \sigma _{t}^{2} X_{t}^{2}\frac{\partial ^{2} V}{\partial X^{2}}\right) & =r_{c}( t) V( t) \end{align*} $${#eq-pricing-pde-collateralized-derivative} This is the pricing PDE. Applying [Feynman-Kac](https://quantdev.blog/posts/the_markov_property/#the-feynman-kac-formula), the solution to this PDE for the boundary condition: $$ V(T,x) = g(x) $$ has the stochastic representation: $$ V(t,x) = \mathbb{E}^{\mathbb{Q}^f}[e^{-\int_t^T r_c(t) dt } g(X_T)|\mathcal{F}_t] $${#eq-pricing-formula} where $\mathbb{Q}^f$ is the measure associated with the funding account numeraire $B_f(t)$ and the underlying risky asset has the dynamics: $$ dX_t = r_f(t)X(t)dt+ \sigma(t)X(t)dW^{\mathbb{Q}^f}(t) $$ # References {.appendix} - *[Cooking with collateral](http://janroman.dhis.org/finance/Kreditrisk/Risk_Cooking%20with%20collateralpdf.pdf), Vladimir Piterbarg* - *[Funding beyond Discounting](https://www.researchgate.net/profile/Vladimir-Piterbarg-2/publication/284078682_Funding_beyond_Discounting_Collateral_Agreements_and_Derivatives_Pricing/links/614c85a9a595d06017e55b0e/Funding-beyond-Discounting-Collateral-Agreements-and-Derivatives-Pricing.pdf), Vladimir Piterbarg*