An overview of the valuation of collateralized derivative contracts

Abstract

We consider the valuation of collateralized derivative contracts such as interest rate swaps or forward FX contracts. We allow for posting securities or cash in different currencies. In the latter case, we focus on using overnight index rates on the interbank market. Using time varying haircuts, we provide an intuitive way to derive the basic discounting results, keeping in line with the most standard theoretical and market views. In a number of cases associated with margining with major central counterparties, pricing rules for collateralized trades remain linear, thus the use of (multiple) discount curves. We also show how to deal with partial collateralization, involving haircuts, asymmetric CSA, counterparty risk and funding costs. We therefore intend to provide a unified view. Mathematical or legal details are not dealt with and we privilege financial insights and easy to grasp concepts and tools.

Appendix: Convexity adjustments

$E_t^{Q_T^B } \left[ {X(T)} \right] ={E_t^{Q_T^A } \left[ {\frac{dQ_T^B }{dQ_T^A }X(T)} \right] }/{E_t^{Q_T^A } \left[ {\frac{dQ_T^B }{dQ_T^A }} \right] }$. Thus,

$$\begin{aligned} E_t^{Q_T^B } \left[ {X(T)} \right] =E_t^{Q_T^A } \left[ {\frac{\exp \left( {-\int \limits _t^T {\left( {r_B -r_A } \right) (s)ds} } \right) }{E_t^{Q_T^A } \left[ {\exp \left( {-\int \limits _t^T {\left( {r_B -r_A } \right) (s)ds} } \right) } \right] }X(T)} \right] \end{aligned}$$

It can easily be shown that: $E_t^{Q_T^A } \left[ {\exp \left( {-\int \limits _t^T {\left( {r_B -r_A } \right) (s)ds} } \right) } \right] =\frac{B_B \left( {t,T} \right) }{B_A \left( {t,T} \right) }$. Thus,

$$\begin{aligned} E_t^{Q_T^B} \left[ {X(T)} \right]&= E_t^{Q_T^A } \left[ {X_A \left( T \right) } \right] \\&\quad +\, \frac{B_A \left( {t,T} \right) }{B_B \left( {t,T} \right) }\hbox {Cov}_t^{Q_T^A } \left( {\exp \left( {-\int \limits _t^T {\left( {r_B -r_A } \right) (s)ds} } \right) ,X(T)} \right) \end{aligned}$$

which involves a covariance term between the payoff $X(T)$ and the stochastic spread terms $r_B -r_A$.

Let us consider the special case associated with a Brownian filtration. Since $\xi _T^A$ is a positive $Q^{\beta }$- martingale, we have: $\frac{d\xi _T^A (t)}{\xi _T^A (t)}=\sigma _A \left( {t,T} \right) dW_A^\beta (t),\, \xi _T^A (t)=\varepsilon \left( {\int \nolimits _0^t {\sigma _A \left( {s,T} \right) dW_A^\beta (s)} } \right)$ where $W_A^\beta$ is a $Q^{\beta }$- Brownian motion. Since $\xi _T^{B,A} \left( t \right) =\frac{\xi _T^B \left( t \right) }{\xi _T^A \left( t \right) }$ is a $Q_T^A$ - martingale and using Girsanov theorem, we have:

$$\begin{aligned} \xi _T^{B,A} (t)=\varepsilon \left( {\int \limits _0^t {\sigma _B \left( {s,T} \right) dW_B^{A,T} (s)-\sigma _A \left( {s,T} \right) dW_A^{A,T} (s)} } \right) , \end{aligned}$$

where $W_B^{A,T},W_A^{A,T}$ are $Q_T^A$ - Brownian motions.

$$\begin{aligned} E_t^{Q_T^B } \left[ {X(T)} \right]&= E_t^{Q_T^A } \left[ {\frac{\xi _T^{B,A} \left( T \right) }{\xi _T^{B,A} \left( t \right) }X(T)} \right] \\&\quad =E_t^{Q_T^A } \!\left[ \! {\varepsilon \!\left( {\int \limits _t^T\! {\sigma _B \left( {s,T} \right) dW_B^{A,T} (s)\!-\!\sigma _A \left( {s,T} \!\right) dW_A^{A,T} (s)} } \right) X(T)} \!\right] \!. \end{aligned}$$

$\sigma _A \left( {t,T} \right)$ is related to collateralized discount bond prices since from $B_A \left( {t,T} \right) =B_A \left( {0,T} \right) \xi _T^A \left( t \right) \exp \left( {\int \limits _0^t {r_A (s)ds} } \right)$, we get $\frac{dB_A (t,T)}{B_A (t,T)}=r_A (t)dt+\sigma _A \left( {t,T} \right) dW_A^\beta (t)$.

Let us assume that $X(T)>0$, thus $p_{A,T} (t)>0$. $\frac{dp_{A,T} (t)}{p_{A,T} (t)}=\sigma _X \left( {t,T} \right) dW_A^{X,T} (t)$, where $W_A^{X,T}$ is a $Q_T^A$- martingale and $X(T)\!=\!E_t^{Q_T^A } \left[ {X(T)} \right] \varepsilon \left( {\int \limits _t^T \!{\sigma _X \left( {s,T} \right) dW_A^{X,T} (s)} } \right)$. This leads to the following relation between forward prices under collaterals $A,B$:

$$\begin{aligned} E_t^{Q_T^B } \left[ {X(T)} \right]&= E_t^{Q_T^A } \left[ {X(T)} \right] E_t^{Q_T^A } \left[ \varepsilon \left( {\int \limits _t^T {\sigma _X \left( {s,T} \right) dW_A^{X,T} (s)} } \right) \right. \\&\left. \varepsilon \left( {\int \limits _t^T {\sigma _B \left( {s,T} \right) dW_B^{A,T} (s)-\sigma _A \left( {s,T} \right) dW_A^{A,T} (s)} } \right) \right] \end{aligned}$$