Dual curve pricing

Andersen and Piterbarg (2010).

Ametrano and Marco Bianchetti (2013).

OIS discounting is used in conjunction with Libor rates for pricing in the dual-curve setup. If we consider a fixed tenor structure

\begin{align} 0=T_0<T_1<\ldots<T_M, \label{eq:tau}\end{align}

where in Eq. \eqref{eq:tau} the intervals $\tau_i=T_{i}-T_{i-1}$, $i=0,\ldots,M$ is the year fraction between the accrual dates $T_i$ and $T_{i-1}$ depending on day count convention, then Libor and OIS zero-coupon bonds (discount factors) can be obtained from the continuously-compounded forward Libor and forward OIS curves in the following manner. Let $f(t,T_{i},T_{i-1})$ be the continuously-compounded forward rate and let $P(t,T_0)=1$ and $P(t,T_1)=\exp(-f(t,T_1)\cdot\tau_1)$ then

\[P(t,T_n) = P(t,T_{n-1})\exp\left(-f(t,T_n)\cdot\tau_n\right),\hspace{.25pc}n=2,\ldots M.\]

Let $P(t,T_n)$ and $P_{OIS}(t,T_n)$ be the Libor and OIS discount factors respectively for a given $t$ and $T_n$, then define the simply-compounded Libor and OIS forward rate as

\[L(t,T_{n-1},T_{n}) = L_n(t) = \frac{1}{\tau_n}\left(\frac{P(t,T_{n-1})}{P(t,T_{n})}-1\right),\]

and

\[F(t,T_{n-1},T_{n}) = F_n(t) = \frac{1}{\tau_n}\left(\frac{P_{OIS}(t,T_{n-1})}{P_{OIS}(t,T_{n})}-1\right).\]

References

1. Leif B.G. Andersen and Vladimir V. Piterbarg. Interest Rate Modelling, Volume I: Foundations and Vanilla Models. Atlantic Financial Press, 1st edition, 2010a.

2. Ferdinando M. Ametrano and Marco Bianchetti, Everything You Always Wanted to Know About Multiple Interest Rate Curve Bootstrapping but Were Afraid to Ask (April 2, 2013). Available at SSRN: https://ssrn.com/abstract=2219548