Interest Rates Basic

Suppose $b>a>0$ . Let $P(a,b)$ be the discount factor discounting 1 dollar paid at $b$ to time $a$ . Let $T_1, T_2, \ldots, T_{N+1}$ be the floating rate reset date of the floating leg in a swap. For the case of LIBOR 3M, $T_i = \frac{i}{4}$ if we assume the day count convention is that 3 months is $\frac{1}{4}$ of a year. Define $\alpha_n = T_{n+1} - T_{n}$ .

Forward LIBOR Rate

Let L_n(t) be the forward LIBOR rate for period $[T_n, T_{n+1}]$ seen at time $t$ for $t < T$ . Then,

$L_n(t,T) := \frac{1}{\alpha_n}\left(\frac{P(t,T_n)}{P(t,T_{n+1})}-1\right).$

This definition makes sense if we seen it as:

$1+\alpha_nL_n(t) = \frac{1}{P(T_n, T_{n+1})} = \frac{P(t,T_n)}{P(t,T_{n+1})}.$

Instantaneous Forward Rate

Instantaneous forward rate is the forward rate with time interval goes to infinity.

$\begin{aligned}f(t,T)&=\lim_{\delta\rightarrow 0}\frac{1}{\delta}\left(\frac{P(t,T)}{P(t,T+\delta)}-1\right) \\ & = -\frac{\partial \log P(t,T)}{\partial T}\end{aligned}.$

Short Rate

Short rate $r(t)$ is an instantaneous interest rate such that if $r$ is deterministic

$P(t,T) = e^{-\int_t^Tr(s)\textnormal{d}s}.$

Or if $r$ is stochastic

$P(t,T) = \mathbf{E}\left[e^{-\int_t^Tr(s)\textnormal{d}s}|\mathcal{F}_t\right].$

Forward Par Swap Rate

In finance, ‘par’ usually means ‘equal’. Par swap rate is the fixed rate such that the floating leg and the fixed leg of the swap make the contract has value 0. We denote $S_n(t)$ by the forward par swap rate on tenor $[T_n, T_{N+1}]$ . Suppose the pay frequency of the fixed leg is the same as the floating leg (otherwise redefine the below annuity as one that matches the fixed leg pay frequency). Let $A_n(t)$ represents the present value of annuity, i.e.,

$A_n(t) = \sum_{k=n+1}^{N+1} \alpha_{k-1}P(t,T_k).$

Then the present value of the fixed leg is

$PV_{\mbox{fix}} = S_n(t)A_n(t)$

The present value of the floating leg is

$\begin{aligned}& PV_{\mbox{float}} \\ = & \sum_{k=n}^N L_k(t)\alpha_k P(t,T_{k+1}) \\ =& \sum_{k=n}^N \frac{1}{\alpha_k}\left(\frac{P(t,T_k)}{P(t,T_{k+1})}-1\right) \alpha_k P(t,T_{k+1}) \\ =& \sum_{k=n}^N \left(P(t,T_k) - P(t,T_{k+1})\right) \\ =& P(t,T_n) - P(t,T_{N+1}) \end{aligned}$

By letting $PV_{\mbox{fix}} = PV_{\mbox{float}}$ , we have that

$S_n(t) = \frac{P(t,T_n) - P(t,T_{N+1})}{A_n(t)}.$