## Glossary

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# Swap

The Swap contract swaps the floating LIBOR rates to a fixed rate $K$ over the future time intervals $[S_0,S_1],~[S_1,S_2],\ldots,[S_{n-1},S_n]$
The times $\bar{S}=[S_0,S_1,S_2,\ldots,S_n]$ is called the tenor structure of the SWAP.
The SWAP contract can be seen as a sum of $n$ FRA:s. Value at time $t>S_0$
Put $\tau_i=(S_i-S_{i-1}),~i=1,2,\ldots,n$. \begin{eqnarray*} \Pi^{SWAP}_t[\bar{S}]&=&\sum_{i=1}^n \Pi^{FRA}_t[S_{i-1},S_i]\\ &=&\sum_{i=1}^n\tau_ip(t,S_i)\left(\frac{p(t,S_{i-1})-p(t,S_i)}{\tau_ip(t,S_i)}-K\right)\\ &=&\sum_{i=1}^n(p(t,S_{i-1})-p(t,S_i))-K\sum_{i=1}^n\tau_ip(t,S_i)\\ &=&p(t,S_o)-p(t,S_n)-K\sum_{i=1}^n\tau_ip(t,S_i)\\ &=&\left(\sum_{i=1}^n\tau_ip(t,S_i)\right)\left(\frac{p(t,S_o)-p(t,S_n)}{\sum_{i=1}^n\tau_ip(t,S_i)}-K\right) \end{eqnarray*} The annuity of the SWAP is denoted $p(t,\bar{S})$. It is given by $p(t,\bar{S})=\sum_{i=1}^n\tau_ip(t,S_i)$ If we put the fixed rate $K$ as $\frac{p(t,S_o)-p(t,S_n)}{\sum_{i=1}^n\tau_ip(t,S_i)}=\frac{p(t,S_o)-p(t,S_n)}{p(t,\bar{S})}$ then $\Pi^{SWAP}_t[\bar{S}]=0$. This rate is called the SWAP rate and we denote it $y_t[\bar{S}]$.