Question

. Let $\delta>0$ be given. Consider an interest rate swap paying a fixed interest rate $K$ and receiving backset LIBOR $L\left(T_{j-1}, T_{j-1}\right)$ on a principal of 1 at each of the payment dates $T_j=\delta j, j=1,2, \ldots, n+1$. Show that the value of the swap is $$ \delta K \sum_{j=1}^{n+1} B\left(0, T_j\right)-\delta \sum_{j=1}^{n+1} B\left(0, T_j\right) L\left(0, T_{j-1}\right) . $$ Remark 10.7.1. The swap rate is defined to be the value of $K$ that makes the initial value of the swap equal to zero. Thus, the swap rate is $$ K=\frac{\sum_{j=1}^{n+1} B\left(0, T_j\right) L\left(0, T_{j-1}\right)}{\sum_{j=1}^{n+1} B\left(0, T_j\right)} $$

    . Let $\delta>0$ be given. Consider an interest rate swap paying a fixed interest rate $K$ and receiving backset LIBOR $L\left(T_{j-1}, T_{j-1}\right)$ on a principal of 1 at each of the payment dates $T_j=\delta j, j=1,2, \ldots, n+1$. Show that the value of the swap is
$$
\delta K \sum_{j=1}^{n+1} B\left(0, T_j\right)-\delta \sum_{j=1}^{n+1} B\left(0, T_j\right) L\left(0, T_{j-1}\right) .
$$

Remark 10.7.1. The swap rate is defined to be the value of $K$ that makes the initial value of the swap equal to zero. Thus, the swap rate is
$$
K=\frac{\sum_{j=1}^{n+1} B\left(0, T_j\right) L\left(0, T_{j-1}\right)}{\sum_{j=1}^{n+1} B\left(0, T_j\right)}
$$

Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve 1st Edition

Chapter 10, Problem 11

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In an interest rate swap, one party agrees to pay a fixed rate of interest (denoted as $K$) on a notional principal (assumed to be 1 for simplicity) and receives a floating rate of interest, which in this case is the backset LIBOR rate $L(T_{j-1}, T_{j-1})$ at Show more…

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. Let $\delta>0$ be given. Consider an interest rate swap paying a fixed interest rate $K$ and receiving backset LIBOR $L\left(T_{j-1}, T_{j-1}\right)$ on a principal of 1 at each of the payment dates $T_j=\delta j, j=1,2, \ldots, n+1$. Show that the value of the swap is $$ \delta K \sum_{j=1}^{n+1} B\left(0, T_j\right)-\delta \sum_{j=1}^{n+1} B\left(0, T_j\right) L\left(0, T_{j-1}\right) . $$ Remark 10.7.1. The swap rate is defined to be the value of $K$ that makes the initial value of the swap equal to zero. Thus, the swap rate is $$ K=\frac{\sum_{j=1}^{n+1} B\left(0, T_j\right) L\left(0, T_{j-1}\right)}{\sum_{j=1}^{n+1} B\left(0, T_j\right)} $$

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