Question

In the proof of Theorem 10.4.1, we showed by an arbitrage argument that the value at time 0 of a payment of backset LIBOR $L(T, T)$ at time $T+\delta$ is $B(0, T+\delta) L(0, T)$. The risk-neutral price of this payment, computed at time zero, is $$ \widetilde{\mathbb{E}}[D(T+\delta) L(T, T)] $$ Use the definitions $$ \begin{aligned} L(T, T) & =\frac{1-B(T, T+\delta)}{\delta B(T, T+\delta)}, \\ B(0, T+\delta) & =\widetilde{\mathbb{E}}[D(T+\delta)], \end{aligned} $$ and the properties of conditional expectations to show that $$ \widetilde{\mathbb{E}}[D(T+\delta) L(T, T)]=B(0, T+\delta) L(0, T) . $$

     In the proof of Theorem 10.4.1, we showed by an arbitrage argument that the value at time 0 of a payment of backset LIBOR $L(T, T)$ at time $T+\delta$ is $B(0, T+\delta) L(0, T)$. The risk-neutral price of this payment, computed at time zero, is
$$
\widetilde{\mathbb{E}}[D(T+\delta) L(T, T)]
$$

Use the definitions
$$
\begin{aligned}
L(T, T) & =\frac{1-B(T, T+\delta)}{\delta B(T, T+\delta)}, \\
B(0, T+\delta) & =\widetilde{\mathbb{E}}[D(T+\delta)],
\end{aligned}
$$
and the properties of conditional expectations to show that
$$
\widetilde{\mathbb{E}}[D(T+\delta) L(T, T)]=B(0, T+\delta) L(0, T) .
$$

Stochastic Calculus for Finance II : Continuous-Time Models

Steven E. Shreve 1st Edition

Chapter 10, Problem 12

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Step 1

- $B(t, T)$ is the discount factor from time $t$ to time $T$. - $D(T+\delta)$ is the discount factor from time 0 to time $T+\delta$, which can be expressed as $D(T+\delta) = B(0, T+\delta)$. - $\widetilde{\mathbb{E}}[\cdot]$ denotes the expectation under the Show more…

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In the proof of Theorem 10.4.1, we showed by an arbitrage argument that the value at time 0 of a payment of backset LIBOR $L(T, T)$ at time $T+\delta$ is $B(0, T+\delta) L(0, T)$. The risk-neutral price of this payment, computed at time zero, is $$ \widetilde{\mathbb{E}}[D(T+\delta) L(T, T)] $$ Use the definitions $$ \begin{aligned} L(T, T) & =\frac{1-B(T, T+\delta)}{\delta B(T, T+\delta)}, \\ B(0, T+\delta) & =\widetilde{\mathbb{E}}[D(T+\delta)], \end{aligned} $$ and the properties of conditional expectations to show that $$ \widetilde{\mathbb{E}}[D(T+\delta) L(T, T)]=B(0, T+\delta) L(0, T) . $$

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