Suppose that the risk-neutral probabilities are equal to 1/2 in every state. Given the following short rates, find the prices of a bond maturing at time 3 (with a one-month time step, τ = 1/12 ):

Question

Suppose that the risk-neutral probabilities are equal to [latex]\frac{1}{2} [/latex] in every state. Given the following short rates, find the prices of a bond maturing at time 3 (with a one-month time step, [latex] au =\frac{1}{12} [/latex] ):

[latex]r(0) =9.5\%[/latex]

[latex]\begin{matrix} / \ \setminus \end{matrix} \begin{matrix}r(1; u) = 8.5\%\ \ r(1; d) = 9.8\% \end{matrix}[/latex]

[latex]\begin{matrix} \lt \ \ \lt \end{matrix} \begin{matrix} r(2; uu) = 8.3\%\r(2; ud) = 8.9\% \r(2; du) = 9.1\%\r(2; dd) = 9.3\% \end{matrix}[/latex]

The next proposition gives an important result, which simplifies the model significantly.

Accepted Answer

Using formula (11.5) and the short rates given, we find the following structure of bond prices:

[latex]B(n,N; sn) = [p_∗B(n + 1,N; s_nu) + (1 − p_∗)B(n + 1,N; s_nd)] imes exp\left\{−τr(n; s_n) ight\} .[/latex] (11.5)

B(0, 3) = 0.9773

[latex]\begin{matrix} / \ \setminus \end{matrix} \begin{matrix}B(1, 3; u) = 0.9859\ \ B(1, 3; d) = 0.9843 \end{matrix}[/latex]

[latex]\begin{matrix} \lt \ \ \lt \end{matrix} \begin{matrix} B(2, 3; uu) = 0.9931\B(2, 3; ud) = 0.9926 \B(2, 3; du) = 0.9924\B(2, 3; dd) = 0.9923\end{matrix}[/latex]

Question 11.9: Suppose that the risk-neutral probabilities are equal to 1/2...

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