Let's break down the problem and fill out the tree step by step. --- ## **Given Information** - **3-year annual coupon bond** - **Coupon rate:** 6.% (so annual coupon = \$6.00) - **Face value:** \$100 - **Initial one-year forward rate \(i_{}\):** 3.00% -One-year forward rate at (1, L):** \(i_{1, L} = 3.50\%\) - **One-year forward rate at (2, LL):** \(i_{2, LL} = 4.10\%\) - **Volatility (\(\sigma\)) = 20% (.20)** - **Tree is binomial.** --- ## **Step 1: Fill in Missing Interest Rates** ### **Year 1** - **At node H (up move):** - Use the formula for binomial tree forward rates: \[ i_{1,H} = i_ \cdot e^{\sigma} \] \[ i_{1,L} = i_ \cdot e^{-\sigma} \] Plug in values: \[ i_{1,H} = 3.00\% \cdot e^{.2} \approx 3.00\% \cdot 1.2214 \approx 3.6642\% \] \[ i_{1,L} = 3.00\% \cdot e^{-.2} \approx 3.00\% \cdot .8187 \approx 2.4561\% \] But the tree says \(i_{1,L} = 3.50\%\), so let's use the provided value for L and keep the calculated value for H. --- ### **Year 2** - **At node (2, HH):** \[ i_{2,HH} = i_{1,H} \cdot e^{\sigma} \] \[ i_{2,HL} = i_{1,H} \cdot e^{-\sigma} \] \[ i_{2,LL} = 4.10\% \text{ (given)} \] First, calculate \(i_{1,H}\) as above: \(3.6642\%\). - **Calculate:** \[ i_{2,HH} = 3.6642\% \cdot e^{.2} \approx 3.6642\% \cdot 1.2214 \approx 4.478\% \] \[ i_{2,HL} = 3.6642\% \cdot e^{-.2} \approx 3.6642\% \cdot .8187 \approx 3.002\% \] - **At node (2, LL):** Given as \(4.10\%\). --- ## **Step 2: Bond Valuation at Each Node** At maturity (Time 3), the value is always the face value plus last coupon: \[ V = 100 + 6 = 106 \] ### **Time 2** - At each node, the value is: \[ V_2 = \frac{V_{3,up} + V_{3,down}}{2} \cdot \frac{1}{1+i_2} + 6 \] But since the tree is recombining, at each node at time 2 there is only one possible next value. At each node: \[ V_2 = \frac{106}{1+i_2} \] - \(V_{2,HH} = \frac{106}{1+.04478} = \frac{106}{1.04478} \approx 101.48\) - \(V_{2,HL} = \frac{106}{1+.03002} = \frac{106}{1.03002} \approx 102.91\) - \(V_{2,LL} = \frac{106}{1+.041} = \frac{106}{1.041} \approx 101.92\) --- ### **Time 1** - At each node: \[ V_1 = \frac{V_{2,up} + V_{2,down}}{2} \cdot \frac{1}{1 + i_1} + 6 \] #### **Node H:** \[ V_{1,H} = \frac{101.48 + 102.91}{2} \cdot \frac{1}{1+.036642} + 6 \] \[ = \frac{204.39}{2} \cdot \frac{1}{1.036642} + 6 \] \[ = 102.195 \cdot .9646 + 6 \approx 98.55 + 6 = 104.55 \] #### **Node L:** \[ V_{1,L} = \frac{102.91 + 101.92}{2} \cdot \frac{1}{1+.035} + 6 \] \[ = 102.415 \cdot \frac{1}{1.035} + 6 = 102.415 \cdot .9662 + 6 \approx 98.93 + 6 = 104.93 \] --- ### **Time ** \[ V_ = \frac{V_{1,H} + V_{1,L}}{2} \cdot \frac{1}{1+i_} \] \[ = \frac{104.55 + 104.93}{2} \cdot \frac{1}{1.03} \] \[ = 104.74 \cdot .9709 = 101.67 \] --- ## **Step 3: Fill in the Tree** ### **Interest Rates:** - \(i_ = 3.00\%\) - \(i_{1,H} \approx 3.66\%\) - \(i_{1,L} = 3.50\%\) (given, use this for the tree) - \(i_{2,HH} \approx 4.48\%\) - \(i_{2,HL} \approx 3.00\%\) - \(i_{2,LL} = 4.10\%\) (given) ### **Bond Values:** - \(V_ \approx 101.67\) - \(V_{1,H} \approx 104.55\) - \(V_{1,L} \approx 104.93\) - \(V_{2,HH} \approx 101.48\) - \(V_{2,HL} \approx 102.91\) - \(V_{2,LL} \approx 101.92\) - \(V_3 = 106\) at each terminal node --- ## **Final Tree (Filled):** ```plaintext V = 101.67 i = 3.000% | --------------------- | | V1,H = 104.55 V1,L = 104.93 i1,H = 3.66% i1,L = 3.50% | | ----------- ----------- | | | | V2,HH=101.48 V2,HL=102.91 V2,LL=101.92 i2,HH=4.48% i2,HL=3.00% i2,LL=4.10% | | | V=106 V=106 V=106 ``` --- ### **Summary Table** | Node | Rate (%) | Value | |--------|----------|---------| | | 3.00 | 101.67 | | 1H | 3.66 | 104.55 | | 1L | 3.50 | 104.93 | | 2HH | 4.48 | 101.48 | | 2HL | 3.00 | 102.91 | | 2LL | 4.10 | 101.92 | | 3 | - | 106.00 | --- ### **Supporting Work** All calculations shown above. If you need to show detailed work for each node, you can use the formulas above and plug in the numbers step by step. --- **If you need the tree in a more compact graphical form or with equations for each node, just let me know!**