How do I value a callable bond using backward induction on a binomial tree?

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AcadiFi · Accepted Answer

Backward induction values a callable bond by computing node values from maturity back to today, applying the call rule at each callable node. Procedure: ```mermaid flowchart TD T5[Year 5 Nodes
Pay 104.80] --> T4[Year 4 Nodes
Discount+Coupon] T4 --> T3[Year 3 Nodes
Apply Call] T3 --> T2[Year 2 Nodes
Apply Call] T2 --> T1[Year 1 Nodes
No Call] T1 --> T0[Year 0
Bond Value] ``` Step-by-step for our 5-year bond: (1) At year 5, every node pays 100 principal + 4.80 coupon = 104.80. (2) At year 4, compute value at each node as [0.5 * (V_up + coupon) + 0.5 * (V_down + coupon)] / (1 + r_node), where r_node is the 1-year rate at that node. For a year-4 node with rate 3.2%, if V_up at year 5 is 104.80 and V_down is 104.80, value = [0.5*104.80 + 0.5*104.80] / 1.032 = 101.55. Add the 4.80 coupon received at year 4? No — coupon is already incorporated by discounting future cash flows. Actually the recursion is: Value_t = [0.5*(V_up + C) + 0.5*(V_down + C)] / (1 + r_t), where C is the coupon paid at t+1. (3) At year 3 (callable), compute the backward value as above, then apply the call rule: if computed value exceeds call price (100), cap it at 100. Same for year 2. (4) At year 1 (non-callable), no capping. (5) At year 0, discount year-1 values back to get price. The callable bond will always be worth less than an otherwise-identical straight bond because the issuer's call option is subtracted: V_callable = V_straight - V_call_option.

How do I value a callable bond using backward induction on a binomial tree?

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