How do I compute individual node values in a binomial bond tree?

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AcadiFi · Accepted Answer

At any non-terminal node, the value equals the expected discounted value of the next period's cash flows, using risk-neutral probabilities of 0.5 up and 0.5 down. Formula: V_t = [0.5 * (V_up + C) + 0.5 * (V_down + C)] / (1 + r_t), where V_up and V_down are the values at the two nodes one period later, C is the coupon paid at t+1, and r_t is the 1-year rate at the current node. Example: a 3-year 5% annual callable bond (call at 100 starting year 1), face 100, at a year-1 up node with rate r_up1 = 6.2%, where year-2 up-up node has V_uu = 98.65 and up-down node has V_ud = 101.30. Step 1: Compute backward value = [0.5 * (98.65 + 5) + 0.5 * (101.30 + 5)] / (1 + 0.062) = [0.5 * 103.65 + 0.5 * 106.30] / 1.062 = [51.825 + 53.15] / 1.062 = 104.975 / 1.062 = 98.846. Step 2: Apply the call rule since this is a callable node. Call price is 100. Backward value 98.846 is less than 100, so the issuer does not call. Final node value = 98.846. At a different node where backward value = 102.10 and the bond is callable at 100, the value is capped to 100 because the issuer would call. Note also the coupon handling convention: some texts include the coupon at the current node in V, others treat coupon as paid at the next node. The formula above assumes coupons are paid at the end of each period (at t+1), which is the common CFA convention. Always verify tree convention before reconciling with a vendor's valuation model.

How do I compute individual node values in a binomial bond tree?

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