How do you value an option-free bond using a binomial interest rate tree?

Great question. You're right that for a simple option-free bond, discounting at spot rates gives the same answer. But the binomial tree approach is essential because (1) it ensures arbitrage-free pricing and (2) it extends naturally to bonds with embedded options where simple discounting doesn't work.

Why Use a Binomial Tree for Option-Free Bonds?

The tree is calibrated to match the current term structure of interest rates. This ensures prices are consistent with observable market rates. Once calibrated, the same tree can price bonds with calls, puts, or other features.

Step-by-Step: Pricing a 3-Year 5% Annual Coupon Bond

Assume we've calibrated a one-year rate tree:

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Step 1: Start at maturity (Year 3). At every terminal node, the bond pays its final coupon ($50) plus par ($1,000) = $1,050.

Step 2: Work backward to Year 2. At each Year 2 node, discount the expected Year 3 value:

Node UU: V_uu = $1,050 / (1 + 0.071) =$980.39. Add the Year 2 coupon: $980.39 +$50 = $1,030.39

Node UD: V_ud = $1,050 / (1 + 0.050) =$1,000.00. Add coupon: $1,000.00 +$50 = $1,050.00

Node DD: V_dd = $1,050 / (1 + 0.036) =$1,013.51. Add coupon: $1,013.51 +$50 = $1,063.51

Step 3: Work backward to Year 1. At each node, take the average of the two possible next-period values and discount:

Node U: V_u = [0.5 x $1,030.39 + 0.5 x$1,050.00] / (1 + 0.052) = $1,040.20 / 1.052 =$988.78. Add coupon: $988.78 +$50 = $1,038.78

Node D: V_d = [0.5 x $1,050.00 + 0.5 x$1,063.51] / (1 + 0.038) = $1,056.76 / 1.038 =$1,018.07. Add coupon: $1,018.07 +$50 = $1,068.07

Step 4: Discount to Year 0:

V_0 = [0.5 x $1,038.78 + 0.5 x$1,068.07] / (1 + 0.035) = $1,053.43 / 1.035 =$1,017.80

The bond's arbitrage-free value is $1,017.80.

Key Principles:

1. Risk-neutral probabilities of 0.5 are used (not real-world probabilities)
2. Discount at the one-period rate at each node
3. The tree must be calibrated so that it correctly prices on-the-run benchmark bonds

Exam Tip: CFA Level II frequently gives you a pre-built rate tree and asks you to price a bond. Practice the backward induction until it's mechanical.

Master binomial trees with our CFA Level II Fixed Income resources.