How do you price a bond using a binomial interest rate tree?

Binomial interest rate trees are used to price bonds — especially those with embedded options — by modeling the possible evolution of short-term interest rates and working backward to determine present value.

Building the Tree

The tree is calibrated to match the current term structure of interest rates and an assumed volatility. At each node, the rate can move up or down:

r_up = r × e^(2σ) (approximately)

r_down = r

The tree is typically calibrated so that it correctly prices on-the-run benchmark bonds (this ensures arbitrage-free pricing).

Step-by-Step: Pricing a 3-Year 5% Annual Coupon Bond (Par = $100)

Assume the following calibrated 1-year rates at each node:

Year 0:     3.50%
Year 1:     4.80% (up)  /  3.20% (down)
Year 2:     6.20% (uu)  /  4.50% (ud)  /  3.10% (dd)

Working Backward from Year 3:

At Year 2, the bond has one year left. At each node, the value = (Coupon + Par) / (1 + rate):

• Node uu: $105 / 1.062 = $98.87
• Node ud: $105 / 1.045 = $100.48
• Node dd: $105 / 1.031 = $101.84

At Year 1:

At each node, value = [0.5 × V_up + 0.5 × V_down + Coupon] / (1 + rate):

• Node u: [0.5 × $98.87 + 0.5 × $100.48 + $5] / 1.048 = $99.50
• Node d: [0.5 × $100.48 + 0.5 × $101.84 + $5] / 1.032 = $100.68

At Year 0:

Value = [0.5 × $99.50 + 0.5 × $100.68 + $5] / 1.035 = $101.52

Pricing a Callable Bond

For a callable bond (callable at par), at each node you apply:

Value = min(Calculated Value, Call Price)

If the bond is callable at $100 at Year 1:

• Node u: min($99.50, $100) = $99.50 (not called — below par)
• Node d: min($100.68, $100) = $100.00 (called — above par, issuer exercises)

Reworking Year 0:

Callable value = [0.5 × $99.50 + 0.5 × $100.00 + $5] / 1.035 = $101.21

Option value = Option-free bond - Callable bond = $101.52 - $101.21 = $0.31

For the FRM exam, practice building trees, pricing straight bonds, and then adjusting for call or put provisions. The key insight is that the call option benefits the issuer and reduces the bond's value to the investor. Check our question bank for more tree-based problems.