How is an arbitrage-free binomial interest rate tree calibrated to match market prices?

An arbitrage-free binomial interest rate tree is constructed so that it correctly prices all on-the-run benchmark bonds (typically Treasuries). Calibration is the process of adjusting the tree's rates until this condition holds.

Calibration Process (Step by Step):

Step 1: Set the 1-Year Rate

The current 1-year rate (r0) is directly observable. Suppose r0 = 4.00%.

Step 2: Calibrate Year 1 Rates

We need r1u and r1d such that the tree prices the 2-year on-the-run Treasury correctly.

Given: 2-year on-the-run coupon bond priced at par with coupon = 4.50%

Assume log-normal model: r1u = r1d x e^(2σ), where σ is the assumed volatility.

With σ = 15% and trial-and-error:

• r1d = 4.15%, r1u = 4.15% x e^(0.30) = 5.60%

Verification: Price the 2-year 4.50% bond using the tree:

• At node r1u: Value = (100 + 4.50) / 1.0560 = $99.05
• At node r1d: Value = (100 + 4.50) / 1.0415 = $100.34
• At node r0: Value = [0.5 x (99.05 + 4.50) + 0.5 x (100.34 + 4.50)] / 1.04
• = [103.55 x 0.5 + 104.84 x 0.5] / 1.04 = 104.195 / 1.04 = $100.19

If this does not equal par ($100), adjust r1d and repeat until it does.

Step 3: Calibrate Year 2 Rates

Repeat the process using the 3-year on-the-run bond, now solving for r2uu, r2ud, and r2dd with the constraint that adjacent rates are related by the volatility parameter.

Key Properties of the Calibrated Tree:

1. Equal probability (0.5 up, 0.5 down) at each node
2. Adjacent rates separated by e^(2σ)
3. Every on-the-run bond prices to par
4. The tree is 'arbitrage-free' because it matches observable market prices

Exam Tip: CFA Level II often asks you to use a given calibrated tree, not build one from scratch. But understand the calibration logic and know that the tree must reprice benchmark bonds correctly.

Practice binomial tree problems in our CFA Level II bank.