How is a binomial interest rate tree calibrated?

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AcadiFi · Accepted Answer

Calibration ensures that valuing on-the-run Treasury zeros using the tree reproduces observed prices exactly — the tree is arbitrage-free with respect to the curve. Procedure for a 3-year tree at 18% volatility: (1) Start with today's 1-year rate, which is set equal to the 1-year spot rate (say 3.40%). (2) For year 1, the tree branches to r_up and r_down. Under a lognormal model with volatility sigma, r_up = r_down * e^(2*sigma). So r_up / r_down = e^(0.36) = 1.4333. (3) Find r_down such that the tree values the 2-year zero-coupon Treasury at the price implied by the 2-year spot rate. If the 2-year spot is 3.70%, the 2-year zero is worth 100 / 1.037^2 = 92.995. Set up the equation: 92.995 = [0.5 * (100/(1+r_up)) + 0.5 * (100/(1+r_down))] / 1.0340. Solve for r_down (iteratively). If r_down = 3.05%, then r_up = 3.05% * 1.4333 = 4.37%. Check: [0.5*(100/1.0437) + 0.5*(100/1.0305)] / 1.0340 = [0.5*95.82 + 0.5*97.04] / 1.0340 = 96.43 / 1.0340 = 93.26 — not matching 92.995; iterate until equation balances. (4) For year 2, branches are rates r_uu, r_ud, r_dd with ratios e^(2*sigma) between adjacent. Solve for r_dd such that the 3-year zero prices correctly given the now-calibrated year-0 and year-1 rates. (5) Continue outward for additional maturities. Result: a tree where every node is consistent with the observed curve at the chosen volatility, ready for pricing bonds with embedded options and computing OAS.

How is a binomial interest rate tree calibrated?

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