Employ a risk-neutral rate tree to assess how the term structure is shaped by volatility.
The idea
A risk-neutral rate tree is a lattice of possible one-period rates with a model-set probability on each branch, chosen so that the tree is arbitrage-free and calibrated to market prices, and some trees use one half up and one half down as a convention. A zero-coupon bond is valued by backward induction, starting from the face value at maturity. Let Vup and Vdown be the values one step later, and r the rate at the node. With one-half branch probabilities, Value = (0.5 × Vup + 0.5 × Vdown) ÷ (1 + r). Because price is a convex function of the rate, a wider spread of up and down rates with the same average rate raises the expected price and lowers the spot yield. Volatility therefore pulls the curve below the path of expected rates, and the effect grows with maturity. Common errors: discounting at the average rate, skipping the discount at the first node, and weighting the nodes of a recombining tree equally instead of by path counts.
Formulas
Node value in a risk-neutral tree with one-half branch probabilities
Value = (0.5 × Vup + 0.5 × Vdown) ÷ (1 + r)
- Vup
- the value at the up node one step later
- Vdown
- the value at the down node one step later
- r
- the one-period rate at the current node
Applied backwards from the face value at maturity, discounting at each node's own rate. The trap is discounting at the average rate, which ignores convexity and understates the price.
A practice question
In a risk-neutral binomial tree the one-year rate is 6.00% today and moves to 12.00% or 2.00% next year with probability 0.5 each. The price of a two-year zero-coupon bond per 100 face is closest to:
- 89.00Why not: Discounts at the expected rate of 6% in both years, 100 / 1.06², ignoring the convexity from rate volatility.
- 93.66Why not: Averages the two year-1 node values but forgets to discount them one more year at 6%.
- 88.36 - correct
- 84.23Why not: Follows only the up path, 100 / (1.06 x 1.12), instead of averaging both branches.
Node values at year 1: 100/1.12 = 89.29 and 100/1.02 = 98.04; average 93.66; discount at 6%: 93.66 / 1.06 = 88.36.
A harder one
In a risk-neutral tree the one-year rate is 4.00% today, and next year it is either 8.00% or 2.00% with probability 0.5 each, so the expected rate is 5.00%. The one-year forward rate for the period from year 1 to year 2 implied by the tree is closest to:
- 5.00%Why not: Sets the forward equal to the expected rate, ignoring the convexity effect.
- 5.09%Why not: Applies the convexity adjustment with the wrong sign, adding it to the expected rate.
- 4.00%Why not: Uses the current one-year rate instead of computing P(1)/P(2) - 1.
- 4.91% - correct
P(1) = 100/1.04 = 96.154. P(2) = [0.5 x 100/1.08 + 0.5 x 100/1.02] / 1.04 = 91.650. Forward = 96.154 / 91.650 - 1 = 4.91%, below the 5.00% expected rate.
The hardest tier
A recombining risk-neutral tree has a one-year rate of 3% today, 5% or 3% after one year (up, down), and 7%, 5% or 3% after two years (up-up, middle, down-down). Each branch has probability 0.5, and the middle node is reached by two paths. The price of a three-year zero-coupon bond per 100 face is closest to:
- 88.94 - correct
- 93.36Why not: Stops after two years and prices a two-year zero; the year-3 discounting is missing.
- 91.61Why not: Averages the year-1 values without discounting at the first node's 3% rate.
- 86.42Why not: Follows only the up-up path, 100 / (1.03 x 1.05 x 1.07).
Year-3 payoff discounted at year-2 nodes: 93.46 (7%), 95.24 (5%), 97.09 (3%). Year-1 up node (rate 5%) = 0.5(93.46 + 95.24)/1.05 = 89.86; down node (rate 3%) = 0.5(95.24 + 97.09)/1.03 = 93.36. Today = 0.5(89.86 + 93.36)/1.03 = 88.94.
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