Two-Step Binomial Tree

This is the basics page for term-structure intuition: start from the zero curve, turn it into a short-rate tree, and then use risk-neutral pricing to value cashflows. No HJM fireworks yet—just the essential mechanics.

Pedagogical learning path

Step 1 of 8: Binomial Tree

Start with discrete-tree intuition, move into curve representations, then graduate to continuous-time short-rate models and finally the full forward-rate framework.

Next: Nelson-Siegel →

1Binomial Tree 2Nelson-Siegel 3Svensson 4PCA Yield 5Vasicek 6CIR 7Hull-White 1F 8HJM

Core pricing equation

V_{0} = \frac{1}{1 + r _{0}} (q V_{u} + (1 - q) V_{d})

Calibration target

P (0, 2) = \frac{1}{( 1 + y _{2} ) ^{2}}, r_{u} = m e^{σ}, r_{d} = m e^{- σ}

1Y zero yield3.50%

2Y zero yield4.00%

σ tree step0.18

q risk-neutral0.50

Coupon rate4.00%

Root short rate

3.50%

fitted from 1Y zero

Up / down nodes

5.31% / 3.71%

t = 1 short rates

P(0,2) error

0.000 bp

target vs repricing

Par coupon

3.99%

2Y annual bond

1. The short-rate tree

Recombining tree calibrated to the 1Y and 2Y zero curve

The tree is drawn out to two steps for intuition, but only the first future layer is needed to fit the 2Y zero. That is the whole point: the tree gives you a pricing engine, not just a curve fit.

2. Repricing the zero curve

Target zero prices vs model repricing

What the tree implies numerically

Observed zero prices

$P (0, 1) = 0.9662$ , $P (0, 2) = 0.9246$

Risk-neutral continuation

$q / (1 + r_{u}) + (1 - q) / (1 + r_{d}) = 0.9569$

Coupon bond example

2Y 4.00% annual coupon bond price = 100.02

P (0, 2) = \frac{1}{1 + r _{0}} (\frac{q}{1 + r _{u}} + \frac{1 - q}{1 + r _{d}})

3. State prices and Arrow–Debreu intuition

Arrow–Debreu prices at t = 2

Read them like probabilities with discounting attached

State prices are discounted risk-neutral probabilities. They tell you what one unit paid in each state is worth today.

$p i_{uu} = 0.2294$
$p i_{u d} = 0.4623$
$p i_{dd} = 0.2329$

Their sum equals the 2Y discount factor: 0.9246.

This is the real beginner-level unlock: once you can compute state prices, pricing becomes weighted addition instead of mystery. Callable bonds, swaptions, mortgage prepay trees—they all start here.

4. Where to go next

From tree to calibration engine

Extend the two-step example into Black-Derman-Toy or Hull-White lattices that fit an entire curve and volatility term structure.

From discrete to continuous

Vasicek and CIR replace the discrete tree with diffusion dynamics but preserve the same risk-neutral pricing logic.

From short rates to forward surfaces

HJM generalises the idea to the entire forward curve, where no-arbitrage determines the drift once volatility is chosen.

Next: Vasicek →Then: Hull-White 1F →Later: HJM →