FRM Exam Part II · Arbitrage Pricing with Term Structure Models

Binomial Interest Rate Trees and Backward Induction

Updated 11 October 2026 · Fact-checked

A binomial interest rate tree models the short rate moving up or down each period, with recombining nodes. To price a zero-coupon bond, start at maturity with the face value, then work backward. At each node, take the 0.5-0.5 average of the two next-period values and discount it at that node's one-period rate.

Understand Binomial Interest Rate Trees

A binomial interest rate tree is a map of possible future short-term rates. Today there is one rate, r0. After one period the rate can go up or down. After the next period each of those can again go up or down.

The tree is recombining: an up move followed by a down move gives the same rate as a down move followed by an up move. So after n periods there are only n + 1 distinct nodes, not 2^n. This keeps the tree small and the maths manageable.

In the standard FRM setup, the up and down moves have risk-neutral probabilities of 0.5 each. The tree is used to price instruments as if investors were risk-neutral. Any risk premium is already built into the rates on the tree.

You price a bond by backward induction. At maturity a zero-coupon bond is worth its face value at every node. One step earlier, each node's value is the expected next-period value, discounted at the rate at that node. Repeat until you reach time 0. The result is today's price, and it is arbitrage-free given the tree.

Note that the discount rate at a node is known at the start of that period. The uncertainty is only about which node comes next.

Key formulas to remember

Backward induction at a node: V(node) = [0.5 × V(up) + 0.5 × V(down)] ÷ (1 + r(node) × Δt); Use Δt = 1 for annual steps. For semiannual steps with an annualised rate, Δt = 0.5, so you divide by (1 + r/2). Probabilities are risk-neutral.
Terminal condition for a zero-coupon bond: V(maturity) = Face value at every node; The bond pays the face value regardless of the rate path.
Recombining property: Nodes at step n = n + 1; paths to step n = 2^n; An up-then-down move equals a down-then-up move, so the tree has fewer nodes than paths.
Spot rate from a bond price (annual compounding): z(T) = (Face ÷ P)^(1 ÷ T) − 1; Use this to turn the tree price into a yield. For semiannual compounding, the annualised yield is 2 × [(Face ÷ P)^(1 ÷ 2T) − 1].

How to solve Binomial Interest Rate Trees questions

Use this method for any question that gives a rate tree and asks for a bond price, node value or yield.

1Read the time step. Is it annual or semiannual? Check whether rates are quoted as annual rates with a half-year step.
2Draw the tree and label every node with its short rate. Confirm the rates recombine.
3Write the face value at every terminal node of the zero-coupon bond.
4Move back one period. For each node, average the two successor values using the stated probabilities (normally 0.5 each).
5Discount that average at the rate sitting at the node you are valuing, not the successor rates. Use 1 + r × Δt.
6Repeat until you reach time 0. Keep at least four decimals in intermediate values.
7If asked, convert the price to a spot yield or compare it with a given discount factor.
8Sanity check: the price must be below face value for positive rates, and the answer should sit near face ÷ (1 + average rate)^T.

Quickest way: Node-by-node mental shortcut

When to use it: Use it when the tree is small (two or three steps) and the answer options are well separated.

Estimate each discounted value with 1 ÷ (1 + r) ≈ 1 − r for small rates, then compute exactly only the final step.
Value the last column first. Average the two values, then divide by the node rate.
Eliminate options that are above face value or far from face ÷ (1 + r0)^T.
For a one-period-ahead question, you only need the two successor nodes. Do not build the whole tree.

Common mistakes in Binomial Interest Rate Trees

Discounting at the successor node's rate instead of the current node's rate.
Students look at the next column when averaging and keep using those rates.
Fix: Average the values first, then divide by 1 plus the rate at the node you are standing on.
Forgetting to halve the rate for semiannual steps.
Quoted rates are annual, and the step length is easy to miss in the question.
Fix: Check Δt before calculating. For half-year steps divide by (1 + r ÷ 2) at every node.
Treating the tree as having 2^n nodes.
Students confuse the number of paths with the number of distinct nodes.
Fix: Remember that recombining trees have n + 1 nodes at step n. Compute each node only once.
Averaging the rates and discounting once.
It looks like a shortcut and gives a number close to the right one.
Fix: Discount step by step. The expected-value order matters because discounting is non-linear in the rate. A bond price from average rates is not the tree price.
Using a bond's face value at an intermediate node.
Students forget that only the terminal column is set to face value.
Fix: Set face value only at maturity. Earlier nodes carry discounted values.
Rounding too early.
Prices are close together in answer options.
Fix: Carry four or five decimals, and round only the final answer.

Worked examples

Example 1

A two-year annual tree has a current short rate of 4%. After one year the rate is 5% in the up state and 3% in the down state. Risk-neutral probabilities are 0.5 each. Find the price of a two-year zero-coupon bond with face value 100, and its annual spot yield.

Show the solution

Terminal value at year 2 is 100 at both nodes.
Up node at year 1 (rate 5%): 100 ÷ 1.05 = 95.2381.
Down node at year 1 (rate 3%): 100 ÷ 1.03 = 97.0874.
Expected value at year 1: 0.5 × 95.2381 + 0.5 × 97.0874 = 96.1628.
Discount at today's rate of 4%: 96.1628 ÷ 1.04 = 92.4642.
Spot yield: (100 ÷ 92.4642)^(1/2) − 1 = 1.081506^0.5 − 1 ≈ 3.995%.

Answer: Price ≈ 92.46 per 100 of face value, with a two-year spot yield of about 3.995% a year.

Example 2

A three-year annual recombining tree starts at 3%. Each year the rate moves up or down by 1 percentage point, with probability 0.5 each. Rates at year 1 are 4% and 2%. Rates at year 2 are 5%, 3% and 1%. Price a three-year zero-coupon bond with face value 100.

Show the solution

Value at year 3 is 100 at every node.
Year 2 nodes: 100 ÷ 1.05 = 95.2381; 100 ÷ 1.03 = 97.0874; 100 ÷ 1.01 = 99.0099.
Year 1 up node (rate 4%): average of 95.2381 and 97.0874 = 96.1628; divide by 1.04 = 92.4642.
Year 1 down node (rate 2%): average of 97.0874 and 99.0099 = 98.0487; divide by 1.02 = 96.1261.
Year 0: average of 92.4642 and 96.1261 = 94.2952; divide by 1.03 = 91.5489.

Answer: The three-year zero-coupon bond price is about 91.55 per 100 of face value.

Exam tips

Check the step size first. A semiannual tree with annual quoted rates is a common trap.
Questions often give risk-neutral probabilities other than 0.5. Read them before averaging.
Answer options are often close, so keep four decimals until the last step.
If a question asks for a one-period value, you only need the two successor nodes, not the full tree.
Use the sanity check: a zero-coupon price must be below face value when rates are positive.

Practice questions from Arbitrage Pricing with Term Structure Models

Binomial Interest Rate Trees in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Binomial Interest Rate Trees: frequently asked questions

What does recombining mean in a binomial interest rate tree?

It means an up move followed by a down move leads to the same node as a down move followed by an up move. The tree has n + 1 nodes after n steps, instead of 2^n. This keeps calculations short.

Why are the probabilities usually 0.5 in FRM trees?

They are risk-neutral probabilities chosen for simplicity, so the tree prices as if investors are risk-neutral. They are not forecasts of real-world rate moves. Exam questions may give different probabilities, so always read the question.

What is backward induction in bond pricing?

You start with the known payoff at maturity, then step back one period at a time. At each node you average the two next values and discount at that node's rate. You end at time 0 with the price.

How do I handle semiannual steps?

Use Δt = 0.5, so you discount by (1 + r ÷ 2) at each node when the rate is quoted as an annual rate. After pricing, convert to yield with the right compounding. Always check the question for the compounding convention.