What it means
Start with today's value at the left of the page. Draw two branches to the next period, one for an up move and one for a down move, then repeat from each of those points until you reach the end of the period you care about.
Most trees are built so that an up move followed by a down move lands on the same value as a down followed by an up. This "recombining" property matters enormously in practice, because it means a ten-step tree has eleven end points rather than more than a thousand separate paths.
The tree is valued backwards, not forwards. You write the payoff at each end point, then step back one column at a time, replacing each pair of branches with their probability-weighted average discounted for one period, until a single number remains at the starting point.
Trees are used well beyond share options. A drug developer can map stage-gate decisions, a miner can map the choice of when to open a pit, and a property developer can map the option to sell land rather than build, all with the same backward-induction technique.
The size of the up and down moves is usually set from the volatility of the underlying value and the length of a step, so a more uncertain asset produces a wider fan. Shorter steps mean more nodes and a smoother, more accurate answer, at the cost of more computation.
The main limitation is that a tree needs a clean two-outcome structure at each node and a stable set of assumptions. When the decision has many possible outcomes at once, or when the value depends on the path taken rather than the end point, a simulation is usually the better tool.
In practice
Real-world examples.
Example
A utility is deciding whether to keep a mothballed power station available for another three years. Mapping annual up and down moves in the spark spread shows the option to restart is worth more than the standby cost, so the plant stays on the books. The tree makes the argument concrete for a sceptical board.
Example
A bank prices a mortgage that borrowers may repay early at any time. It builds a tree of future interest rates and, at every node, compares the value of continuing with the value of the borrower refinancing, which puts a number on the early repayment right.
Example
A biotechnology firm faces three sequential trial stages, each of which it can abandon. Laying the stages out as a tree and valuing backwards shows that the freedom to stop after stage two is worth more than the entire stage-three budget, which changes how the programme is funded.
Formula
Calculation
Risk-neutral probability per step: p = ((1 + r) - d) / (u - d). At each node, value = [p x (value if up) + (1 - p) x (value if down)] / (1 + r).
Worked example: a project is worth $100 million today. Each year for two years its value either rises 20% (u = 1.20) or falls 20% (d = 0.80), and the risk-free rate is 4% a year. The owner can pay $100 million at the end of year two to develop it, or walk away.
p = (1.04 - 0.80) / (1.20 - 0.80) = 0.24 / 0.40 = 0.60
Year-two values: $144 million, $96 million and $64 million. Payoffs after the $100 million cost: $44 million, $0 and $0.
Upper year-one node: (0.60 x $44m + 0.40 x $0) / 1.04 = $26.4m / 1.04 = $25.38 million
Lower year-one node: both branches pay nothing, so the value is $0
Today: (0.60 x $25.38m + 0.40 x $0) / 1.04 = $15.23m / 1.04 = $14.64 million
The right to develop is therefore worth about $14.64 million even though the project breaks exactly even at today's value.Case study
Seen in the real world.
Kestrel Mining is an illustrative and completely fictional exploration company holding a licence over an undeveloped copper deposit. An engineering study put the present value of the developed mine at $100 million and the cost of building it at $100 million, so the internal recommendation was to let the licence lapse because the project merely broke even.
The corporate finance team instead built a two-step binomial tree using the figures above: a 20% annual move up or down in the value of the deposit, a 4% risk-free rate and a two-year window before the licence expired. Only the top branch, at $144 million, justified spending the $100 million build cost, giving a payoff of $44 million; the other two outcomes were worth nothing because Kestrel could simply walk away.
Discounting backwards gave the licence a value of roughly $14.64 million. In this illustrative story Kestrel renewed the licence for a fraction of that amount and later sold it to a larger miner, on the strength of an argument that a simple break-even calculation would never have produced.
Watch out
Common mistakes.
- Valuing a tree forwards by guessing which branch will happen, instead of working backwards from every possible ending.
- Using real-world probabilities and a risk-adjusted discount rate together with risk-neutral logic, which double-counts risk.
- Building so many steps that the model becomes unauditable, when the answer stopped changing materially many steps earlier.
Questions
People also ask.
What makes a tree "recombining"?
An up move followed by a down move must land on the same value as a down move followed by an up move, which keeps the number of nodes manageable.
Can a tree handle more than two outcomes per step?
Trinomial and multinomial versions exist, but adding steps to a binomial tree usually achieves the same accuracy with less complexity.
Is a binomial tree the same as a decision tree?
They look alike, but a decision tree branches on management choices with real probabilities while a binomial tree branches on market moves and uses risk-neutral pricing.
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