What it means
Options are tricky to value because their payoff depends on a price that moves unpredictably. A lattice model breaks the problem into small time steps, in which the price can rise or fall by set amounts.
Together, these steps form a branching grid that looks like a lattice. At the end of the tree, you calculate what the option would be worth in each possible outcome.
For a call option (the right to buy at a fixed price), that is the share price minus the strike price, or zero if the price is lower. You then move backwards through the tree, weighting each outcome by its probability and discounting for the time value of money.
A key idea is risk-neutral valuation. Rather than guessing real-world probabilities, the model uses special probabilities that make the expected return on the share equal to the risk-free interest rate.
This lets you value the option without needing to know how investors feel about risk. Lattice models are very flexible.
They can handle American options, which can be exercised at any time before expiry, and options on assets paying dividends. They are also used to value employee share options, real options in project decisions and other contracts with early exercise features.
The more steps you use, the closer the lattice result comes to the answer from the well-known Black-Scholes formula for ordinary options. For finance teams, the model is valuable because it is transparent and can be built in a spreadsheet.
The trade-off is that the results depend on assumptions about volatility (how much the price moves) that are never known for certain.
In practice
Real-world examples.
Example
A bank trader values a six-month option on a share by building a lattice with 100 steps. She compares the result with the market price to decide whether the option is cheap or expensive.
Example
A company needs to put a value on employee share options for its accounts. The finance team uses a lattice because the options can be exercised early and the shares pay dividends.
Example
A mining company considers a project that can be expanded if metal prices rise. The finance director models the option to expand with a lattice, to see how much the flexibility is worth.
Formula
Calculation
Up factor u and down factor d describe the moves in each step.
Risk-neutral probability of an up move: p = (1 + r - d) / (u - d)
Option value today = [p x value if up + (1 - p) x value if down] / (1 + r)
Worked example (one step): a share is priced at $100 and can move to $120 (u = 1.2) or $80 (d = 0.8) in one year. The risk-free rate r is 5%, and you want to value a call option with a strike price of $100.
Step 1: Payoff if up = 120 - 100 = $20. Payoff if down = 0 (the price is below the strike).
Step 2: p = (1 + 0.05 - 0.8) / (1.2 - 0.8) = 0.25 / 0.4 = 0.625.
Step 3: Expected payoff = 0.625 x 20 + 0.375 x 0 = $12.50.
Step 4: Discount one year = 12.50 / 1.05 = about $11.90.
The call option is worth about $11.90 today. Real models use many more steps, but they follow the same logic.Case study
Seen in the real world.
Orchard Bay Energy is a fictional company deciding whether to buy the right to build a second gas plant in three years, for a fee of $5 million. The finance team was unsure how to value the right, since the decision would depend on future electricity prices.
An analyst built a lattice of possible electricity prices over three years and calculated the payoff of building in each scenario. Working backwards, she found that the right was worth about $7.4 million.
In this illustrative story, the board agreed to buy the right because its value exceeded the fee. The analyst also showed how the value changed with different volatility assumptions, which helped the board see the range of possible outcomes.
Watch out
Common mistakes.
- Treating the model output as exact, when it depends on assumptions such as volatility and interest rates.
- Using too few steps, which gives a rough answer for long-dated options.
- Confusing risk-neutral probabilities with the real chance of a price rise.
Questions
People also ask.
What is a binomial lattice?
It is a lattice in which the price can move to only one of two values, up or down, at each step.
Why use a lattice instead of Black-Scholes?
A lattice can handle early exercise and varying inputs, which the simple formula cannot.
How many steps should a lattice have?
More steps give greater accuracy, and many practical models use dozens or hundreds, as spreadsheets and software make this easy.
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