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FRM Part I · FRM Exam Part I

Binomial Trees for FRM Part I: Chapter Guide

A binomial tree models an asset price as moving up by factor u or down by factor d each step. You price an option by working backward from the final payoffs, discounting the risk-neutral expected value at the risk-free rate at each node. Risk-neutral probability is p = (e^(rΔt) − d) ÷ (u − d).

What this chapter covers

This chapter teaches you to value options with a simple lattice. The stock price can go up or down in each short period. You compute the option payoff at the end, then step back through the tree to today's value. The method is easy to apply by hand, and it handles early exercise, which Black-Scholes-Merton cannot.

The chapter has two linked ideas. First, you can build a portfolio of stock and borrowing that replicates the option, so no-arbitrage fixes the price. Second, the same price comes from discounting expected payoffs under risk-neutral probabilities. Later topics add multiple steps, American exercise, volatility matching with u and d, and underlyings that pay a yield, such as indices, currencies and futures.

It connects to the rest of Valuation and Risk Models and to Financial Markets and Products. It builds on option payoffs, put-call parity and the properties of options. It leads into Black-Scholes-Merton, the Greeks and delta hedging. It also uses probability and exponentials from Quantitative Analysis. Treat it as the place where option pricing logic becomes concrete.

Binomial questions are calculation-based, which suits an exam of 100 multiple-choice questions in 4 hours. Each one follows a fixed recipe, so you can win marks reliably once the steps are automatic. The ideas also support nearby topics: risk-neutral pricing, delta, and the link to Black-Scholes. Time spent here pays off across the option-pricing material. It is also a chapter where small slips, such as the wrong p or a missed early-exercise check, cost whole questions, so practice matters more than reading.

Binomial Trees: topics in the order to study them

  1. 1One-Step Binomial Tree Option PricingStart here, because the replicating portfolio and no-arbitrage logic underpin everything else.
  2. 2Risk-Neutral Valuation and ProbabilitiesIt gives the faster shortcut for the same price, and you need p for every later calculation.
  3. 3Two-Step and Multi-Step Binomial TreesIt extends the one-step method by repeating backward induction node by node.
  4. 4American Options and Early ExerciseYou need a finished tree first, because you add a comparison with intrinsic value at each node.
  5. 5Matching Volatility: Cox-Ross-Rubinstein u and dOnce the mechanics work, learn how u = e^(σ√Δt) and d = 1/u tie the tree to volatility.
  6. 6Options on Indices, Currencies and FuturesThese only change the growth rate in p, so they are easy after the core method.
  7. 7Delta and Binomial Tree Convergence to Black-ScholesIt ties delta, hedging and the Black-Scholes limit together, so it works best as the capstone.

How to prepare Binomial Trees

Work by drilling the same recipe until it is automatic. Reading alone will not give you speed or accuracy.

  1. Learn the one-step replication argument: buy Δ shares and borrow so the portfolio matches the option payoff in both states. Compute Δ = (fu − fd) ÷ (S0u − S0d).
  2. Memorize p = (e^(rΔt) − d) ÷ (u − d) and the price f = e^(−rΔt) × [p fu + (1 − p) fd]. Check that 0 < p < 1.
  3. Solve two-step trees on paper. Write the stock tree first, the payoffs second, then step backward. Do this for both calls and puts.
  4. For American options, compare the continuation value with intrinsic value at every node and keep the larger. Practice a put on a non-dividend stock, where early exercise can matter.
  5. Practice swapping the growth rate: use e^((r − q)Δt) for an index or foreign currency (q = foreign rate), and e^(0·Δt) = 1 for a futures price, so p = (1 − d) ÷ (u − d).
  6. Use your financial calculator's e^x and square root keys. Keep at least four decimals in p and u, and round only at the end.
  7. Finish with mixed timed sets that combine delta, convergence and early exercise, and review every miss.

Common mistakes in Binomial Trees

  • Using the real-world probability of an up move instead of the risk-neutral p.

    Fix: Always compute p from r, u and d. Ignore any real-world probability unless the question asks about expected returns.

  • Forgetting to discount at each step or discounting the wrong number of periods.

    Fix: Discount by e^(−rΔt) at every backward step, with Δt equal to the step length in years.

  • Skipping the early-exercise check at interior nodes of an American option.

    Fix: At every node, set the value equal to the larger of intrinsic value and discounted expected value, then carry it back.

  • Using the wrong growth rate for indices, currencies and futures.

    Fix: Use a = e^((r − q)Δt) for an index or currency, and a = 1 for a futures price. Then p = (a − d) ÷ (u − d).

  • Rounding u, d and p too early.

    Fix: Keep four or more decimals through the calculation and round only the final value.

  • Mixing up Δ in the delta formula with the time step Δt.

    Fix: Label them clearly: delta is the share count in the hedge, Δt is the length of a step in years.

Last-day revision: Binomial Trees

  • u = e^(σ√Δt) and d = 1/u in the Cox-Ross-Rubinstein setup.
  • Risk-neutral p = (e^(rΔt) − d) ÷ (u − d) for a non-dividend stock.
  • Option value = e^(−rΔt) × [p fu + (1 − p) fd], discounted one step at a time.
  • One-step delta = (fu − fd) ÷ (S0u − S0d), the shares held in the hedge.
  • For an index or currency, replace e^(rΔt) with e^((r − q)Δt) in p.
  • For a futures option, p = (1 − d) ÷ (u − d), since the futures price has no cost of carry.
  • No-arbitrage requires d < e^(rΔt) < u.
  • American options: value at each node = max(intrinsic value, continuation value).
  • An American call on a non-dividend stock is not exercised early, so it equals the European call.
  • Build the stock tree first, then the payoffs, then work backward.
  • More steps make the tree price converge to the Black-Scholes-Merton value for European options.
  • Real-world probabilities of up and down moves do not enter the price.

Binomial Trees practice questions

Binomial 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 Trees: frequently asked questions

How many steps do I need to solve by hand in the FRM Part I exam?

Expect one-step and two-step trees, since they can be solved in a few minutes. Larger trees are discussed conceptually, mainly for convergence to Black-Scholes-Merton. Practice two-step trees until they take only a few minutes.

What is the difference between real-world and risk-neutral probability?

Real-world probability describes how likely the price is to rise in practice. Risk-neutral probability is the weight that makes the discounted expected stock price equal today's price, and it is used only for pricing. The option price does not depend on the real-world probability.

Do I need to know how to derive u and d?

You should know the Cox-Ross-Rubinstein formulas u = e^(σ√Δt) and d = 1/u and how to apply them. A full derivation is not needed. Focus on using them to compute the tree and p.

When is early exercise optimal in a binomial tree?

Early exercise is optimal at a node when intrinsic value exceeds the continuation value. For a put, this is more likely when it is deep in the money and interest rates are high. For a call, it is more relevant when the underlying pays a dividend or yield.

Can I use a financial calculator for these questions?

Yes. You need the e^x and square root functions to compute u, d, p and the discount factors. Store p and the discount factor in memory so you can reuse them across nodes.