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IAI Actuarial Core Principles · Economic Modelling

Binomial Option-Pricing Model: Replicating Portfolios and Risk-Neutral Valuation

The binomial model prices an option by letting the share price move up or down at each step. You build a replicating portfolio of shares and cash, or use risk-neutral probabilities, and discount the expected payoff at the risk-free rate. Work backwards from expiry, checking early exercise for American options.

What this chapter covers

This chapter builds the binomial option-pricing model from first principles. You start with what options pay at expiry. Then you use no-arbitrage to link prices. Then you value an option in one step, extend that to several steps, and finally test whether early exercise helps the holder.

The core idea is simple. If you can copy an option's payoff with shares and cash, the option must cost the same as the copy. This gives a price that does not depend on anyone's view of how likely an up-move is. The same price comes out of risk-neutral valuation, which is the faster method for trees with many steps.

The chapter sits in the Option theory part of CM2, which carries a large share of the 2026 syllabus weighting. It links to the rest of the paper through no-arbitrage ideas, discounting at the risk-free rate, and the move from discrete trees to continuous-time models such as Black-Scholes. CM2 has a Paper A written exam and a Paper B computer-based exam, so you should be able to do the tree by hand and also set it up in R or Excel.

Option theory is one of the heaviest syllabus areas in CM2, and the binomial model is the most testable method within it. Questions are structured: you set up a tree, find a price, then add a twist such as a put, a dividend, an American feature or a change in u and d. Each step earns marks if your method and notation are clear. The same skills also help in Paper B, where you build trees in a spreadsheet or code, and the chapter reinforces no-arbitrage reasoning that you use elsewhere in the paper.

Binomial option-pricing model: topics in the order to study them

  1. 1Options Basics and Payoff DiagramsYou need to read call and put payoffs fluently before pricing anything.
  2. 2No-Arbitrage Principle and Put-Call ParityEvery binomial price rests on no-arbitrage, and parity gives you a quick check on your answers.
  3. 3One-Period Binomial Model and Replicating PortfolioThis is the building block, and the replicating argument shows why the price is unique.
  4. 4Risk-Neutral Valuation and Risk-Neutral ProbabilitiesIt gives the same price as replication but is much faster, so learn it once the one-period case is clear.
  5. 5Multi-Period Binomial TreesYou apply the one-period step repeatedly, working backwards from the final nodes.
  6. 6American Options and Early Exercise in Binomial TreesIt adds a comparison at each node, so you must be secure on the standard backward method first.
  7. 7Calibrating u and d and Link to Black-ScholesIt ties the tree to volatility and the continuous-time model, so it comes last.

How to prepare Binomial option-pricing model

Treat this as a method chapter. Marks come from setting up the tree correctly and carrying the same steps through every node, so practise by doing, not by reading.

  1. Draw payoff diagrams for long and short calls and puts until you can sketch them without help. State the payoff formula next to each, such as max(S − K, 0) for a call.
  2. Learn the no-arbitrage argument and put-call parity with its exact conditions, including how dividends or the stated assumptions change it. Check every price you find against parity where it applies.
  3. Solve one-period problems twice: once by replicating portfolio (find the share holding and the cash), and once by risk-neutral probability. Confirm the two prices agree.
  4. Memorise the risk-neutral probability for your stated setup, for example q = (e^(r) − d) ÷ (u − d) with one-step discounting at the risk-free rate, and check that d < e^r < u. Write the formula and assumptions at the start of every answer.
  5. Build two- and three-step trees. Fill in the share prices first, then the payoffs at expiry, then roll back node by node. Keep a neat layout so you can find slips quickly.
  6. For American options, compare the continuation value with the immediate exercise value at every node. Practise puts, where early exercise is more often worthwhile, and note when it is not.
  7. Rebuild a tree in Excel or R for Paper B. Then practise the written link to Black-Scholes: how u and d relate to volatility and why a finer tree approaches the continuous-time price.

Common mistakes in Binomial option-pricing model

  • Using the real-world up-probability to discount expected payoffs.

    Fix: Use the risk-neutral probability q computed from u, d and the risk-free rate. Ignore real-world probabilities for pricing.

  • Forgetting to discount, or discounting by the wrong number of periods.

    Fix: Discount by one step at each node when rolling back. Write the discount factor beside the formula before you start.

  • Ignoring early exercise in American option trees or applying it only at the first node.

    Fix: At every node, take the larger of exercise value and continuation value, and carry that value back to earlier nodes.

  • Getting the sign or size of the replicating portfolio wrong.

    Fix: Solve the two payoff equations at the up and down nodes carefully. Check by confirming the portfolio matches the option at both nodes.

  • Not checking that d < e^(rh) < u, or computing q without stating assumptions.

    Fix: State the assumptions first and check the inequality. If q is not between 0 and 1, the tree admits arbitrage.

  • Mixing up step length and annual rates when setting u, d and r.

    Fix: Write h, the step length in years, and convert r and σ to that step before computing u, d and q.

Last-day revision: Binomial option-pricing model

  • Call payoff at expiry = max(S − K, 0); put payoff = max(K − S, 0).
  • No arbitrage means two portfolios with the same payoff must have the same price.
  • Put-call parity for European options on a non-dividend share: C − P = S₀ − K·e^(−rT).
  • One-period replication: hold Δ = (Vu − Vd) ÷ (S₀u − S₀d) shares, financed by cash.
  • Option value = S₀·Δ + cash, where the cash amount may be negative (borrowing).
  • Risk-neutral probability of an up-move: q = (e^(rh) − d) ÷ (u − d) for step length h.
  • Option value = e^(−rh) × [q·Vu + (1 − q)·Vd].
  • The real-world probability of an up-move does not enter the price.
  • No-arbitrage requires d < e^(rh) < u; otherwise q falls outside 0 to 1.
  • In multi-period trees, work backwards from expiry, applying the one-step formula at each node.
  • American option value at a node = max(exercise value, continuation value).
  • In a Cox-Ross-Rubinstein style tree, u = e^(σ√h) and d = 1/u; the price tends to the Black-Scholes value as steps increase.

Binomial option-pricing model practice questions

Binomial option-pricing model in other exams

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

Binomial option-pricing model: frequently asked questions

Is the binomial model examined in both Paper A and Paper B of CM2?

Option theory is part of the CM2 syllabus, and you should be ready for it in both the written Paper A and the computer-based Paper B. Paper A tests your method and notation by hand. Paper B tests whether you can build and check a tree in a spreadsheet or code.

Do I need to learn both replication and risk-neutral valuation?

Yes. Replication explains why the price is unique, which written questions often ask you to show. Risk-neutral valuation is faster for multi-period trees. Learn both and check that they give the same price.

Why does the real-world probability not matter for the option price?

The replicating portfolio matches the option's payoff in both the up and down states, so no-arbitrage fixes the price whatever the real probabilities are. The risk-neutral probability is just a tool that reproduces that price.

When is early exercise worth considering in a binomial tree?

You must test it at every node for an American option. Whether it is optimal depends on the option type, interest rate and any dividends. For a non-dividend share, an American call is not exercised early, but an American put often can be.