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FRM Exam Part I · Binomial Trees

Risk-Neutral Valuation and Probabilities in Binomial Trees

Updated 11 October 2026 · Fact-checked

Risk-neutral valuation prices a derivative by using an artificial up-move probability p = (e^(rΔt) − d) ÷ (u − d), under which the stock earns the risk-free rate. You compute the expected payoff with p, then discount at the risk-free rate. The result equals the no-arbitrage price.

Understand Risk-Neutral Valuation and Probabilities

A binomial tree says that over one short period a stock either moves up by a factor u or down by a factor d. A derivative on that stock then has two possible payoffs. You can build a portfolio of stock and cash that copies those payoffs exactly. Because the copy has the same payoff in both states, its cost must equal the derivative's price. If not, there is arbitrage.

The key insight is that this price does not depend on how likely you think the up-move is. It depends only on u, d and the risk-free rate. So you are free to pick any probability that gives the right price. The convenient choice is the one under which the stock's expected return equals the risk-free rate. This is the risk-neutral probability p.

The probability p is set so that the expected stock price at the end of the step, discounted at the risk-free rate, equals today's price. That gives p = (e^(rΔt) − d) ÷ (u − d). The down probability is 1 − p. For p to be a valid probability you need d < e^(rΔt) < u. If this fails, there is an arbitrage in the tree.

So the method is: find p, compute the expected option payoff using p, then discount at the risk-free rate. The real-world probability is the actual chance of an up-move. It reflects investors' risk premium, and it is not used for pricing. The risk-neutral p is not a forecast. It is a pricing tool. This does not mean investors are risk-neutral. It means the price is the same as it would be in a world where they were.

For multi-step trees, you apply the same p at every node and roll back from the end. Discount one step at a time, or discount the whole expected payoff over the full term if the option is European.

Key formulas to remember

Risk-neutral up probability (no dividends)
p = (e^(rΔt) − d) ÷ (u − d)
r is the continuously compounded risk-free rate. Δt is the step length in years. Down probability is 1 − p.
Risk-neutral up probability (continuous yield q)
p = (e^((r − q)Δt) − d) ÷ (u − d)
Use for an index or currency. For a currency, q is the foreign risk-free rate. For a futures price, q = r, so p = (1 − d) ÷ (u − d).
One-step option value
f = e^(−rΔt) × [p × f_u + (1 − p) × f_d]
f_u and f_d are the option values after an up-move and a down-move.
Stock-price condition for the stock itself
S₀ = e^(−rΔt) × [p × S₀u + (1 − p) × S₀d]
This is how p is derived. Use it to check your p.
CRR factors
u = e^(σ√Δt), d = 1 ÷ u
Cox-Ross-Rubinstein matching of volatility σ.
Annual compounding version
p = ((1 + r) − d) ÷ (u − d)
Use only if the question gives r with annual compounding and Δt = 1 year.

How to solve Risk-Neutral Valuation and Probabilities questions

Use this sequence for any one-step or multi-step risk-neutral pricing question.

  1. 1Write down S₀, u, d, r, Δt, and any yield q. Check whether r is continuous or annual.
  2. 2Compute the growth factor e^(rΔt), or e^((r − q)Δt) if there is a yield.
  3. 3Compute p = (growth − d) ÷ (u − d). Check that 0 < p < 1.
  4. 4Build the tree of stock prices and compute the option payoff at each final node.
  5. 5Roll back: at each earlier node, value = e^(−rΔt) × [p × up value + (1 − p) × down value].
  6. 6For American options, compare the rolled-back value with the exercise value at each node and keep the larger.
  7. 7State the final price at time 0 and check it is sensible, for example within no-arbitrage bounds.

Quickest way: One-step shortcut with p and discounting

When to use it: Use it for one-step or two-step European options when the question gives u, d and r directly.

  1. Compute p once. Keep 4 decimals.
  2. Find only the payoffs that are positive. A zero payoff drops out of the sum.
  3. For a European option over several steps, compute the terminal payoffs and their probabilities, such as p², 2p(1 − p), (1 − p)². Sum the products, then discount once by e^(−rT).
  4. Check the answer against the sign of the option value: it must be positive and below the stock price for a call.

Common mistakes in Risk-Neutral Valuation and Probabilities

  • Using the real-world probability to price the option.

    The question gives a probability of an up-move, and it feels like it must be used.

    Fix: Price with p from the formula. Real-world probabilities are ignored for pricing a derivative on a traded asset.

  • Forgetting to discount, or discounting at the stock's expected return.

    Students compute the expected payoff and stop, or use a risk-adjusted rate.

    Fix: Always discount at the risk-free rate. Use e^(−rΔt) per step.

  • Writing p = (u − e^(rΔt)) ÷ (u − d) or swapping u and d.

    The formula is memorised without the logic.

    Fix: Remember that p is the weight on the up-state, so p = (growth − d) ÷ (u − d). Check that p × u + (1 − p) × d equals the growth factor.

  • Ignoring the dividend or foreign rate when pricing options on indices or currencies.

    Students use the stock formula for every underlying.

    Fix: Replace r with r − q in the growth factor, but still discount the payoff at r.

  • Using e^(rΔt) with the full-year rate on a multi-step tree.

    Δt is forgotten.

    Fix: Compute Δt = T ÷ number of steps first, and use it in both p and the discount factor.

  • Believing that p is the true probability of an up-move, or that investors are risk-neutral.

    The name suggests a real belief.

    Fix: Treat p as a device that makes the discounted expected stock price equal today's price. It gives the same price as replication.

Worked examples

Example 1

A stock is at $50. Over the next 6 months it can rise to $55 or fall to $46. The continuously compounded risk-free rate is 4% per year. Find the risk-neutral probability of an up-move and the value of a 6-month European call with strike $50.

Show the solution
  1. u = 55 ÷ 50 = 1.10 and d = 46 ÷ 50 = 0.92. Δt = 0.5.
  2. Growth factor = e^(0.04 × 0.5) = e^0.02 = 1.020201.
  3. p = (1.020201 − 0.92) ÷ (1.10 − 0.92) = 0.100201 ÷ 0.18 = 0.55667.
  4. Call payoffs: f_u = max(55 − 50, 0) = 5 and f_d = max(46 − 50, 0) = 0.
  5. Expected payoff = 0.55667 × 5 + 0.44333 × 0 = 2.78335.
  6. Discount: 2.78335 × e^(−0.02) = 2.78335 × 0.980199 = 2.7282.

Answer: p ≈ 0.5567 and the call is worth about $2.73.

Example 2

A stock is at $40. Each step is 1 year, with u = 1.20 and d = 0.90. The risk-free rate is 5% continuously compounded. Price a 2-year European put with strike $42.

Show the solution
  1. Growth factor = e^0.05 = 1.051271.
  2. p = (1.051271 − 0.90) ÷ (1.20 − 0.90) = 0.151271 ÷ 0.30 = 0.50424. So 1 − p = 0.49576.
  3. Terminal prices: uu = 40 × 1.44 = 57.6, ud = 40 × 1.08 = 43.2, dd = 40 × 0.81 = 32.4.
  4. Put payoffs with K = 42: uu = 0, ud = max(42 − 43.2, 0) = 0, dd = 42 − 32.4 = 9.6.
  5. Probability of dd = 0.49576² = 0.24578.
  6. Expected payoff = 0.24578 × 9.6 = 2.3595.
  7. Discount over 2 years: e^(−0.10) = 0.904837. Value = 2.3595 × 0.904837 = 2.135.

Answer: The put is worth about $2.13.

Exam tips

  • Questions often give a real-world probability as a distractor. Ignore it for pricing.
  • Always check whether the question gives r as continuous. If it does, use e^(rΔt), not 1 + r.
  • On a calculator, store p and 1 − p in memory so you can reuse them across nodes.
  • For options on currencies or indices, the change is only in p. The discount rate is still r.
  • Sanity-check p lies between 0 and 1. If it does not, you have mixed up u and d or the rate.

Practice questions from Binomial Trees

Risk-Neutral Valuation and Probabilities in other exams

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

Risk-Neutral Valuation and Probabilities: frequently asked questions

How do you calculate the risk-neutral probability of an up-move?

Compute the growth factor e^(rΔt), subtract d, and divide by u − d. For an asset with yield q, use e^((r − q)Δt). The down probability is 1 minus the result.

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

The real-world probability is the actual chance of an up-move and reflects investors' required risk premium. The risk-neutral probability is chosen so that the stock earns the risk-free rate on average. Only the risk-neutral one is used to price derivatives by discounting at r.

Why can we discount at the risk-free rate?

Under risk-neutral probabilities, the expected return on every asset is the risk-free rate. The option's price is the same as the cost of the replicating portfolio, so no risk premium needs to be added in the discount rate.

Does the risk-neutral price depend on investors' risk preferences?

No. The price depends on u, d, r and the payoff. Any investor who rules out arbitrage must agree on it.