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FRM Exam Part II · Arbitrage Pricing with Term Structure Models

Risk-Neutral Pricing and Probabilities for FRM Part II

Updated 11 October 2026 · Fact-checked

Risk-neutral pricing values a derivative as the expected payoff under risk-neutral probabilities, discounted at the risk-free rate. These probabilities are chosen so the tree reprices the traded securities, which rules out arbitrage. They are not real-world probabilities. They bake in risk premiums. On a one-step tree, solve for p, then discount.

Understand Risk-Neutral Pricing and Probabilities

A derivative can be replicated by a portfolio of traded securities. If so, its price must equal the cost of that portfolio, or an arbitrage exists. This is the no-arbitrage principle. It means you can price the derivative without knowing how likely each outcome really is.

Risk-neutral probabilities are a shortcut for this replication. They are the probabilities that make the expected discounted value of each traded security equal to its current price, when you discount at the risk-free rate. Once you find them, any derivative on the same tree is priced by taking its expected payoff under them and discounting. You never need to build the replicating portfolio.

Real-world probabilities (also called physical or objective probabilities) describe how often outcomes actually occur. Investors want a higher expected return on risky assets. So the real-world expected return on a risky bond exceeds the risk-free rate. Under risk-neutral probabilities, the expected return equals the risk-free rate. When the asset carries a positive risk premium, risk-neutral probabilities shift weight toward the bad states, such as higher rates for a bond or defaults for a credit. The gap between the two sets of probabilities reflects the risk premium.

On an interest rate tree, the risk-neutral probability is set so the tree prices the observed bond (or the spot curve) exactly. The same idea underlies term structure models: the drift used for pricing is the risk-neutral drift, not the real-world drift. Use real-world probabilities for forecasting and risk measurement such as VaR. Use risk-neutral ones for pricing and hedging.

Key formulas to remember

Risk-neutral pricing, one step
V₀ = [p × V_up + (1 − p) × V_down] ÷ (1 + r × Δt)
Discount at the risk-free (tree) rate for that node and period, using simple compounding. With continuous compounding use e^(−r × Δt) instead.
Risk-neutral probability for a security with a known price
P₀ = [p × P_up + (1 − p) × P_down] ÷ (1 + r × Δt), solved for p
Use a traded security, such as a zero-coupon bond or a stock, to find p. Then reuse p for the derivative.
Risk-neutral probability for a stock (simple compounding)
p = ((1 + r × Δt) − d) ÷ (u − d)
Applies when the stock pays no dividends. u and d are the up and down gross return factors. This matches the (1 + r × Δt) convention used in the other formulas and in the worked examples. The exponential form p = (e^(rΔt) − d) ÷ (u − d) applies only under continuous compounding, and it gives a slightly different p.
Rates tree with equal 50% probabilities
Price at node = 0.5 × (V_up + V_down) ÷ (1 + r × Δt)
Many FRM tree questions state p = 0.5. Then the tree values already are risk-neutral.
Risk premium link
Real-world expected return − risk-free rate = risk premium
Under risk-neutral probabilities the expected return on every traded asset equals the risk-free rate.

How to solve Risk-Neutral Pricing and Probabilities questions

Use this order for any tree or risk-neutral pricing question.

  1. 1Identify what is given: the tree of rates or prices, the time step, and any probabilities. Check whether the probabilities are stated as risk-neutral or real-world.
  2. 2If p is not given, pick a traded security with a known price today, such as a zero-coupon bond or the stock.
  3. 3Write its one-step pricing equation using up and down values and the risk-free rate for that step. Solve for the risk-neutral probability p.
  4. 4Check that 0 < p < 1. If not, the tree allows arbitrage.
  5. 5Work backward from maturity. At each node, compute the expected value under p of the next-step values, then discount at that node's rate.
  6. 6Add any cash flows (coupons, strikes, exercise value) at the right nodes. For options, take the exercise or hold maximum at each node where exercise is allowed.
  7. 7State the answer and, if asked, interpret it: compare with the real-world expectation and explain the risk premium.

Quickest way: Solve for p, then roll back

When to use it: Use for one- or two-step tree questions with a stated risk-free rate and a known security price.

  1. Write p × up + (1 − p) × down = price × (1 + r). Solve p in one line.
  2. Check p is between 0 and 1.
  3. Compute the derivative payoffs at the end nodes.
  4. Roll back once: p × up payoff + (1 − p) × down payoff, divided by (1 + r).
  5. If the question gives p = 0.5, skip step 1.

Common mistakes in Risk-Neutral Pricing and Probabilities

  • Using real-world probabilities to price a derivative.

    The question gives a 'probability of an up move' that sounds like the one to use.

    Fix: Check the wording. Price with risk-neutral p solved from a traded security. Use real-world probabilities only for expected return or risk measurement.

  • Discounting at the expected return or a risky rate.

    Students think risky payoffs need a higher discount rate.

    Fix: Under risk-neutral probabilities, the risk is already in p. Discount at the risk-free rate for that node.

  • Thinking risk-neutral probabilities mean investors are risk-neutral.

    The name is misleading.

    Fix: It is a pricing device. Real investors are risk-averse. The probabilities are adjusted so that the math works as if they were not.

  • Mixing up p and 1 − p when solving the equation.

    Rushing and reversing up and down values.

    Fix: Always write p next to the up value. Substitute your p back into the security price as a check.

  • Forgetting that the discount rate changes by node in a rates tree.

    Students reuse the first-step rate.

    Fix: Discount each step at the rate at the node you are rolling back to, not the initial rate.

  • Not checking that p lies between 0 and 1.

    Students stop once they get a number.

    Fix: A p outside 0 to 1 signals an arbitrage in the tree. Recheck inputs.

Worked examples

Example 1

A two-year zero-coupon bond costs 90.70 per 100 of face value today. One year from now it will be a one-year zero. Its price then is either 96.0 (up state, lower one-year rate) or 93.0 (down state, higher one-year rate). The one-year risk-free rate today is 4%. Using a single step, find the risk-neutral probability of the up state. If the real-world probability of the up state were 0.5, what would the real-world expected return on the bond be?

Show the solution
  1. Pricing equation: 90.70 = [p × 96.0 + (1 − p) × 93.0] ÷ 1.04.
  2. Multiply: 90.70 × 1.04 = 94.328.
  3. So 94.328 = 93.0 + 3p, giving p = 1.328 ÷ 3 = 0.4427.
  4. Check: 0 < 0.4427 < 1, and 94.328 lies between 93.0 and 96.0, so the tree has no arbitrage. The end prices imply one-year rates of 100 ÷ 96 − 1 = 4.17% (up) and 100 ÷ 93 − 1 = 7.53% (down). The 2-year price implies a forward one-year rate of 1.05² ÷ 1.04 − 1 = 6.0%, which sits between them.
  5. Check by substitution: 0.4427 × 96.0 + 0.5573 × 93.0 = 94.328. Discounted at 1.04 this gives 90.70.
  6. Real-world comparison with p = 0.5: expected price = 0.5 × 96.0 + 0.5 × 93.0 = 94.5.
  7. Expected return = 94.5 ÷ 90.70 − 1 = 4.19%. This is above the 4% risk-free rate, so the difference of about 0.19% is the risk premium.
  8. Under the risk-neutral probability the expected return is exactly 4%. The weight on the up state falls from 0.5 to 0.4427, so it shifts toward the bad state (higher rate, lower bond price). This tilt appears because the bond carries a positive risk premium.

Answer: The risk-neutral probability of the up state is about 0.4427. With a real-world probability of 0.5, the expected return is about 4.19%, which is above 4%. The gap of about 0.19% is the risk premium.

Example 2

A one-period tree has a risk-free rate of 4% per period. A security costs 100 now and is worth 112 in the up state or 96 in the down state. A call option on the security with a strike of 100 expires at the end of the period. Find the risk-neutral probability and the call price.

Show the solution
  1. Solve: 100 × 1.04 = 104 = p × 112 + (1 − p) × 96.
  2. 104 = 96 + 16p, so p = 8 ÷ 16 = 0.5.
  3. Call payoff: up = max(112 − 100, 0) = 12. Down = max(96 − 100, 0) = 0.
  4. Expected payoff: 0.5 × 12 + 0.5 × 0 = 6.
  5. Discount: 6 ÷ 1.04 = 5.769.

Answer: p = 0.5 and the call price is about 5.77.

Exam tips

  • Read whether the question gives risk-neutral or real-world probabilities. The wording decides which method applies.
  • Expect conceptual MCQs: which probability is used for pricing, which for VaR, and why risk-neutral weights tilt toward bad states when the asset carries a positive risk premium.
  • Always discount at the risk-free rate for that node. Many wrong options use a risky rate.
  • Do a quick sanity check: a price cannot exceed the best payoff discounted, and p must be between 0 and 1.
  • Be ready to explain that the difference between the two probability measures reflects the risk premium.

Practice questions from Arbitrage Pricing with Term Structure Models

Risk-Neutral Pricing 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 Pricing and Probabilities: frequently asked questions

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

Real-world probabilities describe how likely outcomes actually are. Risk-neutral probabilities are chosen so that traded assets earn the risk-free rate on average. When an asset carries a positive risk premium, they shift weight toward bad states, and the gap reflects the risk premium.

How do I calculate a risk-neutral probability from a bond?

Write the pricing equation: price today = [p × up value + (1 − p) × down value] ÷ (1 + r). Solve for p. Confirm it lies between 0 and 1. Then reuse it for any derivative on the same tree.

Why do we use risk-neutral probabilities if investors are risk-averse?

Derivative prices come from replication and no-arbitrage, not from investor preferences. The risk-neutral measure is a convenient way to compute that price. Risk aversion is already reflected in the observed prices used to solve for p.

Should I use risk-neutral probabilities for VaR?

No. VaR and other risk measures aim to forecast actual losses, so they use real-world probabilities. Risk-neutral probabilities are for pricing and hedging.