Skip to content

Economic Modelling · Binomial option-pricing model

Risk-Neutral Valuation and Risk-Neutral Probabilities in the Binomial Model

Updated 11 October 2026 · Fact-checked

Risk-neutral valuation prices a derivative as the discounted expected payoff, using probabilities q chosen so the share's expected growth equals the risk-free rate. In a binomial model, q = (e^(rΔt) − d) ÷ (u − d). Value = e^(−rΔt) × [q × Vu + (1 − q) × Vd].

Understand Risk-Neutral Valuation and Risk-Neutral Probabilities

A derivative can be priced without knowing the real chance that a share goes up. You build a portfolio of shares and cash that copies the derivative's payoff in every state. If two things pay the same, they must cost the same, or there is an arbitrage. That is the no-arbitrage argument.

Working this out gives a neat result. The price equals a discounted expected payoff, but the expectation uses special probabilities, not the real ones. These are the risk-neutral probabilities, q for an up move and 1 − q for a down move. They are chosen so that the share itself grows, on average, at the risk-free rate.

The name comes from a thought experiment. In a world where investors did not care about risk, every asset would have an expected return equal to the risk-free rate. q is the probability that makes the share behave that way. It is a pricing tool, not a forecast. The real-world probability p reflects actual expected return and investors' risk premium. It does not appear in the option price.

The method works because the option payoff can be replicated. The real-world p and the risk premium are already built into the share price S0. You only need S0, u, d, r and the payoff. Use the same q for every step in a multi-period tree when u, d and r are constant.

Note that q is only a probability if d < e^(rΔt) < u. If this fails, there is an arbitrage in the model. Always state the assumptions: no arbitrage, no transaction costs, share moves to only two values each step, and a constant continuously compounded risk-free rate r.

Key rules to remember

Risk-neutral probability (continuous compounding)
q = (e^(rΔt) − d) ÷ (u − d)
Here u and d are the up and down factors per step, and Δt is the step length in years. Probability of a down move is 1 − q.
Risk-neutral probability (annual compounding rate i)
q = ((1 + i)^Δt − d) ÷ (u − d)
Use when the question gives an effective annual rate. For one yearly step, q = (1 + i − d) ÷ (u − d).
One-period option value
V0 = e^(−rΔt) × [q × Vu + (1 − q) × Vd]
Vu and Vd are the option values after an up and a down move.
Martingale check on the share
S0 = e^(−rΔt) × [q × S0u + (1 − q) × S0d]
Use it to check your q. The discounted expected share price must equal today's price.
No-arbitrage condition
d < e^(rΔt) < u
Ensures 0 < q < 1.
Multi-period value
V0 = e^(−rnΔt) × Σ P(j ups) × V(n, j), with P(j ups) = C(n, j) q^j (1 − q)^(n−j)
Valid for European options on a recombining tree with constant u, d and r.

How to solve Risk-Neutral Valuation and Risk-Neutral Probabilities questions

Use this method for any question on risk-neutral valuation in a binomial tree.

  1. 1Write down S0, u, d, r, Δt and the payoff (call: max(S − K, 0); put: max(K − S, 0)). State your assumptions.
  2. 2Check that d < e^(rΔt) < u. If not, there is an arbitrage and q is not a probability.
  3. 3Calculate q = (e^(rΔt) − d) ÷ (u − d) and 1 − q. Keep at least four decimal places.
  4. 4Build the share price tree to the expiry date and work out the payoff at each final node.
  5. 5For a European option, discount: V = e^(−rΔt) × [q × Vu + (1 − q) × Vd], or use the binomial weights over all steps and discount once.
  6. 6For an American option, compare the discounted value with the exercise value at every node and take the larger.
  7. 7Verify with the martingale check on the share, then state the answer with units.

Quickest way: Compute q once, then roll back

When to use it: Use for MCQs and for multi-step trees where u, d and r are constant.

  1. Compute q straight away and check it lies between 0 and 1.
  2. For a European option over n steps, find the payoffs only at expiry. Skip the middle nodes.
  3. Weight each payoff by C(n, j) q^j (1 − q)^(n−j), add, then discount once by e^(−rnΔt).
  4. For two steps, the three weights are q², 2q(1 − q) and (1 − q)².
  5. If the payoff is zero at some nodes, ignore them. This often removes half the arithmetic.

Common mistakes in Risk-Neutral Valuation and Risk-Neutral Probabilities

  • Using the real-world probability p in the pricing formula.

    The tree is often given with p, and it feels natural to use it as the weight.

    Fix: Real-world p never enters the price. Always compute q from u, d and r. Use p only if asked for a real-world expected payoff.

  • Writing q = (u − e^(rΔt)) ÷ (u − d), which is the down probability.

    The numerator is swapped from memory.

    Fix: The up probability has e^(rΔt) − d on top. Check that q × u + (1 − q) × d = e^(rΔt).

  • Forgetting Δt, or using the annual rate on a half-year step.

    Students copy the one-year formula into a multi-step tree.

    Fix: Write Δt beside every step. Use e^(rΔt) for the growth factor per step, not e^r.

  • Discounting at every step and also at the end for a European option.

    Mixing the rollback method with the one-shot expectation method.

    Fix: Choose one method. Rollback discounts each step. The expectation method discounts once over the full term.

  • Not checking that 0 < q < 1.

    Students trust the formula without testing the inputs.

    Fix: Check d < e^(rΔt) < u first. If it fails, say there is an arbitrage opportunity.

  • Saying risk-neutral valuation assumes investors are risk neutral.

    The name is misleading.

    Fix: Say it is a mathematical device. Investors may be risk averse. The price is the same because it comes from replication.

Worked examples

Example 1

A share is priced at ₹100. Each year it either rises by 20% or falls by 10%. The effective annual risk-free rate is 5%. Find the risk-neutral probability q and the value of a one-year European call with strike ₹105.

Show the solution
  1. u = 1.2, d = 0.9, growth factor 1.05. Check: 0.9 < 1.05 < 1.2, so no arbitrage.
  2. q = (1.05 − 0.9) ÷ (1.2 − 0.9) = 0.15 ÷ 0.30 = 0.5.
  3. Share prices at expiry: up ₹120, down ₹90.
  4. Call payoffs: Vu = max(120 − 105, 0) = ₹15. Vd = max(90 − 105, 0) = 0.
  5. V0 = [0.5 × 15 + 0.5 × 0] ÷ 1.05 = 7.5 ÷ 1.05 = 7.142857.
  6. Check on the share: (0.5 × 120 + 0.5 × 90) ÷ 1.05 = 105 ÷ 1.05 = 100. Correct.

Answer: q = 0.5 and the call is worth about ₹7.14.

Example 2

A share is priced at ₹80. Over each of two successive six-month periods it rises by a factor u = 1.25 or falls by a factor d = 0.8. The continuously compounded risk-free rate is 0, so e^(rΔt) = 1. Find the value of a one-year European put with strike ₹80.

Show the solution
  1. Check: 0.8 < 1 < 1.25, so no arbitrage.
  2. q = (1 − 0.8) ÷ (1.25 − 0.8) = 0.2 ÷ 0.45 = 4/9. So 1 − q = 5/9.
  3. Expiry prices: uu: 80 × 1.5625 = ₹125. ud: 80 × 1.25 × 0.8 = ₹80. dd: 80 × 0.64 = ₹51.20.
  4. Put payoffs: uu: 0. ud: max(80 − 80, 0) = 0. dd: 80 − 51.20 = ₹28.80.
  5. Probability of dd = (5/9)² = 25/81.
  6. Discount factor = 1 because r = 0.
  7. V0 = 28.80 × 25/81 = 720 ÷ 81 = 8.8889.

Answer: The put is worth about ₹8.89.

Exam tips

  • Show q first, with its formula and the no-arbitrage check. Marks are usually given for the method even if later arithmetic slips.
  • State clearly that p does not affect the price. Examiners often ask you to explain the difference between p and q in words.
  • In written answers, name the assumptions: no arbitrage, no transaction costs, two outcomes per step, constant r, and the ability to trade any amount of shares and cash.
  • In the computer-based paper, set q, the payoff vector and the discount factor in separate cells or variables so you can re-use them if the strike changes.
  • Use the martingale check on the share to catch errors in q in under ten seconds.

Practice questions from Binomial option-pricing model

Risk-Neutral Valuation and Risk-Neutral 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 Risk-Neutral Probabilities: frequently asked questions

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

Real-world probability p is the actual chance of an up move and reflects the share's expected return, including a risk premium. Risk-neutral probability q is the value that makes the share grow at the risk-free rate on average. You use q for pricing and p for real-world projections.

How do I calculate risk-neutral probability q?

Use q = (e^(rΔt) − d) ÷ (u − d). Here r is the continuously compounded risk-free rate and Δt is the step length. If the question gives an annual effective rate i, replace e^(rΔt) with (1 + i)^Δt.

Does the option price depend on the real-world probability?

No. The price comes from replicating the payoff with shares and cash, which depends only on S0, u, d, r and the payoff. The real-world probability and investors' risk attitudes are already reflected in S0.

Can q be negative or above 1?

Only if the model allows arbitrage, meaning e^(rΔt) lies outside the range from d to u. In that case q is not a probability and the model is invalid. Check this before you price anything.