Binomial Treesintermediate

Risk-neutral Probability

The risk-neutral probability $p$ is the synthetic weight that makes the discounted stock price a martingale. In a one-step binomial model with up and down factors $u$ and $d$ , $p = (e^{r\Delta t} - d)/(u - d)$ . Under this measure, every traded asset is expected to grow at the risk-free rate $r$ , which lets us price any derivative as a discounted expectation without estimating real-world drift or investor risk preferences.

Why it matters

Real investors demand extra return for holding risky stock. Risk-neutral pricing sidesteps that argument with a clever trick. We reweight the up and down states until the stock's expected return collapses to $r$ , then price the option as if the world were populated by indifferent investors. The trick works because the derivative can be replicated by a hedge portfolio of stock plus bond, and that portfolio's cost does not depend on anyone's risk appetite. The replication argument, not the probability story, is what gives the formula its bite.

Formulas

Risk-neutral probability

p = \frac{e^{r\Delta t} - d}{u - d}

Requires

d < e^{r\Delta t} < u

to avoid arbitrage (Hull 2022, §13.2).

Martingale condition for stock

E^{\mathbb{Q}}[S_{\Delta t}] = p\, S_0 u + (1-p)\, S_0 d = S_0 e^{r\Delta t}

Derivative price

f_0 = e^{-r\Delta t}\bigl[p f_u + (1-p) f_d\bigr]

Worked examples

Scenario

Stock $S_0$ at A$50 over one year, $u = 1.3$ , $d = 0.7$ , and continuously compounded risk-free rate $r = 0.10$ .

Solution

Risk-neutral probability $p = (e^{0.10} - 0.7)/(1.3 - 0.7) = (1.1052 - 0.7)/0.6 = 0.6753$ . Check the martingale condition. Expected stock $= 0.6753 \times 65 + 0.3247 \times 35 = A\$ 55.26 $, which equals$ 50 e^{0.10}$. The probability is purely the weight that makes the discounted stock a fair bet.

Common mistakes

✗Risk-neutral probabilities are forecasts of where the stock will actually go. They are not. They are artificial weights that make the discounted stock a martingale. A stock with strong positive drift in reality (say expected return 15%) can have $p$ above or below 0.5 depending purely on $u$ , $d$ , and $r$ .
✗The formula assumes investors are risk-neutral. It does not. The replicating portfolio argument holds for any risk preference. The risk-neutral measure is a pricing convenience, not a behavioural claim (Harrison & Kreps 1979).

Revision bullets

•Risk-neutral probability $p = (e^{r\Delta t} - d)/(u - d)$
•Martingale condition makes $E^{\mathbb{Q}}[S]$ grow at $r$
•Artificial weights, not real-world forecasts
•No-arbitrage requires $d < e^{r\Delta t} < u$
•Replication argument is what justifies the formula
•Core of all derivative pricing under Hull (2022) §13

Quick check

Under the risk-neutral measure $\mathbb{Q}$ , the expected return on any traded asset equals:

Connected topics

1-Step Tree Backward Ind.

In learning paths

Pricing Models Path

Sources

Hull (2022), §13.2
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Chapter 13 introduces risk-neutral valuation through the one-step binomial tree, including the formula for $p$ and the no-arbitrage condition.
Cox, Ross & Rubinstein (1979)
Cox, John C., Stephen A. Ross and Mark Rubinstein. "Option Pricing: A Simplified Approach." Journal of Financial Economics 7(3), 1979, pp. 229–263.
Original derivation of the binomial model showing that the up-state probability is set by no-arbitrage rather than investor beliefs.
Harrison & Kreps (1979)
Harrison, J. Michael and David M. Kreps. "Martingales and Arbitrage in Multiperiod Securities Markets." Journal of Economic Theory 20(3), 1979, pp. 381–408.
Foundational paper establishing the equivalent martingale measure as the rigorous justification for risk-neutral pricing.

How to cite this page

Dr. Phil's Quant Lab. (2026). Risk-neutral Probability. Derivatives Atlas. https://phucnguyenvan.com/concept/risk-neutral-probability

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