FRM Part 1 – Financial Markets and Products · FRM
Risk-Neutral Probabilities: A Weight, Not a Forecast
The risk-neutral probability is the single most misread object in derivatives pricing, and almost all of the confusion comes from the word probability.
It is not anyone’s estimate of how likely a price rise is. Nobody believes it. It is a weight, constructed so that a particular calculation gives the right answer, and its defining property is that the actual probability of the underlying going up plays no part in the price of the derivative at all. Once that is accepted the topic becomes mechanical, and this article works through why it is true rather than asserting it.
Key Takeaways
- A derivative’s value is the expected payoff under risk-neutral weights, discounted at the risk-free rate.
- The weight is the risk-free growth factor minus d, divided by u minus d, and it is a pricing device rather than anyone’s forecast.
- It works because the derivative can be replicated by the underlying and cash, so no-arbitrage fixes the price and preferences are already in the underlying.
- In the worked example the risk-neutral calculation and the replicating portfolio both give 8.2052.
- Using the real probability with the risk-free rate gives 12.84, and reconciling it needs a 49.8% discount rate nobody could know in advance.
- The requirement that the growth factor lies between d and u is the no-arbitrage condition, so a weight outside zero to one means inconsistent inputs.
The idea, stated precisely
Risk-neutral valuation says a derivative can be priced in two steps. Compute the expected payoff using a set of probabilities under which every asset is expected to earn the risk-free rate. Discount that expectation at the risk-free rate.
f = e−rT [ p × fu + (1 − p) × fd ]
f is the derivative’s value today, fu and fd are its payoffs in the up and down states, and p is the risk-neutral probability of the up state
The world this describes does not exist. In it, no investor demands compensation for bearing risk, so every asset earns the risk-free rate and every expected payoff is discounted at that rate. Real investors are not risk neutral and real risky assets are expected to earn more.
That does not matter, and the reason it does not matter is the whole content of the topic. The derivative can be exactly replicated by a portfolio of the underlying and cash. Once that is true, its price is pinned by no-arbitrage to the cost of the replicating portfolio, and the price of the underlying already contains whatever risk premium investors demand. Preferences have been priced in one level down, so they cannot enter again.
Where the formula comes from
Take one period and two outcomes. The underlying starts at S0 and moves to either S0u or S0d. Build a portfolio that is long Δ units of the underlying and short one derivative, and choose Δ so the portfolio is worth the same in both states.
Δ = (fu − fd) / (S0u − S0d)
the hedge ratio that makes the portfolio’s value identical whichever way the underlying moves
A portfolio worth the same in every state is riskless, so under no-arbitrage it must earn exactly the risk-free rate. Setting its certain future value equal to its present value grown at r, and rearranging for the derivative’s price, produces the expression in Section 1 with
p = (erT − d) / (u − d)
for discrete compounding, replace erT with (1 + r)
Notice what did not appear anywhere in that derivation: the real probability of an up move. It was never used, because the argument never needed to know it. The hedge removes the uncertainty rather than averaging over it.
It is worth pausing on why the hedge is possible at all, since everything rests on it. With two future states and two instruments to trade, the underlying and cash, the system has exactly as many unknowns as equations, so any payoff pattern across the two states can be reproduced. This is what completeness means in this setting. Add a third possible outcome without adding a third tradable instrument and the equations no longer close, replication fails, and there is no longer a single arbitrage-free price. The binomial model is not simplified for teaching convenience. Two branches is the largest number that keeps the argument working.
The quantity p is not a probability that was discovered. It is the number that makes a discounted expectation reproduce a price already determined by replication. It behaves like a probability, since it lies between zero and one and the two weights sum to one, which is why the term is used. Reading it as a forecast is reading a piece of algebra as a belief.
A worked valuation, and the replication check
A share trades at 100. In one year it will be worth either 120 or 85. The continuously compounded risk-free rate is 5%. Value a one year European call with a strike of 105, then verify the answer by replication.
- u = 120 / 100 = 1.20 and d = 85 / 100 = 0.85. Growth factor erT = e0.05 = 1.0513.
- p = (1.0513 − 0.85) / (1.20 − 0.85) = 0.2013 / 0.35 = 0.5751, so 1 − p = 0.4249.
- Payoffs: fu = max(120 − 105, 0) = 15 and fd = max(85 − 105, 0) = 0.
- f = e−0.05 [0.5751 × 15 + 0.4249 × 0] = 0.9512 × 8.6259 = 8.2052.
- Replication check: Δ = (15 − 0) / (120 − 85) = 0.4286 shares. Borrowing B such that 0.4286 × 120 − 1.0513B = 15 gives B = 34.6519.
- Cost of the replicating portfolio = 0.4286 × 100 − 34.6519 = 8.2052.
Answer: 8.21, by both routes, to four decimal places. That agreement is not a coincidence and it is the point of the exercise. The risk-neutral calculation is a shortcut for building the replicating portfolio, and it exists because the shortcut scales to many periods while drawing the portfolio out each time does not.
| Risk-neutral expectation | Replicating portfolio | |
|---|---|---|
| What you compute first | The weight p from r, u and d | The hedge ratio from the two payoffs and the two prices |
| The number | p = 0.5751 | Δ = 0.4286 shares |
| Second step | Weight the payoffs and discount | Solve for the borrowing that makes both states match |
| The number | e−0.05 × 8.6259 | B = 34.6519 |
| Price | 8.2052 | 0.4286 × 100 − 34.6519 = 8.2052 |
| Scales to many periods | Yes, the weight is unchanged at every node | Poorly, the portfolio must be rebuilt at each node |
The last row is why the risk-neutral form is the one that survives into practice. Both routes are equally correct on one period, and only one of them is workable on fifty.
Why the real probability never appears
Suppose you are confident the share will rise, and put the real probability of the up state at 0.9. Value the option the way instinct suggests: take the real expected payoff and discount it at the risk-free rate.
Using a real-world probability of 0.9 for the up state, what does the instinctive calculation give, and why is it wrong?
- Real expected payoff = 0.9 × 15 + 0.1 × 0 = 13.50.
- Discounted at the risk-free rate: 13.50 × 0.9512 = 12.84.
- The true no-arbitrage price is 8.21, so this overstates it by more than half.
- To reconcile the two, the option would have to be discounted at a rate satisfying 13.50 × e−kT = 8.21, which gives k = ln(13.50 / 8.2052) = 0.498, that is 49.8%.
Answer: 12.84, and it is wrong because it discounts a risky payoff at a riskless rate. The correct discount rate for this option is about 49.8%, which is enormous and which nobody could have known in advance. Risk-neutral valuation exists precisely to avoid having to know it: it replaces one unknown, the risk-adjusted discount rate, with a construction that removes the need for it.
There is a neat consistency check available on the same numbers. Under the risk-neutral weights the underlying itself must be expected to earn the risk-free rate, and it does: 0.5751 times 120 plus 0.4249 times 85 comes to 105.13, which is exactly 100 grown at 5% for one year. That is not a coincidence, it is the property p was constructed to have, and it is a quick way to confirm a weight before using it in a valuation rather than discovering the error in the answer.
That is the trade the method makes. You give up any claim that p means something about the world, and in exchange you get to discount at a rate you can observe. Anyone insisting on real probabilities has to supply a risk premium for the derivative, and that premium depends on the derivative’s own leverage, which changes continuously as the underlying moves.
The no-arbitrage condition hiding in the formula
For p to behave like a probability it must lie strictly between zero and one. Applying that to the formula gives a condition on the inputs.
d < erT < u
the risk-free growth factor must lie strictly between the down and up moves
This is not a technical footnote. It is the no-arbitrage condition itself, arriving in disguise. If erT were above u, the underlying would earn less than the risk-free rate in every state, so shorting it and lending the proceeds would be a riskless profit. If erT were below d, the underlying would beat the risk-free rate in every state, so borrowing to buy it would be. In the worked example, 0.85 < 1.0513 < 1.20, so the market is coherent and p comes out at 0.5751.
Notice also that the condition involves no reference to the derivative at all. It is a statement about the underlying, the rate and the time step, so it either holds for every derivative written on that underlying or for none of them. That is consistent with what the model claims: the arbitrage-free price of a derivative is determined by the market it lives in, not by the contract’s own features.
A p outside the zero to one range in an exam answer is therefore not a rounding problem. It is a signal that the stated inputs describe a market containing an arbitrage, and the usual cause is a mistake in u, d or the rate.
Risk-neutral is not real-world
The distinction between the two probability measures runs well beyond options, and mixing them is a common and consequential error.
| Risk-neutral, market implied | Real-world, physical | |
|---|---|---|
| Where it comes from | Backed out of observed prices through a pricing model | Estimated from historical experience or from a statistical model |
| What it is used for | Pricing and hedging | Risk management, expected loss, capital, forecasting |
| Does it contain a risk premium | Yes, and it cannot be separated out without further assumptions | No, it is meant to describe frequency |
| Typical relationship for credit | Higher | Lower |
The credit example is the one that catches people. A default probability implied by a CDS spread is risk-neutral, and it is systematically higher than the historical default frequency for the same rating, because it contains compensation for bearing default risk as well as the risk itself. Using it as a forecast of how often the borrower will default overstates the expected loss, sometimes by a large multiple. Using a historical rate to price a CDS understates the spread. The two answer different questions and neither substitutes for the other.
Where this goes next
The one period tree is a teaching device, and everything built on top of it inherits the same logic.
Multi-period binomial trees apply the same p at every node and work backwards from the payoffs, which is where the method’s practical advantage appears: replication would have to be recomputed at every node, while the weights do not change. American options are handled inside the same recursion by comparing the continuation value at each node with the immediate exercise value and taking the larger.
Let the number of periods grow and the step size shrink, and the binomial model converges to Black-Scholes-Merton, which is the same risk-neutral expectation written in continuous time. The N(d2) term in that formula is a risk-neutral probability of finishing in the money, which is why it is not the real probability of exercise and why quoting it as one is wrong.
One caution to carry forward, because it is where the method’s limits sit. Everything above assumed the derivative can be replicated by trading the underlying and cash, continuously and without cost. Where that assumption weakens the conclusion weakens with it. An underlying that cannot be shorted, a market where trading moves the price, transaction costs, or a payoff that depends on something untradable all break the replication argument to some degree, and the price stops being a single number determined by no-arbitrage and becomes a range. The mathematics does not announce this. It is worth knowing which assumption is doing the work.
The reliable questions are: compute p from u, d and r; value a one or two period option; and state why the real probability is irrelevant. For the last one, the answer that earns the mark is that the derivative can be replicated by the underlying and cash, so no-arbitrage fixes its price, and the underlying’s price already reflects investors’ risk preferences. An answer that says “because investors are assumed to be risk neutral” has repeated the name of the method rather than explained it.
The short version
- A derivative’s value is the expected payoff under risk-neutral weights, discounted at the risk-free rate.
- p = (erT − d) / (u − d), and it is a pricing weight rather than anyone’s forecast.
- It works because the derivative can be replicated by the underlying and cash, so no-arbitrage fixes the price and preferences are already in the underlying.
- In the worked example the risk-neutral calculation and the replicating portfolio both give 8.2052.
- Using the real probability and the risk-free rate gives 12.84, and reconciling it would need a discount rate of 49.8% that nobody could know in advance.
- d < erT < u is the no-arbitrage condition, so a p outside zero to one means the inputs are inconsistent.
Frequently Asked Questions
What is a risk-neutral probability?
A weight used in pricing, constructed so that discounting the expected payoff at the risk-free rate reproduces the derivative’s no-arbitrage price. In a one period binomial model it is (e to the rT, minus d) divided by (u minus d). It lies between zero and one and the two weights sum to one, which is why it is called a probability, but nobody believes it as a forecast.
Why does the real probability of an up move not affect the option price?
Because the option can be exactly replicated by a portfolio of the underlying and cash. Once replication is possible, no-arbitrage fixes the option’s price at the cost of that portfolio, and the derivation never needs to know how likely each branch is. The underlying’s own price already contains whatever risk premium investors demand, so preferences cannot enter a second time.
What happens if I use the real probability instead?
You get the wrong answer, because you would be discounting a risky payoff at a riskless rate. In the worked example a real probability of 0.9 gives 12.84 against a true price of 8.21. Reconciling the two requires a discount rate of about 49.8%, which is the option’s actual risk-adjusted required return and is not something anyone could estimate in advance. Avoiding that is the whole purpose of the method.
What does it mean if p comes out negative or above one?
That the inputs describe a market containing an arbitrage. The condition for a valid p is that the risk-free growth factor lies strictly between the down and up moves. If the growth factor exceeds the up move, shorting the asset and lending is riskless profit; if it is below the down move, borrowing to buy is. In an exam it almost always means an error in u, d or the rate.
How does this relate to Black-Scholes?
Directly. Increase the number of periods and shrink each step and the binomial model converges to Black-Scholes-Merton, which is the same risk-neutral expectation written in continuous time. The N of d two term in that formula is a risk-neutral probability of finishing in the money, which is why it should not be quoted as the real probability of exercise.
Why are risk-neutral default probabilities higher than historical ones?
Because a probability implied by a CDS or bond spread contains compensation for bearing default risk as well as the risk itself, and the two cannot be separated without further assumptions. A historical default frequency is meant to describe how often default occurs. Using the implied figure as a forecast overstates expected loss, and using the historical figure to price protection understates the spread.
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