FRM Exam Part I · Binomial Trees

Cox-Ross-Rubinstein Up and Down Factors from Volatility

Q: What is the Cox-Ross-Rubinstein formula for u and d?

u = e^(σ√Δt) and d = 1/u = e^(−σ√Δt). σ is annual volatility and Δt is the step length in years. These factors depend only on volatility and time step.

Q: Why is d equal to 1/u in the CRR tree?

It makes the tree recombine, because an up move followed by a down move returns the price to its starting level. The number of nodes then grows by one at each step, which keeps the calculation manageable.

Q: How does the CRR tree differ from the Jarrow-Rudd tree?

In CRR, the factors use only σ√Δt and the probability carries the drift. In Jarrow-Rudd, the drift is inside u and d and the probability is 0.5 on each branch. Both converge to the same lognormal behavior as Δt shrinks.

Q: Does the expected return of the stock enter u and d?

No. In risk-neutral valuation, the real-world drift is irrelevant. Volatility sets u and d, and the risk-free rate (less any yield) sets the probability p.

Updated 11 October 2026 · Fact-checked

In the Cox-Ross-Rubinstein tree, you match the tree to volatility by setting u = e^(σ√Δt) and d = 1/u. Here σ is annual volatility and Δt is the step length in years. The factors do not depend on the drift. You then find the risk-neutral probability p = (e^(rΔt) − d) ÷ (u − d).

Understand Matching Volatility: Cox-Ross-Rubinstein u and d

A binomial tree moves the stock price up by a factor u or down by a factor d in each short time step Δt. To price options well, the tree must reproduce the volatility of the stock. That is why u and d are tied to σ.

The Cox-Ross-Rubinstein (CRR) choice is u = e^(σ√Δt) and d = 1/u = e^(−σ√Δt). Over one step, the log of the stock price moves by +σ√Δt or −σ√Δt. The standard deviation of that move is about σ√Δt. Scaling by √Δt matches the rule that variance grows in proportion to time, so volatility grows with the square root of time.

Because d = 1/u, an up move followed by a down move (or the reverse) returns the price to its start. The tree recombines. After n steps the number of nodes is n + 1, not 2ⁿ. This keeps the tree small and easy to compute.

Volatility is built into u and d. The expected return is built into the probability. In risk-neutral valuation, you pick p so that the expected stock price grows at the risk-free rate. For a non-dividend stock, p = (e^(rΔt) − d) ÷ (u − d). The real-world drift never appears in the pricing.

A related alternative is the Jarrow-Rudd tree. It uses u = e^((r − q − σ²/2)Δt + σ√Δt) and d = e^((r − q − σ²/2)Δt − σ√Δt), with p = 0.5 on each branch. CRR puts the drift in the probability and keeps u × d = 1. Jarrow-Rudd puts the drift in the factors and keeps the probabilities equal. Both match volatility as Δt gets small.

Key formulas to remember

CRR up factor: u = e^(σ√Δt); σ is annualized volatility. Δt is the step length in years, for example 1/12 for a month or 0.25 for a quarter.
CRR down factor: d = 1/u = e^(−σ√Δt); This makes the tree recombine, because u × d = 1.
Risk-neutral probability (no dividends): p = (e^(rΔt) − d) ÷ (u − d); r is the continuously compounded risk-free rate. The down probability is 1 − p.
Probability with yield q or foreign rate: p = (e^((r − q)Δt) − d) ÷ (u − d); Use q for the dividend yield on an index, or the foreign rate rf for a currency. For a futures option, the growth term is 1, so p = (1 − d) ÷ (u − d).
No-arbitrage condition: d < e^(rΔt) < u; If this fails, p falls outside 0 to 1. It can happen when Δt is large and σ is small.
Jarrow-Rudd factors: u = e^((r − q − σ²/2)Δt + σ√Δt); d = e^((r − q − σ²/2)Δt − σ√Δt); p = 0.5; Equal probabilities, with the drift placed in the factors. Here u × d is not 1 in general.

How to solve Matching Volatility: Cox-Ross-Rubinstein u and d questions

Use this method for any question that asks you to build or check the factors of a volatility-matched tree.

1Identify σ as an annual figure. If you are given monthly or daily volatility, convert it to annual first, or adjust Δt to match.
2Compute Δt in years from the step length. For example, 3 months is 0.25 and 1 week is 1/52.
3Calculate σ√Δt. Do the square root of Δt before multiplying by σ.
4Find u = e^(σ√Δt), then d = 1/u. Keep four to five decimals.
5If asked, compute p = (e^((r − q)Δt) − d) ÷ (u − d) using the continuously compounded rates, and 1 − p for the down branch.
6Build the nodes: Su, Sd, then Su², S (since ud = 1), Sd², and so on.
7Check that d < e^(rΔt) < u and that p lies between 0 and 1, then answer in the units asked.

Quickest way: Shortcut with small σ√Δt

When to use it: Use when the options are far apart, or to check which choice is plausible. Use the full calculation when options are close.

Compute x = σ√Δt. If x is small, u ≈ 1 + x and d ≈ 1 − x.
Better: u ≈ 1 + x + x²/2, and d = 1/u.
For a recombining node after an up and a down move, remember the price equals S0 in CRR, so you need no calculation.
Estimate p by taking (e^(rΔt) − d) ÷ (u − d). The result is close to 0.5. It is above 0.5 only if r − q exceeds about σ²/2. Otherwise it is slightly below 0.5.
On a financial calculator, use the e^x key for the factor. Store u, then use 1/x to get d.

Common mistakes in Matching Volatility: Cox-Ross-Rubinstein u and d

Using σΔt instead of σ√Δt in the exponent.
Students treat volatility like a rate that scales linearly with time.
Fix: Volatility scales with the square root of time. Always take √Δt first.
Using a monthly Δt of 1 or 12 with an annual σ.
The time unit is not converted to years.
Fix: Write Δt in years first, such as 1/12 for monthly steps, then apply the formula.
Putting the interest rate or expected return into u and d for the CRR tree.
Students mix up CRR with Jarrow-Rudd.
Fix: In CRR, u and d depend only on σ and Δt. The rate enters through p.
Using p = 0.5 in a CRR tree.
Students recall equal probabilities from Jarrow-Rudd or simple coin-flip trees.
Fix: CRR uses p = (e^(rΔt) − d) ÷ (u − d), which is close to 0.5 but not equal to it.
Forgetting dividend yield or foreign rate in p.
The no-dividend formula is memorized and applied to every underlying.
Fix: Replace r by r − q for an index or r − rf for a currency, and use 1 for a futures underlying.
Setting d = 1 − (u − 1) instead of 1/u.
Students want a symmetric percentage move.
Fix: d = 1/u, so the down move is slightly smaller in percentage terms than the up move.

Worked examples

Example 1

A stock has an annual volatility of 30%. A binomial tree uses 3-month steps. Using the Cox-Ross-Rubinstein approach, find u, d, and the risk-neutral probability of an up move when the continuously compounded risk-free rate is 4% and there are no dividends.

Show the solution

Δt = 0.25, so √Δt = 0.5.
σ√Δt = 0.30 × 0.5 = 0.15.
u = e^0.15 = 1.16183.
d = 1/u = 0.86071.
e^(rΔt) = e^(0.04 × 0.25) = e^0.01 = 1.01005.
p = (1.01005 − 0.86071) ÷ (1.16183 − 0.86071) = 0.14934 ÷ 0.30112 = 0.4959.

Answer: u ≈ 1.1618, d ≈ 0.8607, p ≈ 0.496 (down probability ≈ 0.504).

Example 2

A stock trades at USD 100 with annual volatility of 24%. A tree uses monthly steps. Find u and d, the two possible prices after one step, and the price after an up move followed by a down move.

Show the solution

Δt = 1/12 = 0.083333, so √Δt = 0.288675.
σ√Δt = 0.24 × 0.288675 = 0.069282.
u = e^0.069282 ≈ 1.07174.
d = 1/u ≈ 0.93306.
After one step: up = 100 × 1.07174 = 107.17; down = 100 × 0.93306 = 93.31.
Up then down: 100 × u × d = 100 × 1 = 100, because ud = 1 in the CRR tree.

Answer: u ≈ 1.0717, d ≈ 0.9331. One-step prices are USD 107.17 and USD 93.31. After an up and a down move the price returns to USD 100.

Exam tips

Questions usually give σ, Δt (or the step length) and ask for u, d or p. Lock in the annual time unit first.
In a CRR tree, d = 1/u, so you can quickly reject answer choices where u × d is clearly not 1.
For index, currency or futures options, check which growth term goes into p before calculating.
If asked CRR versus Jarrow-Rudd, remember: CRR has u × d = 1 and p ≠ 0.5, Jarrow-Rudd has p = 0.5.
Check that p is between 0 and 1. If not, the step is too large for the inputs.

Practice questions from Binomial Trees

Matching Volatility: Cox-Ross-Rubinstein u and d: frequently asked questions

What is the Cox-Ross-Rubinstein formula for u and d?

u = e^(σ√Δt) and d = 1/u = e^(−σ√Δt). σ is annual volatility and Δt is the step length in years. These factors depend only on volatility and time step.

Why is d equal to 1/u in the CRR tree?

It makes the tree recombine, because an up move followed by a down move returns the price to its starting level. The number of nodes then grows by one at each step, which keeps the calculation manageable.

How does the CRR tree differ from the Jarrow-Rudd tree?

In CRR, the factors use only σ√Δt and the probability carries the drift. In Jarrow-Rudd, the drift is inside u and d and the probability is 0.5 on each branch. Both converge to the same lognormal behavior as Δt shrinks.

Does the expected return of the stock enter u and d?

No. In risk-neutral valuation, the real-world drift is irrelevant. Volatility sets u and d, and the risk-free rate (less any yield) sets the probability p.