Outline how the binomial model captures volatility.

FRM® Part I · Valuation and Risk Models · Binomial Trees · FRM1.VR.14.b

The idea

The binomial model captures volatility through the size of the up and down moves. In the Cox-Ross-Rubinstein setup, the up factor is u = e^(σ × √dt). The down factor is d = 1 ÷ u. Here σ is annual volatility and dt is the step length in years. Higher volatility makes u larger and d smaller, so the tree fans out. The square root appears because variance grows in proportion to time. The probability follows from no-arbitrage, with growth between d and u. The risk-neutral probability, which then lies between 0 and 1, is p = (e^(r × dt) − d) ÷ (u − d). So the stock's real-world expected return plays no part in the price. Because d = 1 ÷ u, an up move followed by a down move returns to the start and the tree recombines. Common mistakes are scaling by dt instead of its square root and treating σ as drift. If a question gives u and dt, work backwards. Implied volatility = ln(u) ÷ √dt.

Formulas

Up factor in a Cox-Ross-Rubinstein tree

u = e^(σ × √dt)

σ: the annual volatility
dt: the step length in years

The down factor is d = 1 ÷ u, so the tree recombines. Scaling σ by dt instead of √dt, or using 1 + σ√dt, gives the wrong move.

Volatility implied by a CRR up factor

σ = ln(u) ÷ √dt

u: the up factor of the tree
dt: the step length in years

Works backwards from a given tree. Dividing by dt rather than its square root, or using u − 1 instead of ln(u), are the traps.

A practice question

From the question bank, with the reason each wrong option is wrong.

In the Cox-Ross-Rubinstein binomial tree, how does the volatility of the underlying asset enter the model?

Through the risk-neutral probability, which is set equal to 0.5 + sigma / 2Why not: Invents a direct link between sigma and p; p is fixed by the risk-free rate, u and d.
Through the discount factor, which is raised in proportion to sigmaWhy not: Confuses volatility with discounting, which uses the risk-free rate only.
Through the stock's real-world expected return, which sets the up probabilityWhy not: Uses the real-world expected return, which drops out of risk-neutral valuation.
Through the size of the moves, u = e^(sigma x sqrt(dt)) and d = 1/u - correct

Volatility determines how far the price can move in one step: u = e^(sigma x sqrt(dt)) and d = 1/u. The probability p then follows from no-arbitrage using the risk-free rate, and the real-world drift does not matter.

A harder one

A risk manager builds a tree with three-month steps (dt = 0.25 years) for a stock with annual volatility of 25%, using the Cox-Ross-Rubinstein convention. What is the up factor u?

1.1250Why not: Uses 1 + sigma x sqrt(dt), a simple-return approximation, instead of the exponential.
1.0645Why not: Scales volatility by dt instead of by the square root of dt.
1.1331 - correct
1.2840Why not: Uses the annual volatility with no adjustment for the step length, e^sigma.

u = e^(sigma x sqrt(dt)) = e^(0.25 x 0.5) = e^0.125 = 1.1331, and d = 1/u = 0.8825. Volatility scales with the square root of time.

The hardest tier

A tree uses steps of six months (dt = 0.5 years) with u = e^(sigma x sqrt(dt)) and d = 1/u, for annual volatility 20% and a continuously compounded risk-free rate of 5%. What is the risk-neutral probability of an up move?

46.5%Why not: Sets the growth factor to 1, leaving out the risk-free growth over the step.
50.0%Why not: Assumes up and down are equally likely because d = 1/u looks symmetric.
64.5%Why not: Uses a full year of growth, e^r, instead of e^(r x dt).
55.4% - correct

u = e^(0.2 x sqrt 0.5) = 1.1519, d = 0.8681. p = (e^0.025 - d) / (u - d) = (1.0253 - 0.8681) / 0.2838 = 0.5539. The volatility fixes u and d, and the risk-free rate then fixes p.

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