The volatility (sigma) of a stock is 69% per annum.We will model stock-price movements using a Binomial tree, where each branch of the tree has a step length (delta t) of 4 months. Calculate the proportion up movement (u) that applies to each branch of the tree. Your answer should have at least two decimal places.

Question

BlackTom AI · Accepted Answer

In the Cox–Ross–Rubinstein (CRR) binomial model the up-factor is chosen to match instantaneous volatility: u = exp(sigma * sqrt(delta t)).

1) Relevant knowledge points and concepts
- sigma is annual volatility (0.69 = 69%).
- delta t must be in years: 4 months = 4/12 = 1/3 year.
- CRR prescription: u = exp(sigma * sqrt(delta t)). This ensures the binomial tree reproduces the lognormal variance over that time step. Typically d = 1/u.

2) Step-by-step reasoning and calculations
- Convert delta t to years: delta t = 4/12 = 1/3 ≈ 0.3333333333.
- Compute sqrt(delta t): sqrt(1/3) = 0.5773502692.
- Multiply by sigma: sigma * sqrt(delta t) = 0.69 * 0.5773502692 = 0.3983716873.
- Exponentiate: u = exp(0.3983716873).

3) Intermediate numeric steps
- sqrt(1/3) ≈ 0.5773502692
- 0.69 × 0.5773502692 ≈ 0.3983716873
- exp(0.3983716873) ≈ 1.4893

4) Why the correct answer is correct
Using the CRR formula directly matches the model’s requirement that the single-step variance equals sigma^2 * delta t. The numerical exponentiation gives u ≈ 1.4893, which rounded to two decimal places is 1.49. That value means an “up” branch multiplies the stock price by about 1.49.

5) Why common alternatives are incorrect
- Using u = 1 + sigma * sqrt(delta t) would be a linear approximation and underestimates multiplicative returns (wrong because returns are lognormal).
- Using u = exp(sigma * delta t) (without the square root) overstates the per-step move because volatility scales with sqrt(time), not linearly with time.
- Using u = exp((r - 0.5 sigma^2) delta t + sigma sqrt(delta t)) would incorporate drift; CRR chooses the simpler volatility-matching form for u (drift is handled via risk-neutral probabilities).

6) Conclusion
Therefore the up-factor u = exp(0.69 * sqrt(1/3)) ≈ 1.49, which matches the provided answer.

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