FRM Part I · FRM Exam Part I · Binomial Trees
A stock trades at 100 with annual volatility of 20%. A two-step CRR tree uses Δt = 0.5 years with d = 1/u. What is the difference between the stock price at the top node (two up moves) and the bottom node (two down moves) at the end of the tree?
The difference is 57.33. With σ√Δt of 0.14142, u is 1.1519 and d is 0.8681. The top node is 100 × u² = 132.70 and the bottom node is 100 × d² = 75.36, so the gap is about 57.33.
- A40.27
- B57.33Correct
- C80.00
- D82.15
Explanation
σ√Δt = 0.2 × 0.70711 = 0.14142. u = e^0.14142 = 1.15191 and d = 0.86812. Top node = 100 × 1.15191² = 132.70. Bottom node = 100 × 0.86812² = 75.36. The difference is 57.33. Using u = 1+σ gives 80.00. Using Δt = 1 for the step gives 82.15. Using σΔt gives 40.27.
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