A stock is priced at $27.50 per share and has a volatility of 30%. The interest rate is 2.5%. An American put option on the stock has a strike price of $30 per share and expires and 6 months. Using a binomial tree with two time steps, what is the price per share of the option?

Question

BlackTom AI · Accepted Answer

Use the Cox-Ross-Rubinstein two-step binomial model with Δt = 0.5/2 = 0.25 years. Volatility σ = 0.30, so u = e^{σ√Δt} = e^{0.30·0.5} = e^{0.15} = 1.16183424, d = 1/u = e^{-0.15} = 0.86070798. Risk-free growth per step R = e^{rΔt} = e^{0.025·0.25} = e^{0.00625} = 1.00626942. Risk-neutral probability p = (R - d)/(u - d) = (1.00626942 - 0.86070798)/(1.16183424 - 0.86070798) = 0.48365972.

Step-by-step valuation:
1) Stock node prices:
- S0 = 27.50
- Su = S0·u = 27.50·1.16183424 = 31.95009
- Sd = S0·d = 27.50·0.86070798 = 23.66897
- Suu = S0·u^2 = 27.50·e^{0.30} = 27.50·1.34985881 = 37.12111
- Sud (or Sdu) = S0·u·d = S0 = 27.50
- Sdd = S0·d^2 = 27.50·e^{-0.30} = 20.37249

2) Terminal put payoffs (K=30): 
- Vuu = max(30 - 37.12111, 0) = 0
- Vud = max(30 - 27.50, 0) = 2.50
- Vdd = max(30 - 20.37249, 0) = 9.62751

3) Backwards induction for American option (check early exercise):
- At node u: immediate exercise = max(30 - 31.95009, 0) = 0. Continuation = (1/R)[p·Vuu + (1-p)·Vud] = (1/1.00626942)[0.48365972·0 + 0.51634028·2.50] = (1/1.00626942)[1.29085069] = 1.28278. So Vu = 1.28278 (do not exercise).
- At node d: immediate exercise = 30 - 23.66897 = 6.33103. Continuation = (1/R)[p·Vud + (1-p)·Vdd] = (1/1.00626942)[0.48365972·2.50 + 0.51634028·9.62751] = (1/1.00626942)[6.18021526] = 6.14109. Since 6.33103 > 6.14109, early exercise is optimal, so Vd = 6.33103.

4) Initial node value:
Continuation = (1/R)[p·Vu + (1-p)·Vd] = (1/1.00626942)[0.48365972·1.28278 + 0.51634028·6.33103] = (1/1.00626942)[3.88944] = 3.86553. Immediate exercise at time 0 would give 2.50, so we keep continuation value. Rounding gives 3.87.

Why the correct answer is 3.87: the full binomial calculation above yields ≈3.8655, rounded to 3.87. Other choices are inconsistent with these arithmetic steps: 4.03 and 3.93 are too large (they would imply larger continuation values or different p/u/d), while 3.65 and 3.79 are too small (they neglect either the correct u/d/R or the early exercise at the down node). Final answer: 3.87 (matches the provided correct answer).

A stock is priced at $27.50 per share and has a volatility of 30%. The interest rate is 2.5%. An American put option on the stock has a strike price of $30 per share and expires and 6 months. Using a binomial tree with two time steps, what is the price per share of the option?单项选择题

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