Question text 4Marks An American put option has a strike price of $95 and expires 12 months from now.The following diagram shows a 2-step Binomial tree modelling share-price movements over the 12-month life of this put option. Key details are:the proportional up (u) and down (d) movements on each branch are u = 1.1934 and d = 0.8380 the branch length (δt) is 0.50 the riskfree interest rate is 6% pa continuously compounded. On the upper-right branch labelled with a big number 1, the value of the put option is Answer 2[input]On the lower-right branch labelled with a big number 2, the value of the put option is Answer 3[input].Do not enter a dollar sign in your answer. Enter answers to 2 decimal places.Notes Report question issue Question 19 Notes多项填空题
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Question

Question text 4Marks An American put option has a strike price of $95 and expires 12 months from now.The following diagram shows a 2-step Binomial tree modelling share-price movements over the 12-month life of this put option. Key details are:the proportional up (u) and down (d) movements on each branch are u = 1.1934 and d = 0.8380 the branch length (δt) is 0.50 the riskfree interest rate is 6% pa continuously compounded. On the upper-right branch labelled with a big number 1, the value of the put option is Answer 2[input]On the lower-right branch labelled with a big number 2, the value of the put option is Answer 3[input].Do not enter a dollar sign in your answer. Enter answers to 2 decimal places.Notes Report question issue Question 19 Notes多项填空题

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BlackTom AI · Accepted Answer

1) Relevant concepts:
- Binomial option pricing with risk‑neutral probability p = (e^{rΔt} − d)/(u − d).
- For an American put, the node value = max(immediate exercise K − S_node, continuation value).
- Continuation value = e^{-rΔt}[p·V_up + (1−p)·V_down] using option values at the two terminal branches.

2) Parameters and preliminary numbers:
- K = 95, S0 = 60, u = 1.1934, d = 0.8380, Δt = 0.5, r = 6% pa continuous.
- e^{rΔt} = e^{0.06·0.5} = e^{0.03} ≈ 1.03045453, e^{-rΔt} ≈ 0.97044553.
- Node prices: S_up = 60·u = 71.60 (node 1), S_down = 60·d = 50.28 (node 2).
- Terminal prices from node1: up→85.45 (put 9.55), down→60.00 (put 35.00).
  From node2: up→60.00 (put 35.00), down→42.13 (put 52.87).

3) Risk‑neutral probability:
p = (e^{rΔt} − d)/(u − d) = (1.03045453 − 0.8380)/(1.1934 − 0.8380)
  = 0.19245453 / 0.3554 ≈ 0.54161.

4) Node 1 (S = 71.60) continuation value:
Expected terminal value = p·9.55 + (1−p)·35.00
  = 0.54161·9.55 + 0.45839·35.00
  = 5.17239 + 16.04365 = 21.21604.
Discounted continuation = e^{-rΔt}·21.21604 ≈ 0.97044553·21.21604 ≈ 20.59.
Immediate exercise value = K − S = 95 − 71.60 = 23.40.
Since 23.40 > 20.59, early exercise is optimal at node 1, so put value = 23.40.

5) Node 2 (S = 50.28) continuation value:
Expected terminal value = p·35.00 + (1−p)·52.87
  = 0.54161·35.00 + 0.45839·52.87
  = 18.95635 + 24.23510 = 43.19145.
Discounted continuation = 0.97044553·43.19145 ≈ 41.93.
Immediate exercise value = K − S = 95 − 50.28 = 44.72.
Since 44.72 > 41.93, early exercise is optimal at node 2, so put value = 44.72.

6) Why other possibilities are incorrect:
- The continuation values (≈20.59 at node1 and ≈41.93 at node2) are less than the immediate exercise values, so the option holder would exercise early; any answer equaling the continuation values would be incorrect for an American put at these nodes.