1. We will derive a two-state put option value in this problem. Data: ; (i.e. r=
ID: 2761948 • Letter: 1
Question
1. We will derive a two-state put option value in this problem. Data: ; (i.e. r=10%). T=1 year. The two possibilities for are 130 and 80.
a. Show that the range of is 50 while that of put option is 30 across the two states. What is the hedge ratio of the put?
b. Form a portfolio by shorting three shares of stock and writing five puts. What is the (nonrandom) pay-off to this portfolio? What is the present value of the portfolio?
c. Given that the stock currently is selling at 100, show that the value of the put must be 10.91.
Explanation / Answer
Solution:
a. Given that uS0 = 130, dS0 = 80
Pu = 0 and Pd = 30
Hedge ratio = (Pu – Pd)/ (uS0 – dS0)
Hedge ratio = (30 – 0)/ (130 – 80)
Hedge ratio = 30/50
Hedge ratio = 3/5
b.
Riskless portfolio
ST = 80
ST = 130
Buy 3 shares
240
390
Buy 5 puts
150
0
Total
390
390
Present value = $390/1.10
Present value = $354.545
c. Portfolio cost : 3S + 5P = 300 + 5P
Value of the portfolio = $354.545
300 + 5P = $354.545
5P = $54.545
P = $10.91
Hence, the value of the put must be $10.91
Riskless portfolio
ST = 80
ST = 130
Buy 3 shares
240
390
Buy 5 puts
150
0
Total
390
390
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