We will derive a two-state put option value in this problem. Data: S 0 = 240; X
ID: 2763448 • Letter: W
Question
We will derive a two-state put option value in this problem. Data: S0 = 240; X = 250; 1 + r = 1.1. The two possibilities for ST are 270 and 170.
The range of S is 100 while that of P is 80 across the two states. What is the hedge ratio of the put? (Negative value should be indicated by a minus sign. Round your answer to 2 decimal places.)
Form a portfolio of 4 shares of stock and 5 puts. What is the (nonrandom) payoff to this portfolio? (Round your answer to 2 decimal places.)
Given that the stock currently is selling at 240, calculate the put value. (Round your answer to 2 decimal places.)
We will derive a two-state put option value in this problem. Data: S0 = 240; X = 250; 1 + r = 1.1. The two possibilities for ST are 270 and 170.
Explanation / Answer
Two-state put option
S = 240; X=250; 1+r = 1.1
The stock price today is $240, At the end of the year, stock price will be either $270 or $100
If the stock price increase to $270, put option will not be exercised so payoff =0
If the stock price decreases to $100, put option will pay $20 (i.e. buy the stock in the open market for $100 and exercise the put option to sell the stock for X=250)
The hedge ratio (ratio of put option payoffs to stock payoffs)
= (0-80)/(270-170) = -80/100 = -8/10
So I will create the following portfolio
CF today CF one year from today
If S=270 If S=170
Buy 2 Shares -420 4*270 = $1080 4*170 = $680
Buy 4 puts -5P 0 5*80 = $400
TOTAL -(420+5P) $1080 $1080
Since the payoff is the same in either outcome, this is a riskless portfolio which should earn 10% rate of return. So the most I would be willing to pay for it today is the present value of $390 discounted at 10%
= 1080/(1.1) = $981.82
In equilibrium, 420+5P = 981.82 So P = $112.36
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