You would like to be holding a protective put position on the stock of XYZ Co. t

Question

You would like to be holding a protective put position on the stock of XYZ Co. to lock in a guaranteed minimum value of $108 at year-end. XYZ currently sells for $108. Over the next year the stock price will increase by 12% or decrease by 12%. The T-bill rate is 4%. Unfortunately, no put options are traded on XYZ Co.

a. Suppose the desired put option were traded. How much would it cost to purchase?(Do not round intermediate calculations and round your final answer to 2 decimal places. Omit the "$" sign in your response.) Purchase cost $

b. What would have been the cost of the protective put portfolio? (Do not round intermediate calculations and round your final answer to 2 decimal places. Omit the "$" sign in your response.) Cost $

c. What portfolio position in stock and T-bills will ensure you a payoff equal to the payoff that would be provided by a protective put with X = 108? Show that the payoff to this portfolio and the cost of establishing the portfolio matches that of the desired protective put. Portfolio S = 95.04 S = 120.96 Buy 0.5 shares Invest in T-bills Total

Explanation / Answer

a)Exercise price of put (X) = $108
Expiration time (T) = 1 year
Dividend = 0
Stock Price (So) = $108
Probability of stock increasing to $120.96 = 0.5
Probability of stock price decreasing to $95.04 = 0.5
Stock increases by factor u
Stock decreases by factor d
So, the two possible option prices are $120.96 (S1) and $95.04 (S2)
Since (X) i.e. the exercise price is $108, the values if put will be:-
Pd = 0 (out of the money) and Pu =$12.96 (in the money)

Calculation of hedge ratio:
H = (Pd – Pu) / uS0 – dS0

Where, Pu = value of call option in the money
Pd = value of call option out of the money
u = Stock increases by this factor
d= Stock decreases by this factor
uS0 = stock price in case it increases
dS0 = stock price in case it decreases

Substituting the value,
Pu= $12.96 Pd = $0 uS0 =$120.96, dS0 =$95.04 u = 1.1 d =0.9

H = (0 – 12.96) / (120.96 – 95.04)
     = -12.96 / 25.92 = -0.5
The hedge ratio is -.5

Calculation of the present value of certain stock price $120.96 with a one year interest rate of 4%:-

Present value of stock price = S1 / (1+r) ^1    
                                             = $120.96 / 1.04
                                              = $116.3077

Setting the value of hedged position as per the present value of certain payoff:

(S1+2P) = $116.3077, where S1 = 120.96
therefore, P = 116.31 – 108/ 2
                     = $4.155

Therefore, cost of purchase if option is traded is $4.16

b) Cost of protective put = S0+P, where S0 = $95.04 and P = $4.15
                                        = $95.04 + $4.15
                                         = $99.19

Therefore, Cost of protective put is $99.19

c) Since the hedge ratio is -0.5 investor will hold (1-0.5) = 0.5 which costs $54 and invest the rest if money. This means $54+$4.16 = $58.16 in bills earning 4% interest = 58.16*1.04 = $60.48

Stock Price

S = 95.04

S = 120.96

Share

47.52

60.48

T-bills

60.48

60.48

Total

108

120.96

As per the above calculation, it is similar to the supposed protective put portfolio. Therefore, the plan of investing in T-bills and stocks is quite similar to both the strategies i.e. cost of supposed protective put and payoff.

Stock Price

S = 95.04

S = 120.96

Share

47.52

60.48

T-bills

60.48

60.48

Total

108

120.96

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