We will derive a two-state put option value in this problem. Data: Sq = 110; X =

Question

We will derive a two-state put option value in this problem. Data: Sq = 110; X = 120; 1 + r= 1.1. The two possibilities for S7 are 140 and 100. The range of S is 40 while that of P is 20 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 2 shares of stock and 4 puts. What is the (nonrandom) payoff to this portfolio? (Round your answer to 2 decimal places.) What is the present value of the portfolio? (Round your answer to 2 decimal places.) Given that the stock currently is selling at 110, calculate the put value. (Round your answer to 2 decimal places.) Use the Black-Scholes formula to find the value of a call option on the above stock: Calculate the value of a call option. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Explanation / Answer

Answer-1:

Two-state put option

          S = 110;    X=120;       1+r = 1.1

The stock price today is $110, At the end of the year, stock price will be either $140 or $100

         

If the stock price increase to $140, 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=120)

          The hedge ratio (ratio of put option payoffs to stock payoffs)

                             =    (0-20)/(140-100)   = -20/40 = -2/4

          So I will create the following portfolio

                                      CF today              CF one year from today

                                                                   If S=140               If S=100

          Buy 2 Shares                 -220                      2*140 = $280               2*100 = $200

          Buy 4 puts           -4P                         0                       4*20 = $80

                   TOTAL       -(220+4P)                               $280                      $280

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%

                             = 280/(1.1) = $254.54

          In equilibrium,    220+4P = 254.54          So       P = $8.63

Answer-2:

Using the Black-Scholes option pricing model to find the price of the call option, we find:

d1 = [ln($50/$50) + (0.03 + 0.502/2) x (6/12)] / (0.50 x (6/12)1/2 ) = 0.02121

d2 = 0.02121 – (0.50 x (6/12)1/2 ) = –0.332396

N(d1) = 0.5080

N(d2) = 0.3707

Putting these values into the Black-Scholes model, we find the call price is:

C = $50(0.5080) – ($50e–0.03(0.5))(0.3707) = $6.583

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