Question

You are attempting to value a put option with an exercise price of $106 and one year to expiration. The underlying stock pays no dividends, its current price is $106, and you believe it has a 50% chance of increasing to $125 and a 50% chance of decreasing to $87. The risk-free rate of interest is 8%. Calculate the value of a put option with exercise price $106. (Do not round intermediate calculations. Round your answer to 2 decimal places.)

Answer 1

Two-state put option:

S = 106;    X=106;    1+r = 1.08

The stock price today is $106, At the end of the year, stock price will be either $125 or $87

If the stock price increase to $125, put option will not be exercised so payoff =0

If the stock price decreases to $87, put option will pay $19  

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

= (0-19)/(125-87) = -19/38 = -1/2

So I will create the following portfolio

                                                CF today                     CF one year from today

                                                                                    If S=125 If S=87

            Buy 1 Shares               -106 1*125 = $125 1*87 = $87

            Buy 2 puts                   -2P                                  0 2*19 = $38

                        TOTAL           -(106+2P)                   $125 $125

Since the payoff is the same in either outcome, this is a riskless portfolio which should earn 8% rate of return. So the most I would be willing to pay for it today is the present value of $125 discounted at 8%

= $125/(1.08) = $115.74

In equilibrium,

$106 + 2P = $115.74

2P = $115.74 - $106

P = $9.74 / 2 = $4.87