Question

The current price of a non-dividend-paying stock is $50. The risk-free interest rate is 1%. Over the next year, it is expected to rise to $52 or fall to $47. An investor buys a European put option with a strike price of $53. What is the value of the option?

Group of answer choices:

A: $0.93

B: $1.93

C: $1.95

D: $2.47

Answer 1

Two-state put option:

S = 50;    X=53;    1+r = 1.01

The stock price today is $50, At the end of the year, stock price will be either $52 or 47

If the stock price increase to $52, put option will be exercised so payoff =1

If the stock price decreases to $47, put option will pay $6

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

= (1-6)/(52-47) = -5/5 = -1

So I will create the following portfolio

                                                CF today                     CF one year from today

                                                                                    If S=52 If S=47

            Buy 1 Shares               -50 1*52 = $52 1*47 = $47

            Buy 1 puts                   -1P 1*1 = 1 1*6 = $6

                        TOTAL           -(50+1P)                   $53 $53

Since the payoff is the same in either outcome, this is a riskless portfolio which should earn 1% rate of return. So the most I would be willing to pay for it today is the present value of $52 discounted at 1%

= $53/(1.01) = $52.48

In equilibrium,

$50 + 1P = $52.47

1P = $52.47 - $50

P = $2.47

So, Option "D" is correct.