Question

You would like to be holding a protective put position on the stock of XYZ Co. to lockin a guaranteed minimum value of \(100 at year-end. XYZ currently sells for \)100. Overthe next year, the stock price will either increase by 10% or decrease by 10%. The T-billrate is 5%. Unfortunately, no put options are traded on XYZ Co.

a. Suppose the desired put option were traded. How much would it cost to purchase?

b. What would have been the cost of the protective put portfolio?

c. What portfolio position in stock and T-bills will ensure you a payoff equal to thepayoff that would be provided by a protective put with X = $100? Show that thepayoff to this portfolio and the cost of establishing the portfolio matches that ofthe desired protective put.

Short answer

a. $2.38

b. $102.38

c. Identical to protective put portfolio.

Step-by-step solution

Given information

Stock Price	110	90
Put payoff	0	10

Calculation of hedge ratio

H = P_u - P_d/ uS₀– dS₀

=-10 / 20

= .5

Calculation of present value ‘a’

Portfolio having 1 share and 2 puts gives guaranteed payoff of 110

Hence present value = $110 / 1.05

= $104.76

S + 2P = $104.76

$100 + 2P = $104.76

P = $2.38

Calculation of cost of protective put ‘b’

Cost of 1 share + cost of 1 put = cost of protective put

$100 + $2.38

= $102.38

Calculation of portfolio position ‘c’

Since the hedge ratio of the put = -.5, let’s hold (1-.5 = .5) shares of stock, costing $50 and place remaining $52.38 in bills earning 5% interest.

Stock price	S = 90	S = 110
Half share	45	55
Bills	55	55
Total	100	110

Hence identical to protective put portfolio

This implies that stock plus bills strategy replicate both cost and payoff of the protective put