Question

Suppose you have just purchased one share of Proctor & Gamble stock (PG) for $123. You have forecasted that in one year the stock price will either rise to $140 or fall to $108 Suppose further that you can either buy or sell a call option on PG stock with a strike price of $120. Assume that this is a European style contract that expires exactly in one year and that the risk-free interest rate is 2.8% (a) Calculate the hedge ratio (H). [3 Points) (b) Based on your answer from Part (a), fill in the following table showing that you have created a riskless portfolio. [7 Points] Stock Price = $140 Stock Price = $108 Value of _ Shares Value of_Written Call Options Total Payoff (c) Calculate the price of the call option [10 Points)

Answer 1

Current worth=$123

It can go up =U=$140

It can go down=D=$108

Probability of going up=p

Probability of going down=1-p

Call strike price=$120(In the money)

Assume ,we buy (long)D Shares and Sell(short) one call option.

Payoff on shorting   one Call option,if the share price goes up to U=$140:

(120-140)=-$20

Value of D shares=D*140

Payoff of shorting one Call option , if shares goes down to D=$108,

Payoff on option =$0

Value of D shares=D*108

Portfolio is riskless if,

D*140-20=D*108

32D=20

D=20/32=0.625

(a) Hedge Ratio=0.625

Riskless portfolio :

Long 0.625 shares and short 1 Call Option

Value of portfolio in one month:

140*0.625-20=$67.5

Value of Portfolio today=67.5/(e^rt)

r=Risk free interest rate=2.8%=0.028

t=1 year

Value of the portfolio today=67.5/(e^0.028)=$65.64

Value of Share=0.625*123=$76.88

Value of Option =76.88-65.64=$11.24

.(c)Price of the Call option =$11.24

(b).

	Stock Price	Stock Price
	$140	$108
Value of 0.625 shares	$87.50	$67.50
Value of Written Call Option	($20)	$0
Total Payoff	$67.50	$67.50