Question

The stock of Delta, Inc. is selling now for $40. A year from today it will sell for either $30 or $120. The risk-free interest rate is 4%, and a call option on Delta, Inc. with an exercise price of $50 is available.

a. What is the fair (i.e. arbitrage-free) value of the call?

b. The call is selling at $10. Show that you can create an arbitrage position to take advantage of the difference between the price of the call and its arbitrage-free value.

Answer 1

(a)Current worth=$40

It can go up =U=$120

It can go down=D=$30

Probability of going up=p

Probability of going down=1-p

Call strike price=$50

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=$120:

(50-120)=-$70

Value of D shares=D*120

Payoff of shorting one Call option , if shares goes down to D=$30,

Payoff on option =$0

Value of D shares=D*30

Portfolio is riskless if,

D*120-70=D*30

90*D=70

D=70/90=0.77778

Riskless portfolio :

Long 0.77778 shares and short 1 Call Option

Value of portfolio in one month:

120*0.77778-70=$23.33

Value of Portfolio today=23.33/(e^r)

r=interest rate=4%=0.04

Value of the portfolio today=23.33/(e^0.04)=$22.42

Value of Share=0.77778*40=$31.11

Value of Option =31.11-22.42=$8.69

Current Price of European Call option

$8.69

(b) If the Call is trading at $10

Arbitrage Strategy:

(i)Buy one share at $40

(ii)Sell one Call Option (Strike=$50) at $10

Net amount required=40-10=$30

(iii) Borrow $30

At the end of the year, amount to be paid back on borrowing with interest=30*(e^0.04)=$31.22

At the end of Year you will receive $50 by selling the share.

Net Profit=50-31.22