Question

A stock currently has a market value of $10. The risk-free rate of return is 3%. In one year, the stock is expected to sell for either $14 or $9.

a) What is the value of a twelve-month call option with a strike price of $12? (6 points)

b) If the twelve-month call option is traded at $1.2, Recommend a riskless strategy by showing the positions in both shares and calls. (4 points).

Answer 1

a) Let a riskless portfolio be constructed by taking a long position in X unit of stock and short position in One unit of Call option

X= (Value of option in upmove - value of option in downmove) / (stock price in upmove - stock price in downmove)

=(2-0)/(14-9) = 0.4

Value of portfolio after one year = 14*X - 2 = 9*X = $3.6 at expiry

As the portfolio is riskless , value of portfolio today = value of portfolio at expiry/ 1.03

=> X*10-C= 3.6/1.03

=> C= 4-3.6/1.03 = 0.504854 or $0.50

The value of a twelve-month call option with a strike price of $12 is $0.50

b) If the call option is traded at $1.2 , the arbotrage strategy is as follows :

1. Short 5 call options for $1.2 each and get $6. Take a long position in 2 shares for $20. . Fund the requirement of $14 by taking a riskfree loan (assuming it is possible) at a rate of 3% for one year

2. After one year , if the stock price = $14, the call options will be exercised, buy 3 shares from the market at $14 each and deliver the 5 shares to get $60 - $42 = $18. Pay the loan amount of 14*1.03 = $14.42 and take the remaining amount = $18-$14.42 = $3.58 as arbitrage profit

3.  After one year , if the stock price = $9, the call options will be worthless. Sell the 2 shares to get $18. Pay the loan amount of 14*1.03 = $14.42 and take the remaining amount = $18-$14.42 = $3.58 as the arbitrage profit

So, in both situations, one can make arbitrage profit of $3.58

(Please note that the ratio of long stocks to short calls is 0.4 - same as the one derived above )