Question

Question 1

Consider a 1-year forward contract on a stock with a price of $51. The current price of the stock is $50. A cash dividend payment of $2 per share is anticipated in 9 months. The interest rate is 3% per annum with continuously compounding. Assume that there are no transaction costs.

(a) Determine the fair price and the initial value of the forward contract today.

(b) Is there any arbitrage opportunity? Verify your trading positions taken at each point in time.

(c) After 10 months, the stock price falls 2% and the interest rate remains unchanged. Calculate the value of the forward contract. What is the mark-to-market?

Answer 1

Answer a)

Initial Value will be 0.

Fair Value = (Spot price - Present Value of Dividend) * e^(r*t)

= ((50 - Dividend * e^(-r*t)) * e^(r*t)

= ((50 - 2 * e^(-0.03*9/12)) * e^(0.03*1)

= ((50 - 2 * 0.97775123719) * 1.03045453395

= ((50 - 1.95550247438) * 1.03045453395

= 48.0444975257 * 1.03045453395

= 49.50

Answer b)

Yes Arbitrage opportunity is there since the Fair Forward Value is less than actual Forward Price.

Step 1: - Buy Stock @ 50 and borrow 50 @ 3%

Step 2: - Sell Forward Contract @ 51

Step 3 : - Receive Dividend after 9 months i.e. $2  and invest it for 3 months @ 3%

Now on Settlement Date

1. Sell Stock as per forward Contract @ 51

2. Receive Dividend with Returns = 2 * e^(r*t) i.e. 2 * e^(0.03* 3/12) = 2 * 1.00752819544 = $ 2.02

Total Amount t be received = 51 + 2.02 = 53.02 _ _ _ _ _ (A)

Loan repaid with interest 50 * e^(r*t) i.e. 50 * e^(0.03*1) = 50 * 1.03045453395 = 51.52 _ _ _ _ _ _(B)

Arbitrage = A - B

= 53.02 - 51.52

= $ 1.5

Answer c)

Stock Price after 20 months = 50 - 2% = $ 49

After 10 months Future value of 2 - months Forward will be = 49 * e^(r*t) = 49 * e^(0.03*2/12) = 49.25

Value of Forward on expiry of 1 year = 51 - 49.25 = $1.75

Marked to market = 51 - 49 = 2$