THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MATH4210 (2016/17 Term 1) Financial Mathematics Assignment 1 solution Note: If you have any questions about the solution or assignment score, please let me know by sending an Email to kckchan@math.cuhk.edu.hk In all of the questions we ignore the effect of interest rate, for simplicity. 1. An investor enters into a short cotton futures contract when the futures price is 50 cents per pound. The contract is for the delivery of 50, 000 pounds. How much does the investor gain or lose if the cotton price at the end of the contract is (a) 48.20 cents per pound; and (b) 51.30 cents per pound? Solution: When you short a future contracts, you have the obligation to sell the underlying asset at a particular time and particular price. Let P, S be the profit and the cotton price at the day of deiivery respectively. We then have P = 50000(0.5 − S) (a) P = 50000(0.5 − 0.482) = $900 (b) P = 50000(0.5 − 0.513) = −$650 2. You would like to speculate on a rise in the price of a certain stock. The current stock price is $29 and a 3-month call with a strike of $30 costs $2.90. You have $5, 800 to invest. Identify two alternative strategies, one involving an investment in the stock and the other involving investment in the option. What are the potential gains and losses from each? Solution: Let Pts , Pto St be the profit of investing stocks, profit of investing options and the stock price at any time t respectively. Method 1: Investing stocks Pts = 5800 (St − 29) = 200(St − 29) 29 Methond 2: Investing call options Pto = max{ 5800 (St − 30), 0} − 5800 = max{2000St − 65800, −5800} 2.9 There are some points we can derive from here: 1. St ≤ 30 ⇒ Pto = −5800 65800 2. St ∈ (30, ] = (30, 32.9] ⇒ Pto = 2000St − 65800 ≤ 0 2000 3. St > 32.9 ⇒ Pto = 2000St − 65800 > 0 4. Pto ≥ Pts iff St ≥ 33.33 1 3. The price of gold is currently $500 per ounce. The futures price for the delivery in one year is $700. An arbitrageur can borro