Outline Forward contracts Futures contracts Hedging strategies using futures Futures and Forward Contracts Haipeng Xing Department of Applied Mathematics and Statistics Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Outline 1 Forward contracts Forward contracts and their payoffs Forward price Valuing forward contracts 2 Futures contracts Futures contracts and their prices The operation of margin accounts 3 Hedging strategies using futures Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward contracts and their payoffs Forward contracts A forward contract,orsimplyforward,isanagreementbetweentwo counterparties to trade a specific asset, for example a stock, at a certain future time T and at a certain price K. At the current time t T , one counterparty agrees to buy the asset at T ,andislong the forward contract. The other counterparty agrees to sell the asset, and is short the forward contract. The specified price K is known as the delivery price. The specified time T is known as the maturity. It can be contrasted with a spot contract, which is an agreement to buy or sell almost immediately. A forward contract is traded in the over-the-counter (OTC) market. Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward contracts and their payoffs Payoffs from forward contracts We define V (t, T ) to be the value at current time t T of being K long a forward contract with delivery price K and maturity T , that is, how much the contract itself is worth at time t to the counterparty which is long. Since the counterparty long the forward contract must pay K at T to buy an asset which is worth ST , we immediately have V (T,T)=S K. Similarly, the payoff at maturity from a short K T − forward contract is K S . − T Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward contracts and their payoffs Payoffs from forward contracts: An example Forward contracts can be used to hedge foreign currency risk. Suppose that, on May 6, 2013, the treasurer of a US corporation knows that the corporation will pay £1 million in 6 months (i.e., on November 6, 2013) and wants to hedge against exchange rate moves. Using the quotes in Table 1.1 (Hull, 2014), the treasurer can agree to buy £1 million 6 months forward at an exchange rate of 1.5532. Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward contracts and their payoffs Payoffs from forward contracts: An example The corporation then has a long forward contract on GBP. It has agreed that on November 6, 2013, it will buy £1 million from the bank for $1.5532 million. The bank has a short forward contract on GBP. It has agreed that on November 6, 2013, it will sell £1million for $1.5532 million. Both sides have made a binding commitment. If the spot exchange rate rose to, say, 1.6000, at the end of the 6 months, the forward contract would be worth $1, 600, 000 $1, 553, 200 = $46, 800 − to the corporation. If the spot exchange rate fell to 1.5000 at the end of the 6 months, the forward contract would be worth $1, 553, 200 $1, 500, 200 = $53, 200. − to the corporation. Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward price Forward price Define the forward price F (t, T ) at current time t T to be the delievery price K such that VK (t, T )=0, that is, such that the forward contract has zero value at time t. In particular, by definition one can at t go long—or ‘buy’—a forward contract with delivery price K = F (t, T ) for no upfront cost. From above, we immediately have F (T,T)=ST . Assume that the market participants are subject to no transaction costs when they trade, are subject to the same tax rate on all net trading profits, can borrow money at the same risk-free rate of interest as they can lend money, take advantage of arbitrage opportunities as they occur. Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward price Forward price: Non-divident-paying stocks Theorem 1 (Forward price) Let r be the constant interest rate with continuous compounding. For an asset paying no income, r(T t) F (t, T )=Ste − (1) rT When t =0we have F (0,T)=S0e . The replication proof. At current time t, consider two portfolios A (one unit of stock) and B (long one forward contact with delivery r(T t) price K,plusKe− − of cash with intereste rate r). At time T , both portfolios have value ST . Then at time t, r(T t) St = VK (t, T )+Ke− − .TheforwardpriceF (t, T ) is the value r(T t) of K such that VK (t, T )=0, and thus St = F (t, T )e− − . Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward price Forward price: Non-divident-paying stocks r(T t) The no-arbitrage proof. Suppose that F (t, T ) >Ste− − .At current time t we execute three transactions. (a) Short one forward contract (i.e., sell the stock forward) at its forward price F (t, T ) (b) Borrow S cash at interest rate r for time T t, (c) buy the stock t − at price St at time t. At time T , sell the stock for F (t, T ) and pay r(T t) back the loan Ste − . Hence we obtain an amount of cash at T , r(T t) F (t, T ) Ste − > 0. − r(T t) Now suppose F (t, T ) <Ste − . At time t,(a)longoneforward contract with delivery price F (t, T ), (b) sell the stock for St,(c) deposit the proceeds at r for time T t. At time T , we receive cash r(T t) − Ste − , buy back the stock via the forward contract for F (t, T ). r(T t) Hence we have a profit of Ste − F (t, T ) > 0. − r(T t) Under the assumption of no-arbitrage, we have F (t, T )=Ste − . Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward price Forward price: Non-divident-paying stocks Consider a long forward contract to purchase a non-dividend-paying stock in 3 months. Assume the current stock price is $40 and the 3-month risk-free interest rate is 5% per annum. Suppose that the forward price is relatively high at $43. An arbitrageur can (a) borrow $40 at the risk-free interest rate of 5% per annum, (b) buy one share, and (c) short a forward contract to sell one share in 3 months. At the end of the 3 months, the arbitrageur delivers the share and receives $43, and the net gain of 0.05 3/12 the arbitrageur is $43 $40e ⇥ = $43 $40.50 = $2.50. − − Suppose next that the forward price is relatively low at $39. An arbitrageur can short one share, invest the proceeds of the short sale at 5% per annum for 3 months, and take a long position in a 3-month forward contract. The net gain of the arbitrageur at the 0.05 3/12 end of the 3 months is $40e ⇥ $39 = $40.50 $39 = $1.50. − − Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward price Forward price: Non-divident-paying stocks (Example 5.1) Consider a 4-month forward contract to buy a zero-coupon bond that will mature 1 year from today. (This means that the bond will have 8 months to go when the forward contract matures.) The current price of the bond is $930. We assume that the 4-month risk-free rate of interest (continuously compounded) is 6% per an- num. Because zero-coupon bonds provide no income, we can use equation (1) with T =4/12, r =0.06,andS0 = 930.Theforward price, F0, is given by 0.06 4/12 F (0,T) = 930e ⇥ = $948.79. This would be the delivery price in a contract negotiated today. Haipeng Xing, AMS320, Textbook: Hull (2009) Futures and Forward Contracts Outline Forward contracts Futures contracts Hedging strategies using futures Forward price Forward price: Known income Consider a long forward contract to purchase a coupon-bearing bond whose current price is $900. Suppose that the forward contract matures in 9 months, and that a coupon payment of $40 is expected after 4 months. Assume that the 4-month and 9-month risk-free interest rates (continuously compounded) are, respectively, 3% and 4% per annum. Suppose that the forward price is relatively high at $910. An arbitrageur can (a) borrow $900 to buy the bond, (b) short a forward contract, and (c) short a forward contract to sell one share in 9 months. The coupon payment has a present value of 0.03 4/12 $40e− ⇥ = $39.60.Theremaining$860.40isborrowedat4% per annum for 9 months. The amount owing at the end of the 9-month period is 0.04 9/12 0.01 (900 39.60)e ⇥ = 860.50e = 886.60.