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 )