Futures Contracts and Pricing

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Futures Contracts and Pricing

• A. Basics of futures contracts
• Differences between futures and forward contracts
• Forwards not market to market
• Delivery common on forwards
• Forwards not traded on organized exchanges

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Futures Contracts and Pricing

• B. Workings of futures markets
• 1. What do you do with futures contracts
• a. Speculation
• b. Hedging

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Futures Contracts and Pricing

• B. Workings of futures markets
• 2. Major types of futures contracts
• There are essentially two basic types of futures
  contracts, those written on commodities and those
  written on financial contacts

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Futures Contracts and Pricing

• B. Workings of futures markets
• 2. Major types of futures contracts
• a. Commodity futures
• 1. Grains -- Corn, wheat, oats
• 2. Oil and Meal -- Soybeans, soymeal, soyoil,
  sunflower seed, sunflower oil
• 3. Livestock -- Hogs, cattle, pork bellies,
  fryer chickens
• 4. Foodstuffs -- Cocoa, coffee, orange juice,
  rice, sugar
• 5. Manufacturing goods-- Lumber, plywood, cotton
• 6. Metals -- Gold, silver, platinum, palladium,
  copper
• 7. Petroleum-- Crude oil, heating oil, gasoline,
  propane

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Futures Contracts and Pricing

• B. Workings of futures markets
• 2. Major types of futures contracts
• b. Financial futures
• 1. Interest earning assets -- Treasury bills,
  notes, and bonds Eurodollar deposits, munis
• 2. Currencies -- Pound, Canadian dollar, yen,
  franc, mark
• 3. Indexes --- SP 500, MMI, NYSE, Value Line,
  Nikkei 225, TOPIX

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Futures Contracts and Pricing

• B. Workings of futures markets
• 3. Margins for futures contracts
• A margin account is the amount of money, held by
  a brokerage house, an investor must be put up to
  fund investment in a futures contract.
• The initial margin is the amount deposited with a
  broker at the time of the purchase of the futures
  contract. Exchanges require a minimum initial
  margin for all contracts.
• Generally, for futures contracts, the minimum
  initial margin is between 5 - 10 of the initial
  value of the contract. Brokerage houses may
  require more than the minimum initial margin.

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Futures Contracts and Pricing

• B. Workings of futures markets
• 3. Margins for futures contracts
• Any changes in the value of the futures contract
  are added or subtracted from the margin account.
• A maintenance margin is the minimum amount that
  must be maintained a margin account at any time.
  Exchanges require that the maintenance margin be
  roughly 75 - 80 of the initial margin.
  Brokerage houses may require higher maintenance
  margins.
• If the value of the margin account falls below
  the maintenance margin a margin call is made. The
  investor must put enough money into the account
  to replenish the margin account back to the
  initial margin.

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Futures Contracts and Pricing

• B. Workings of futures markets
• 4. Closing a futures account
• a. Delivery -- In this instance, you hold the
  futures contract until the expiration date and
  you actually take physical delivery of the good.
  Some contracts, particularly index contracts,
  allow for cash settlement, as opposed to physical
  delivery, at the end of the contract. Physical
  delivery is rare, however.

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Futures Contracts and Pricing

• B. Workings of futures markets
• 4. Closing a futures account
• b. Offset -- The most common mechanism for
  closing a futures account is to take a reversing
  or offset position. A reversing or offset
  position is simple taking a position exactly
  opposite to the position that you currently have.

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Futures Contracts and Pricing

• B. Workings of futures markets
• 4. Closing a futures account
• c. Exchange for physicals (EFP) -- In an
  exchange for physicals, two traders agree to a
  simultaneous exchange of a cash commodity and
  futures contracts based on that commodity.
  Although the EFP is similar to an offsetting
  position, it differs in several ways. First, the
  traders actually exchange the physical good.
  Second, the futures contract was not closed by a
  transaction on the floor of the exchange. Third,
  the traders privately negotiated the price and
  other terms of the transaction.

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Futures Contracts and Pricing

• B. Workings of futures markets
• 5. Functioning of markets
• Standardization of contracts
• Daily price limits -- Restrict the range in which
  futures prices can vary on a daily basis. A
  contract that reaches the upper price limit is
  said to be limit up and a contract that reaches
  the lower price limit is said to be limit down.

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Futures Contracts and Pricing

• C. Example of valuing futures and forwards

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Futures Contracts and Pricing

• D. Pricing futures contracts and the opportunity
  for arbitrage
• 1. Notation
• T time when the futures contract matures
• t current time
• S current spot price of the underlying asset
• ST spot price of asset at time T
• f current value of futures contract
• F current futures price
• r current, annual, continuously compounded
  risk-free rate of interest for an investment
  maturing in time T

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Futures Contracts and Pricing

• D. Pricing futures contracts and the opportunity
  for arbitrage
• 2. Assumptions
• There are no transactions costs
• All trading profits are subject to the same tax
  rate
• Market participants can borrow and lend at the
  same risk free rate
• Market participants take advantage of arbitrage
  opportunities as they exist
• Additionally, it is assumed that interest rates
  are constant. If interest rates are constant,
  forward prices and futures prices are equal.

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Futures Contracts and Pricing

• D. Pricing futures contracts and the opportunity
  for arbitrage
• The value of a financial futures contract that
  provides no income is

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Futures Contracts and Pricing

• D. Pricing futures contracts and the opportunity
  for arbitrage
• To see that the previous pricing relationship is
  true, we will examine the potential for abitrage
  when

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Futures Contracts and Pricing

• D. Pricing futures and the opportunity for
  arbitrage
• 3. The expectations hypothesis for pricing
  financial futures
• The result that F Ser(T-t) is know as the
  expectations hypothesis for pricing financial
  futures. Basically, we are assuming that the
  futures price is an unbiased expectation of the
  future spot price. That is, F E(ST)
  Ser(T-t). Thus, we are assuming that the
  expected future spot price is just the future
  value of the current spot price.

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Futures Contracts and Pricing

• D. Pricing futures and the opportunity for
  arbitrage
• Pricing T-bill Futures
• Since T-bill futures are a risk-free, pure
  discount instrument we can use the basic futures
  pricing relationship to price T-bill futures

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Futures Contracts and Pricing

• D. Pricing futures and the opportunity for
  arbitrage
• Pricing Currency Futures
• With some slight modification we can use the
  expectations hypothesis to price currency futures
  and show when potential arbitrage opportunities
  may exist. Here we must consider an old economic
  concept interest rate parity.

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Interest rate parity suggests that
With some rearrangement we can see that
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Futures Contracts and Pricing

• Pricing Stock Index futures
• When pricing index futures we must consider the
  fact that dividends may be paid. If a dividend
  is paid, that will reduce the value of the asset,
  making the futures value of the asset less.
  Hence, holders of futures contracts must be
  compensated for future dividends.
• If the dividend is considered to be a constant
  proportion of the stock price (i.e. dividend
  yield) we can use

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Futures Contracts and Pricing

• D. Pricing futures and the opportunity for
  arbitrage
• 4. Arbitrage in the futures markets
• The following pricing relationships that are
  developed are arbitrage relationships. This
  implies that if the pricing relationships do not
  hold there would be a riskless profit opportunity
  available. In reality, arbitrage opportunities
  do exist in futures markets and people do try to
  exploit those opportunities.

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Futures Contracts and Pricing

• D. Pricing futures and the opportunity for
  arbitrage
• 4. Arbitrage in the futures markets
• a. Comments on T-bill Arbitrage -- Rendelman and
  Cabrini (1979) indicate that after accounting for
  brokerage costs, bid-ask spreads, and borrowing
  costs there were no significant arbitrage
  opportunities in the Treasury bill futures
  market. Capozza and Cornell (1979) found that
  near-term Treasury bill contracts were
  efficiently priced, but longer-term contracts
  tended to exhibit some under-pricing (F lt S ).

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Futures Contracts and Pricing

• Comments on Currency Arbitrage
• If F lt S you would want to buy the futures
  contract, sell the spot foreign currency, lend
  dollars, and borrow the foreign currency. If F gt
  S you would want to sell the futures contract,
  buy the spot foreign currency, borrow dollars,
  and lend the foreign currency.

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Futures contracts and pricing

• Comments on Stock Index Arbitrage
• If F gt S, arbitrage profits can be made by
  buying the stocks underlying the index and
  shorting the futures contract. Conversely, if F
  lt S, arbitrage profits can be made by shorting
  the stocks underlying the index and taking a long
  position in the futures contract.

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Stock Index Futures and the 1987 Stock Market
Crash

• At the time of the crash, both hedgers and
  speculators were using computerized transactions
  in index futures.
• Hedgers short futures contracts to protect
  portfolios against declines in value. When the
  market started dropping, a massive amount of sell
  orders were initiated pushing futures prices
  below their rational level. Hence, F lt S.
• An arbitrage play existed by shorting the index
  and buying the futures contract. This pushed the
  index lower, creating a downward spiral.

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Pricing commodity futures

• Pricing commodity futures is very difficult.
  Commodity prices are highly variable and accurate
  spot market prices are often difficult to obtain.
  Additionally, factors such as storage and
  convenience yield need to be factored in to the
  futures price.

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Pricing commodity futures

• 1. Pricing a futures contract on a commodity
  used for an investment
• When pricing futures contracts on commodities it
  is necessary to determine if the commodity is to
  be used for investment or consumption.
• Storage costs include the interest foregone on
  the value of the commodity being stored, the
  warehousing cost, and any shrinkage. Storage
  costs can be considered as negative income and
  must be reflected in the futures price.

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Pricing commodity futures

• 1. Pricing a futures contract on a commodity
  used for an investment
• If U represents the present value of all storage
  costs, the futures price of an investment
  commodity would be
• F (S U)er(T-t). If storage costs are
  proportional to the price of the commodity we can
  write the futures price of an investment
  commodity as F Se(rU)(T-t).

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Pricing commodity futures

• 1. Pricing a futures contract on a commodity
  used for an investment
• Arbitrage exists if F ? (S U)er(T-t). If
  F gt (S U)er(T-t), it would pay to borrow
  an amount equal to SU at the risk-free rate to
  buy one unit of the commodity and pay storage
  costs and then short the futures contract. If
  F lt (S U)er(T-t) it would pay to sell the
  commodity, save the storage costs, and invest the
  proceeds at the risk-free rate and then buy the
  futures contract.

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Pricing commodity futures

• 2. Pricing a futures contract when the commodity
  is used for consumption
• When a commodity is used for consumption,
  arbitrage opportunities like those described
  above may not exist. For example, if F lt (S
  U) and the underlying commodity is being used for
  consumption, the investor holding the commodity
  may not want to sell. Holding the asset may be
  of some value.

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Pricing commodity futures

• 2. Pricing a futures contract when the commodity
  is used for consumption
• In this case there is a convenience yield
  associated with holding the commodity. The
  convenience yield is defined roughly as the
  benefit to holding a commodity.

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Pricing commodity futures

• Convenience yield should be low when inventories
  are plentiful and should be high when shortages
  of the commodity occur. The convenience yield
  serves to reduce the futures price. Thus, if the
  convenience yield is defined as y, the futures
  price of a commodity used for consumption would
  be F Se(ru-y)(T-t) .

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Relationship between expected spot price and
futures price

• Recall that early in our discussion we assumed
  that E(ST) F. What happens if this is not the
  case? One of the earliest theories on the
  pricing of futures contracts, presented by Keynes
  (1930) suggested that F lt E(ST), which is known
  as normal backwardation.

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Relationship between expected spot price and
futures price

• The origin of this idea came from the fact that
  farmers (producers) want to hedge their risk by
  shorting the commodity. To attract speculators
  in to the market they have to offer futures
  contracts at a discount from the expected spot
  price. Thus, futures contracts should earn a
  rate of return higher than the risk-less rate and
  their prices should increase as the maturity of
  the contract nears.

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Relationship between expected spot price and
futures price

• It is also possible that hedgers may want to go
  long, in which case F gt E(ST), which is known as
  contango. In this situation, hedgers have to
  offer futures contracts at a premium above the
  expected spot rate to attract speculators.

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