Futures: Pricing and Hedging Strategies

1
Futures Pricing and Hedging Strategies

• Lecture Notes for FIN 353
• Yea-Mow Chen
• Department of Finance
• San Francisco State University

2
I. Futures Market Structure

•  1. A futures contract is an agreement that the
  buyer (seller) will accept (make) delivery of a
  particular asset on a future date at a price
  pre-determined today.  
• Example Entering a December gold futures at
  350/oz entitles the futures buyer to purchase
  100 oz. of gold on the December maturity date for
  350/oz, disregard the spot market price of gold
  in December. If the spot price on the maturity
  date is 370/oz, the buyer still pays 350/oz for
  the gold, making a 20/oz profit. On the other
  hand, if the spot price in December is 340/oz,
  the buyer would be losing 10/oz profit.

3
I. Futures Market Structure

•  Some points to know
• By entering into a futures contract, you know
  today the price you will pay for the purchase of
  gold in December (350/oz). In such a way, you
  have lock in the price for a future
  transaction.
• At the time of contract agreement, the contract
  is typically designed to be of zero value to
  either the buyer or the seller. Therefore none
  of the buyer or the seller will have to pay to
  the other party for the contract.
• Both the buyer and the seller, however, will have
  to deposit into a margin account, which is used
  to guarantee the fulfillment of the contract.

4
I. Futures Market Structure

• 2. When the forward contract is for a
  standardized amount of a carefully defined asset
  for delivery on a specific date and subject to
  the terms and conditions established by the
  organized market on which is traded, it becomes a
  future contract. Financial futures are
  standardized in
• 1) assets being exchanged
• 2) settlement dates
• 3) face value and
• 4) price quotation.

5
I. Futures Market Structure

• Assets being exchanged
• The underlying asset 91-day T-bills for a
  T-bill futures T-bonds with 20 years to maturity
  and an 8 coupon rate for a T-bond futures and
  .999 contents for a gold futures.
• 2) Settlement dates
• March, June, September, and December cycle for
  most stock index futures
• Face value
• 1m for a T-bill futures
• 100,000 for a T-bond futures
• (Index)(500) for a SP 500 index futures
• Price quotation
• of discount for short-term futures contracts,
  such as 8 discount on a T-bill futures
• of par for long-term futures contracts, such
  as T-bond futures are quoted at 96-24

6
I. Futures Market Structure

• 3. Most markets prescribe daily settlement of
  any gains and losses on the futures contract to
  minimize the risk of default at its maturity.
• This is also called Marking-to-market because,
  with daily settlement for any gains or losses,
  the value of the futures position is kept to
  equal to the current market value.
•  
•  4. The existence of organized futures markets
  provides a secondary market for the trading of
  contracts before maturity. In fact most of
  contracts are offset before they become mature.

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I. Futures Market Structure

• 5. Types of Orders
• Market Order
• Limited Order
• Stop-loss order
• Spread order

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I. Futures Market Structure

• 7. Forward Contracts vs. Futures Contracts
•   

9
II. Trading Mechanism

• Three steps to a futures trading 
• 1. Agreeing To Trade creates long and short
  positions.
• The role of the Futures Clearing Corporation The
  clear house is critical to the trading of futures
  because it settles and guarantees the contracts.
  After a contract is agreed to, the clearing house
  puts itself between buyer and seller and, in
  effect, becomes the party to whom delivery is
  made and from whom delivery is taken.

10
II. Trading Mechanism

• 2. Margin requirements initial margin and
  maintenance margin.
•  
• Initial Margin
• Contract Exchange Multiple Speculator Hedger
•  _________________________________________________
  _____________________SP500 CBOE
  500 22,000 9,000
• NYSE Index NYSE 500 9,000 4,000
• Major Market Index AMSE 250 21,000 7,500
• Value Line Index KCBT 500 7,000 5,000
•  _________________________________________________
  _____________________

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II. Trading Mechanism

• EX Suppose an investor purchases one December
  1999 gold futures at 400/oz and the initial
  margin 2,000/contract and maintenance margin is
  1,500/contract. The margin account is marked on
  a daily basis (daily settlement). The following
  table summarizes the changes in the margin
  account until the close of the contract.

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II. Trading Mechanism
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II. Trading Mechanism

• 3. Offsetting Contracts The majority of futures
  contracts are offset before maturity. This is
  because it is costly to take delivery.

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III. Leverage with Futures

• On futures trading, the only out-of-pocket
  payment is the margin deposited as a security
  performance bond. No payments are required for
  the contract, nor for the underlying assets until
  the settlement of the contract. This provides an
  opportunity for leverage.
• The gold futures buyer is leveraging his/her
  2,000 initial margin into a contract to buy 100
  oz of gold in the future, which amounts to
  40,000 in today's value. This provides 20 times
  leverage as compare to buying gold in the spot
  market. This leverage, however, increases the
  return volatility. It only takes a small change
  on the gold price to wipe out the 2,000 initial
  investment.

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III. Leverage with Futures

• Example Initial investment required on the gold
  futures 2,000
• Initial investment required for a
  spot market purchase
• 40,000
•  
•  

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VI. FINANCIAL FUTURES PRICING

•  I. Commodity Futures Prices and the Cost of
  Carry
•  A. Two important characteristics of futures
  prices
• 1. The futures price of a commodity or asset, F,
  is greater than the spot price, P i.e., F ?P.
• 2. The futures price rises as the time to
  maturity increases.
•  These characteristics reflect the cost of carry
  for a futures contract and illustrate a critical
  arbitrage relation.

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VI. FINANCIAL FUTURES PRICING

• Ex Suppose that the spot price of No. 2 Wheat in
  a Chicago warehouse is 300 cents per bushel, the
  yield a one-month T-bill is 6, and the cost of
  storing and insuring one bushel of wheat is 4
  cents per month. What is the price of a futures
  contract that has one-month to maturity?

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VI. FINANCIAL FUTURES PRICING

• Two ways to have wheat in one month
• 1. Purchase a one-month wheat futures contract at
  F/bushel
• Costs in one month F
• 2. Purchase in spot today and carry it over for
  one month
• Costs in one month (300 4)(1 6/12)
  305.5
• For the two alternatives to be indifferent, two
  costs would have to be the same, i.e.,
• F 305.5 or
• F P C

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VI. FINANCIAL FUTURES PRICING

• This is an equilibrium condition. It this is not
  true, then market adjustments will bring back the
  equilibrium.
• If F gt PC, a trader could make a riskless profit
  by taking a long position in the asset and a
  short position in the futures contract.
• If F lt PC, the arbitrage strategy would be to
  buy the futures and sell the commodity short.

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VI. FINANCIAL FUTURES PRICING

• B. Two implications on the movements of futures
  prices
•  
• 1.      The convergence of the futures price to
  the spot price is implied by the cost of carry
  relation.
• 2.      The convergence of the futures price to
  the cash price at expiration of the futures
  contract.
•  

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VI. FINANCIAL FUTURES PRICING

• C. Determinants of the Basis (Risk)
• 1.      The Convergence of the Future Price to
  the Cash Price If the future position is
  unwinded prior to contract maturity, the return
  from the futures position could differ from the
  return on the asset due to the basis risk.
• 2.      Changes in Factors Affecting the Cost of
  Carry the most significant determinant of the
  cost of carrying is the interest rate. As the
  interest rate increases, the opportunity cost of
  holding the asset rises, so the cost of carry-
  and therefore the basis-rises.

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VI. FINANCIAL FUTURES PRICING

• 3.      Mismatches between the Exposure Being
  Hedged and the Futures Contract Being Used as the
  Hedge
• In a cross-hedge, there is an additional source
  of basis risk. Basis results not only from
  differences between the futures price and the
  prevailing spot price of the deliverable asset,
  but also from differences between the spot Prices
  of the deliverable asset and the exposure being
  hedged. Major factors responsible for variation
  in the basis for a cross-hedge
• (1) Maturity mismatch
• (2) Liquidity differences
• (3) Credit Risk Differences
• 4.      Random Deviations from the Cost-of-Carry
  Relation "White noises", but there are canceled
  out in the long run.

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VI. FINANCIAL FUTURES PRICING

• II. Futures Prices and Expected Futures Spot
  Prices
• The expectation model states that the current
  futures price is equal to the market's expected
  value of the spot price at period T
•   Ft E(PT)
• If this model is correct, a speculator can expect
  neither to gain nor to lose from a position in
  the futures market
• E(Profit) E(PT)- FT 0
•  

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VI. FINANCIAL FUTURES PRICING

• EX Suppose that at time period 0, a speculator
  purchases a futures contract at a price of F, and
  posts 100 margin in the form of riskless
  securities.
•  
• At contract maturity T, the value of the margin
  account will have grown to F0 (1rf)
• At maturity, the value of the futures contract
  itself will be (PT - F0).
•  

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VI. FINANCIAL FUTURES PRICING

• The actual rate of return the speculator will
  earn is
•  
• (1rf)F0 (PT - F0) (PT - F0)
• r --------------------------- -1 rf
  --------
• F0 F0
•  The expected rate of return r is
•  
• E(PT) - F0
• E(R) rf ------------- rf
• F0
• If the expectation model is correct.

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• Hedging Principle
• To hedge against falling interest rates or rising
  prices (on a spot position), take a long position
  by buying futures
• To hedge against rising interest rates or falling
  prices (on a spot position), take a short
  position by selling futures

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• Long Hedging

value
Futures position value
gain
Spot price
0
loss
Spot position value
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V. Financial Futures Interest Rate Futures as a
Hedging Device

• I. Long Hedge
• A long hedge is chosen in anticipation of
  interest rate declines and requires the purchase
  of interest rate futures contract. If the
  forecast is correct, the profit on the hedge
  helps to offset losses in the cash market.
• Example In April 2005, the manager of a money
  market portfolio expects interest rates to
  decline. New funds, to be received invested in
  90 days, will suffer from the drop in yields. The
  manager expects an inflow of 10m in July. The
  discount yield currently available on 91-day
  T-bills is 10, and the goal is to establish a
  yield of 10 on the anticipated funds.

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• Cash Market Futures
  Market
• __________________________________________________
  _________________
•  April T-bill discount yield at 10 April buy
  10 T-bill Contracts for
• Price of 91-day T-bills,
  September delivery at 10 discount
• 10m par 9,747,222 yield.
  Value of contracts 9,750,000
•  
• July T-bill discount yield at 8 July Sell 10
  Sept. T-bill contracts
• Price of 91-day T-bills, at
  8 discount yield.
• 10m par 9,797,778 Value of
  contracts 9,800,000
• __________________________________________________
  _____________________
• Opportunity Loss Gain 50,000
• 50,556
• Effective Discount Yield with the Hedge
•  
• 10,000,000- (9,797,778- 50,000) 360
• ------------------------------------------
  -- ---- 9.978
• 10,000,000 91

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• Even if the expectation on future interest rates
  for the cash market is incorrect, the position is
  still hedged. The cost is that the potential
  profitable opportunities in the cash market is
  foregone.
• EX Assume the T-bill discount yield rises to 12
  , instead of declining to 8 as expected.
•   
• Cash Market Futures Market
• __________________________________________________
  _____________________April T-bill discount
  yield at 10 April Buy 10 T-bill Contracts for
  
• Price of 91-day T-bills,
  September delivery at 10 discount
• 10m par 9,747,222 yield.
  Value of contracts 9,750,000
•  
• July T-bill discount yield at 12 July Sell 10
  Sept. T-bill contracts
• Price of 91-day T-bills, at 12
  discount yield.
• 10m par 9,696,667 Value of
  contracts 9,700,000
• __________________________________________________
  _____________________Opportunity gain Loss
  50,000
• 50,555

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• Long speculation Instead of expecting new funds
  to arrive invest in July, the manager could
  speculate on the direction of interest rates. If
  he/she speculates on a declining interest rate,
  and the speculation is materialized
•  
• Cash Market Futures Market
• __________________________________________________
  _____________________
•  April T-bill discount yield at 10 April buy
  10 T-bill Contracts for
• Price of 91-day T-bills,
  September delivery at 10 discount
• 10m par 9,747,222 yield.
  Value of contracts 9,750,000
•  
• July T-bill discount yield at 8 July Sell 10
  Sept. T-bill contracts
• Price of 91-day T-bills, at 8
  discount yield.
• 10m par 9,797,778 Value of
  contracts 9,800,000
• __________________________________________________
  _____________________
• Gain 50,000

32
V. Financial Futures Interest Rate Futures as a
Hedging Device

• If he/she speculates on a declining interest
  rate, but market rate rises in September instead
•  
• Cash Market Futures Market____________________
  __________________________________________________
  _
• April T-bill discount yield at 10 April Buy 10
  T-bill Contracts for
• Price of 91-day T-bills,
  September delivery at 10 discount
• 10m par 9,747,222 yield.
  Value of contracts 9,750,000
•  
• July T-bill discount yield at 12 July Sell 10
  Sept. T-bill contracts
• Price of 91-day T-bills, at 12
  discount yield.
• 10m par 9,696,667 Value of
  contracts 9,700,000
• __________________________________________________
  _____________________
•   Loss 50,000

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• II. Short Hedge
• A short hedge is chosen in anticipation of
  interest rate increases and requires the sale of
  interest rate futures. If the forecast is correct
  the profit on the hedge helps to offset losses in
  the cash market.
•  

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• Example A saving institution in April 2005 wants
  to hedge 5m in short-term CDs whose owners are
  expected to roll them over in 90 days. If market
  yields go up, the thrift must offer a higher rate
  on its CDs to remain competitive, reducing the
  net interest margin. If the CD rare rises from 7
  to 9, the interest cost will increase by 25,000
  for the 3-month period. The asset/liability
  manager can reduce these by the sale of T-bill
  futures contracts.

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• Cash Market Futures Market
• __________________________________________________
  _____________________April. CD rate 7
  April. Sell 5 Sept. T-bill
  contracts at
• interest cost on 5m 3-month 7 discount
  yield
• interest costs 87,500 Value
  of contract 4,912,500
• July CD rate 9 July Buy 5 Sept.
  T-bill contracts at
• interest cost on 5m 3-month
  9 discount
• deposits Value of contracts
  4,887,500
• 112,500
•  _________________________________________________
  _____________________
• Opportunity Loss 25,000 Gain 25,000
•  
• Net result of hedge 0
•  
• 112,500 -25,000 360
• Effective CD Rate ----------------------
  ----- 7
• 5,000,000 90
•  

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• Basis Risk Using the Long Hedge Example
• Long Hedge Example as previously stated
• Cash Market Futures Market
• __________________________________________________
  _________________
•  April T-bill discount yield at 10 April buy
  10 T-bill Contracts for
• Price of 91-day T-bills,
  September delivery at 10 discount
• 10m par 9,747,222 yield.
  Value of contracts 9,750,000
•  
• July T-bill discount yield at 8 July Sell 10
  Sept. T-bill contracts
• Price of 91-day T-bills, at
  8 discount yield.
• 10m par 9,797,778 Value of
  contracts 9,800,000
• __________________________________________________
  _____________________
• Opportunity Loss 50,556 Gain 50,000
• Effective Discount Yield with the Hedge
•  
• 10,000,000- (9,797,778- 50,000) 360
• ------------------------------------------
  -- ---- 9.978
• 10,000,000 91

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V. Financial Futures Interest Rate Futures as a
Hedging Device

• Revised Example Rather than using T-bill
  contract for hedging, a long-term T-bond futures
  contract is used for hedging which is price at
  96-12. If the T-bill rate drops to 8 in
  September as expected, the T-bond futures will
  have it price increased to 98-16.
• Cash Market Futures Market
• __________________________________________________
  _____________________April T-bill discount
  yield at 10 April Buy 100 T-bond Contracts for
  
• Price of 91-day T-bills,
  September delivery at 96-12 which
• 10m par 9,747,222 gives the
  value of contracts 9,637,500
•  
• July T-bill discount yield at 8 July Sell 100
  Sept. T-bond contracts
• Price of 91-day T-bills, at
  98-16 for a value 9,850,000
• 10m par 9,797,778
• __________________________________________________
  _____________________Opportunity Loss Gain
  212,500
• 50,556

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VI. Macrohedging with Futures for a Financial
Institution

• Suppose a FI's balance sheet structure is as
  follows Assets 100m, Liabilities 90m, and
  equity 10m. The average duration of assets and
  liabilities is 5 and 3 years, respectively. If
  interest rates are expected to rise from 10 to
  11, then
•  
• ?E (DA - kDL) A (?R/1R)
• - (5 - .9 3) 100m (.01/1.1) - 2.09m
•  

39
VI. Macrohedging with Futures for a Financial
Institution

• The manager's objective is to fully hedge the
  balance sheet exposure by constructing a futures
  position to make a gain to just offset the loss
  of 2.09m on equity.
•  
• When interest rates rise, the price of futures
  contracts falls. The sensitivity of the price of
  a futures contracts depends on the duration of
  the deliverable bond underlying the contract, or
•  ?F/F - DF (?R/1R), or
• ?F - DF F (?R/1R)
• - D (NF PF) (?R/1R)

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VI. Macrohedging with Futures for a Financial
Institution

• Fully hedging can be defined as selling
  sufficient number of futures contracts so that
  the loss of net worth on the balance sheet is
  just offset by the gain from off-balance-sheet
  selling of futures
•   ?F ?E
•  which implies
•  N F (DA - kDL) A / DF PF
• (5-.93)100m/(9.597,000)
• 249.59 contracts
• if a T-bond futures contract is used for hedging.
  The futures is quoted 97 per 100 of face value
  for the benchmark 20-yr., 8 coupon bond that has
  a duration of 9.5 yrs.

41
VI. Macrohedging with Futures for a Financial
Institution

• Suppose instead of using the 20-yr. T-bond
  futures to hedge it had used the 3-month T-bill
  futures that has a price of 97 per 100 par
  value and a duration of.25 yrs. Then
•  
• NF (5 - .93)1 00m/.2597,000 948.45
  contracts
•  
• In general fewer T-bond contracts need to be sold
  because of its greater interest rate sensitivity.
  This suggests a simple transaction cost basis,
  the FI might normally prefer to use T-bond
  futures. 

42
VI. Macrohedging with Futures for a Financial
Institution

• The Problem of Basis Risk
•  
• Because spot bonds and futures on bonds are
  traded in different markets, the shift in yields
  (?R/1R) affecting the value of the
  on-balance-sheet cash portfolio may differ from
  the shift (?RF/1RF) in yields affecting the
  value of the underlying bond in the futures
  contracts i.e., spot and futures prices or
  values are not perfectly correlated. To take this
  basis risk into account
•  ?E -(DA - kDL) A (?R/1R)
• ?F - DF (N FP F ) (?R F /1R F)

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VI. Macrohedging with Futures for a Financial
Institution

• Setting ?E ?F , we have
• N F (DA - kDL) A / DF PF b,
•  
• Where b (?R/1R)/ (?RF/1RF)
•  
• where b measures the degree to which the futures
  price yields move more or less than spot price
  yields. For example, if b 1.1, this implies
  that for every 1 change in discounted spot rate
  (?R/1R), the implied rate on the deliverable
  bond in the futures market moves by 1.1.
•  NF (5 -.93) 100m/9.5497,000 1.1
• 226.9 contracts

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VII. Risks In Futures Transactions

• 1. Basis risk The "basis" is difference between
  the spot price of an instrument and the price of
  that asset in the futures market. Basic risk
  results from the fact that this price
  relationship may change overtime. However this
  basis risk is stable and predictable.
• 2. Related-Contract Risk Hedges can also fail
  because of default in the contract being hedged.

45
VII. Risks In Futures Transactions

• 3. Manipulation Risk Most manipulation involved
  "short Squeezes"' whereby an individual of group
  tries to make in difficult on impossible for
  short sellers in the futures markets to liquidate
  their contracts through delivery of acceptable
  commodities. The "short" will have to buy back
  their contracts as inflated prices.
• 4. Margin Risk An illiquid individual can also
  encounter difficulty by hedging in the futures
  markets if the future prices moves adversely and
  the individual must constantly pose more
  maintenance margin funds.