Futures Pricing and Strategies by fionan

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									Futures: Pricing and Hedging
         Strategies

      Lecture Notes for FIN 353

           Yea-Mow Chen
       Department of Finance
    San Francisco State University
    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.
      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.
        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.
     I. Futures Market Structure
1)   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
3)   Face value:
     $1m for a T-bill futures
     $100,000 for a T-bond futures
     (Index)*($500) for a S&P 500 index futures
4)   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
      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.
    I. Futures Market Structure
 5. Types of Orders:
 Market Order
 Limited Order
 Stop-loss order
 Spread order
         I. Futures Market Structure
   7. Forward Contracts vs. Futures Contracts:

                             Forward                                 Futures

     Nature of Transaction   Both buyers and sellers are obligated   Same
                             to buy or sell a given amount of an
                             asset at a set price at a future date
     Size of Contracts       Negotiable                              Standardized

     Delivery Date           Negotiable                              Standardized

     Method of Transaction   Prices are determined in private by     Prices are determined by Open
                             the buyer and seller                    Outcry in a auction-type market
                                                                     at registered exchange
     Security Deposit        Very high                               Very low

     Regulation              State or Federal laws of commerce       Commodity Futures Trading
                                                                     Commission
                                                                     National Futures Association

                                                                     Self-regulation by exchanges
       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.
         II. Trading Mechanism
   2. Margin requirements: initial margin and
    maintenance margin.

                                                   Initial Margin
   Contract               Exchange Multiple Speculator            Hedger
   ______________________________________________________________________
    S&P500                 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
   ______________________________________________________________________

                                    
      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.
           II. Trading Mechanism
Day         Futures Price   Daily Gain or   Cumulative     Margin Account   Margin Call
                            Loss            Gain or Loss
August 1    $400            -               -              $2,000
August 1    397.00          -$300           -$300          1,700
2           396.10          -90             -390           1,610
3           398.20          210             -180           1,820
4           397.10          -110            -290           1,710
5           396.70          -40             -330           1,670
6           395.40          -130            -460           1,540
7           393.30          -210            -670           1,330            $670
8           393.60          30              -640           2,030
9           391.80          -180            -820           1,850
10          392.70          90              -730           1,940
11          387.00          -570            -1,300         1,370            630
12          387.00          0               -1,300         2,000
13          388.10          110             -1,190         2,110
14          388.70          60              -1.130         2,170
15          391.00          230             -900           2,400
16          391.30          130             -770           2,530
       II. Trading Mechanism
   3. Offsetting Contracts: The majority of
    futures contracts are offset before maturity.
    This is because it is costly to take delivery.
      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.
    III. Leverage with Futures
   Example :      Initial investment required on the gold futures = $2,000
                  Initial investment required for a spot market purchase
                         = $40,000

     Spot               Spot Market                 Futures Market
     Price      Gain or Loss   % Return     Gain or Loss      % Return

     $420       $2,000         5%           $2,000            100%
     $410       $1,000         2.5%         $1,000            50%

     $400       0              0%           0                 0%
    $390       -$1,000        -2.5%        -$1,000            -50%
     $380       -$2,000        -5%          -$2,000           -100%
    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.
    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?
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
       VI. FINANCIAL FUTURES
               PRICING
   This is an equilibrium condition. It this is not true,
    then market adjustments will bring back the
    equilibrium.
    – If F > P+C, 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 < P+C, the arbitrage strategy would be to buy the
      futures and sell the commodity short.
      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.

      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.
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.
    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

    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* (1+rf)
 At maturity, the value of the futures contract itself
  will be: (PT - F0).

    VI. FINANCIAL FUTURES PRICING
   The actual rate of return the speculator will earn is

   (1+rf)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.
      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
            V. Financial Futures: Interest Rate
               Futures as a Hedging Device
    Long Hedging:
    value

                        Futures position value



                      gain
0                              Spot price
                      loss

                        Spot position value
     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.
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
    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
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
    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
     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.

      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.
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

      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%
    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
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/1+R)
 = - (5 - .9 * 3) * $100m * (.01/1.1)= - $2.09m

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/1+R), or
 F = - DF * F * (R/1+R)
          = - D *(NF* PF)* (R/1+R)
    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-.9*3)*$100m]/(9.5*$97,000)
     = 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
    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 - .9$3)$1 00m/.25*$97,000 = 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.
    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/1+R)
  affecting the value of the on-balance-sheet cash
  portfolio may differ from the shift (RF/1+RF) 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/1+R)
 F = - DF * (N F*P F ) * (R F /1+R F)
    VI. Macrohedging with Futures for
          a Financial Institution
   Setting : E = F , we have
   N F = [(DA - kDL) * A] / DF * PF * b,

   Where b =(R/1+R)/ (RF/1+RF)

 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/1+R), the implied rate on the deliverable bond in the
  futures market moves by 1.1%.
 NF = (5 -.9*3) * $100m/9.5*497,000* 1.1
    = 226.9 contracts
         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.
        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.

								
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