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Determination of Forward and Futures Prices(2)

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					Determination of Forward
      and Futures Prices
                                                      Chapter 5

Note: In this chapter forward and futures contracts are treated identically.




                                                                               5.1
Consumption vs. Investment
Assets
l   Investment assets are assets held by
    significant numbers of people purely for
    investment purposes (Examples: gold,
    silver); no convenience yield
l   Consumption assets are assets held
    primarily for consumption/usage in
    production process (Examples: copper,
    oil); positive convenience yield

                                               5.2
4 Underlying Assets in this Course
l   Common stock: financial, investment asset
l   Debt: financial, investment asset
l   Foreign Exchange: financial, investment asset
l   Commodities: nonfinancial, may be investment
    or consumption assets
l   Note: Financial assets are investment assets.
    Nonfinancial assets (i.e. commodities) may be
    investment or consumption assets
Short Selling

   l   Short selling involves selling
       securities you do not own
   l   Your broker borrows the
       securities from another client and
       sells them in the market
   l   So what? Short selling of
       consumption assets is
       problematic

                                            5.4
Short Selling
(continued)

        l   At some stage you must buy the
            securities back so they can be
            replaced in the account of the
            client
        l   You must pay dividends and
            other benefits the owner of the
            securities receives; the income
            flow of the shorted asset is the
            short-seller’s liability

                                               5.5
Notation

       S0: Spot price today
       F0: Futures or forward price today
           T: Time until delivery date
           r: Risk-free interest rate, quoted
              per annum cc (default
              assumption), for maturity T

                                                5.6
1. Gold: An Arbitrage
Opportunity?
  l   Suppose that:
      l The spot price of gold is US$600

      l The 1-year futures price of gold is US$650

      l The 1-year US$ interest rate is 5%

      l No income or storage costs for gold

  l   Is there an arbitrage opportunity? 650>630.76
      Now: Sell futures, borrow & buy spot, “carry” to
      yearend to satisfy futures obligation



                                                         5.7
2. Gold: Another Arbitrage
Opportunity?
   l   Suppose that:
       l The spot price of gold is US$600
       l The 1-year gold futures price = US$590
       l The 1-year US$ interest rate is 5%
       l No income or storage costs for gold
   l   Is there an arbitrage opportunity? 590<630.76
       Now: short gold spot, deposit proceeds, long
       futures. Later: close out deposit, cover short
       position via futures contract.



                                                        5.8
The Futures Price of Gold

If the spot price of gold is S & the futures price is
for a contract deliverable in T years is F, then
                    F0 = S0erT
where r is the 1-year cc (domestic currency) risk
-free rate of interest.
In our examples, S=600, T=1, and r=0.05 so that
              F = 600e0.05x1 = 630.76


                                                    5.9
Interest Rates Measured with
Continuous Compounding (default)


              F0 = S0erT

 This equation relates the forward price
 and the spot price for any investment
 asset that provides no income and has
 no storage costs

                                           5.10
When an Investment Asset
Provides a Known Dollar Income
                F0 = (S0 – I )erT
   where I is the present value of the
   income during life of futures contract;
   underlying asset income flow depicted
   as a present value. If the underlying
   asset generates an income flow, the
   cost of carrying the asset is thereby
   reduced. Thus, the negative sign.

                                             5.11
Investment Asset Provides a Known Yield


               F0 = S0 e(r–q )T
  where q is the average yield during the
  life of the contract (expressed with
  continuous compounding); underlying
  asset income flow depicted as a cc rate.
  Consistency requires: S0 (1-e–qT ) = I



                                          5.12
Valuing a Forward/Futures Contract: post-inception or
when contractual price differs from prevailing
forward/futures price


l   Suppose that K is delivery/contractual price in
    a forward contract & F0 is forward price that
    would apply to the contract today
l   The value of a long forward contract, ƒ, is
                 ƒ = (F0 – K )e–rT
l   Similarly, the value of a short forward contract
    is
                     ƒ = (K – F0 )e–rT
In the above expressions, the cost-of-carry model can be substituted for F0.



                                                                               5.13
Post-inception value of forward/futures = f

l   K = original forward/futures rate
l   F0 = prevailing forward/futures rate
l   For F0 can substitute cost-of-carry model
l   F0=Se^(r+u-q-y)T: r=interest rate,
    u=storage cost (u>0 for commodities),
    q=income yield (q=0 for commodities),
    y=convenience yield (y>0 for consumption
    assets, i.e. commodities used in
    production).
Forward vs. Futures Prices

l   Forward and futures prices are usually assumed to be
    the same. When interest rates are uncertain they are, in
    theory, slightly different:
l   A strong positive correlation between interest rates and
    the asset price implies the futures price is higher than
    the forward price. Long position experiences increases
    in his MAB when interest rates are rising. Thus, he
    prefers futures to forward.
l   A strong negative correlation implies the futures price is
    lower than the forward price. Long position experiences
    margin calls when interest rates rise. Thus, he prefers
    forward to futures.
                                                          5.15
Stock Index
l   Can be viewed as an investment asset
    paying a dividend yield
l   The futures price and spot price
    relationship is therefore
              F0 = S0 e(r–q )T
    where q is the dividend yield on the
    portfolio represented by the index
    during life of contract

                                           5.16
Stock Index
(continued)


      l   For the formula to be true it is
          important that the index represent an
          investment asset
      l   In other words, changes in the index
          must correspond to changes in the
          value of a tradable portfolio
      l   CME’s Nikkei 225 futures contract
          (quanto): underlying asset = $5x
          Nikkei225 index is not traded
                                                  5.17
Index Arbitrage

l   When F0 > S0e(r-q)T an arbitrageur buys the
    stocks underlying the index and sells
    futures
l   When F0 < S0e(r-q)T an arbitrageur buys
    futures and shorts or sells the stocks
    underlying the index



                                              5.18
Index Arbitrage
(continued)


    l   Index arbitrage involves simultaneous
        computer-generated trades in futures
        and many different stocks
    l   Occasionally (e.g., on Black Monday)
        simultaneous trades are not possible
    l   Result: theoretical no-arbitrage
        relationship between F0 and S0 fails to
        hold at times

                                                  5.19
Futures and Forwards on
Currencies

l   A foreign currency is analogous to a security
    providing a dividend yield
l   The continuous dividend yield is the foreign risk
    -free interest rate
l   It follows that if rf is the foreign risk-free interest
    rate




                                                        5.20
Why the Relation Must Be True: 2 different
ways of accumulating $’s at time T




                                         5.21
Futures on assets that incur storage
costs, i.e., nonfinancial assets
                 F0 £ S0 e(r+u)T
   where u is the storage cost per unit time as a
   cc percent of the asset value.
   Alternative way of depiction:
            F0 £ (S0+U )erT
   where U is the present value of the storage
   costs. Consistency: S0 (euT -1) = U

                                                    5.22
The Cost of Carry

 l   The cost of carry, c, is the storage cost plus the
     interest costs less the income earned
 l   For an investment asset F0 = S0ecT
 l   For a consumption asset F0 £ S0ecT . Cannot
     arbitrage: Short spot and long futures is problematic
 l   The convenience yield on the consumption
     asset, y, is defined so that
     F0 = S0 e(c–y )T
 l   Note: y is not observed, it is inferred.


                                                             5.23
Futures Prices & Expected Future
Spot Prices
 l   Suppose k is the expected return required by
     investors on an asset
 l   We can deposit F0e–r T and undertake long
     futures now, obtaining ST at maturity of the
     futures contract.
 l   NPV = - F0e–rT +ST e–kT
 l   Profit seeking drives E(NPV) = 0 which
     implies
        F0 = E (ST )e(r–k )T

                                                5.24
Futures Prices & Future Spot
Prices (continued)
  l   If the asset has
      l no systematic risk, then k = r and F0 is an
          unbiased estimator of ST
      l positive systematic risk, then
         k > r and F0 < E (ST ) Normal Backwardation
           Implication: profit by long futures then sell spot.
      l negative systematic risk, then
         k < r and F0 > E (ST ) Contango
          Implication: profit by short futures then buy spot.
      CAPM: k is positive function of systematic risk.



                                                                 5.25

				
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posted:7/22/2013
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Lingjuan Ma Lingjuan Ma MS
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