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					    Forward and futures contracts

    (Long position) Boths are agreements to buy
      an underlying asset at future (delivery) date
      for a specific price.
    Futures contracts have some added
      institutional features – most noticably daily
      settlement.
    Material: Hull Chapters 2,3 and 5. We’ll skip
      and jump. (Thus: no need to read Hull like
      CT1.)
1                November 23, 2010    MATH 2510: Fin. Math. 2
    Futures contracts

    Available on a wide range of underlyings
    Exchange traded
    Specifications need to be defined:
      –   What can be delivered,
      –   Where it can be delivered
      –   When it can be delivered (often a period)
    Settled daily: On long position you receive the
      day-to-day change in the futures price.

2                  November 23, 2010        MATH 2510: Fin. Math. 2
    Convergence of Futures to Spot

    If the futures price is above the spot price
       during the delivery period, arbitrage is
       possible
       –   Short futures contract
       –   Buy underlying asset
       –   Make delivery
    And vice versa if the futures price is below the
      spot price.

3                   November 23, 2010   MATH 2510: Fin. Math. 2
                 Futures
                                              Spot Price
                  Price
    Spot Price                       Futures
                                      Price

                        Time                              Time

           (a)                              (b)
4                November 23, 2010     MATH 2510: Fin. Math. 2
    Futures vs. forward prices

    If you think that the definition of the futures
       contract/price seems circular --- I’m inclined
       to agree.

    Let’s look a Appendix A in Hull’s chapter 5.
      This shows that if interest rates are constant,
      then futures and forward prices are equal.


5                November 23, 2010      MATH 2510: Fin. Math. 2
    If interest rates are stochastic, then forward
        and futures prices are not (necessarily)
        equal.
    If the price of the underlying is positively
        correlated w/ interest rates, then we receive
        futures payments in high interest rate
        scenarios -> futures above forward.
    But differences are typically small.
6                November 23, 2010      MATH 2510: Fin. Math. 2
    Margins

    A margin is cash or marketable securities
      deposited by an investor with his or her
      broker; serves as collateral.
    The balance in the margin account is adjusted
      to reflect daily settlement.
    Margins minimize the possibility of a loss
      through a default on a contract.


7               November 23, 2010   MATH 2510: Fin. Math. 2
    Collateral

    It is becoming increasingly common (post
        Subprime Crisis/Credit Crunch/Financial
        Meltdown in particular) for contracts to be
        collateralized/have margins.

    They are then similar to futures contracts in
      that they are settled regularly (e.g. every day
      or every week)
8                November 23, 2010      MATH 2510: Fin. Math. 2
    Delivery

    Closing out a futures position involves entering
      into an offsetting trade; most contracts are
      closed out before maturity.

    Though sometimes spectacularly not:
     Business Snapshot 2.1: Live cattle
     Amajaro summer 2010: Cocoa


9               November 23, 2010     MATH 2510: Fin. Math. 2
     When there are alternatives about what is
      delivered, where it is delivered, and when it
      is delivered, the party with the short position
      chooses.

     A few contracts (for example, those on stock
       indices and Eurodollars) are settled in cash.


10                November 23, 2010     MATH 2510: Fin. Math. 2
     Accounting and Taxation

     This is really tricky stuff.
      Ideally hedging profits (losses) should be
       recognized at the same time as the losses (profits)
       on the item being hedged
      Ideally profits and losses from speculation should be
       recognized on a mark-to-market basis
      Roughly speaking, this is what the accounting and
       tax treatment of futures in the U.S.and many other
       countries attempts to achieve.

11                 November 23, 2010        MATH 2510: Fin. Math. 2
     Long & Short Futures Hedges

     A long futures hedge is appropriate when you
       know you will purchase an asset in the
       future and want to lock in the price

     A short futures hedge is appropriate when
       you know you will sell an asset in the future
       & want to lock in the price


12               November 23, 2010     MATH 2510: Fin. Math. 2
     Arguments in Favor of Hedging


     Companies should focus on the main
     business they are in and take steps to
     minimize risks arising from interest rates,
     exchange rates, and other market variables.




13              November 23, 2010   MATH 2510: Fin. Math. 2
     Arguments Against Hedging

        Shareholders are usually well diversified and
         can make their own hedging decisions
        It may increase risk to hedge when
         competitors do not
        Explaining a situation where there is a loss
         on the hedge and a gain on the underlying
         can be difficult


14                 November 23, 2010    MATH 2510: Fin. Math. 2
     Basis Risk

      Basis is the difference between spot and
        futures prices.

      Basis risk arises because of the uncertainty
        about the basis when the hedge is closed
        out. (Say you can’t match w/ exact delivery
        date and/or underlying asset for futures.)


15              November 23, 2010    MATH 2510: Fin. Math. 2
     Long Hedge

     Suppose that
          F1 : Initial Futures Price
          F2 : Final Futures Price
          S2 : Final Asset Price
     You hedge the future purchase of an asset by
       entering into a long futures contract
     Cost of Asset=S2 – (F2 – F1) = F1 + Basis


16               November 23, 2010    MATH 2510: Fin. Math. 2
     Choice of Contract

     Choose a delivery month that is as close as
      possible to, but later than, the end of the life
      of the hedge
     When there is no futures contract on the asset
      being hedged, choose the contract whose
      futures price is most highly correlated with
      the asset price. This is known as cross
      hedging.

17                November 23, 2010     MATH 2510: Fin. Math. 2
     Optimal Hedge Ratio

     Proportion of the exposure that should optimally be hedged is

                                  sS
                                r
                                  sF
     where
     sS is the standard deviation of DS, the change in the spot price
         during the hedging period,
     sF is the standard deviation of DF, the change in the futures
         price during the hedging period
     r is the coefficient of correlation between DS and DF.


18                   November 23, 2010              MATH 2510: Fin. Math. 2

				
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