Payoff by yaofenji

VIEWS: 23 PAGES: 3

									                        Graphical Approach to Forward Contracts

       In this note we examine the relationship between forward contracts, bonds and the
underlying asset. For simplicity we use the example given in class where we derived the
price of a forward on a non-dividend paying stock. This stock trades today for $25 and
we consider a forward contact that expires in 3 months from now ("the maturity").

        We begin by plotting payoff diagrams for various assets. These diagrams show
the payoff to the owner of the asset at maturity. These payoffs do not include any costs
or gains earned when purchasing the assets today.

Long/Short the security:

                                Payoff to long and short positions in the stock

                         60                                                   long stock
                         40                                                   short stock
                         20
               Payoff




                          0
                        -20 0                   20                40                  60
                        -40
                        -60
                                          Price of security at maturity


Long/Short Forward:

                              Payoff to long and short position in Forward Contract

                                                                          long forward
                         30
                                                                          short forward
                         20
             Payoff




                         10
                          0
                        -10 0                   20                 40                  60

                        -20
                        -30
                                             Price of security at maturity

        Note that both the long and short forward payoff positions break even when the
price of the stock at maturity is equal to the forward price (25.375 in our example).
Buy/sell a bond for $25 with 1.5% quarterly return:

                                         Buy or Sell a Bond


                      30                                                     buy bond
                      20                                                     sell bond
                      10
            Payoff


                       0
                     -10 0                 20                  40                60
                     -20
                     -30
                                       Price of security at maturity



       By buying a bond (lending) today we know that we are going to get a fixed payoff
equal to 25*1.015=25.375. By selling a bond today (borrowing) we know that we are
committed to repay 25.375.


The previous plots enable us to achieve the following goals:

1. Construct a forward contract using only the bond and stock.
2. Construct a stock payoff using only the forward contract and the bond.
3. Construct a bond payoff using only the forward contract and the stock.


1. Construction of a long forward contract using the stock and bond:

The payoff of the long forward can be replicated by borrowing $25 and buying the stock.
At maturity the payoff is just the sum of the payoffs of the constituent assets:


                             Replicating the Payoff of a forward contract
                                                                            long stock
                     60                                                     long forward
                     40                                                     sell bond
            Payoff




                     20
                      0
                     -20 0      10        20         30        40       50         60

                     -40
                                       Price of security at maturity
The payoff diagram when we sell a forward contract can be obtained by reversing the
above actions.

2. Construction of a long stock payoff using the forward contract and the bond.

The payoff of the long stock can be replicated by lending $25 and entering into a long
forward position. Again, at maturity the payoff is just the sum of the payoffs of the
constituent assets:

                                     Replicating the payoff of a long stock position
                       60                                                            long stock
                       50                                                            long forward
                       40                                                            buy bond
                       30
          Payoff




                       20
                       10
                            0
                      -10 0             10        20        30        40        50          60
                      -20
                      -30
                                              Price of security at maturity


The payoff diagram of a short stock position can be obtained by reversing the above
actions.

3. Constructing a long bond payoff (namely the position of an investor who lent $25)
   using the forward contract and the stock:


                                          Replicating the payoff of a long Bond

                                60                                                     long stock
                                40                                                     short forward
                                                                                       buy bond
                   Payoff




                                20
                                 0
                            -20 0                      20               40                   60
                            -40
                                                Price of security at maturity

The payoff of a short bond (borrowing) can be obtained by reversing the above actions.

								
To top