# Payoff by yaofenji

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```									                        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:

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
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
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.

```
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