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Excel代写-1FIXED

时间：2021-03-05

1FIXED INCOME SECURITIES

FRE : 6411

Sassan Alizadeh, PhD

Tandon School of Engineering

NYU

2021

2FORWARD AND FUTURE CONTRACT

Forward and Future are contract for a deferred delivery of an

asset at a fix price, at specific time in future. Both price and

time is agreed today.

Forward Contract:

An investor who buys a forward contract, agrees to buy one unit

of the underlying asset at a specified future time (Maturity

Date) and a specified price

The agreed price is set when the contract is written and the

price doesn’t change through the life of the contract

If the agreed price is set such that the value of contract at time

when the contract is initiated, is 0, (i.e. neither party, pays or

receive anything), then the agreed price is called “Forward

Price”

3FORWARD CONTRACT

During the life of the contract, neither party have ANY cash

flow BUT at maturity the long position receives one unit of

the asset or its cash value, and pays the forward price to

the seller.

Example:

Let G(t,T) be a forward price at time t ,maturing at time T. Let p(t)

be the spot price of the asset at time t. The cash flow from buyer

side is:

Date Forward Price Cash Price Cash Flow

0 G(0,T) P(0) 0

1 G(1,T) P(1) 0

..

T-1 G(T-1,T) P(T-1) 0

T P(T) P(T) P(T)-G(0,T)

4FORWARD CONTRACT

At 0the value of the forward contract is 0 0 = 0 . However

as time passes, although there is no cash flow during the life of

contract the value of the forward contract fluctuates, due to

fluctuation of the value of the underlying asset ≥ 0 ≤

5FORWARD PRICE

G(t,T1,T2) : Forward Price at time t, maturing at T1 on a T2

zero-coupon bond.

V(t) is the time t value of this forward contract. No Cash Flow at

time t, therefore the value of this forward contract at time ‘t’

must be zero. We set the forward price such that the value of

the contract is 0.

=

1,2 − 0,1,2

1,1−1

, −1

=

1,2

1,1−1

, −1 +

0, 1, 2

1

1,1−1

, −1

= , 2, + 0, 1, 2 , 1,

FORWARD PRICE

0 = 0, 2, 0 + 0, 1, 2 0, 1, 0 = 0

0, 1, 2 =

0,2,0

0,1,0

Using Mimicking Portfolio:

1. Short G(0,T1,T2) units of the T1 maturity bond Collect

$G(0,T1,T2) * p(0,T1)

2. Buy T2 maturity Bond, pay p(0,T2)= G(0,T1,T2) * p(0,T1)

Note at time 0, cash Flow is G(0,T1,T2) * p(0,T1)-p(0,T2) =0

3. At time T1 , you deliver the T2 maturity Bond for the forward

contract and collect G(t,T1,T2) which is exactly your obligation

from shorting G(t,T1,T2)* p(T1,T1)

6

7FORWARD PRICE

Summary:

1. To find the forward price at time t, G(t,T1,T2), set the value of

the forward contract to zero and solve for the forward price ,

using the risk-neutral expectation.

2. To find the price of time t forward contract at time m , find

the value of the contract such that the forward price is

G(t,T1,T2). Here we have the forward price and we need to

solve for the value of the contract, where as in 1 we have to

set the value of the contract to 0 and we solve for the

forward price.

8Figure 12.1: An Example of a One-Factor Bond Price Curve Evolution. The Money Market Account Values

and Spot Rates are Included on the Tree. Pseudo-Probabilities Are Along Each Branch of the Tree.

.923845

.942322

.961169

.980392

1

.947497

.965127

.982699

1

.937148

.957211

.978085

1

1/2

1/2

1/2

1/2

1/2

1/2

.967826

.984222

1

.960529

.980015

1

.962414

.981169

1

.953877

.976147

1

.985301

1

.981381

1

.982456

1

.977778

1

.983134

1

.978637

1

.979870

1

.974502

1

1

1

1

1

1

1

1

1

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

P(0,4)

P(0,3)

P(0,2)

P(0,1)

P(0,0)

=

B(0) 1

1.02

1.02

1.037958

1.037958

1.042854

1.042854

r(0) = 1.02

1.017606

1.022406

1.016031

1.020393

1.019193

1.024436

1.054597

1.054597

1.059125

1.059125

1.062869

1.062869

1.068337

1.068337

time 0 1 2 3 4

9Example - Forward Price

F(0,3,4)= p(0,4)/P(0,3) = 0.923845 /0.942322 = 0.980392

F(1,3,4,u) = p(1,4,u)/p(1,3,u)=0.947497/0.965127= 0.981733

F(1,3,4,d) = P(1,4,u)/p(1,3,d)=0.937148/0.957211= 0.979041

F(2,3,4,uu) = P(2,4,uu)/p(2,3,uu)=0.967826/0.98422= 0.983341

F(2,3,4,ud) = P(2,4,ud)/p(2,3,ud)=0.960529/0.9800150= 0.980117

F(2,3,4,du) = P(2,4,du)/p(2,3,du)=0.9624147/0.981169= 0.980886

F(2,3,4,dd) = P(2,4,dd)/p(2,3,dd)=0.953877/0.9761470= 0.977186

Note that :

F(3,3,4,St) = P(3,4,S3)/ P(3,3,S3) =P(3,4,S3) /1

So we know the price for all forward Contracts

10

Example - Forward Price

To find the forward price: F(t,T1,T2)

1) Set the value of forward contract to zero

2) Solve for forward price subject to the value of the

forward contract is zero; get F(t,T1,T2)

To find the value of the time ‘’t” forward contract at time “m” :

V(m,Sm)=p(m,T2; Sm) – F(t,T1,T2) P(m,T1; Sm)

Here we have the forward price and we want to solve for the value

of contract.

V(1;u)=P(1,4,u) – F(0,3,4) * P(1,3;u)

=0.947497 - 0.980392 * 0.965127 = 0.0001294

11

Example - Forward Price

V(0)=1/1.02*E[V(1;S1)]=

1/2 * 1/1.02 * V(1,u) + 1/2 *1/1.02 * V(1,d) = 0 so V(1,d)= -V(1,u)

V(3;dud)=P(3,4;dud)-F(o,3,4)* P(3,3;dud)

=0.978637-0.980392= - 0.001755

V(3,duu) = P(3,4,duu) – F(0,3,4) * p(3,3,duu)

= 0.983134 – 0.980392 = + 0.002742

V(2;du) = 1/ r(2,du) * E [ v( 3;S3)| S2=du ]

= 1/019193 * [1/2 * ( 0.002742 - 0.001755) ] = 0.00048175

12

Future Contract

Future contracts are like Forward contract EXCEPT they are

marked to market everyday. (For every period there is a cash

flow)

As the contract matures, investor will make or receive daily

installment payment toward the eventual purchase of the

underlying asset, which will be at spot price of the underlying

asset. The total value of daily installments and the final

payment at the maturity will be equal to the future prices set

when the contract was initiated.

The daily installment is determined by the daily change in the

Future price. When the future price goes up then the investor

who is long will receive a payment from the investor who is

short. This is called MARKING-TO-MARKET. The daily Future

price is set such that the value of contract reset to 0.

13

Future Contract

Date Future Price Cash Price Cash Flow

0

H(0,T) P(0) 0

1

H(1,T) P(1) H(1,T) - H(0,T)

2 H(2,T)

P(2) H(2,T) - H(1,T)

3

H(3,T) P(3) H(3,T) - H(2,T)

…

T-2

H(T-2,T) P(T-2) H(T-2,T) - H(T-3,T)

T-1

H(T-1,T) P(T-1)

H(T-1,T) - H(T-2,T)

T

P(T) P(T) P(T) - H(T-1,T)

14

FUTURE PRICE

Future price is marked to market, that is in every period there is a

cash flow, to reset the value of Future contract to 0. The

cash flow is such that at each state of the world the value of the

future contract is 0 . The value of each state depends not only on

the final payoff , but the value of the future contract the next

period, you will either receive or pay cash. We need to use the

back-ward induction technique.

H(t,T1,T2) : The future price at time t , maturing at time T1 , on a

T2 zero coupon bond.

H(T1-1,T1,T2) is a one period future price.

15

FUTURE PRICE

),(),,(

),,1(),,2(

zero!* is termsecond the*

))2(

)(

);,1();,(

(

)2(

)1(

);,2();,1(

0

),,(),();,1(

)(

)1(

1

);,1(),(

0

),();,(

)1(

)(

);,1(),(

0

2121

2112211

1

1

211211

2

1

1

211211

2

2111211211

1

121121

11

21211

1

1

21121

1

1

1

1

11

1

TTpETTtH

TTTHETTTH

TB

TB

TTTHTTTH

E

TB

TB

TTTHTTTH

E

TTTHETTpETTTH

TB

TBTTTHTTp

E

TTpTTTH

TB

TB

TTTHTTp

E

t

T

T

T

TT

T

T

The future price is the time t expectation of the underlying T2- maturity

zero-coupon bonds’ price trading at time T1

16

Example-Future Price

H(3,3,4;uuu)=p(3,4,uuu)=0.985301

H(3,3,4;uud)=p(3,4,uud)= 0.981381

H(2,3,4;uu)=E[ H(3,3,4,uu?)]= ½ * [0.985301+ 0.981381]=0.983341

Cash flow at state uuu = 0.985301-0.983341= + 0.001960

Cash flow at state uud = 0.981381-0.983341= - 0.001960

What would the mimicking portfolio look like?

M(2,uu)*B(3,uuu)+N1(2,uu)* P(3,4,uuu) =

M(2,uu)*1.05459 + N1(2,uu)*0.985301 = + 0.001960

M(2,uu)*B(3,uud)+ N1(2,uu)*P(3,4,uud) =

M(2,uu)*1.05459 + N1(2,uu)*0.981381 = - 0.001960

So N1(2,uu) = 1 M(2,uu) = - 0.932439

17

Example-Future Price

Another method:

From delta:

1

981381.0985301.0

00196.0*2

uud)P(3,4, - uuu)P(3,4,

uud)(3, flowcash -uuu)(3, flowcash

uu)N1(2,

980383.0

974502.0979870.0978637.0983134.0

97778.0982456.0981381.0985301.0

8

1

)dddP(3,4, )(s )dduP(3,4, )(s ... uud)P(3,4, )(s

)uuuP(3,4, )(s ]Sp(3,4, [ E H(0,3,4)

333

33

ddddduuud

uuu

18

Relation of Forward and future price

We can now use the theory of the preceding sections to relate

forward and futures prices.

)

1

,(

)(

)

1

(

2

,

1~

2

,

1

,

2

,

1

~

2

,

1

,

)

1

(

1~

)

1

(

2

,

1~

2

,

1

,

TtP

tB

TB

TTP

tETTtH

TTPtETTtH

TBt

E

TB

TTP

t

E

TTtG

19

Relation between Forward and future price

COV(x,y)=E[(x-E(x)) * (y-E(y))] =

E[xy]-E[x E(y)]- E[y E(x)]+ E[E(x) E(y)]

= E [xy] – E[x] E[y]= COV(x,y) , E [xy] = COV(x,y) + E[x] E[y]

)

1

(

1

0

~

)

1

(

1

),

2

,

1

((

)

1

(

1

0

~

2

,

10

~

)

1

(

1

0

~

)

1

(

2

,

1

0

~

)

2

,

1

0(

)

1

,(

)(

)

1

(

1

),

2

,

1

((

)

1

,(

)(

)

1

(

~

.

2

,

1

~

2

,

1

,

,

1

TB

E

TB

TTPCOV

TB

ETTPE

TB

E

TB

TTP

E

TTG

TtP

tB

TB

TTPCOV

TtP

tB

TBt

ETTPtETTtG

20

Relation between Forward and future

price

21110

21

21110

21

,

1

..

1

),,((,,

,

1

..

1

),,((,

)

2

,

1

0()

2

,

1

0(

2

,

10

~

)

2

,

1

0(

TTPrrr

TTPCOV

TTPrrr

TTPCOV

T

T

TTHTTG

TTPETTG

110 ..

1

Trrr

So if interest rates goes up , Goes down

If interest rate goes up , 21,TTP Goes down

Cov ( , ) >0 , then G(0,T1,T2) > H( 0,T1,T2) , which means

when covariance is positive then forward price exceeds

future prices.

21

Options on Futures

European call option T* < T1 exercise K ,

You are entitled to buy a future contract with excersie price of

H(T*,T1,T2) at K .

Payoff = Max( H(T*,T1,T2)-k ,0)

),(]

)*,(

)0,),*,((

[),( 1

1*

21

t

t

tt StB

STB

kTTThMax

EStC

K=0.981 H(2,3,4)

C(2,uu)= Max( 0.983341 – 0.981 , 0 ) = 0.002341

C(2,ud) = Max ( 0.980117 – 0.981 , 0 ) = 0

C(2,du ) = Max ( 0.980886- 0,981 , 0 ) = 0

C(2,dd ) = Max ( 0.977186 – 0.981 , 0 ) =0

22

Options on Futures

C( 1,u ) = 1 / (2*1.017606) * 0.002341= 0.001150

C(1,d) =0

C(0) = ½ * 1/ 1.02 * 0.001150 = 0.00564

To ways to hedge:

01127.0

02.1

001150.0

)1(

0),1(1),1(

),1(

726117.0

)001612.(001612.0

002341.0

)udflow(2,cash )uuflow(2,cash

),2(),2(

),1(

29686.0

02.1

947497.0*3208.0001150.0

)1(

),4,1(),1(1),1(

),1(

3208.0

960529.0967826.0

002341.0

),4,2(),4,2(

),2(),2(

),1(

0

1

0

1

contract future theuse )2

Asset underlying the Use)1

B

uNuC

uN

udCuuC

uN

B

upuNuC

uN

udPuuP

udCuuC

uN

23

EURODOLLAR FUTURE CONTRACT

Eurodollar deposits are deposits that are maintained outside the

United states. Generally they are exempt from federal Reserve

regulation that applied to the domestic deposits.

The rate that are applied to Eurodollar deposits in Interbank

transaction are known as London Interbank Offered Rates (

LIBOR). The LIBOR spot is active in maturities ranging from few

days to 10 years. The depth of the market is great at 3 months

and six month maturates.

The Eurodollar future contracts trades at Merc in Chicago. the

contract settles to 90 days LIBOR , which is the yield derived

from the underlying asset that is the 90-day Eurodollar time

deposit.

24

EURODOLLAR FUTURE CONTRACT

Each contract is for $1 million face value of euro deposits, with

delivery each month for several months in the future, after

which the delivery occurs only in quarterly periods ( March ,

June , September and December) . Currently the market is for

up to 10 years in the future

Future contracts settles by cash , on the maturity date. On

expiration which is the second London business day before the

third Wednesday of the maturity month, the contract settles by

cash to LIBOR

How?

25

EURODOLLAR FUTURE CONTRACT

On Expiration date the clearing house determines the three

month LIBOR, by selecting at random 12 reference banks from

a list of no less than 20 participating banks, and asks for their

three months LIBOR rate.

Then the highest and lowest quotes are eliminated and the

average of the remaining rates are calculated. The final

settlement price is 100-Average LIBOR.

Margin Requirement

An investor has to set up margin as a security deposit.

Exchange will set a minimum margin requirements , but each

trading house will set their own margin requirements. Currently

for Eurodollars future contract the margin requirement is around

$2,000. Most clearing house accept a very high percentage of

the margin as interest bearing securities such as T-bils .

26

EURODOLLAR FUTURE CONTRACT- Margin

In addition to the initial margin, we also have a maintenance

margin which is set around 75-80% of the initial margin. This is

how the maintenance margin works:

Suppose we bought a June 02 Eurodollar future at 97.47 ( this is

to say that in June 02 ,the three months LIBOR is 2.53%) . We

set up the initial margin of $2000 for our contract. Now we

know each thick is equivalent to basis point move in the interest

rate and is equal to $25. The minimum price fluctuation is 0.5

basis points or $12.50.

27

EURODOLLAR FUTURE CONTRACT- Margin

Now the maintenance margin for this is $1500, so if the future

interest rates goes up by 20 basis point from 2.53% to 2.73%

the future price will fall from 97.47 to 97.27 . If on any day the

final settlement price falls at or below 97.27 , which will result

in a loss of $500 or more, and we hit the maintenance margin,

the exchange will call us and ask us for an additional cash to

bring the account balance to $2000 again.

If the customer can not meet the margin call, the brokerage

house will sell the position and give whatever is left to the

customer.

28

EURODOLLAR FUTURE CONTRACT

From Eurodollar contracts we can obtain the yield curve

Important factor:

1. Actively traded for wide range of maturity , up to seven years.

2. Contract is heavily used for hedging and synthesizing interest

rate swaps

What is Swap?

Assume that I have floating rate liabilities index of 90-day LIBOR .

From Dec 2002 , through 2010.Now if I sell all the future

Eurodollar contract. I converted all the floating rate liabilities

into a stream of currently known liabilities.

29

EURODOLLAR FUTURE CONTRACT- Swap

Example:

If the LIBOR on March 2005 where 7%. Now I am locking in the

rate of 6.18%. My liabilities on my loan is : 0.82% higher than

what I lock in.

0.82*$1,000,000*1/4

On the other hand I am making :

100-7=93 , 100-6.18 =93.82

93.820-93=0.82*250,000

So I swap my floating liabilities into known liabilities.

30

EURODOLLAR FUTURE CONTRACT- Swap rate

Swap rate:

The discount weighted average of the short term

Eurodollar rates:

),0(...),0()

1

,0(

),0()(...)

2

,0()

2

()

1

,0()

1

(

),0()(...)

2

,0()

2

()

1

,0()

1

()],0(...),0()

1

,0([

2

2

nTpTpTp

nTpnTrTpTrTpTr

x

nTpnTrTpTrTpTrnTpTpTpx

For following slide data, the numerator is 164.90 , and the

denominator is 29.538 or the swap rate is 5.583%

31

32

33

Yield Curve

学霸联盟

FRE : 6411

Sassan Alizadeh, PhD

Tandon School of Engineering

NYU

2021

2FORWARD AND FUTURE CONTRACT

Forward and Future are contract for a deferred delivery of an

asset at a fix price, at specific time in future. Both price and

time is agreed today.

Forward Contract:

An investor who buys a forward contract, agrees to buy one unit

of the underlying asset at a specified future time (Maturity

Date) and a specified price

The agreed price is set when the contract is written and the

price doesn’t change through the life of the contract

If the agreed price is set such that the value of contract at time

when the contract is initiated, is 0, (i.e. neither party, pays or

receive anything), then the agreed price is called “Forward

Price”

3FORWARD CONTRACT

During the life of the contract, neither party have ANY cash

flow BUT at maturity the long position receives one unit of

the asset or its cash value, and pays the forward price to

the seller.

Example:

Let G(t,T) be a forward price at time t ,maturing at time T. Let p(t)

be the spot price of the asset at time t. The cash flow from buyer

side is:

Date Forward Price Cash Price Cash Flow

0 G(0,T) P(0) 0

1 G(1,T) P(1) 0

..

T-1 G(T-1,T) P(T-1) 0

T P(T) P(T) P(T)-G(0,T)

4FORWARD CONTRACT

At 0the value of the forward contract is 0 0 = 0 . However

as time passes, although there is no cash flow during the life of

contract the value of the forward contract fluctuates, due to

fluctuation of the value of the underlying asset ≥ 0 ≤

5FORWARD PRICE

G(t,T1,T2) : Forward Price at time t, maturing at T1 on a T2

zero-coupon bond.

V(t) is the time t value of this forward contract. No Cash Flow at

time t, therefore the value of this forward contract at time ‘t’

must be zero. We set the forward price such that the value of

the contract is 0.

=

1,2 − 0,1,2

1,1−1

, −1

=

1,2

1,1−1

, −1 +

0, 1, 2

1

1,1−1

, −1

= , 2, + 0, 1, 2 , 1,

FORWARD PRICE

0 = 0, 2, 0 + 0, 1, 2 0, 1, 0 = 0

0, 1, 2 =

0,2,0

0,1,0

Using Mimicking Portfolio:

1. Short G(0,T1,T2) units of the T1 maturity bond Collect

$G(0,T1,T2) * p(0,T1)

2. Buy T2 maturity Bond, pay p(0,T2)= G(0,T1,T2) * p(0,T1)

Note at time 0, cash Flow is G(0,T1,T2) * p(0,T1)-p(0,T2) =0

3. At time T1 , you deliver the T2 maturity Bond for the forward

contract and collect G(t,T1,T2) which is exactly your obligation

from shorting G(t,T1,T2)* p(T1,T1)

6

7FORWARD PRICE

Summary:

1. To find the forward price at time t, G(t,T1,T2), set the value of

the forward contract to zero and solve for the forward price ,

using the risk-neutral expectation.

2. To find the price of time t forward contract at time m , find

the value of the contract such that the forward price is

G(t,T1,T2). Here we have the forward price and we need to

solve for the value of the contract, where as in 1 we have to

set the value of the contract to 0 and we solve for the

forward price.

8Figure 12.1: An Example of a One-Factor Bond Price Curve Evolution. The Money Market Account Values

and Spot Rates are Included on the Tree. Pseudo-Probabilities Are Along Each Branch of the Tree.

.923845

.942322

.961169

.980392

1

.947497

.965127

.982699

1

.937148

.957211

.978085

1

1/2

1/2

1/2

1/2

1/2

1/2

.967826

.984222

1

.960529

.980015

1

.962414

.981169

1

.953877

.976147

1

.985301

1

.981381

1

.982456

1

.977778

1

.983134

1

.978637

1

.979870

1

.974502

1

1

1

1

1

1

1

1

1

1/2

1/2

1/2

1/2

1/2

1/2

1/2

1/2

P(0,4)

P(0,3)

P(0,2)

P(0,1)

P(0,0)

=

B(0) 1

1.02

1.02

1.037958

1.037958

1.042854

1.042854

r(0) = 1.02

1.017606

1.022406

1.016031

1.020393

1.019193

1.024436

1.054597

1.054597

1.059125

1.059125

1.062869

1.062869

1.068337

1.068337

time 0 1 2 3 4

9Example - Forward Price

F(0,3,4)= p(0,4)/P(0,3) = 0.923845 /0.942322 = 0.980392

F(1,3,4,u) = p(1,4,u)/p(1,3,u)=0.947497/0.965127= 0.981733

F(1,3,4,d) = P(1,4,u)/p(1,3,d)=0.937148/0.957211= 0.979041

F(2,3,4,uu) = P(2,4,uu)/p(2,3,uu)=0.967826/0.98422= 0.983341

F(2,3,4,ud) = P(2,4,ud)/p(2,3,ud)=0.960529/0.9800150= 0.980117

F(2,3,4,du) = P(2,4,du)/p(2,3,du)=0.9624147/0.981169= 0.980886

F(2,3,4,dd) = P(2,4,dd)/p(2,3,dd)=0.953877/0.9761470= 0.977186

Note that :

F(3,3,4,St) = P(3,4,S3)/ P(3,3,S3) =P(3,4,S3) /1

So we know the price for all forward Contracts

10

Example - Forward Price

To find the forward price: F(t,T1,T2)

1) Set the value of forward contract to zero

2) Solve for forward price subject to the value of the

forward contract is zero; get F(t,T1,T2)

To find the value of the time ‘’t” forward contract at time “m” :

V(m,Sm)=p(m,T2; Sm) – F(t,T1,T2) P(m,T1; Sm)

Here we have the forward price and we want to solve for the value

of contract.

V(1;u)=P(1,4,u) – F(0,3,4) * P(1,3;u)

=0.947497 - 0.980392 * 0.965127 = 0.0001294

11

Example - Forward Price

V(0)=1/1.02*E[V(1;S1)]=

1/2 * 1/1.02 * V(1,u) + 1/2 *1/1.02 * V(1,d) = 0 so V(1,d)= -V(1,u)

V(3;dud)=P(3,4;dud)-F(o,3,4)* P(3,3;dud)

=0.978637-0.980392= - 0.001755

V(3,duu) = P(3,4,duu) – F(0,3,4) * p(3,3,duu)

= 0.983134 – 0.980392 = + 0.002742

V(2;du) = 1/ r(2,du) * E [ v( 3;S3)| S2=du ]

= 1/019193 * [1/2 * ( 0.002742 - 0.001755) ] = 0.00048175

12

Future Contract

Future contracts are like Forward contract EXCEPT they are

marked to market everyday. (For every period there is a cash

flow)

As the contract matures, investor will make or receive daily

installment payment toward the eventual purchase of the

underlying asset, which will be at spot price of the underlying

asset. The total value of daily installments and the final

payment at the maturity will be equal to the future prices set

when the contract was initiated.

The daily installment is determined by the daily change in the

Future price. When the future price goes up then the investor

who is long will receive a payment from the investor who is

short. This is called MARKING-TO-MARKET. The daily Future

price is set such that the value of contract reset to 0.

13

Future Contract

Date Future Price Cash Price Cash Flow

0

H(0,T) P(0) 0

1

H(1,T) P(1) H(1,T) - H(0,T)

2 H(2,T)

P(2) H(2,T) - H(1,T)

3

H(3,T) P(3) H(3,T) - H(2,T)

…

T-2

H(T-2,T) P(T-2) H(T-2,T) - H(T-3,T)

T-1

H(T-1,T) P(T-1)

H(T-1,T) - H(T-2,T)

T

P(T) P(T) P(T) - H(T-1,T)

14

FUTURE PRICE

Future price is marked to market, that is in every period there is a

cash flow, to reset the value of Future contract to 0. The

cash flow is such that at each state of the world the value of the

future contract is 0 . The value of each state depends not only on

the final payoff , but the value of the future contract the next

period, you will either receive or pay cash. We need to use the

back-ward induction technique.

H(t,T1,T2) : The future price at time t , maturing at time T1 , on a

T2 zero coupon bond.

H(T1-1,T1,T2) is a one period future price.

15

FUTURE PRICE

),(),,(

),,1(),,2(

zero!* is termsecond the*

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t

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TT

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T

The future price is the time t expectation of the underlying T2- maturity

zero-coupon bonds’ price trading at time T1

16

Example-Future Price

H(3,3,4;uuu)=p(3,4,uuu)=0.985301

H(3,3,4;uud)=p(3,4,uud)= 0.981381

H(2,3,4;uu)=E[ H(3,3,4,uu?)]= ½ * [0.985301+ 0.981381]=0.983341

Cash flow at state uuu = 0.985301-0.983341= + 0.001960

Cash flow at state uud = 0.981381-0.983341= - 0.001960

What would the mimicking portfolio look like?

M(2,uu)*B(3,uuu)+N1(2,uu)* P(3,4,uuu) =

M(2,uu)*1.05459 + N1(2,uu)*0.985301 = + 0.001960

M(2,uu)*B(3,uud)+ N1(2,uu)*P(3,4,uud) =

M(2,uu)*1.05459 + N1(2,uu)*0.981381 = - 0.001960

So N1(2,uu) = 1 M(2,uu) = - 0.932439

17

Example-Future Price

Another method:

From delta:

1

981381.0985301.0

00196.0*2

uud)P(3,4, - uuu)P(3,4,

uud)(3, flowcash -uuu)(3, flowcash

uu)N1(2,

980383.0

974502.0979870.0978637.0983134.0

97778.0982456.0981381.0985301.0

8

1

)dddP(3,4, )(s )dduP(3,4, )(s ... uud)P(3,4, )(s

)uuuP(3,4, )(s ]Sp(3,4, [ E H(0,3,4)

333

33

ddddduuud

uuu

18

Relation of Forward and future price

We can now use the theory of the preceding sections to relate

forward and futures prices.

)

1

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TB

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t

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19

Relation between Forward and future price

COV(x,y)=E[(x-E(x)) * (y-E(y))] =

E[xy]-E[x E(y)]- E[y E(x)]+ E[E(x) E(y)]

= E [xy] – E[x] E[y]= COV(x,y) , E [xy] = COV(x,y) + E[x] E[y]

)

1

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2

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TB

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20

Relation between Forward and future

price

21110

21

21110

21

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1

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TTPrrr

TTPCOV

TTPrrr

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T

T

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110 ..

1

Trrr

So if interest rates goes up , Goes down

If interest rate goes up , 21,TTP Goes down

Cov ( , ) >0 , then G(0,T1,T2) > H( 0,T1,T2) , which means

when covariance is positive then forward price exceeds

future prices.

21

Options on Futures

European call option T* < T1 exercise K ,

You are entitled to buy a future contract with excersie price of

H(T*,T1,T2) at K .

Payoff = Max( H(T*,T1,T2)-k ,0)

),(]

)*,(

)0,),*,((

[),( 1

1*

21

t

t

tt StB

STB

kTTThMax

EStC

K=0.981 H(2,3,4)

C(2,uu)= Max( 0.983341 – 0.981 , 0 ) = 0.002341

C(2,ud) = Max ( 0.980117 – 0.981 , 0 ) = 0

C(2,du ) = Max ( 0.980886- 0,981 , 0 ) = 0

C(2,dd ) = Max ( 0.977186 – 0.981 , 0 ) =0

22

Options on Futures

C( 1,u ) = 1 / (2*1.017606) * 0.002341= 0.001150

C(1,d) =0

C(0) = ½ * 1/ 1.02 * 0.001150 = 0.00564

To ways to hedge:

01127.0

02.1

001150.0

)1(

0),1(1),1(

),1(

726117.0

)001612.(001612.0

002341.0

)udflow(2,cash )uuflow(2,cash

),2(),2(

),1(

29686.0

02.1

947497.0*3208.0001150.0

)1(

),4,1(),1(1),1(

),1(

3208.0

960529.0967826.0

002341.0

),4,2(),4,2(

),2(),2(

),1(

0

1

0

1

contract future theuse )2

Asset underlying the Use)1

B

uNuC

uN

udCuuC

uN

B

upuNuC

uN

udPuuP

udCuuC

uN

23

EURODOLLAR FUTURE CONTRACT

Eurodollar deposits are deposits that are maintained outside the

United states. Generally they are exempt from federal Reserve

regulation that applied to the domestic deposits.

The rate that are applied to Eurodollar deposits in Interbank

transaction are known as London Interbank Offered Rates (

LIBOR). The LIBOR spot is active in maturities ranging from few

days to 10 years. The depth of the market is great at 3 months

and six month maturates.

The Eurodollar future contracts trades at Merc in Chicago. the

contract settles to 90 days LIBOR , which is the yield derived

from the underlying asset that is the 90-day Eurodollar time

deposit.

24

EURODOLLAR FUTURE CONTRACT

Each contract is for $1 million face value of euro deposits, with

delivery each month for several months in the future, after

which the delivery occurs only in quarterly periods ( March ,

June , September and December) . Currently the market is for

up to 10 years in the future

Future contracts settles by cash , on the maturity date. On

expiration which is the second London business day before the

third Wednesday of the maturity month, the contract settles by

cash to LIBOR

How?

25

EURODOLLAR FUTURE CONTRACT

On Expiration date the clearing house determines the three

month LIBOR, by selecting at random 12 reference banks from

a list of no less than 20 participating banks, and asks for their

three months LIBOR rate.

Then the highest and lowest quotes are eliminated and the

average of the remaining rates are calculated. The final

settlement price is 100-Average LIBOR.

Margin Requirement

An investor has to set up margin as a security deposit.

Exchange will set a minimum margin requirements , but each

trading house will set their own margin requirements. Currently

for Eurodollars future contract the margin requirement is around

$2,000. Most clearing house accept a very high percentage of

the margin as interest bearing securities such as T-bils .

26

EURODOLLAR FUTURE CONTRACT- Margin

In addition to the initial margin, we also have a maintenance

margin which is set around 75-80% of the initial margin. This is

how the maintenance margin works:

Suppose we bought a June 02 Eurodollar future at 97.47 ( this is

to say that in June 02 ,the three months LIBOR is 2.53%) . We

set up the initial margin of $2000 for our contract. Now we

know each thick is equivalent to basis point move in the interest

rate and is equal to $25. The minimum price fluctuation is 0.5

basis points or $12.50.

27

EURODOLLAR FUTURE CONTRACT- Margin

Now the maintenance margin for this is $1500, so if the future

interest rates goes up by 20 basis point from 2.53% to 2.73%

the future price will fall from 97.47 to 97.27 . If on any day the

final settlement price falls at or below 97.27 , which will result

in a loss of $500 or more, and we hit the maintenance margin,

the exchange will call us and ask us for an additional cash to

bring the account balance to $2000 again.

If the customer can not meet the margin call, the brokerage

house will sell the position and give whatever is left to the

customer.

28

EURODOLLAR FUTURE CONTRACT

From Eurodollar contracts we can obtain the yield curve

Important factor:

1. Actively traded for wide range of maturity , up to seven years.

2. Contract is heavily used for hedging and synthesizing interest

rate swaps

What is Swap?

Assume that I have floating rate liabilities index of 90-day LIBOR .

From Dec 2002 , through 2010.Now if I sell all the future

Eurodollar contract. I converted all the floating rate liabilities

into a stream of currently known liabilities.

29

EURODOLLAR FUTURE CONTRACT- Swap

Example:

If the LIBOR on March 2005 where 7%. Now I am locking in the

rate of 6.18%. My liabilities on my loan is : 0.82% higher than

what I lock in.

0.82*$1,000,000*1/4

On the other hand I am making :

100-7=93 , 100-6.18 =93.82

93.820-93=0.82*250,000

So I swap my floating liabilities into known liabilities.

30

EURODOLLAR FUTURE CONTRACT- Swap rate

Swap rate:

The discount weighted average of the short term

Eurodollar rates:

),0(...),0()

1

,0(

),0()(...)

2

,0()

2

()

1

,0()

1

(

),0()(...)

2

,0()

2

()

1

,0()

1

()],0(...),0()

1

,0([

2

2

nTpTpTp

nTpnTrTpTrTpTr

x

nTpnTrTpTrTpTrnTpTpTpx

For following slide data, the numerator is 164.90 , and the

denominator is 29.538 or the swap rate is 5.583%

31

32

33

Yield Curve

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