Excel代写-1FIXED

1FIXED INCOME SECURITIES
FRE : 6411
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  