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
