The Term Structure of Interest Rates
Chapter 3
Modeling Fixed-Income Securities and Interest Rate Option, 2nd Edition,
Copyright © Robert A. Jarrow 2002
報告者
張富昇
陳郁婷
指導教授 戴天時 博士
Outline
•
•
•
•
•
•
The economy
The traded securities
Interest rates
Forward contracts
Futures contracts
Option contracts
The Economy
• Frictionless：
-no transaction costs, no bid/ask spreads, no
restrictions on trade, no taxes
-If these traders determine prices, then this
model approximates actual pricing and
hedging well
-frictionless markets v.s friction-filled markets
The Economy
• Competitive：
-perfectly (infinitely) liquid
-organized exchanges v.s over-the-counter
markets
• discrete trading：{0, 1, 2, ..., τ}
-Continuous trading
The Traded Securities
• Money Market Account-shortest term zero-coupon bond
0
T
T
B(0)=$1
rdt

0
B(t )  e
• Zero-coupon bond price
t
T
P(t,T)
$1
-default free , strictly positive prices
Table 3.1: Hypothetical Zero-Coupon Bond Prices, Forward Rates and Yields
PANEL A:
FLAT
TERMSTRUCTURE
PANEL B:
DOWNWARD
SLOPING
TERMSTRUCTURE
PANEL C:
UPWARD
SLOPING
TERMSTRUCTURE
Time to
Maturity (T)
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
Zero-Coupon Bond Forward Rates
Prices P(O,T)
f(O,T)
1
1.02
.980392
1.02
.961168
1.02
.942322
1.02
.923845
1.02
.905730
1.02
.887971
1.02
.870560
1.02
.853490
1.02
.836755
1
1.024431
.976151
1.023342
.953885
1.022701
.932711
1.022319
.912347
1.022025
.892686
1.021794
.873645
1.021627
.855150
1.021544
.837115
1.020748
.820099
1
1.016027
.984225
1.016939
.967831
1.017498
.951187
1.017836
.934518
1.018102
.917901
1.018312
.901395
1.018465
.885052
1.018542
.868939
1.019267
.852514
Yields
y(O,T)
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.02
1.024431
1.023886
1.023491
1.023198
1.022963
1.022768
1.022605
1.022472
1.022281
1.016027
1.016483
1.016821
1.017075
1.017280
1.017452
1.017597
1.017715
1.017887
Panel A：flat term structure
1.024
Interest rates(%)
1.023
1.022
Forward Rates f(0,T)
1.021
Yields y(0,T)
1.02
1.019
0
1
2
3
4
5
6
7
Time to Maturity (T)
8
9
Panel B：downward-sloping term structure
1.025
Interest rates (%)
1.024
1.023
Forward Rates f(0,T)
1.022
Yields y(0,T)
1.021
1.02
0
1
2
3
4
5
6
Time to maturity (T)
7
8
9
Panel C：upward-sloping term structure
1.02
Interest rates (%)
1.019
1.018
Forward Rates f(0,T)
1.017
Yields y(0,T)
1.016
1.015
0
1
2
3
4
5
6
Time to maturity (T)
7
8
9
Term Structure of Interest Rates
• The interest rates vary with maturity.
• Concerned with how interest rates change
with maturity.
• The set of yields to maturity for bonds forms
the term structure.
-The bonds must be of equal quality.
-They differ solely in their terms to maturity.
Yield
The yield (holding period return) at time t on
a T-maturity zero-coupon bond is
1/(T t )


1
y(t ,T )  

P
(
t
,
T
)


<=> P(t ,T ) 
with y(t,T)>0
1
 y(t ,T )(T t )
(3.1)
(3.2)
The yield is the internal rate of return on the
zero-coupon bond.
Forward rate
The time t forward rate for the period [T,T+1] is
f ( t ,T ) 
P ( t ,T )
P (t ,T  1) .
--Implicit rate earned on the longer maturity
bond over this last time period
--One can contract at time t for a riskless loan
over the time period [T,T+1]
(3.3)
TIME
t
T
buy bond
with maturity T
P(t,T )
1
sell


 P(t ,T ) P(t ,T 1) 


bonds with
maturity T 1
TOTAL
CASH
FLOW
 P(t,T ) P(t,T 1)
P(t,T 1)
0
T 1
 P(t,T )
P(t,T 1)
1
 P(t,T )
P(t,T 1)
Table 3.2: A Portfolio Generating a Cash Flow
Equal to Borrowing at the Time t Forward Rate for Date T, f(t,T).
Forward rate
f ( t ,T ) 
P ( t ,T )
P (t ,T  1)
(3.3)
Drive an expression for the bond’s price in terms
of the various maturity forward rates：
P ( t ,T ) 
1
T 1
 f (t , j )
j t
(3.4)
Derivation of Expression (3.4)
step1.
P (t , t )
1
f (t , t ) 

( P (t , t )  1)
P (t , t  1) P(t , t  1)
1
 P (t , t  1) 
f (t , t )
step2. Next
P (t , t  1)
f (t , t  1) 
P (t , t  2)
P (t , t  1)
1
 P (t , t  2) 

f (t , t  1) f (t , t ) f (t , t  1)
1
 P (t , t  j ) 
f (t , t ) f (t , t  1) f (t , t  2)
f (t , t  j  1)
Spot rate
The spot rate is the rate contracted at time t on a one-period
riskless loan starting immediately.
P(t,t )
r (t )  f (t, t ) 
 y(t,t 1)
P(t,t 1)
( P(t , t )  1, P(t , T ) 
1
 y(t , T )
T t
 P(t , t  1) 
 3.53.6 
1
 y(t , t  1)
t 1t
)
Return to the money market account：
t 1
B(t )  B(t 1)r (t 1)   r ( j )
j 0
(3.7)
Interest rates
Mark
Name
Meaning
Zero-coupon bond
price
到期日T的零息債券在時間t的價格
Money market
account
時間t到T，以利率r(t)投資1元至到
期時的金額。在此表示，將1元投
入極短期zero-coupon bond
y(t,T )
Yield
Internal rate of return；時間t到T的
平均利率
f (t,T )
Forward rate
在時間點t下，將來時間點T的瞬間
利率
Spot rate；Zero rate
時間t的瞬時利率
P(t,T )
B(t )
r (t )
Forward Contracts
• Forward contract
– forward price
a prespecified price that determined at the time
the contract is written)
– delivery or expiration date
a prespecified date.
– The contract has zero value at initiation.
Forward Contracts
• forward contracts on zero-coupon bonds:
– the date the contract is written (t)
– the date the zero-coupon bond is purchased or
delivered (T1)
– the maturity date of the zero-coupon bond (T2)
– The dates must necessarily line up as t  T1  T2
Forward Contracts
– We denote the time t forward price of a contract
with expiration date T1 on the T2-maturity
zero-coupon bond as F(t,T1:T2)
–
F (T ,T :T )  P(T ,T )
1 1 2
1 2
– The boundary condition or payoff to the forward
contract on the delivery date is
P(T ,T )  F (t,T :T )
1 2
1 2
P(T1, T2) - F(t, T1: T2)
0
P(T1, T2)
F(t, T1: T2)
Figure 3.1: Payoff Diagram for a Forward Contract with Delivery Date T1
on a T2-maturity Zero-coupon Bond
Futures Contracts
• Futures contract
– futures price
A given price at the time the contract is written.
The futures price is paid via a sequence of random
and unequal installments over the contract's life.
– delivery or expiration date
a prespecified date.
– The contract has zero value at initiation.
Futures Contracts
• futures contracts on zero-coupon bonds:
– the date the contract is written (t)
– the date the zero-coupon bond is purchased or
delivered (T1)
– the maturity date of the zero-coupon bond (T2)
– The dates must necessarily line up as t  T1  T2
Futures Contracts
– We denote the time t futures price of a contract with
expiration date T1 on the T2-maturity zero-coupon
bond as F (t,T :T )
1 2
–
F (T ,T :T )  P(T ,T )
1 1 2
1 2
– The cash flow to the futures contract at time t+1 is
the change in the value of the futures contract over
the preceding period [t,t+1], i.e
F (t 1,T :T ) F (t,T :T )
1 2
1 2
Futures Contracts
– This payment occurs at the end of every period
over the futures contract’s life.
– This cash payment to the futures contract is called
marking to the market.
Time
Forward Contract
Futures Contract
t
0
0
t+1
0
F t 1,T1: T2  – F t,T1: T2 
t+2
0
F t  2,T1: T2  – F t 1,T1: T2 
T1 1
0
F T1 1,T1: T2  – F T1  2,T1: T2 
T1
PT1,T2   Ft,T1: T2 
PT1,T2   F T1 1,T1: T2 
SUM
PT1,T2   F t,T1: T2 
PT1,T2   F t,T1: T2 
Table 3.3: Cash Flow Comparison of a Forward and Futures Contract
• Let us decide whether a long position in a
forward contract is preferred to a long
position in a futures contract with delivery
date on the same -maturity bond. If the
forward contract is preferred, then the
forward price should be greater than the
futures price. i.e.
F (t,T :T )  F (t ,T :T )
1 2
1 2
F (t,T :T )  F (t ,T :T )
1 2
1 2
• IF spot rate
zero-coupon bond price
the current futures price
the change in the futures price is negative
we need to borrow cash to cover the
loss, and spot rates are high.
F (t,T :T )  F (t ,T :T )
1 2
1 2
• This is a negative compared to the forward
contract that has no cash flow and an implicit
borrowing rate set before rates increased.
F (t,T :T )  F (t ,T :T )
1 2
1 2
• IF spot rate
zero-coupon bond price
the current futures price
the change in the futures price is positive
after getting this cash profit, we need to
invest it and spot rates are low.
F (t,T :T )  F (t ,T :T )
1 2
1 2
• This is a negative compared to the forward
contract that has no cash flow and an implicit
investment rate set before rates decreased.
Option Contracts
• A call option of the European
a financial security that gives its owner the right to
purchase a commodity at a prespecified price (strike
price or exercise price) and at a predetermined
date(maturity date or expiration date).
• A call option of the American
it allows the purchase decision to be made at any
time from the date the contract is written until the
maturity date.
Option Contracts
• A put option of the European
a financial security that gives its owner the right to
sell a commodity at a prespecified price (strike price
or exercise price) and at a predetermined
date(maturity date or expiration date).
• A put option of the American
it allows the sell decision to be made at any time
from the date the contract is written until the
maturity date.
Option Contracts
• a European call option with strike price K and
maturity date T1  T2 written on this zerocoupon bond. Its time t price is denoted
C (t ,T , K :T )
1
2
• At maturity its payoff is:
C(T1, T1, K: T2) = max [P(T1, T2) - K, 0]
In-the-money
Out-of-themoney
K
P(T1, T2)
Figure 3.2: Payoff Diagram for a European Call Option on the T2-maturity
Zero-coupon Bond with Strike K and Expiration Date T1
35
Option Contracts
• a European put option with strike price K and
maturity date T1  T2 written on this zerocoupon bond. Its time t price is denoted
P (t,T , K :T )
1
2
• At maturity its payoff is:
P (T ,T , K :T )  max[ K  P(T ,T ),0]
1 1
2
1 2
In-the-money
Out-of-the-money
K
K
P(T1, T2)
Figure 3.3: Payoff Diagram for a European Put Option on the T2-maturity
Zero-coupon Bond with Strike K and Expiration Date T1
37
Option Contracts
• Put-call parity
c  KP(t , T1 )  p  P(t , T2 )
• Protfolio A : European call + cash KP(t,T1 )
• Protfolio B: European put + bond (maturity at T2 )
P(T1 ,T2 )>K
A
B
[P(T1 ,T2 )-K]+K
0+P(T1 ,T2 )
P(T1 ,T2 )
P(T1 ,T2 )<K
0+K
[K-P(T1 ,T2 )]+P(T1 ,T2 )
K