The Term Structure of Interest Rates Chapter 3

報告者 張富昇 陳郁婷

指導教授 戴天時 博士

Modeling Fixed-Income Securities and Interest Rate Option, 2nd Edition, Copyright © Robert A. Jarrow 2002

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‑

• Competitive ： -perfectly (infinitely) liquid -organized exchanges v.s over the counter ‑ ‑ markets• discrete trading ： {0, 1, 2, ..., τ} -Continuous trading

The Economy

The Traded Securities

• Money Market Account-shortest term zero-coupon bond

• Zero-coupon bond price

-default free , strictly positive prices

0 T

B(0)=$1 0( )T

rdtB t e

P(t,T)

t T

$1

Table 3.1: Hypothetical Zero-Coupon Bond Prices, Forward Rates and Yields Time to

Maturity (T) Zero-Coupon Bond Prices P(O,T)

Forward Rates f(O,T)

Yields y(O,T)

PANEL A: FLAT

TERM-STRUCTURE

0 1 2 3 4 5 6 7 8 9

1 .980392 .961168 .942322 .923845 .905730 .887971 .870560 .853490 .836755

1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02

1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02 1.02

PANEL B: DOWNWARD

SLOPING TERM-

STRUCTURE

0 1 2 3 4 5 6 7 8 9

1 .976151 .953885 .932711 .912347 .892686 .873645 .855150 .837115 .820099

1.024431 1.023342 1.022701 1.022319 1.022025 1.021794 1.021627 1.021544 1.020748

1.024431 1.023886 1.023491 1.023198 1.022963 1.022768 1.022605 1.022472 1.022281

PANEL C: UPWARD SLOPING

TERM-STRUCTURE

0 1 2 3 4 5 6 7 8 9

1 .984225 .967831 .951187 .934518 .917901 .901395 .885052 .868939 .852514

1.016027 1.016939 1.017498 1.017836 1.018102 1.018312 1.018465 1.018542 1.019267

1.016027 1.016483 1.016821 1.017075 1.017280 1.017452 1.017597 1.017715 1.017887

0 1 2 3 4 5 6 7 8 91.019

1.02

1.021

1.022

1.023

1.024Panel A： flat term structure

Forward Rates f(0,T)Yields y(0,T)

Time to Maturity (T)

Inte

rest

rate

s(%

)

0 1 2 3 4 5 6 7 8 91.02

1.021

1.022

1.023

1.024

1.025

Panel B： downward-sloping term structure

Forward Rates f(0,T)Yields y(0,T)

Time to maturity (T)

Inte

rest

rate

s (%

)

0 1 2 3 4 5 6 7 8 91.015

1.016

1.017

1.018

1.019

1.02Panel C： upward-sloping term

structure

Forward Rates f(0,T)Yields y(0,T)

Time to maturity (T)

Inte

rest

rate

s (%

)

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.

YieldThe yield (holding period return) at time t on a T-maturity zero-coupon bond is

)/(1

),(1),(

tT

TtPTty

with y(t,T)>0 (3.1)

<=> )(),(

1),(tTTty

TtP (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

)1,(),(),(

TtPTtPTtf . (3.3)

--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]

1

( , ) 1

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

1

( , )0 1( , 1)

TIME t T T

buy bond P t Twith maturity T

sell

P t T P t T P t T P t TP t TP t T P t Tbonds with

maturity T

TOTALP t TCASH

P t TFLOW

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

1

),(

1),(T

tjjtf

TtP

)1,(),(),(

TtPTtPTtf

(3.3)

Drive an expression for the bond’s price in terms of the various maturity forward rates ： (3.4)

Derivation of Expression (3.4)step1.

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

( , 1) ( , 1)

1( , 1)

( , )

step2.

( , 1)( , 1)

( , 2)

( , 1) 1( , 2)

( , 1) ( , ) ( , 1)

1( , )

( , ) ( , 1)

Nex

, )

t

( 2

P t tf t t P t t

P t t P t t

P t tf t t

P t tf t t

P t t

P t tP t t

f t t f t t f t t

P t t jf t t f t t f t t

( , 1)f t t j

Spot rateThe spot rate is the rate contracted at time t on a one-period riskless loan starting immediately.

1

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

1, ) )

( , ) ( , )

( ) ( , )( , ) ( , 1) 3.5 3.6

( , 1)

T t ttP t t P t T P ty t T y

ttt

r t f t tP t t y t t

P t t

Return to the money market account：

1( ) ( 1) ( 1) ( )

0

tB t B t r t r j

j

(3.7)

Interest ratesMark 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

Yield Internal rate of return ；時間 t 到 T的平均利率

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

( , )y t T

( , )f t 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

1 2t T T

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)

–

– The boundary condition or payoff to the forward contract on the delivery date is

( , : ) ( , )1 1 2 1 2F T T T P T T

( , ) ( , : )1 2 1 2P T T F t T T

Figure 3.1: Payoff Diagram for a Forward Contract with Delivery Date T1 on a T2-maturity Zero-coupon Bond

P(T1, T2)

P(T1, T2) - F(t, T1: T2)

0 F(t, T1: T2)

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

1 2t T T

Futures Contracts

– We denote the time t futures price of a contract with expiration date T1 on the T2 maturity zero coupon bond ‑ ‑as

–

– 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

( , : ) ( , )1 1 2 1 2T T T P T TF

( , : )1 2t T TF

( 1, : ) ( , : )1 2 1 2t T T t T T F F

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

t+1

t+2

T1 1

T1

0

0

0

0

P T1,T2 F t,T1: T2

0

F t 1,T1: T2 – F t,T1: T2

F t 2,T1: T2 – F t 1,T1: T2

F T1 1,T1: T2 – F T1 2,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.

( , : ) ( , : )1 2 1 2t T T F t T TF

• 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.

( , : ) ( , : )1 2 1 2t T T F t T TF

• This is a negative compared to the forward contract that has no cash flow and an implicit borrowing rate set before rates increased.

( , : ) ( , : )1 2 1 2t T T F t T TF

• 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.

( , : ) ( , : )1 2 1 2t T T F t T TF

• This is a negative compared to the forward contract that has no cash flow and an implicit investment rate set before rates decreased.

( , : ) ( , : )1 2 1 2t T T F t T TF

Option Contracts

• A call option of the Europeana 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 Europeana 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 written on this zero-coupon bond. Its time t price is denoted

• At maturity its payoff is:C(T1, T1, K: T2) = max [P(T1, T2) - K, 0]

1 2T T

( , , : )1 2C t T K T

35

Figure 3.2: Payoff Diagram for a European Call Option on the T2-maturity Zero-coupon Bond with Strike K and Expiration Date T1

KP(T1, T2)

In-the-moneyOut-of-the- money

Option Contracts

• a European put option with strike price K and maturity date written on this zero-coupon bond. Its time t price is denoted

• At maturity its payoff is:

1 2T T

( , , : )1 2t T K TP

( , , : ) max[ ( , ),0]1 1 2 1 2T T K T K P T T P

37

Figure 3.3: Payoff Diagram for a European Put Option on the T2-maturity Zero-coupon Bond with Strike K and Expiration Date T1

KP(T1, T2)

KOut-of-the-money

In-the-money

Option Contracts

• Put-call parity

1 2( , ) ( , )c KP t T p P t T

• Protfolio A :• Protfolio B:

1European call + cash KP(t,T )

2European put + bond (maturity at T )

A

B

1 2P(T ,T )>K 1 2P(T ,T )<K

1 2[P(T ,T )-K]+K

1 20+P(T ,T )

1 2P(T ,T )

1 2 1 2[K-P(T ,T )]+P(T ,T )

0+K

K