ch 21 hull fundamentals 8 the d

7/25/2019 Ch 21 Hull Fundamentals 8 the d

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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"

Interest Rate Options

Chapter 21

1


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Exchange-Traded Interest Rate

Options

Treasury bond futures options

Eurodollar futures options

2


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Treasury Bond Futures Option

Suppose March T-bond call futures optionwith strike price of 110 is 2-06 (This is 2

This !eans one contract costs "2#0$%&')n e*ercise it pro+ides a payoff e,ual to

1000 ti!es the e*cess of the March T-

bond futures ,uote o+er 110

Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"3

64

6


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Eurodollar Futures Option

Suppose that the ,uote for a uneEurodollar put futures option with a strike

price of $6&2 is $ basis points)ne contract costs $."2 / "1#')n e*ercise it pro+ides a payoff e,ual to

the nu!ber of basis points by which $6&2e*ceeds the une Eurodollar futures ,uoteti!es "2



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Embedded Bond Options (page 46!

Callable bonds ssuer has option to buybond back at the 3call price&4 The call price

!ay be a function of ti!e5uttable bonds older has option to sell

bond back to issuer

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Blac"#s $odel % Its Extensions

7lack8s !odel is used to +alue !anyinterest rate options

t assu!es that the +alue of an interest

rate# a bond price# or so!e other +ariable ata particular ti!e Tin the future has alo9nor!al distribution

The payoff is discounted fro! the ti!e ofthe payoff to today at today8s risk-free rate

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European Bond Options

:hen +aluin9 European bondoptions it is usual to assu!e that

the future bond price is lo9nor!al

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European Bond Options

continued

F0 forward bond price todayK strike price

r interest rate for !aturity T

T life of the optionB +olatility of forward

bond price

8

TddT

TKFd

dNFdKNep

dKNdNFec

B

B

B

rT

rT

=

+=

=

=

12

2

01

102

210

;2/)/ln(

)]()([

)]()([


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&ield 'ols s )rice 'ols

The chan9e in forward bond price is related tothe chan9e in forward bond yield by

whereD is the (!odified duration of the

forward bond at option !aturity

9

y

yDy

B

ByD

B

B

or


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&ield 'ols s )rice 'ols

continued

This relationship i!plies the followin9appro*i!ation

where yis the yield +olatility and Bis the

price +olatilityMarket practice to ,uote ywith the

understandin9 that this relationship will beused to calculate B

10

yB Dy=


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*ash s +uoted Bond )rice

The bond price and strike price used in7lack8s !odel should be cash (i&e& dirtyprices not ,uoted (i&e& clean prices

The cash price is the ,uoted price plusaccrued interest&

The forward bond price#F0# is (B ;IerT

whereBis the cash bond price andI is thepresent +alue of coupons recei+ed durin9the option8s life



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*aps and Floors

< cap pro+ides payoffs to co!pensate theholder for situations when =7)> is abo+eabo+e a certain le+el (the cap rate

< floor pro+ides payoffs to co!pensate theholder for situations where =7)> is belowa certain le+el (the floor rate



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Example

The principal is "10 !illion and in a 2 yearcap with ,uarterly resets the cap rate is?

:hat are the payoffs if %-!onth =7)> insuccessi+e ,uarters are %?# %?# %&?#&?# ?# ?# %&?# and %&?

:hen are the payoffs reali@edA



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*aplet< caplet is one ele!ent of a cap Suppose that the reset dates in a cap are t1#t2#

B&tn andk= tk+1tk

SupposeRKis the cap rate#Lis the principal#andRkis the actual =7)> rate for the period

between ti!e tkand tk+1& The caplet pro+ides a

payoff at ti!e tk+1ofLk!a*(RkRK# 0

for k/1# 2&&&n& ote there is no payoff for the firstperiod between ti!e @ero and t

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*aps

< cap is a portfolio of capletsEach caplet can be re9arded as a call

option on a future interest rate with thepayoff occurrin9 in arrears:hen usin9 7lack8s !odel we assu!e

that the interest rate underlyin9 eachcaplet is lo9nor!al

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Blac"#s $odel ,or *aps(Euation ./012 page 43.!

The +alue of a caplet# for period Dtk, tk+1is

Fk forward interest rate

for (tk, tk+1

k forward interest rate +ol&

rk interest rate for !aturity tk

L# principalRK cap rate

k=tk+1-tk

16

-/andwhere

k

kk

kkKk

Kk

tr

k

tddt

tRFd

dNRdNFeL kk

+=

++

12

2

1

21

2/)/ln(

)]()([11


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hen 5pplying Blac"#s $odel

To *aps e $ust 000

ETE> Fse forward +olatilities Golatility different for each caplet

)> Fse flat +olatilities Golatility sa!e for each caplet within a

particular cap but +aries accordin9 tolife of cap

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European 7ap Options

< European swap option 9i+es the holder theri9ht to enter into a swap at a certain future ti!e

Either it 9i+es the holder the ri9ht to pay aprespecified fi*ed rate and recei+e =7)>

)r it 9i+es the holder the ri9ht to pay =7)> andrecei+e a prespecified fi*ed rate

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European 7aptions

:hen +aluin9 European swap options it isusual to assu!e that the swap rate islo9nor!al

Consider a swaption which 9i+es the ri9ht to

payRK on an n-year swap startin9 at ti!e T&The payoff on each swap pay!ent date is

whereLis principal# m is pay!ent fre,uencyandRis !arket swap rate at ti!e T

19

)0,max( KRR

m

L


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European 7aptions continued(Euation ./0/82 page 433!

The +alue of the swaption is

F0 is the forward swap rateH is the swap rate

+olatilityH ti is the ti!e fro! today until the ith

swap pay!entH and

20

)]()([ 210 dNRdNFLA K

=

=

nm

i

tr iiem

A

1

1

Tdd

T

TRFd K =

+= 12

2

01 ;

2/)/ln(where


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Relationship Bet7een 7aptions

and Bond Options


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Relationship Bet7een 7aptions

and Bond Options (continued!


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Term tructure $odels

This is a !odel of how a particulare ter!structure interest rates !o+es throu9hti!e

The !odel should incorporate the !ean

re+ertin9 property of short-ter! interestrates

23

ch 21 hull fundamentals 8 the d

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