Ch 21 Hull Fundamentals 8 the d
Transcript of 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
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
Exchange-Traded Interest Rate
Options
Treasury bond futures options
Eurodollar futures options
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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
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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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"4
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
&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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
&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
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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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"11
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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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"12
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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
Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"13
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
*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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
*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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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
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-/andwhere
k
kk
kkKk
Kk
tr
k
tddt
tRFd
dNRdNFeL kk
+=
++
12
2
1
21
2/)/ln(
)]()([11
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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
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)0,max( KRR
m
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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
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)]()([ 210 dNRdNFLA K
=
=
nm
i
tr iiem
A
1
1
Tdd
T
TRFd K =
+= 12
2
01 ;
2/)/ln(where
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
Relationship Bet7een 7aptions
and Bond Options
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
Relationship Bet7een 7aptions
and Bond Options (continued!
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Fundamentals of Futures and Options Markets, 8th Ed, Ch 21, Copyright John C. ull 2!1"
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
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