futures and options chapter 16 futures and options relations futures option contracts
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FUTURES AND OPTIONSChapter 16
Futures and Options Relations
Futures Option Contracts
Put-Call-Futures Parity
Conversion:
• Long in futures at fo
• Long in put
• Short in call
At expiration the value of the position will be X-fo regardless of the price of the underlying asset.
Put-Call-Futures Parity
Position S X S X S X
Long Futures S f S f S f
Long Put X S
Short Call S X
X f X f X f
T T T
T T T
T
T
0 0 0
0 0 0
0 0
0 0
Value of the conversion
P C X f R fT
:
( )( )0 0 00 1
Note The Cost of a
futurescontract is zero
:
.
Put-Call-Futures Parity
• Note: If the carrying-Cost Model holds and the futures and option expire at the same time, then put-call-futures parity and put-call parity are the same.
• Proof:
P C X f R
P C X S R R
P C X R S
P C S X R
fT
fT
fT
fT
fT
0 0 0
0 0 0
0 0 0
0 0 0
0 1
0 1 1
0 1
1
( )( )
( ( ) )( )
( )
( )
BOPM Defined in Terms of Futures Contracts
• The replicating portfolio underlying the BOPM can be defined in terms of futures positions instead of the spot
• Consider the example for the single-period BOPM for currency options presented in Chapter 15:
• u = 1.1, d = .95, Rus = .05, RF = .03, X = $1.50, Eo = $1.50, and Co = $0.066.
• Suppose there is a futures contract on the currency that expires in one period and assume that the carrying-cost model (IRPT) holds.
BOPM Defined in Terms of Futures Contracts
E
Er
rEf us
F
0
0 0
50
105
10350 529
$1.
.
.($1. ) $1.
uE
Long futures uE E
C Max uE X
Max
f
u
0
0 0
0
11 50 65
121
0
65 50 0 15
( . )$1. $1.
$0.
[ , ]
[$1. $1. , ] $0.
dE
Long futures dE E
C Max dE X
Max
f
u
0
0 0
0
95 50 425
104
0
425 50 0 0
(. )$1. $1.
$0.
[ , ]
[$1. $1. , ]
BOPM Defined in Terms of Futures Contracts
• Replicating Portfolio:
• Go long in Ho futures contracts and borrow Bo dollars.
H B0 00( ) H uE E r Bf
us0 0 0 0[ ]
H dE E r Bfus0 0 0 0[ ]
BOPM Defined in Terms of Futures Contracts
• Solve for Ho and Bo where:
• Solution:
H uE E r B C
H dE E r B C
fus u
fus d
0 0 0 0
0 0 0 0
[ ]
[ ]
HC C
uE dE
Example H
u d0
0 0
0 6667
: .
BC dE E C uE E
r uE dE
Example H
uf
df
us0
0 0 0 0
0 0
0 066
( )
( )
: .
c h
BOPM Defined in Terms of Futures Contracts
• Equilibrium Price
• The same price obtained with a replicating portfolio using the spot position.
C H B
C
0 0 0
0
0
066 066
*
*
( )
( . ) .
BOPM Defined in Terms of Futures Contracts
• If the call is mispriced, then the arbitrage can be defined in terms of the futures position. For example, if the market price of the currency call were $0.075, an arbitrageur would sell the call at $0.75, go long in Ho = .6667 currency futures at Ef = $1.529, and invest $0.066 in a risk-free security. This would yield an initial CF of .009 and no liabilities at T (see Table 16.3-1).
• This is a much simpler arbitrage strategy than the one using a spot position.
Futures Options
• Futures options give the holder the right to take a futures position:– Futures Call Option gives the holder the right to go long.
When the holder exercises, she obtains a long position in the futures at the current price, ft, and the assigned writer takes the short position and pays the holder ft - X.
– Futures Put Option gives the holder the right to go short. When the holder exercises, she obtains a short position at the current futures price, ft, and the assigned writer takes the long position and pays the put holder X - ft.
• Futures options on Treasuries, stock indices, currency, and commodities.
Futures Options
Call on S&P 500 Futures:• X = 1250
• C = 10, Multiplier = 500
• Futures and options futures have same expiration.
S fT T
5000
1250 1280
10000Exercises at
Obtains a long position
at which can be closed
by going short at
ceives from writer
1280
1280
1280
1280 1250 000
000 000 000
:
.
Re :
( )500 $15,
$15, $5, $10, .
Futures Options
• Put on SP 500 Futures• X = 1250
• P =10, multiplier = 500
• Futures and options futures have same expiration.
S fT T
50001220 1250
10 000,Exercises at
Obtains a short position
at which can be closed
by going long at
ceives from writer
1220
1220
1220
1250 1220 000
000 000 000
:
.
Re :
( )500 $15,
$15, $5, $10, .
Put-Call Parity
• Put-call parity for futures options is formed with a conversion: Long in futures at fo, long in put, and Short in call.
• At expiration the value of the position will be X-fo regardless of the price of the underlying futures.
• If the futures option, spot option, and futures expire at the same time and the carrying-cost model holds, then put-call-futures, put-call spot and put-call on futures option are the same.
BOPM for Futures Option
• BOPM for a futures option is the same as the BOPM for a spot if the futures and option expire at the same time and if the carrying cost model holds.
• If the futures and futures option do not expire at the same time, then the BOPM for futures option will differ.
BOPM for Futures Option
S
f S r
Cfn f
0
0 0
0
uS
f u S r
fu
rf
C Max f X
u fn
uf
u u
f
0
01
0
0
( )
[ , ]
dS
f d S r
fd
rf
C Max f X
d fn
df
d d
f
0
01
0
0
( )
[ , ]
f S r
Sf
r
Substituting
f uS r
f uf
rr
fu
rf
fn
fn
u fn
u
fn f
n
uf
f
f
f
f
f
0 0
00
01
0 1
0
:
Note If futuresoption and
futures ire at the time
i e ires at the end one period
then n and f uS S and
the price of futures option and spot
are the same
f u u
:
exp
. ., exp ,
.
b g 1 0
BOPM for Futures Options
• Replicating Portfolio:
• Go long in Ho futures contract and borrow Bo dollars.
H B0 00( ) H f f r Bu f0 0 0[ ]
H f f r Bd f0 0 0[ ]
BOPM for Futures Options
• Solve for Ho and Bo where:
• Solution:
H f f r B C
H f f r B Cu f u
d f d
0 0 0
0 0 0
[ ]
[ ]
HC C
f fu d
u d0
BC f f C f f
r f fu d d u
f u d0
0 0
( )
( )
b g
r f f ru
rf
d
rf u d f
BOPM for futuresoptions can be defined w o r
f u d ff f
f
( ) ( )
/ .
FHG
IKJ
0 0 0
BOPM for Futures Options
• Equilibrium Price
C H B0 0 00* ( )
Cr
pC p C
where pr d
u dC Max f X C Max f X
If n f S and BOPM for futuresoption
is the same as BOPM for spot
fu d
f
u u d d
f u u
0
11
0 0
1
* [ ( ) ]
: ;
[ , ]; [ , ].
.
Black Model for Futures Options
• Equilibrium Price
• Black Model includes fo instead of So and there is no interest rate.
• If the carrying-cost model holds and the futures and futures option expire at the same time, then the Black futures option model is the same as the B-S OPM for spot.
C f N d X N d0 0 1 2 ( ) ( )