Chapter 16:
Future Options
Presenter: Lida NO
Conte
nts
1.
Nature of Futures Options
2.
Reasons for the Popularity of Futures Options
3.
Put-Call Parity
4.
Bounds for Futures Options
5.
Valuation of Futures Option Using Binomial Trees
6.
Black’s Model for Valuing Futures Options
7.
American Futures Options Vs. American Spot
1. Nature of Futures Options
Future option: the right, but not the
obligation, to enter into a future contract at a
certain futures price by a certain date.
Call future option: the right to enter into a
long futures
contract at a certain price
Put future option: the right to enter into a
1. Nature of Futures Options
When a call futures option is exercised
the holder acquires
1. A long position in the futures
2. A cash amount = Most recent settlement
futures price – Strike price
When a put futures option is exercised
the holder acquires:
1.
A short position in the futures
2.
Cash amount = Strike price – Most recent
1. Nature of Futures Options
Example: On August 15, a Future call option on copper:
Strike price = $2.4/pound
1 future contract = 25,000 pounds of copper
Current future price of copper = $2.51/pound
Last settlement future price = $2.5/pound
If the option is exercised: Investor gets
Cash = 25,000x(2.50 – 2.40) = $2,500 and
A long position in a future contract to buy 25,000 pound of copper in September
The position in future contract can be closed out immediately, leaving the investor with the cash = 25,000x(2.51 – 2.50) = $250.
Total payoff from exercising the option = $2,750
2. Reasons for the Popularity
of Futures Options
Futures contracts may be easier to
trade than underlying asset
Exercise of option does not lead to
delivery of underlying asset
Futures options and futures usually
trade side by side at an exchange
Futures options may entail lower
A European call future option + Amount of cash equals to A European call future option + Amount of cash equals
to rT
Ke
Portfolio A
A European put futures option
+
a long future contract
+
Cash equal to A European put
futures option +
a long future contract
+
Cash equal to KerT
A European call future
option +
Amount of cash equals
to
A European call future
option +
Amount of cash equals
to rT
Ke
Portfolio A Payoff at time T
Call Future Option: Max (FT – K,
0)
Amount of Cash:
K
- FT: Future price - K: Strike price
- r: Risk-free interest rate - T: Maturity date
- c : Price of European call future option
Max (FT,
K) Max
(FT, K)
Cost of portfolio A =
c
Ke
rT+
A European put futures option
+
a long future contract
+
Cash equal to A European put
futures option +
a long future contract
+
Cash equal to
rT
e F0
Portfolio B
Max (FT,
K) Max
(FT, K)
Payoff at time T
Put futures Option: Max(K – FT, 0)
Long Future Contract:
FT – F0
- FT: Future price at time T - F0: Future price at time 0 - K: Strike price
- r: Risk-free interest rate - T: Maturity date
- p: Price of a European put futures option
Cost of portfolio B =
p
F
0e
rTAmount of Cash:
F0
+
+
3. Put-Call Parity (Cont.)
Portfol io A Portfol
io A
Portfolio B
Portfolio B
Max (FT,
K) Max
(FT, K)
Max (FT,
K) Max
(FT, K)
rT
e
F
p
0 rT
Ke
c
=
rT
rT
p
F
e
Ke
4. Bounds for Futures
Options
rT
rT
p
F
e
Ke
c
0 •
We have put-call parity:
•
Since
•
Since
0
p
rT rT
F
e
Ke
c
00
c
rT rT
p
F
e
Ke
0rT
e
K
F
c
(
0)
rT
e
F
K
p
5. Valuation of Futures Options
Using Binomial Tress
Example:
◦ Current future price = 30
◦ The price can either increases to 33 or decreases to 28 next month
◦ 1-month call option on the futures with strike price = 29
30
33
28
-Payoff from option = 4 -Value of future
contract = 3
-Payoff from option = 0
-5. Valuation of Futures Options
Using Binomial Tress
13
Consider a portfolio consists of
◦ A short position in 1 call option contract
◦ A long position in futures contracts
If futures price rises to 33: Value of portfolio is (3 -
4)
If futures price falls to 28: Value of portfolio is (-2)
The portfolio is riskless when these are the same;
That is: 3 - 4 = -2 or = 0.8
The portfolio will be worth: 3x0.8 – 4 = -2x0.8 = -1.6
Value of the portfolio today is:
(r = 6%)
Because the value of the future contract today is
592
.
1
6
.
1
0.06 1/12
5. Valuation of Futures Options
Using Binomial Tress
A current future price = F0
At time T, it can either rises to F0u or falls to F0d
Generalizatio
n
F
0f
F
0u
F
0d
Payoff of option = fu
5. Valuation of Futures Options
Using Binomial Tress
Consider a portfolio consists of
◦ A short position in 1 option contract
◦ A long position in futures contracts
We get:
Value of the portfolio at time T:
Value of the portfolio today:
PV of portfolio is –f, where f is the value of the option
today. Then Or d F u F f fu d
0 0 u
f
F
u
F
)
(
0 0
rTu
e
f
F
u
F
)
(
0 0
rTu
e
f
F
u
F
f
(
0 0)
rTd
u
p
f
e
pf
f
(
1
)
6. Black’s Model for
Valuing Futures Options
Black’s model provides formulas for
European options on futures
T
d
T
T
K
F
d
T
T
K
F
d
d
N
F
d
N
K
e
p
d
N
K
d
N
F
e
c
rT rT
1 0 2 0 1 1 0 2 2 1 02
/
2
)
/
ln(
2
/
2
)
/
ln(
where
)
(
)
(
)
(
)
(
Where is the volatility of the future
6. Black’s Model for
Valuing Futures Options
Example: A European put futures option on
crude oil
◦ Option’s maturity = 4 months
◦ Current future price (F0)= $20
◦ Strike price (K) = $20
◦ Risk-free interest rate (r) = 9% per annum
◦ Volatility of future price ( ) = 25% per annum
What is the put price?
6. Black’s Model for
Valuing Futures Options
07216 . 0 ) 12 / 4 ( 25 . 0 2 / ) 12 / 4 ( ) 25 . 0 ( ) 20 / 20 ln( 2 / 2 ) / ln( 2 0 1 T T K F d 07216 . 0 ) 12 / 4 ( 25 . 0 2 / ) 12 / 4 ( ) 25 . 0 ( ) 20 / 20 ln( 2 / 2 ) /
ln( 0 2
2
T T K F d
12 . 1 ) 4712 . 0 20 5288 . 0 20 ( ) ( ) ( ) 12 / 4 ( 09 . 0 1 0 2 e d N F d N K e p rT 4721 . 0 ) (d1 N5288 .
7. American Futures
Options Vs. Spot Options
If futures prices are higher than spot prices
(normal market),
◦ American call on futures is worth more than a similar American call on spot.
◦ An American put on futures is worth less than a similar American put on spot
