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13.1
Options on
Futures
Chapter 13
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.2
Potential Advantages of Futures
Options over Spot Options
• Futures contract may be easier to trade than
•
•
•
underlying asset
Exercise of the option does not lead to
delivery of the underlying asset
Futures options and futures usually trade in
adjacent pits at exchange
Futures options may entail lower transactions
costs
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
Mechanics of Call Futures
Options
When a call futures option is exercised
the holder acquires
1. A long position in the futures
2. A cash amount equal to the excess of
the futures price over the strike price
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.3
13.4
Mechanics of Put Futures
Option
When a put futures option is exercised
the holder acquires
1. A short position in the futures
2. A cash amount equal to the excess of
the strike price over the futures price
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.5
The Payoffs
If the futures position is closed out
immediately:
Payoff from call = F-X
Payoff from put = X-F
where F is futures price at time of
exercise
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.6
Problem
• An investor buys a July call futures option
•
•
contract on gold. The contract size is 100
ounces. The strike price is 500.
The investor exercise when the gold futures
price is 540 and the most recent settlement
price is 538.
Calculate the payoff from the exercise
decision.
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.7
Problem
• An investor buys a September put
futures option contract on corn. The
contract size is 5,000 bushels. The
strike price is 200 cents. The investor
exercises when the September corn
futures price is 180 and the most recent
settlement price is 179. Calculate the
total payoff from the exercise decision.
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.8
Put-Call Parity for Futures
Options
Consider the following two portfolios:
1. European call plus Xe-rT of cash
2. European put plus long futures plus
cash equal to Fe-rT
They must be worth the same at time T
so that
c+Xe-rT=p+Fe-rT
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.9
Other Relations
Fe-rT-X < C -P < F-Xe-rT
c > (F-X)e-rT
p > (F-X)e-rT
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.10
Problem
• Suppose that a future price is currently
trading at 35. A European call option
and a European put option on the
futures with a strike price of 34 are both
priced at 2 in the market today. The
risk-free interest rate is 10 percent
annum. Identify an arbitrage
opportunity.
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.11
European Options on Stocks
Paying Continuous Dividends
We get the same probability
distribution for the stock price at time
T in each of the following cases:
1. The stock starts at price S &
provides a continuous dividend yield
=q
2. The stock starts at price S e–q T
& provides no income
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.12
Extension of Chapter 11
Results (Eqns 12.4, 12.5, p274)
c  S e  qT N (d1 )  X e  rT N (d 2 )
p Xe
 rT
N ( d2 )  S e
 qT
(12.4)
N (  d1 )
2.5)
ln( S / X )  ( r  q   / 2) T
d1 
 T
2
where
ln( S / X )  ( r  q   2 / 2) T
d2 
 T
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
p.
p. 263
13.13
Valuing European Futures
Options
• We can use
•
the formula for an option on a
stock paying a continuous dividend yield
Set S = current futures price (F )
Set q = domestic risk-free rate (r )
Setting q = r ensures that the expected
growth of F in a risk-neutral world is zero
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.14
Growth Rates For Futures
Prices
• A futures contract requires no initial
•
•
•
investment
In a risk-neutral world the expected return
should be zero
The expected growth rate of the futures
price is therefore zero
The futures price can therefore be treated
like a stock paying a dividend yield of r
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
Black’s Formula
13.15
• The formulas
for European options on
futures are known as Black’s formulas
 F N (d1 )  X N (d 2 )
 rT
p  e  X N (  d1 )  F N (  d 2 )
ce
 rT
ln( F / X )   T / 2
d1 
 T
2
where
ln( F / X )   T / 2
d2 
 d1   T
 T
2
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.16
Futures Option Prices vs Spot
Option Prices
• If futures prices are higher than spot prices
•
(normal market), an 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
When futures prices are lower than spot
prices (inverted market) the reverse is true
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull
13.17
Problem
• Consider a European put futures option
on crude oil. The time to maturity is four
months, the current futures price is $20,
the exercise price is $20, the risk free
interest rate is 9 percent per annum,
and the volatility of the futures price is
25 percent per annum.
Introduction to Futures and Options Markets, 3rd Edition
© 1997 by John C. Hull