FINM2002/7041 (Applied) Derivatives
Lecture 7
Options on Stock Indices and
Currencies;
Options on Futures Contracts
Hull et al: Chapter 15
Review of Previous Lecture
• Last week we examined the evolution of
stock prices, and found the expected price
of a stock based on this evolution.
• We also used the Black-Scholes Model to
price European put and call options on
both dividend and non-dividend paying
stocks.
2
Lecture Overview
• In today’s lecture we will discuss options on
stock indices and currencies. We will focus on
how to treat dividend yields when examining
options. More specifically, with reference to
stock index options and FX options, we will
address necessary alterations to:
– Lower bounds for options;
– Put-call parity;
– Binomial trees;
– Black-Scholes Model for dividend yielding stocks;
and,
– Portfolio insurance.
3
1. Options and Dividend Yields
• How do we treat a stock which pays a
known dividend yield?
– Take two identical stocks. Stock 1 pays no
dividends, and stock 2 pays a dividend yield
at rate q p.a.
– Both stocks will give an identical overall
return. The return from stock 1 will be in the
form of capital gains, and the return from
stock 2 will be in the form of capital gains and
dividends.
4
1. Options and Dividend Yields
• The payment of a dividend will cause stock 2’s price to
drop by an amount equal to the value of the dividend.
• So payment of a dividend yield at rate q causes the
growth rate to be less than that it would otherwise be, by
an amount of q.
• Thus if the price of a stock, with a dividend yield of q,
grows from S0 today to ST at time T:
– In the absence of dividends it would grow from S0 today to STeqT
at time T; or,
– In the absence of dividends it would grow from S0e-qT today to ST
at time T.
5
1. Options and Dividend Yields
• Therefore, we get the same probability
distribution for the stock price at time T in
each of the following cases:
– The stock price starts at price S0 and pays a
dividend yield at rate q; and,
– The stock price starts at price S0e-qT and pays
no dividend yield.
6
1. Options and Dividend Yields
• The rule which stems from the above
example is:
– We can value European options lasting for
time T on a stock paying a known dividend
yield at rate q by reducing the stock price to
S0e-qT , and then valuing the option as though
the stock pays no dividend.
7
1. Options and Dividend Yields
• The above rule can be applied to any
option on an underlying asset which pays
a known dividend yield.
• In this lecture we focus on two particular
options:
– Options on Stock Indices; and,
– Options on Currencies.
8
1. Options and Dividend Yields
• Stock indices paying a known dividend yield:
– Exchange traded stock indices are available in most
exchanges around the world;
– In Australia, options on the ASX200 are traded on the
Australian Securities Exchange; and,
– In this lecture, we assume that dividends on stock
indices are paid in terms of a dividend yield. They
can also be paid in terms of a dollar value.
9
1. Options and Dividend Yields
• Options on currencies:
– A foreign currency is analogous to a stock paying a
known dividend yield.
– The owner of the foreign currency option receives a
yield equal to the risk-free rate of interest in the
foreign currency.
– The options are traded on both OTC markets and on
an exchange, although they are mainly traded in OTC
markets.
– The Philadelphia Stock Exchange is the major
exchange for FX options.
– FX options can be either European or American.
10
2. Lower Bounds for Option Prices
• The payment of a dividend by the underlying asset,
lowers the price of that asset.
• The reduction in price is good for the holder of a put
option as it increases their payoff, while it is bad news
for the holder of a call option as it reduces their payoff.
• Thus, the lower bounds for both call and put options
are affected by the payment of dividends.
11
2. Lower Bounds for Option Prices
• The effect on lower bounds for European calls:
– In lecture 4 we saw that the lower bound for a European
call on a non-dividend paying stock was:
– Substituting S0e-qT for S0, the lower bound for a European
call option on a stock which pays a dividend yield is:
0
rTc S Xe−≥ −
0
qT rTc S e Xe− −≥ −
12
2. Lower Bounds for Option Prices
• We can prove the above result directly by considering
the following two portfolios:
– Portfolio A: one European call option plus an amount of
cash equal to Xe-rT
– Portfolio B: e-qT shares with dividends being reinvested in
additional shares.
– In portfolio A, if the cash is invested at the risk-free interest
rate, it will grow to X at time T. If ST > X, the call option is
exercised at time T, and portfolio A is worth ST.
13
2. Lower Bounds for Option Prices
– If ST < X, the call option expires worthless, and the portfolio is
worth X.
– Hence, at time T portfolio A is worth:
Max(ST,X)
– Due to the reinvestment of dividends, portfolio B becomes one
share at time T. Thus, it is worth ST at this time.
– Therefore, portfolio A must be worth at least as much as
portfolio B at time T.
– Hence:
0
0
rT qT
qT rT
c Xe S e
or
c S e Xe
− −
− −
+ ≥
≥ −
14
2. Lower Bounds for Option Prices
• The effect on lower bounds for European puts:
– In lecture 4 we saw that the lower bound for a European put
on a non-dividend paying stock was:
– Substituting S0e-qt for S0, the lower bound for a European put
option on a stock which pays a dividend yield is:
0
rTp Xe S−≥ −
0
rT qTp Xe S e− −≥ −
15
2. Lower Bounds for Option Prices
• As with the call option, this can be proved directly by
forming the following two portfolios:
– Portfolio C: one European put option plus e-qt shares with
dividends on the shares being reinvested in additional
shares
– Portfolio D: an amount of cash equal to Xe-rT
16
3. Put-Call Parity
• In order to incorporate the effect of dividend yields on
put-call parity:
– In lecture 4 we saw that put-call parity for a European call
and put option on a non-dividend paying stock was:
– Replacing S0 with S0e-qT yields the following put-call parity
relationship on a stock paying a dividend yield at rate q:
0
rTc Xe p S−+ = +
0
rT qTc Xe p S e− −+ = +
17
3. Put-Call Parity
• We can prove the above result directly by considering
the following two portfolios:
– Portfolio A: one European call option plus an amount of cash
equal to Xe-rT
– Portfolio C: one European put option plus e-qT shares with
dividends on the shares being reinvested in additional shares.
– Both portfolios must be worth max(ST,X) at time T. They must
therefore be worth the same today, and the put-call parity
result above holds.
18
3. Put-Call Parity
• The above put-call parity relationship will hold for all
options on underlying assets which pay a dividend
yield.
• A slight alteration to the formula is necessary when
considering foreign currency options. q is replaced by
rf, where rf is the foreign interest rate, and r is the
domestic interest rate.
• Thus, the relationship becomes:
0
fr TrTc Xe p S e−−+ = +
19
4. Binomial Trees
• To examine the effect of a dividend yield
equal to q on the binomial model
discussed in Lecture 5, consider the
following tree diagram:
S0
f
S0u
fu
S0d
fd
20
4. Binomial Trees
• The stock price in this diagram starts at S0 and
either moves up to S0u or down to S0d.
• We define p as the probability of an up
movement in a risk-neutral world.
• The total return provided by the stock in a risk-
neutral world must be the risk-free interest rate r.
• Our stock pays dividends equal to a return of q.
• Therefore, the return from capital gains must be
equal to r-q.
21
4. Binomial Trees
• p must satisfy:
( ) ( )0 0 0
( )
1 r q T
r q T
pS u p S d S e
or
e dp
u d
−
−
+ − =
−
=
−
22
4. Binomial Trees
• As we saw in Lecture 5, the value of the derivative at
time zero is the expected payoff in a risk-neutral world
discounted at the risk free rate:
[ ](1 )rT u df e pf p f−= + −
23
4. Binomial Trees
• Example:
– Consider a six-month put option on a stock
with a strike price of $40. Suppose that the
initial stock price is $50 and the stock price
will either move up to $60 or down to $35
during a six-month period.
• The risk-free interest rate is 6% p.a. and the stock
is expected to provide a dividend yield of 2% p.a.
24
4. Binomial Trees
– Solution:
• u = 1.2 and d = 0.7
• If the stock price moves up the payoff is zero, if it moves
down the payoff is $5. The value of the option is therefore:
(0.06 0.02) 6 /12 0.7 0.6404
1.2 0.7
ep
− × −
= =
−
[ ]0.06 6/12 0.6404 0 0.3596 5 1.74e− × × + × =
25
5. Black-Scholes for Dividend Yielding Stocks
• Substituting S0e-qT for S0 in the Black-Scholes formulas
we covered in lecture 6 yields the following formula’s to
price both European calls and puts on dividend yielding
stocks:
-
0 1 2
-
2 0 1
-
0 0
0
1
0
2 1
( ) ( )
( ) ( )
As: ln =ln
2ln( / ) ( / 2)
2ln( / ) ( / 2)
qT rT
rT qT
qT
c S e N d Xe N d
p Xe N d S e N d
S e S qT
X X
S X r q Td
T
S X r q Td d T
T
σ
σ
σ
σ
σ
−
−
= −
= − − −
−
+ − +
=
+ − −
= = −
26
5. Black-Scholes for Dividend Yielding Stocks
• A slight alteration is necessary to the previous Black-Scholes Model
when considering FX options. q is replaced by rf, where rf is the
foreign interest rate, and r is the domestic interest rate.
• Black-Scholes becomes:
-
0 1 2
-
2 0 1
-
0 0
0
1
0
2 1
( ) ( )
( ) ( )
As: ln =ln
2ln( / ) ( / 2)

2ln( / ) ( / 2)

f
f
f
r T rT
r TrT
r T
f
f
f
c S e N d Xe N d
p Xe N d S e N d
S e S r T
X X
S X r r T
d
T
S X r r T
d d T
T
σ
σ
σ
σ
σ
−
−
= −
= − − −
−
+ − +
=
+ − −
= = −
27
5. Black-Scholes for Dividend Yielding Stocks
• As the forward exchange rate can be written as:
[ ]
[ ]
( )
0 0
-
0 1 2
2 0 1
0
1
0
2 1
BSOPM becomes:
( ) ( )
( ) ( )
Where:
2ln( / ) / 2
2ln( / ) / 2
fr r T
rT
rT
F S e
c e F N d XN d
p e XN d F N d
F X Td
T
F X Td d T
T
σ
σ
σ
σ
σ
−
−
=
= −
= − − −
+
=
−
= = −
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6. Portfolio Insurance
• Portfolio managers can use index options as insurance against the
value of their portfolios falling.
• Consider a manager of a well diversified portfolio which mimics the
ASX 200 Index, and as such has similar returns and dividend yield.
– The percentage changes in the value of the portfolio can be expected to
be approximately the same as the percentage changes in the value of
the index.
– We know that an index option on the ASX 200 is on 25 times the index,
and as such, a long put option on the ASX 200 would allow a manager
to hedge against the risk of the value of their portfolio falling.
• Suppose that the manager’s portfolio is worth $100 million, and the
value of the index is currently 5,000. The volatility of the index is
25% p.a., the risk-free interest rate is 9% p.a., the manager wants to
hedge the portfolio for the next 3 months, the dividend yield on the
index is 5% p.a., and put options can be bought with an exercise
price of 5,000.
– How many options would the manager need to purchase?
– How much would this insurance cost?
– Show that the strategy works?
29
6. Portfolio Insurance
1. The manager would need to buy put options on the
ASX 200. This would give them a payoff on the
option should the value of the index fall.
2. The manager would need to purchase the following
number of put options:
Value to be hedged 100,000,000 800
Value one put option will hedge 5000 25x
= =
30
6. Portfolio Insurance
• The total cost of the insurance to protect the $100
million portfolio is calculated as follows:
0.09 0.25 -0.05 0.25
2 1
1
2 1
5000 ( ) 5000 ( ) 211.36 800 25 $4,227,200
2ln(5000 / 5000) (0.09 0.05 0.25 / 2)0.25 0.14
0.25 0.25
0.25 0.25 0.02
x xp e N d e N d x x
d
d d
−= − − − = =
+ − +
= =
= − =
31
6. Portfolio Insurance
• To illustrate that the insurance works, consider the
situation where the value of the index drops to 4,000 in
three months.
– The portfolio will be worth:
• 4,000/5,000 x $100 million = $80 million
– The payoff from the options will be:
• 800 x (5000-4000) x 25 = $20 million
– This brings the total value of the portfolio up to the insured
value of $100 million (or $95.77 million when the cost of the
options is accounted for).
32
Options on Futures Contracts
Hull et al: Chapter 16
Lecture Overview
• In today’s lecture we will discuss options on
futures. We will focus on the following topics:
– What are futures options?;
– Why are futures options popular?;
– Put-call parity with futures options;
– Bounds for futures options;
– Using binomial trees with futures options (Lecture 8);
and,
– Blacks model for valuing futures options (Lecture 8).
34
1. What are Futures Options?
• A futures option is the right, but not the obligation, to
enter into a futures contract at a certain futures price
by a certain date.
• The underlying security of a futures option is a futures
contract, not a physical commodity. Exercising these
options gives the holder a position in a futures
contract.
35
1. What are Futures Options?
• A call futures option:
– is the right to enter into a long futures contract at a certain
price; and,
– when the holder exercises the option they receive a cash
amount equal to the excess of the futures price over the strike
price.
• A put futures option:
– is the right to enter into a short futures contract at a certain
price; and,
– when the holder exercises the option they receive a cash
amount equal to the excess of the strike price over the futures
price.
36
1. What are Futures Options?
• Example: You have bought a December call futures
option on greasy wool with a strike price of $10 per bale.
– The asset underlying one contract is 100 bales of wool.
– As a result of purchasing this call futures option, you have the
right to enter into a futures contract at the strike price, which in
this case is $10 per bale.
• We know that most futures options are American, which
means you can exercise them before their expiry.
• You decide to exercise your call futures option on greasy
wool and the most recent settlement price for the
December futures contract is $14. This means that the
futures price when you exercise the call futures option is
$14. Hence, F = $14.
37
1. What are Futures Options?
• As a result of exercising this call futures option, you enter into
a long futures position at $14 (the most recent settlement
price).
– You also receive a cash amount equal to the excess of the most
recent settlement price for the December futures contract over
the strike price:
($14-$10)x100 bales=$400. This follows the formula (F-X)
• Now, when you decide to close out your December futures
contract, the December futures price has risen to $15. The
mechanics of your profit from closing out this futures position
are identical to the profits from holding an ordinary futures
contract.
– The gain on the futures contract is the excess of the December
futures price at time of exercise over the price of the long futures
contract you currently hold:
($15-$14)x100 bales=$100.
• Thus you make a total gain of $500. $400 on the exercise of
the option and the $100 on closing out the position.
38
2. Put-Call Parity
• In lecture 4 we saw that put-call parity for a European
call and put option on a non-dividend paying stock was:
• To derive a put-call parity relationship for European
futures options we consider the following two portfolios:
– Portfolio A: a European call futures option plus an amount of
cash equal to Xe-rT
– Portfolio B: a European put futures option plus a long futures
contract plus an amount of cash equal to F0e-rT
0
rTc Xe p S−+ = +
39
2. Put-Call Parity
• The cash in portfolio A is invested at the risk-free
rate, and grows to X at time T.
• Let FT be the futures price at maturity of the option.
– If FT > X, the call option in portfolio A is exercised and
portfolio A is worth FT.
– If FT < X, the call is not exercised and portfolio A is
worth X.
• Therefore, the value of portfolio A at time T is
therefore:
max(FT,X)
40
2. Put-Call Parity
• The cash in portfolio B is invested at the risk-free
rate, and grows to F0 at time T.
• The put option provides a payoff of max(X - FT,0).
• The futures contract provides a payoff of FT - F0.
• The value of portfolio B at time T is therefore:
F0 + (FT - F0) + max(X - FT,0) = max(FT,X)
41
2. Put-Call Parity
• Both portfolios have the same value at time T, namely:
max(FT,X).
• As they are European options and cannot be exercised
early, they must have the same value today.
• The value of portfolio A today is:
• In portfolio B, the futures contract is worth zero today due
to marking to market, so portfolio B is worth:
• Hence, put-call parity for a futures options is:
0
rT rTc Xe p F e− −+ = +
rTc Xe−+
rT
op F e
−+
42
2. Put-Call Parity
• For American futures options, the put-call parity
relation is:
0 0
rT rTF e X C P F Xe− −− < − < −
43
2. Put-Call Parity
• Example:
– Imagine that the price of a European call option on
greasy wool futures for delivery in six-months is $1 a
bale, when the exercise price is $10.
– Assume that the greasy wool futures price for delivery
in six months is currently $9, and the risk-free rate for
an investment that matures in six months is 5% p.a.
– What should be the price of European put option on
greasy wool futures with the same maturity and
exercise date as the call option?
44
2. Put-Call Parity
• Using:
• We have:
1.00 + 10.00e-0.05x6/12 - 9.00e-0.05x6/12 = $1.98
• Therefore, the price of a European put futures
option on greasy wool should be $1.98.
0
rT rTc Xe p F e− −+ = +
45
3. Lower Bounds for Futures Options
• The put-call parity relationship we highlighted above,
provides the lower bounds for European call and put
options.
• As the price of a put option cannot be negative, the
lower bound for a European call must be:
0
0( )
rT rT
rT
c Xe F e
or
c F X e
− −
−
+ ≥
≥ −
46
3. Lower Bounds for Futures Options
• Also, as the price of a call option cannot be negative,
the lower bound for a put option is:
• As American options can be exercised at any time,
the lower bounds for American options are:
0
0( )
rT rT
rT
Xe F e p
or
p X F e
− −
−
≤ +
≥ −
0
0
C F X
and
P X F
≥ −
≥ −
47
Conclusion
• In today’s lecture we have considered options on
underlying assets which pay a dividend yield.
• With particular reference to stock index options and FX
options, we have addressed necessary alterations to:
– Lower bounds for option prices;
– Put-call parity;
– BSOPM pricing formulas;
– Binomial trees; and,
– Portfolio insurance.
• We have also discussed options on futures contracts.
– Put-call parity with futures options;
– Bounds for futures options.
48