Chapter 15:
Options on Stock
Indices and Currencies
Presenter: Lida NO
Conte
nts
1. Options on Stock Indices
2. Currency Options
3. Options on Stocks Paying Known Dividend Yields
4. Valuation of European Stock Index Options
5. Valuation of European Currency Options
1. Options on Stock
Indices
1 index option contract is on 100 times the
index.
Index options are settled in cash.
When exercise the option,
◦ Holder of a call option contract receives (S – K)x100 ◦ Holder of a put option contract receives (K – S)x100 ◦ S is the value of index at the close of trading.
◦ K is the strike price.
Portfolio Insurance
Example 1:
Portfolio has a beta of 1.0
Portfolio value = $500,000
Index value = $1000
What trade is necessary to provide
insurance against the portfolio value
falling below $450,000?
Portfolio insurance when
Beta is not 1.0
Example 2:
Portfolio has a beta of 2.0 Portfolio value = $500,000 Index value = 1000
Risk-free rate = 12% per annum
Dividend yield on portfolio and index = 4% How many put index option contracts to be
purchased to provide insurance against the portfolio value falling below $450,000 in 3 months? What is should be the strike price?
6
Number of put option contracts is: 2x(500,000/1,000x100) = 10 contracts
2. Currency Option
8
A U.S company is to receive 1 million GBP in
3 months. The 3-month forward exchange rate: 1.92 USD= 1GBP
How to hedge?
2 possible ways:
◦ Entering into a contract to sell 1 million GBP in 3 month at 3-month forward rate of
1.92USD = 1GBP.
◦ Buying European put option with a strike
price of K1 and selling a European call option at a strike price of K2 where
2. Currency
Options (Cont.)
Payoff
Asset Price
K1 K2
Payoff
Asset Price
K1 K2
Short Range Forward Contract
- Buy put option at strike price of K1
Long Range Forward Contract
2. Currency
Options (Cont.)
10 K2
K1
K2 K1
Exchange rate in market
Exchange rate realized when range-forward contract is used
Notices: if K1 = K2:
- Range forward contract = Regular forward contract - Short range forward contract = Short forward
contract
3. Options on Stocks Paying
Known Dividend Yields
12
Dividends cause stock prices to reduce on the
ex-dividend date by the amount of the ex-dividend
payment.
We can value this option by reducing the current
stock price from S0 to S0e–q T and then behaving as
though there is no dividend (where q is the
3. Options on Stocks Paying
Known Dividend Yields
rT qT
Ke
e
S
c
0
Lower Bound for calls:
Lower Bound for
puts rT qT
e
S
Ke
p
0 Put Call Parity
qT
rT
p
S
e
Ke
14 T T q r K S d T T q r K S d d N e S d N Ke p d N Ke d N e S c qT rT rT qT ) 2 / 2 ( ) / ln( ) 2 / 2 ( ) / ln( where ) ( ) ( ) ( ) ( 0 2 0 1 1 0 2 2 1 0
3. Options on Stocks Paying
Known Dividend Yields
4. Options on Stocks Paying
Known Dividend Yields
16
Example:
A European call option:
◦ Maturity (T) = 2 months
◦ Current index value (S0)= 930
◦ Exercise price (K)= 900
◦ Risk-free interest rate (r) = 8% per annum
◦ Volatility index ( ) =20% per annum
◦ Dividend yield 0.2% and 0.3% for the 1st and 2nd
month = (0.2%+0.3%)x6 = 3% per annum.
◦ d1
◦ d2
◦ c
◦ One contract would cost $5,183
5444 . 0 12 / 2 2 . 0 12 / 2 ) 2 / 2 . 0 03 . 0 08 . 0 ( ) 900 / 930 ln( 2 4628 . 0 12 / 2 2 . 0 12 / 2 ) 2 / 2 . 0 03 . 0 08 . 0 ( ) 900 / 930 ln( 2 83 . 51 6782 . 0 900 7069 . 0
930 0.03 2/12 0.08 2/12
e e
Forward Price
18
◦ In chapter 5,
◦ Therefore, we can write:
◦ The put-call parity can be written as: T
q r
e S
F0 0 ( )
T
d
d
T
T
K
F
d
d
N
F
d
KN
e
p
d
KN
d
N
F
e
c
rT rT
1 2 2 0 1 1 0 2 2 1 02
/
)
/
ln(
)]
(
)
(
[
)]
(
)
(
[
rTrT
p
F
e
Ke
5. Valuation of
European
5. Evaluation of European
Currency Options
A foreign currency is analogous to a stock
paying a known dividend yield.
Owner of a foreign currency receive a yield
equal to risk-free interest rate rf
We can use the formula for an option on a
stock paying a dividend yield :
5. Evaluation of European
Currency Options (Cont.)
Formula for European currency option:
End of
Chapter
15
