on Options Stock Indices, Currencies, and Futures
Chapter 14
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.1
European Options on Stocks Providing a Dividend Yield
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 S0 and provides a dividend yield = q 2. The stock starts at price S0eq T and provides no income
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.2
European Options on Stocks Providing Dividend Yield continued
We can value European options by reducing the stock price to S0eq T and then behaving as though there is no dividend
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.3
Extension of Chapter 9 Results
(Equations 14.1 to 14.3)
Lower Bound for calls:
c S0e
p Ke
Put Call Parity
- qT
- Ke
- rT
Lower Bound for puts
-rT
- S0e
-qT
c + Ke - rT = p + S 0e - qT
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.4
Extension of Chapter 13 Results
(Equations 14.4 and 14.5)
c = S 0 e -qT N ( d1 ) - Ke -rT N ( d 2 ) p = Ke -rT N (-d 2 ) - S 0 e -qT N ( -d1 ) ln( S 0 / K ) + ( r - q + 2 / 2)T where d1 = T ln( S 0 / K ) + ( r - q - 2 / 2)T d2 = T
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.5
The Binomial Model
S0u u S0d d
S0
p
(1 p)
f=e-rT[pfu+(1-p)fd ]
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.6
The Binomial Model
continued
In
a risk-neutral world the stock price grows at r-q rather than at r when there is a dividend yield at rate q The probability, p, of an up movement must therefore satisfy pS0u+(1-p)S0d=S0e (r-q)T so that ( r- q ) T e -d p= u- d
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.7
Index Options (page 316-321)
The most popular underlying indices in the U.S. are
The Dow Jones Index times 0.01 (DJX) The Nasdaq 100 Index (NDX) The Russell 2000 Index (RUT) The S&P 100 Index (OEX) The S&P 500 Index (SPX)
Contracts are on 100 times index; they are settled in cash; OEX is American and the rest are European.
14.8
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
LEAPS
Leaps
are options on stock indices that last up to 3 years They have December expiration dates They are on 10 times the index Leaps also trade on some individual stocks
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.9
Index Option Example
Consider
a call option on an index with a strike price of 560 Suppose 1 contract is exercised when the index level is 580 What is the payoff?
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.10
Using Index Options for Portfolio Insurance
Suppose the value of the index is S0 and the strike price is K If a portfolio has a of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100S0 dollars held If the is not 1.0, the portfolio manager buys put options for each 100S0 dollars held In both cases, K is chosen to give the appropriate insurance level
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.11
Example 1
Portfolio
has a beta of 1.0 It is currently worth $500,000 The index currently stands at 1000 What trade is necessary to provide insurance against the portfolio value falling below $450,000?
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.12
Example 2
Portfolio
has a beta of 2.0 It is currently worth $500,000 and index stands at 1000 The risk-free rate is 12% per annum The dividend yield on both the portfolio and the index is 4% How many put option contracts should be purchased for portfolio insurance?
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.13
Calculating Relation Between Index Level and Portfolio Value in 3 months
If
index rises to 1040, it provides a 40/1000 or 4% return in 3 months Total return (incl dividends)=5% Excess return over risk-free rate=2% Excess return for portfolio=4% Increase in Portfolio Value=4+3-1=6% Portfolio value=$530,000
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.14
Determining the Strike Price (Table
14.2, page 320)
Value of Index in 3 months 1,080 1,040 1,000 960 920
Expected Portfolio Value in 3 months ($) 570,000 530,000 490,000 450,000 410,000
An option with a strike price of 960 will provide protection against a 10% decline in the portfolio value
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.15
Valuing European Index Options
We can use the formula for an option on a stock paying a dividend yield Set S0 = current index level Set q = average dividend yield expected during the life of the option
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.16
Currency Options
Currency options trade on the Philadelphia Exchange (PHLX) There also exists an active over-the-counter (OTC) market Currency options are used by corporations to buy insurance when they have an FX exposure
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.17
The Foreign Interest Rate
We
denote the foreign interest rate by rf
When
a U.S. company buys one unit of the foreign currency it has an investment of S0 dollars return from investing at the foreign rate is rf S0 dollars shows that the foreign currency provides a "dividend yield" at rate rf
14.18
The
This
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
Valuing European Currency Options
A
foreign currency is an asset that provides a "dividend yield" equal to rf can use the formula an for option on a stock paying a dividend yield : Set S0 = current exchange rate Set q = r
We
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.19
Formulas for European Currency Options
(Equations 14.7 and 14.8, page 322)
- rf T - rT
c = S0e p = Ke
N (d1 ) - Ke - rT N (d 2 ) N (-d 2 ) - S 0e
-rf T
N (- d1 )
ln(S 0 / K ) + (r - r + 2 / 2)T f where d1 = T ln(S 0 / K ) + (r - r - 2 / 2)T f d2 = T
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.20
Alternative Formulas
(Equations 14.9 and 14.10, page 322)
Using
F0 = S0 e
( r -r f ) T
c = e -rT [ F0 N ( d1 ) - KN ( d 2 )] p = e -rT [ KN (-d 2 ) - F0 N ( -d1 )] ln( F0 / K ) + 2T / 2 d1 = T d 2 = d1 - T
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.21
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
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.22
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
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.23
The Payoffs
If the futures position is closed out immediately: Payoff from call = F0 K Payoff from put = K F0 where F0 is futures price at time of exercise
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.24
Put-Call Parity for Futures Options (Equation 14.11, page 329)
Consider the following two portfolios: 1. European call plus Ke-rT of cash 2. European put plus long futures plus cash equal to F0e-rT They must be worth the same at time T so that c+Ke-rT=p+F0 e-rT
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.25
Binomial Tree Example
A 1-month call option on futures has a strike price of 29. Futures Price = $33 Option Price = $4 Futures price = $30 Option Price=?
Futures Price = $28 Option Price = $0
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.26
Setting Up a Riskless Portfolio
Consider the Portfolio: long futures short 1 call option 3 4
-2
Portfolio is riskless when 3 4 = -2 or = 0.8
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.27
Valuing the Portfolio
( Risk-Free Rate is 6% )
The
riskless portfolio is: long 0.8 futures short 1 call option The value of the portfolio in 1 month is -1.6 The value of the portfolio today is -1.6e 0.06/1 2 = -1.592
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.28
Valuing the Option
The
portfolio that is long 0.8 futures short 1 option is worth -1.592 The value of the futures is zero The value of the option must therefore be 1.592
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
14.29
Generalization of Binomial Tree Example (Figure 14.2, page 330)
A
derivative lasts for time T and is dependent on a futures price F0u u F0d d
14.30
F0
Options, Futures, and Other Derivatives, 6th Edition, Copyright John C. Hull 2005
Generalization
(continued)
Consider the portfolio that is long futures and short 1 derivative F0u - F0 u F0d - F0 d
The portfolio is riskless when
u - f d = F0 u - F0 d
Options, Futures, an...