1. Options on Stock Indices, Currencies, and Futures
Chapter 14
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

2. European Options on Stocks Providing a Dividend Yield
If with dividend yield q, the stock price grows from S0 to ST,
then in the absence of dividends it would grow from S0 to S0eq T .
Alternatively, it would grow from S0e-qT to ST.
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 S0e–qT and provides no income
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

3. European Options on Stocks Providing Dividend Yield continued
We can value European options by reducing the stock price to S0e–q T and then behaving as though there is no dividend
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

4. Extension of Chapter 9 Results (Equations 14.1 to 14.3)
Lower Bound for calls:
Lower Bound for puts
Put Call Parity
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

5. Differential Equation
The risk-neutral process for the stock price is
dS = (r-q)Sdt + σS dW,
And the differential equation becomes
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

6. Extension of Chapter 13 Results (Equations 14.4 and 14.5)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

7. The Binomial Model S0u ƒu p S0 ƒ S0d (1 – p ) ƒd f=e-rT[pfu+(1-p)fd ]
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

8. 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
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

9. Index Options (page )
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.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

10. Example
Consider the situation where the S&P 500 two months from the expiration of an option is 930.
The exercise price of the option is 900.
The risk-free interest rate is 8% per annum.
The volatility is 20% per annum.
Dividend yields of 0.2% and 0.3% are expected in the first and second month. Total yield is q= =0.005, or 3% per annum .
So, S0=930, X=900, r=0.08, σ=0.2, q=0.03 and T=1/6.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

11. Example (Continued) Then =0.5444-0.2 (2/12)1/2=0.4628 And c = $51.83
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

12. 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 b of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100S0 dollars held
If the b is not 1.0, the portfolio manager buys b 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

13. 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?
Suppose r=12%, σ=0.22, q=0.04
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

14. Example 1 (continued)
The portfolio has 500 times the value of the index
Buy 5 put option contracts with a strike price of 900
Using the Black-Scholes pricing formula, the value of the put option is $6.48. The cost of the insurance is therefore 5*100*6.48=$3240
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

15. Example 1 (continued) Suppose the index drops to 880 in 3 months.
The portfolio will be worth about $
The payoff from the options will be 5*( )*100=$
The total value is $ $10.000, or $
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

16. 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

17. If index rises to 1040, it provides a 40/1000 or 4% return in 3 months
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)=4%+1%=5%
Excess return over risk-free rate=5-3=2%
Excess return for portfolio=2*2=4%
Dividends from portfolio=0.25*4=1%
Increase in Portfolio Value=4+3-1=6%
Portfolio value=$530,000
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

18. Determining the Strike Price (Table 14.2, page 320)
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

19. Example 2 (continued) S0=1000, K=960, r=0.12, σ=0.22, T=0.25, q=0.04
The put value is calculated as p=$19.21.
In this case, 100S0=$ and beta=2, so 10 put contracts are required for the $
Hedging cost = 10*100*19.21=$19.210
Suppose the index falls to 880. The value of the portfolio is then $
The put options pay off ( )*10*100=$80.000
Total value: $ $80.000=$
The cost of hedging increases as the beta increases
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

20. 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

21. 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
The return from investing at the foreign rate is rf S0 dollars
This shows that the foreign currency provides a “dividend yield” at rate rf
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

22. Valuing European Currency Options
A foreign currency is an asset that provides a “dividend yield” equal to rf
We can use the formula for an option on a stock paying a dividend yield :
Set S0 = current exchange rate
Set q = rƒ
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

23. Formulas for European Currency Options (Equations 14. 7 and 14
Formulas for European Currency Options (Equations 14.7 and 14.8, page 322)
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

24. Example
Consider a 4-month European call option on the British pound. The current exchange rate is 1.60
The exercise price of the option is 1.60.
The risk-free interest rate in the US is 8% per annum.
The risk-free interest rate in the UK is 11% per annum.
The volatility is 20% per annum.
So, S0=1.6, X=1.6, r=0.08, σ=0.2, rf=0.11 and T=1/3.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

25. Example (Continued) Then =-0.0288-0.2 (1/3)1/2=-0.144 And c = $0.0639
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

26. Alternative Formulas (Equations 14.9 and 14.10, page 322)
Using
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

27. 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

28. 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

29. 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

30. Example
It is August 15 and an investor has one September futures call on copper. The strike price is 70 cents.
One contract is on pounds of copper.
Suppose the futures price is currently 81 cents and at the close of trading on August 14 was 80 cents.
If the option is exercised, the investor receives *(80-70)=$2.500
plus a long position in a futures contract to buy pounds of copper in September.
If desired, the position can be closed out immediately, leaving the investor with $2.500 plus *(81-80)=$250
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

31. Options on Interest Rate Futures
1. Options on Treasury bond futures. Treasury bond futures contracts are for delivery of $ of Treasury bonds. The price of the option is quoted as a percentage of the face value of the underlying bond to the nearest sixty-fourth of 1%, i.e., 2-06 means that the contract costs $2.093
2. Options on Eurodollar Futures. When there is a change of 1 basis point, then there is a gain or loss on a Eurodollar Futures contract of $25. Similarly, in the pricing of options on Eurodollars, 1 basis point represents $25.
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

32. 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

33. 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

34. Setting Up a Riskless Portfolio
Consider the Portfolio: long D futures short 1 call option
Portfolio is riskless when 3D – 4 = -2D or D = 0.8
3D – 4
-2D
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

35. 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
The value of the portfolio today is e – 0.06/12 =
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

36. Valuing the Option The portfolio that is
long 0.8 futures short 1 option
is worth
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

37. Generalization of Binomial Tree Example (Figure 14.2, page 330)
A derivative lasts for time T and is dependent on a futures price
F0u ƒu F0 ƒ
F0d ƒd
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

38. Generalization (continued)
Consider the portfolio that is long D futures and short 1 derivative
The portfolio is riskless when
F0u D - F0 D – ƒu
F0d D- F0D – ƒd
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

39. Generalization (continued)
Value of the portfolio at time T is F0u D –F0D – ƒu
Value of portfolio today is – ƒ
Hence ƒ = – [F0u D –F0D – ƒu]e-rT
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

40. Generalization (continued)
Substituting for D we obtain
ƒ = [ p ƒu + (1 – p )ƒd ]e–rT
where
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

41. Valuing European Futures Options
We can use the formula for an option on a stock paying a dividend yield
Set S0 = current futures price (F0)
Set q = domestic risk-free rate (r )
Setting q = r ensures that the expected growth of F in a risk-neutral world is zero
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

42. 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
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

43. Growth Rates For Futures Prices
We enter into a long futures contract today. Its value is zero.
At time Δt, it provides a payoff FΔt-F0.
The risk-neutral valuation gives the value of the contract at time 0 as e-rΔt EQ[FΔt-F0], where EQ denotes expectation in a risk-neutral world
We must have e-rΔt EQ[FΔt-F0]=0, or
EQ(FΔt)=(F0), and for any T, EQ(FT)=(F0).
The drift of the futures price is zero. The futures price behaves like a stock providing a dividend yield r.
The process followed by F is dF=σ F dW
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

44. Black’s Formula (Equations 14.16 and 14.17, page 333)
The formulas for European options on futures are known as Black’s formulas
Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005