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

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

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

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

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

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

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

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

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

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=0.002+0.003=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

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

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

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

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

Example 1 (continued) Suppose the index drops to 880 in 3 months. The portfolio will be worth about $440.000 The payoff from the options will be 5*(900-880)*100=$10.000. The total value is $440.000+$10.000, or $450.000 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

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

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

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

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=$100.000 and beta=2, so 10 put contracts are required for the $500.000 Hedging cost = 10*100*19.21=$19.210 Suppose the index falls to 880. The value of the portfolio is then $370.000. The put options pay off (960-880)*10*100=$80.000 Total value: $370.000+$80.000=$450.000 The cost of hedging increases as the beta increases Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

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

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

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

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

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

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

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

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

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

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

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 25.000 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 25.000*(80-70)=$2.500 plus a long position in a futures contract to buy 25.000 pounds of copper in September. If desired, the position can be closed out immediately, leaving the investor with $2.500 plus 25.000*(81-80)=$250 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

Options on Interest Rate Futures 1. Options on Treasury bond futures. Treasury bond futures contracts are for delivery of $100.000 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

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

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

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

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 – 0.06/12 = -1.592 Options, Futures, and Other Derivatives, 6th Edition, Copyright © John C. Hull 2005

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

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

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

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

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

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

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

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

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