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Options on stock indices, currencies, and futures

Options on stock indices, currencies, and futures. Options On Stock Indices. Contracts are on 100 times index; they are settled in cash

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Options on stock indices, currencies, and futures

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  1. Options on stock indices, currencies, and futures

  2. Options On Stock Indices • Contracts are on 100 times index; they are settled in cash • On exercise of the option ,the holder of a call option receives (S-K)*100 in cash and the writer of the option pays this amount in cash ;the holder of a put option receives (K-S)*100 in cash and the writer of the option pays this amount in cash • S :the value of the index K :the strike price

  3. The most popular underlying indices in the U.S. • The Dow Jones Index (DJX) • The Nasdaq 100 Index (NDX) • The Russell 2000 Index (RUT) • The S&P 100 Index (OEX) • The S&P 500 Index (SPX)

  4. LEAPS • Leaps are options on stock indices that last up to 3 years • They have December expiration dates • Leaps also trade on some individual stocks ,they have January expiration dates

  5. Portfolio Insurance • Consider a manager in charge of a well-diversified portfolio whose b is 1.0 • The dividend yield from the portfolio is the same as the dividend yield from the index • 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

  6. Portfolio InsuranceExample • Portfolio has a b 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?

  7. Portfolio InsuranceExample • Buy 5 three-month put option contracts on the index with a strike price of $900 • The index drops to 880 in three months • the portfolio is worth about 5*880*100 = $440,000 the payoff from the options 5*(900-880)*100 = $10,000 total value of the portfolio 440,000+10,000 = $450,000

  8. When the portfolio beta is not 1.0Example • 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?

  9. When the portfolio beta is not 1.0Example • Value of index in three months 1040 • Return from change in index 40/1000=4% • Dividends from index 0.25*4=1% • Total return from index 4+1=5% • Risk-free interest rate 0.25*12=3% • Excess return from index 5-3=2% • Expected excess return from portfolio 2*2=4% • Expected return from portfolio 3+4=7% • Dividends from portfolio 0.25*4=1% • Expected increase in value of portfolio 7-1=6% • Expected value of portfolio $ 500,000*1.06= $530,000

  10. Relationship between value of index and value of portfolio for beta = 2.0 The correct strike price for the 10 put option contracts that are purchased is 960

  11. 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

  12. Currency options Example • An example of a European call option Buy $1,000,000euros with USD at an exchange rate of 1.2000 USD per euro ,if the exchange rate at the maturity of the option is 1.2500 ,the payoff is 1,000,000*(1.2500-1.2000) = $50,000

  13. Currency options Example • An example of a European put option Sell $10,000,000Australian for USD at an exchange rate of 0.7000 USD per Australian ,if the exchange rate at the maturity of the option is 0.6700 ,the payoff is 10,000,000*(0.7000-0.6700) = $300,000

  14. Range forwardsShort range-forward contract • Buy a European put option with a strike price of K1 and sell a European call option with a strike price of K2

  15. Range forwardsLong range-forward contract • Sell a European put option with a strike price of K1 and buy a European call option with a strike price of K2

  16. Options On Stocks Paying Known Dividend Yields • Dividends cause stock prices to reduce on the ex-dividend date by the amount of the dividend payment • The payment of a dividend yield at rate q therefore cause the growth rate in the stock price to be less than it would otherwise be by an amount q • With a dividend yield of q ,the stock price grow from S0 today to ST • Without dividends it would grow from S0 today to STeqT • Alternatively ,without dividends it would grow from S0e–qT today to ST

  17. European Options on StocksProviding a Dividend Yield • We get the same probability distribution for the stock price at time T in each of the following cases: • The stock starts at price S0 and provides a dividend yield at rate q • The stock starts at price S0e–qTand pays no dividends • We can value European options by reducing the stock price to S0e–qTand then behaving as though there is no dividend

  18. Put-call parity • Put-call parity for an option on a stock paying a dividend yield at rate q • For American options, the put-call parity relationship is

  19. Pricing Formulas • By replacing S0 by S0e–qT in Black-Sholes formulas ,we obtain that

  20. Risk-neutral valuation • In a risk-neutral world ,the total return must be r ,the dividends provide a return of q ,the expected growth rate in the stock price must be r – q • The risk-neutral process for the stock price

  21. Risk-neutral valuation continued • The expected growth rate in the stock price is r –q ,the expected stock price at time T is S0e(r-q)T • Expected payoff for a call option in a risk-neutral world as • Where d1 and d2 are defined as above • Discounting at rate r for the T

  22. Valuation of European Stock Index Options • We can use the formula for an option on a stock paying a dividend yield • S0 : the value of index q : average dividend yield : the volatility of the index

  23. Example • A European call option on the S&P 500 that is two months from maturity • S0 = 930, K = 900, r = 0.08 = 0.2,T = 2/12 • Dividend yields of 0.2% and 0.3% are expected in the first month and the second month

  24. Example continued • The total dividend yield per annum is q = (0.2% + 0.3%)*6 = 3% one contract would cost $5,183

  25. Forward price • Define F0 as the forward price of the index

  26. Implied dividend yields • By

  27. Valuation of European Currency Options • A foreign currency is analogous to a stock paying a known dividend yield • The owner of foreign receives a yield equal to the risk-free interest rate, rf • With q replaced by rf , we can get call price, c, and put price, p

  28. Using forward exchange rates • Define F0 as the forward foreign exchange rate

  29. American Options • The parameter determining the size of up movements, u,the parameter determining the size of down movements, d • The probability of an up movement is • In the case of options on an index • In the case of options on a currency

  30. Nature of Futures Options • A call futures is the right to enter into a long futures contracts at a certain price. • A put futures is the right to enter into a short futures contracts at a certain price. • Most are American; Be exercised any time during the life . 30

  31. Nature of Futures Options • When a call futures option is exercised the holder acquires : • If the futures position is closed out immediately: • Payoff from call = F0–K • where F0 is futures price at time of exercise 1. A long position in the futures 2. A cash amount equal to the excess of the futures priceover the strike price 31

  32. Example Today is 8/15, One September futures call option on copper,K=240(cents/pound), One contract is on 25,000 pounds of copper. Futures price for delivery in Sep is currently 251cents 8/14 (the last settlement) futures price is 250 IF option exercised, investor receive cash: 25,000X(250-240)cents=$2,500 Plus a long futures, if it closed out immediately: 25,000X(251-250)cents=$250 If the futures position is closed out immediately 25,000X (251-240)cents=$2,750 32

  33. Nature of Futures Options • When a put futures option is exercised the holder acquires : • If the futures position is closed out immediately: • Payoff from put =K–F0 • where F0 is futures price at time of exercise 1. A short position in the futures 2. A cash amount equal to the excess of the strike priceover the futures price 33

  34. Reasons for the popularity on futures option • Liquid and easier to trade • From futures exchange, price is known immediately. • Normally settled in cash • Futures and futures options are traded in the same exchange. • Lower cost than spot options 34

  35. European spot and futures options Payoff from call option with strike price K on spot price of an asset: Payoff from call option with strike price K on futures price of an asset: When futures contracts matures at the same time as the option: 35

  36. Put-Call Parity for futures options • Consider the following two portfolios: • A. European call plus Ke-rT of cash • B. 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 36

  37. Example European call option on spot silver for delivery in six month Use equation c+Ke-rT=p+F0 e-rT →P=c+Ke-rT-F0 e-rT P=0.56+8.50e-0.1X0.5-8 e-0.1X0.5 =1.04 37

  38. Bounds for futures options By c+Ke-rT=p+F0 e-rT or Similarly or 38

  39. Bounds for futures options Because American futures can be exercised at any time, we must have 39

  40. Valuation by Binomial trees 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 40

  41. Valuation by Binomial trees 3D – 4 -2D • Consider the Portfolio: long D futures short 1 call option • Portfolio is riskless when 3D – 4 = -2D or D = 0.8 41

  42. Valuation by Binomial trees • 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 42

  43. A Generalization • A derivative lasts for time T and • is dependent on a futures price F0u ƒu F0 ƒ F0d ƒd 43

  44. A Generalization • Consider the portfolio that is long Δ futures and short 1 derivative • The portfolio is riskless when F0uD - F0D – ƒu F0dD- F0D – ƒd 44

  45. A Generalization • Value of the portfolio at time T is F0u Δ –F0 Δ – ƒu • Value of portfolio today is – ƒ • Hence ƒ = – [F0u Δ –F0 Δ – ƒu]e-rT 45

  46. A Generalization • Substituting for Δ we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT where 46

  47. Drift of a futures price in a risk-neutral word 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 47

  48. 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 48

  49. Results 49

  50. Black’s Formula • The formulas for European options on futures are known as Black’s formulas 50

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