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Options on stock indices, currencies, and futures 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 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) 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 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 Portfolio Insurance Example 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? Portfolio Insurance Example 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 When the portfolio beta is not 1.0 Example 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? When the portfolio beta is not 1.0 Example 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= $530,000 Relationship between value of index and value of portfolio for beta = 2.0 Value of Index in 3 Expected Portfolio Value months in 3 months ($) 1,080 570,000 1,040 530,000 1,000 490,000 960 450,000 920 410,000 880 370,000 The correct strike price for the 10 put option contracts that are purchased is 960 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 Currency options Example An example of a European call option Buy $1,000,000 euros 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 Currency options Example An example of a European put option Sell $10,000,000 Australian 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 Range forwards Short 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 Range forwards Long 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 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 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 at rate q 2. The stock starts at price S0e–qT and pays no dividends We can value European options by reducing the stock price to S0e–qT and then behaving as though there is no dividend Put-call parity Put-call parity for an option on a stock paying a dividend yield at rate q c Ke rT p S 0 e qT For American options, the put-call parity relationship is S 0 e qT K C P S 0 Ke rT Pricing Formulas By replacing S0 by S0e–qT in Black-Sholes formulas ,we obtain that c S 0 e qT N ( d1 ) Ke rT N ( d 2 ) p Ke rT N ( d 2 ) S 0 e qT N ( d1 ) S 0 e qT S0 since ln ln qT K K ln( S 0 / K ) ( r q 2 / 2)T w here d1 T ln( S 0 / K ) ( r q 2 / 2)T d2 T 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 dS r q Sdt Sdz 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 e ( r q )TdS 0 N (d1 ) KN (d 2 )as above Where and d are defined 1 2 Discounting at rate r for the T c S 0 e qT N (d1 ) Ke rT N (d 2 ) Valuation of European Stock Index Options We can use the formula for an option on a stock paying a dividend yield c S0e qT N (d1 ) Ke rT N (d 2 ) p Ke rT N (d 2 ) S0e qT N (d1 ) S0 : the value of index q : average dividend yield : the volatility of the index 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 Example continued The total dividend yield per annum is q = (0.2% + 0.3%)*6 = 3% ln(930 / 900) (0.08 0.03 0.2 2 / 2) 2 / 12 d1 0.5444 0.2 2 / 12 ln(930 / 900) (0.08 0.03 0.2 2 / 2) 2 / 12 d2 0.4628 0.2 2 / 12 N (d1 ) 0.7069 N (d 2 ) 0.6782 c 930 0.7069e 0.032 /12 900 0.6782e 0.082 /12 51.83 one contract would cost $5,183 Forward price Define F0 as the forward price of the index F0 S 0 e r q T c F0 e rT N ( d1 ) Ke rT N ( d 2 ) p Ke rT N ( d 2 ) F0 e rT N ( d1 ) ln( F0 / K ) ( 2 / 2)T w here d1 T ln( F0 / K ) ( 2 / 2)T d2 T Implied dividend yields By c Ke rT p S 0 e qT S 0 e qT c p Ke rT c p Ke rT e qT S0 c p Ke rT qT ln S0 1 c p Ke rT q ln T S0 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 rf T c S0e N ( d1 ) Ke rT N ( d 2 ) rf T p Ke rT N ( d 2 ) S 0 e N ( d1 ) ln( S 0 / K ) ( r r f 2 / 2)T w here d1 T ln( S 0 / K ) ( r r f 2 / 2)T d2 T Using forward exchange rates Define F0 as the forward foreign exchange rate r rf T F0 S0e c e rT F0 N (d1 ) KN (d 2 ) p e rT KN ( d 2 ) F0 N ( d1 ) ln( F0 / K ) ( 2 / 2)T where d1 T ln( F0 / K ) ( 2 / 2)T d2 T 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 ad In the casepof options on an index ud a e r q t In the case of options on a currency r r f t ae 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 Nature of 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 If the futures position is closed out immediately: Payoff from call = F0 – K where F0 is futures price at time of exercise 31 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 Nature of Futures Options 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 If the futures position is closed out immediately: Payoff from put =K–F0 where F0 is futures price at time of exercise 33 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 European spot and futures options Payoff from call option with strike price K on spot price of an asset: max(ST K ,0) Payoff from call option with strike price K on futures price of an asset: max(FT K ,0) When futures contracts matures at the same time as the option: FT ST 35 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 Example European call option on spot silver for delivery in six month F0 $8, c $0.56 , K $8.50 , Rf 10 % 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 Bounds for futures options By c+Ke-rT=p+F0 e-rT c Ke rT F0 e rT or c ( F0 K )e rT Similarly Ke rT F0 e rT P rT or P ( K F0 )e 38 Bounds for futures options Because American futures can be exercised at any time, we must have c F0 K P K F0 39 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 Valuation by Binomial trees • Consider the Portfolio: long futures short 1 call option 3 – 4 -2 • Portfolio is riskless when 3 – 4 = -2 or = 0.8 41 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 A Generalization • A derivative lasts for time T and is dependent on a futures price F0u ƒu F0 ƒ F0d ƒd 43 A Generalization • Consider the portfolio that is long Δ futures and short 1 derivative F0u F0 – ƒu F0d F0 – ƒd • The portfolio is riskless when fu f d F0u F0 d 44 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 A Generalization • Substituting for Δ we obtain ƒ = [ p ƒu + (1 – p )ƒd ]e–rT where 1 d p ud 46 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 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 Results 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 49 Black’s Formula • The formulas for European options on futures are known as Black’s formulas c e rT F0 N (d1 ) K N (d 2 ) p e rT K N (d 2 ) F0 N (d1 ) ln( F0 / K ) 2T / 2 where d1 T ln( F0 / K ) 2T / 2 d2 d1 T T 50 Example European put futures option on crude oil,… Fo=20,K=20, r=0.09,T=4/12,σ=0.25,ln(F0=K)=0, So that T d1 0.07216 2 T d2 0.07216 2 N ( d1 ) 0.4712 N ( d 2 ) 0.5288 p e 0.09 X 4 /12 20 0.5288 20 0.4712 $1.12 51 American futures options vs spot options • If futures prices are higher than spot prices (normal market), an American call on futures is worth more than a similar American call on spot. An American put on futures is worth less than a similar American put on spot • When futures prices are lower than spot prices (inverted market) the reverse is true 52

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posted: | 7/19/2012 |

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