1 Options on Stock Indices, Currencies, and Futures Chapters 15-16

2 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 S 0 and provides a dividend yield = q 2.The stock starts at price S 0 e –q T and provides no income

3 European Options on Stocks Providing Dividend Yield continued We can value European options by reducing the stock price to S 0 e –q T and then behaving as though there is no dividend

4 Extension of Chapter 9 Results Lower Bound for calls: Lower Bound for puts Put Call Parity

5 Example 15.3 S 0 = 300; q = 3%; r = 8%; T = 6 months; K = 290. Find the lower bound for the European Call. S 0 = 300; q = 3%; r = 8%; T = 6 months; K = 290. Find the lower bound for the European Call. C ≥ 300e -0.03x1/2 – 290e -0.08x1/2 = C ≥ 300e -0.03x1/2 – 290e -0.08x1/2 = 16.90

6 Example S 0 = 250; q = 4%; r = 6%; T = 3 months; K = 245; c = $10. Find the value of a 3-month European put on the same index and maturing on the same date as the call. S 0 = 250; q = 4%; r = 6%; T = 3 months; K = 245; c = $10. Find the value of a 3-month European put on the same index and maturing on the same date as the call. P=10+245e -0.06x e -0.04x0.25 = 3.84 P=10+245e -0.06x e -0.04x0.25 = 3.84

7 The Binomial Model S0u ƒuS0u ƒu S0d ƒdS0d ƒd S0 ƒS0 ƒ p (1  – p ) f=e -rT [pf u +(1-p)f d ]

8 The Binomial Model Extention to income-paying underlying assets In a risk-neutral world the stock price grows at r-q rather than at r when there is a dividend yield at the rate of q. In a risk-neutral world the stock price grows at r-q rather than at r when there is a dividend yield at the rate of q. The risk-neutral probability, p, of an up movement must therefore satisfy The risk-neutral probability, p, of an up movement must therefore satisfy pS 0 u+(1-p)S 0 d=S 0 e (r-q)T so that

9 Example S 0 = 250; q = 4%; r = 6%; T = 3 months; K = 245. Within the next 3 months the index can either increase by 10% or drop by 10%. Find the value of the 3-month European put. S 0 = 250; q = 4%; r = 6%; T = 3 months; K = 245. Within the next 3 months the index can either increase by 10% or drop by 10%. Find the value of the 3-month European put. f = e -0.06x1/4 ( )x( ) = 9.36 f = e -0.06x1/4 ( )x( ) = 9.36

10 Index Options Both exchange-traded and OTC Both exchange-traded and OTC Indices: DJX, NDX, RUT, OEX, SPX Indices: DJX, NDX, RUT, OEX, SPX CBOE option contracts are on 100 times the index CBOE option contracts are on 100 times the index The most popular underlying indices are The most popular underlying indices are the Dow Jones Industrial (European) DJX the Dow Jones Industrial (European) DJX the S&P 100 (American) OEX the S&P 100 (American) OEX the S&P 500 (European) SPX the S&P 500 (European) SPX Contracts are settled in cash Contracts are settled in cash

11 Index Option Example Consider an American call option on an index with a strike price of 560 Consider an American call option on an index with a strike price of 560 Suppose 1 contract is exercised when the index level is 580 Suppose 1 contract is exercised when the index level is 580 What is the payoff? If the premium is 30, should it be exercised? What is the payoff? If the premium is 30, should it be exercised?

12 Using Index Options for Portfolio Insurance Suppose the value of the index is S 0 and the strike price is K Suppose the value of the index is S 0 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 100 S 0 dollars held If a portfolio has a  of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100 S 0 dollars held If the  is not 1.0, the portfolio manager buys  put options for each 100 S 0 dollars held If the  is not 1.0, the portfolio manager buys  put options for each 100 S 0 dollars held In both cases, K is chosen to give the appropriate insurance level In both cases, K is chosen to give the appropriate insurance level

13 Example 1 Portfolio has a beta of 1.0 Portfolio has a beta of 1.0 It is currently worth $500,000 It is currently worth $500,000 The index currently stands at 1000 The index currently stands at 1000 What trade is necessary to provide insurance against the portfolio value falling below $450,000? What trade is necessary to provide insurance against the portfolio value falling below $450,000?

14 Example 1 Number of puts to buy: Number of puts to buy: The strike price is 450,000 for 5 contracts => 450,000/(100x5) = 900 The strike price is 450,000 for 5 contracts => 450,000/(100x5) = 900

15 Example 2 Portfolio has a beta of 2.0 Portfolio has a beta of 2.0 It is currently worth $500,000 and index stands at 1000 It is currently worth $500,000 and index stands at 1000 The risk-free rate is 12% per annum The risk-free rate is 12% per annum The dividend yield on both the portfolio and the index is 4% per year The dividend yield on both the portfolio and the index is 4% per year How many put option contracts should be purchased for portfolio insurance? How many put option contracts should be purchased for portfolio insurance?

16 If index rises to 1040, it provides a 40/1000 or 4% return in 3 months If index rises to 1040, it provides a 40/1000 or 4% return in 3 months Total return (with dividends)= 4+1 = 5% Total return (with dividends)= 4+1 = 5% Excess return over risk-free rate=2% Excess return over risk-free rate=2% Excess return for portfolio=4% Excess return for portfolio=4% Increase in Portfolio Value=4+3-1=6% Increase in Portfolio Value=4+3-1=6% Portfolio value=$530,000 Portfolio value=$530,000 Calculating Relation Between Index Level and Portfolio Value in 3 months

17 Determining the Strike Price Need to buy 10 put option contracts Cost of hedging increases with  because more options are required and each of them is more expensive.

18 Currency Options Currency options trade on the Philadelphia Exchange (PHLX) Currency options trade on the Philadelphia Exchange (PHLX) There also exists an active over-the-counter (OTC) market 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 are used by corporations to buy insurance when they have an FX exposure

19 The Foreign Interest Rate We denote the foreign interest rate by r f We denote the foreign interest rate by r f When a U.S. company buys one unit of the foreign currency it has an investment of S 0 dollars When a U.S. company buys one unit of the foreign currency it has an investment of S 0 dollars The return from investing at the foreign rate is r f S 0 dollars The return from investing at the foreign rate is r f S 0 dollars This shows that the foreign currency provides a “dividend yield” at rate r f This shows that the foreign currency provides a “dividend yield” at rate r f

20 Valuing European Currency Options A foreign currency is an asset that provides a “dividend yield” equal to r f A foreign currency is an asset that provides a “dividend yield” equal to r f We can use the formula for an option on a stock paying a dividend yield : We can use the formula for an option on a stock paying a dividend yield : Set S 0 = current exchange rate Set S 0 = current exchange rate Set q = r ƒ Set q = r ƒ

21 Example 15.4 A foreign currency is currently worth $0.80. A foreign currency is currently worth $0.80. u = 1.02; d = 0.98; T = 2 months; 2 periods (each is 1 month long) u = 1.02; d = 0.98; T = 2 months; 2 periods (each is 1 month long) R = 6%; r f = 8% R = 6%; r f = 8% What is the value of a 2-month European call option with K = $0.8? What is the value of a 2-month European call option with K = $0.8?

22 Mechanics of Call Futures Options They are American style 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 most recent settlement futures price over the strike price the most recent settlement futures price over the strike price

23 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 most recent settlement futures price the strike price over the most recent settlement futures price

24 The Payoffs If the futures position is closed out immediately: Payoff from call = F 0 – K = =(F -1 – K) + (F 0 – F -1 ) Payoff from put = K – F 0 where F 0 is futures price at the time of the exercise and F -1 is the most recent settlement futures price.

25 Put-Call Parity for Futures Options Consider the following two portfolios: 1. European call plus Ke -rT of cash 2. European put plus long futures plus cash equal to F 0 e -rT 2. European put plus long futures plus cash equal to F 0 e -rT They must be worth the same at time T so that c+Ke -rT =p+F 0 e -rT

26 Example European call on silver futures; T = 6 months; C = $0.56/ounce and K = $8.50; F 0 = $8.00 and r f = 10%. European call on silver futures; T = 6 months; C = $0.56/ounce and K = $8.50; F 0 = $8.00 and r f = 10%. Find the price of a European put with the same T and K. Find the price of a European put with the same T and K. Solution: Solution: p = e -0.1*0.5 – 8e -0.1*0.5 = 1.04 p = e -0.1*0.5 – 8e -0.1*0.5 = 1.04 What if p > 1.04? What if p > 1.04?

27 Example Sell the put and the futures, buy the call and invest the PV of K-F 0 Sell the put and the futures, buy the call and invest the PV of K-F 0

28 Futures Price = $33 Option Price = $4 Futures Payoff = $3 Futures Price = $28 Option Price = $0 Futures Payoff = $-2 Futures price = $30 Option Price=? Binomial Tree Example A 1-month call option on futures has a strike price of 29.

29 Consider the Portfolio:long  futures short 1 call option Consider the Portfolio:long  futures short 1 call option Portfolio is riskless when 3  – 4 = -2  or  = 0.8 Portfolio is riskless when 3  – 4 = -2  or  =  – 4 -2  Setting Up a Riskless Portfolio

30 Valuing the Portfolio ( Risk-Free Rate is 6% ) The riskless portfolio is: The riskless portfolio is: long 0.8 futures short 1 call option The value of the portfolio in 1 month is -2  = -1.6 The value of the portfolio in 1 month is -2  = -1.6 The value of the portfolio today is -1.6e – 0.06  = The value of the portfolio today is -1.6e – 0.06  =

31 Valuing the Option The portfolio that is The portfolio that is long 0.8 futures short 1 option is worth = 0.8f - c is worth = 0.8f - c The value of the futures is zero The value of the futures is zero The value of the option must therefore be The value of the option must therefore be 1.592

32 Generalization of Binomial Tree Example A derivative lasts for time T and is dependent on a futures price A derivative lasts for time T and is dependent on a futures price F0u ƒuF0u ƒu F0d ƒdF0d ƒd F0 ƒF0 ƒ

33 Generalization (continued) Consider the portfolio that is long  futures and short 1 derivative Consider the portfolio that is long  futures and short 1 derivative The portfolio is riskless when The portfolio is riskless when F 0 u  F 0  – ƒ u F 0 d  F 0  – ƒ d

34 Generalization (continued) Value of the portfolio at time T is F 0 u  –F 0  – ƒ u Value of the portfolio at time T is F 0 u  –F 0  – ƒ u Value of portfolio today is  – ƒ Value of portfolio today is  – ƒ Hence ƒ = – [F 0 u  – F 0  – ƒ u ]e -rT Hence ƒ = – [F 0 u  – F 0  – ƒ u ]e -rT

35 Generalization (continued) Substituting for  we obtain Substituting for  we obtain ƒ = [ p ƒ u + (1 – p )ƒ d ]e –rT where

36 Example 16.4 F 0 = 50. At T=6 months it will be either 56 or 46. The risk-free rate is 6%. Find the value of a 6-month European call option on the futures with a strike price of 50. F 0 = 50. At T=6 months it will be either 56 or 46. The risk-free rate is 6%. Find the value of a 6-month European call option on the futures with a strike price of 50. ƒ = [ 0.4* *0 ]e –.06*1/2 = 2.33 ƒ = [ 0.4* *0 ]e –.06*1/2 = 2.33

37 Valuing European Futures Options We can use the formula for an option on a stock paying a dividend yield We can use the formula for an option on a stock paying a dividend yield Set S 0 = current futures price ( F 0 ) Set q = domestic risk-free rate ( r ) Setting q = r ensures that the expected growth of F in a risk-neutral world is zero Setting q = r ensures that the expected growth of F in a risk-neutral world is zero

38 Growth Rates For Futures Prices A futures contract requires no initial investment A futures contract requires no initial investment In a risk-neutral world the expected return should be the risk-free rate In a risk-neutral world the expected return should be the risk-free rate The expected growth rate of the futures price is zero in the risk-neutral world The expected growth rate of the futures price is zero in the risk-neutral world The futures price can therefore be treated like a stock paying a dividend yield of r The futures price can therefore be treated like a stock paying a dividend yield of r

39 Futures Option Prices vs Spot Option Prices 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 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 (see problem 16.12) When futures prices are lower than spot prices (inverted market) the reverse is true (see problem 16.12)

40 Summary of Key Results We can treat stock indices, currencies, and futures like a stock paying a dividend yield of q We can treat stock indices, currencies, and futures like a stock paying a dividend yield of q For stock indices, q = average dividend yield on the index over the option life For stock indices, q = average dividend yield on the index over the option life For currencies, q = r ƒ For currencies, q = r ƒ For futures, q = r For futures, q = r