1. Index, Currency and Futures Options Finance (Derivative Securities) 312 Tuesday, 24 October 2006 Readings: Chapters 13 & 14

2. Known Dividend Yield • Same probability distribution for stock price at time Tif: • Stock starts at price S0 and provides a dividend yield = q • Stock starts at price S0e–qTand provides no income • Reduce current stock price by dividend yield, then value option as though stock pays no dividends

3. Option Pricing • Lower Bound for Calls • cS0e–qT –Ke –rT • Lower Bound for Puts • pKe–rT – S0e–qT • Put-call Parity • p + S0e–qT = c + Ke–rT

4. Black Scholes

5. Binomial Model • In a risk-neutral world the stock price grows at r – qrather than at rwhen there is a dividend yield q • The probability, p, of an up movement must therefore satisfy pS0u + (1 – p)S0d = S0e(r-q)T so that:

6. Index Options • Suppose that: • Current value of index is 930, dividend yields of 0.2% and 0.3% expected in first and second months • European call option with exercise price of 900 expires in two months • Risk-free rate is 8%, volatility is 20% p.a. • What is the price of the option?

7. Index Options • Using Black Scholes: • d1 = 0.5444,d2 = 0.4628 • N(d1)= 0.7069,N(d2)= 0.6782 • c = 930 x 0.7069e–0.03(2/12) – 900 x 0.6782 e–0.082/12 = $51.83

8. 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 b is not 1.0, the portfolio manager buys b put options for each 100S0 dollars held • Kis chosen to give the appropriate insurance level

9. Portfolio Insurance • Suppose that: • Portfolio has a beta of 1.0, worth $500,000 • Index currently stands at 1000 • Risk-free rate is 12%, dividend yield is 4%, volatility is 22% p.a. • Option contract is 100 times the index • What trade is necessary to provide insurance against the portfolio value falling below $450,000 in the next three months?

10. Portfolio Insurance • Using Black Scholes, p = $6.48 • Cost of insurance = 5 x 100 x 6.48 = $3,240 • If index drops to 880: • Portfolio drops to $440,000 • Option payoff = 5 x (900–880) x 100 = $10,000

11. Portfolio Insurance • What if beta was 2.0? • Choose K = 960 (Table 13.2) • p = $19.21 • Since beta is 2.0, two put contracts required for each $100,000 • Cost of insurance = 10 x 100 x 19.21 = $19,210 • If index drops to 880: • Portfolio drops to $370,000 • Option payoff = 10 x (960–880) x 100 = $80,000 • Cost of hedging is higher (more put options, higher K)

12. Currency Options • Denote foreign interest rate by rf • When a U.S. company buys one unit of the foreign currency it has an investment of S0dollars • Return from investing at the foreign rate is rf S0dollars • Foreign currency provides a “dividend yield” at rate rf

13. Currency Option Pricing • Lower Bound for Calls • cS0e–rf T –Ke –rT • Lower Bound for Puts • pKe–rT – S0e–rf T • Put-Call Parity • p + S0e–rf T = c + Ke–rT

14. Black Scholes

15. Black Scholes

16. Futures Options • Call futures option allows holder to acquire: • Long position in futures • Cash amount equal to excess of futures price over strike price at previous settlement • Put futures option enables holder to acquire: • Short position in futures • Cash amount equal to excess of strike price over futures price at previous settlement

17. Payoffs • If 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

18. Advantages over Spot Options • Futures contract may be easier to trade than underlying asset • Exercise of the option does not lead to delivery of the underlying asset • Futures options and futures usually trade in adjacent pits at exchange • Futures options may entail lower transaction costs

19. Put-Call Parity • Strategy I: buy a European call on a futures contract and invest Ke-rT of cash FT≤ K FT > K Buy Call 0FT – K Invest Ke–rTKK Total K FT

20. Put-Call Parity • Strategy II: buy a European put futures option, enter a long futures contract, and invest F0e-rT FT≤ K FT> K Long Futures FT –F0FT – F0 Buy Put K – FT0 Invest F0e-rTF0F0 Total K FT

21. Put-Call Parity • If two portfolios provide the same return, they must cost the same to set up, otherwise an opportunity for arbitrage exists c + Ke-rT = p + F0e-rT

22. Binomial Pricing • Suppose that: • 1-month call option on futures has a strike price of 29 • In one month the futures price will be either $33 or $28 Futures Price = $33 Option Price = $4 Futures price = $30 Option Price = ? Futures Price = $28 Option Price = $0

23. 3D – 4 –2D Binomial Pricing • Consider a portfolio: • Long D futures, short 1 call futures option • Portfolio is riskless when 3D – 4 = – 2D • D = 0.8

24. Binomial Pricing • Riskless portfolio: • Long 0.8 futures, short 1 call futures option • Value of the portfolio in one month: • 3 x0.8 – 4 = –1.6 • Value of portfolio today (r = 6%): • –1.6e–0.06(1/12) = –1.592 • Value of futures is zero, so value of option must be $1.592

25. Generalisation • A derivative lasts for time T and is dependent on a futures contract F0 u ƒu F0 ƒ F0 d ƒd

26. Generalisation • Consider the portfolio that is long D futures and short 1 derivative • The portfolio is riskless when: F0uD - F0D –ƒu F0dD - F0D –ƒd

27. Generalisation • Value of portfolio at time T: • F0uD – F0D – ƒu • Value of portfolio today: • (F0uD – F0D – ƒu)e–rT • Cost of portfolio today: • –f • Hence ƒ = – [F0uD – F0 D – ƒu]e–rT

28. Generalisation • Substituting for D we obtain ƒ = [ pƒu + (1 – p)ƒd ]e–rT where

29. Dividend Yield • Valuing futures is similar to valuing an option on a stock paying a continuous dividend yield Set S0 = current futures price (F0) Set q= domestic risk-free rate (r ) • Setting q = rensures that the expected growth of F in a risk-neutral world is zero

30. Dividend Yield • Futures contracts require no initial investment • In a risk-neutral world the expected return should be zero • Expected growth rate of futures price is thereforezero • Futures price can therefore be treated like a stock paying a dividend yield of r

31. Black’s Model

32. Futures Price v Spot Price • European Options • If a European call (put) futures option matures before futures contract, and futures prices exceed spot prices, it is worth more (less) than the corresponding spot option • When futures prices are lower than spot prices (inverted market) the reverse is true

33. Futures Price v Spot Price • American Options • If futures prices are higher than spot prices, an American call (put) on futures is worth more (less) than a similar American call (put) on spot • When futures prices are lower than spot prices (inverted market) the reverse is true