1. 1 Today Options Option pricing Applications: Currency risk and convertible bonds Reading Brealey, Myers, and Allen: Chapter 20, 21

2. 2 Options Gives the holder the right to either buy (call option) or sell (put option) at a specified price. Exercise, or strike, price Expiration or maturity date American vs. European option In-the-money, at-the-money, or out-of-the-money

3. 3 Option payoffs (strike = $50)

4. 4 Valuation Option pricing How risky is an option? How can we estimate the expected cashflows, and what is the appropriate discount rate? Two formulas Put-call parity Black-Scholes formula * * Fischer Black and Myron Scholes

5. 5 Option Values Intrinsic value - profit that could be made if the option was immediately exercised ( 內在價值、真實價值 ) –Call: stock price - exercise price –Put: exercise price - stock price Time value - the difference between the option price and the intrinsic value

6. 6 Time Value of Options: Call Option value X Stock Price Value of Call Intrinsic Value Time value

7. 7 Factors Influencing Option Values Factor Stock price Exercise price Volatility of stock price Time to expiration Interest rate Dividend Rate

8. 8 Put-call parity Relation between put and call prices P + S = C + PV(X) S = stock price P = put price C = call price X = strike price PV(X) = present value of $X = X / (1+r) t r = riskfree rate

9. 9 Option strategies: Stock + put

10. 10 Put-Call Parity Relationship Long a call and write a put simultaneously. Call and put are with the same exercise price and maturity date.

11. 11 Put-Call Parity Relationship S T X Payoff for Call Owned 0S T - X Payoff for Put Written-( X -S T ) 0 Total Payoff S T - X S T – X PV of (S T -X)= S 0 - X / (1 + r f ) T

12. 12 Payoff of Long Call & Short Put Long Call Short Put Payoff Stock Price Combined = Leveraged Equity

13. 13 Arbitrage & Put Call Parity Since the payoff on a combination of a long call and a short put are equivalent to leveraged equity, the prices must be equal. C - P = S 0 - X / (1 + r f ) T If the prices are not equal, arbitrage will be possible

14. 14 Put Call Parity - Disequilibrium Example Stock Price = 110, Call Price = 17, Put Price = 5 Risk Free = 5% per 6 month (10.25% effective annual yield) Maturity =.5 yr X = 105 C - P > S 0 - X / (1 + r f ) T 17- 5 > 110 - (105/1.05) 12 > 10 Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative

15. 15 Put-Call Parity Arbitrage ImmediateCashflow in Six Months PositionCashflowS T 105 Buy Stock-110 S T S T Borrow 100 X/(1+r) T = 100+100-105-105 Sell Call+17 0-(S T -105) Buy Put -5105-S T 0 Total 2 0 0

16. 16 Example

17. 17 Black-Scholes

18. 18 Cumulative Normal Distribution

19. 19 Example

20. 20 Option pricing

21. 21 Using Black-Scholes Applications Hedging currency risk Pricing convertible debt

22. 22 Currency risk Your company, headquartered in the U.S., supplies auto parts to Jaguar PLC in Britain. You have just signed a contract worth ₤18.2 million to deliver parts next year. Payment is certain and occurs at the end of the year. The $ / ₤ exchange rate is currently S $/₤ = 1.4794. How do fluctuations in exchange rates affect $ revenues? How can you hedge this risk?

23. 23 S $/₤, Jan 1990 – Sept 2001

24. 24 $ revenues as a function of S $/₤

25. 25 Currency risk

26. 26 $ revenues as a function of S $/₤

27. 27 Convertible bonds Your firm is thinking about issuing 10-year convertible bonds. In the past, the firm has issued straight (non- convertible) debt, which currently has a yield of 8.2%. The new bonds have a face value of $1,000 and will be convertible into 20 shares of stocks. How much are the bonds worth if they pay the same interest rate as straight debt? Today’s stock price is $32. The firm does not pay dividends, and you estimate that the standard deviation of returns is 35% annually. Long-term interest rates are 6%.

28. 28 Payoff of convertible bonds

29. 29 Convertible bonds

30. 30 Convertible bonds Call option X = $50, S = $32, σ = 35%, r = 6%, T = 10 Black-Scholes value = $10.31 Convertible bond