Options (2) Class 20 Financial Management, 15.414
Today Options Option pricing Applications: Currency risk and convertible bonds Reading Brealey and Myers, Chapter 20, 21 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
Option payoffs (strike = $50) 25 25 20 15 Buy a call 20 15 Buy a put 10 10 5 5 0 0-5 30 40 50 60 70-5 30 40 50 60 70 Stock price Stock price 5 Stock price 0 30-5 40 50 60 70 5 Stock price 0 30-5 40 50 60 70-10 -15 Sell a call -10-15 Sell a put -20-20 -25-25 4
Option pricing Valuation 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
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 6
Option strategies: Stock + put 70 65 60 55 50 45 40 35 30 Buy stock 0 30 40 50 60 70 30 40 50 60 70-5 Stock price Stock price 25 20 15 10 5 Buy put 70 65 60 55 50 45 40 Stock + put 35 30 30 40 50 60 70 Stock price 7
Option strategies: Tbill + call 70 65 60 55 50 45 40 Buy Tbill with FV = 50 35 0 30 30 40 50 60 70 30 40 50 60 70-5 Stock price Stock price 70 65 60 55 50 45 40 35 30 30 40 50 60 70 Stock price 25 20 15 10 5 Tbill + call Buy call 8
Example On Thursday, Cisco call options with a strike price of $20 and an expiration date in October sold for $0.30. The current price of Cisco is $17.83. How much should put options with the same strike price and expiration date sell for? Put-call parity P = C + PV(X) S C = $0.30, S = $17.83, X = $20.00 r = 1% annually 0.15% over the life of the option Put option = 0.30 + 20 / 1.0015 17.83 = $2.44 9
Black-Scholes Price of a call option C = S N(d 1 ) X e -rt N(d 2 ) S = stock price X = strike price r = riskfree rate (annual, continuously compounded) T = time-to-maturity of the option, in years ln(s/x) + (r + σ 2 /2) T d 1 = σ T d 2 = d 1 σ T N( ) = prob that a standard normal variable is less than d 1 or d 2 σ = annual standard deviation of the stock return 10
Cumulative Normal Distribution 0.5 0.4 0.3 N(-2) = 0.023 N(-1) = 0.159 N(0) = 0.500 N(1) = 0.841 N(2) = 0.977 0.2 0.1 0.0-3.5-3 -2.5-2 -1.5-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 11
Example The CBOE trades Cisco call options. The options have a strike price of $20 and expire in 2 months. If Cisco s stock price is $17.83, how much are the options worth? What happens if the stock goes up to $19.00? 20.00? Black-Scholes S = 17.83, X = 20.00, r = 1.00, T = 2/12, σ 2003 = 36.1% d 1 = ln(s/x) + (r + σ 2 /2)T σ T = -0.694 d 2 = d 1 σ T = -0.842 Call price = S N(d 1 ) X e -rt N(d 2 ) = $0.35 12
$90 80 70 60 50 40 30 20 10 Cisco stock price, 1993 2003 0 Aug- Aug- Aug- Aug- Aug- Aug- Aug- Aug- Aug- Aug- 93 94 95 96 97 98 99 00 01 02 13
Cisco returns, 1993 2003 40% 30% 20% 10% 0% Aug-93 Aug-94 Aug-95 Aug-96 Aug-97 Aug-98 Aug-99 Aug-00 Aug-01 Aug-02-10% -20% -30% -40% 14
Cisco option prices $ 6 5 Payoff (intrinsic value) Today's price (2 months) Option price 4 3 2 1 0 15 16 17 18 19 20 21 22 23 24 25 Stock price 15
Factors affecting option prices Option pricing Call option Put option Stock price (S) Exercise price (X) Time-to-maturity (T) Stock volatility (σ) Interest rate (r) Dividends (D) + + + + + + + + 16
Call option with X = $25, r = 3% Example 2 Time to expire Stock price Std. deviation Call option T = 0.25 T = 0.50 $18 25 32 18 25 32 18 25 32 18 25 32 30% 30 30 50 50 50 30 30 30 50 50 50 $0.02 1.58 7.26 0.25 2.57 7.75 0.14 2.29 7.68 0.76 3.67 8.68 17
Option pricing Option price 18 16 14 12 10 8 6 0 months 1 month 3 months 6 months 4 2 0 9 13 17 21 25 29 33 37 Stock price 41 18
Using Black-Scholes Applications Hedging currency risk Pricing convertible debt 19
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? 20
s $/, Jan 1990 Sept 2001 2.1 1.95 1.8 Volatility Full sample: 9.32% After 1992: 8.34% After 2000: 8.33% After 2001: 7.95% 1.65 1.5 1.35 1.2 J-90 J-91 J-92 J-93 J-94 J-95 J-96 J-97 J-98 J-99 J-00 J-01 21
$ revenues as a function of s $/ $32 30 28 $26.9 million 26 24 22 20 1.30 1.34 1.38 1.42 1.46 1.50 1.54 1.58 1.62 1.66 Exchange rate 22
Currency risk Forwards 1-year forward exchange rate = 1.4513 Lock in revenues of 18.2 1.4513 = $26.4 million Put options* S = 1.4794, σ = 8.3%, T = 1, r = -1.8%* Strike price Min. revenue Option price Total cost ( 18.2 M) 1.35 $24.6 M $0.012 $221,859 1.40 $25.5 M $0.026 $470,112 1.45 $26.4 M $0.047 $862,771 *Black-Scholes is only an approximation for currencies; r = r UK r US 23
$ revenues as a function of s $/ $31 30 29 with put option 28 27 26 with forward contract 25 24 23 22 1.30 1.34 1.38 1.42 1.46 1.50 1.54 1.58 1.62 1.66 Exchange rate 24
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%. 25
$ 1,500 1,400 1,300 1,200 1,100 1,000 900 800 700 600 Payoff of convertible bonds Convertible into 20 shares Convert if stock price > $50 (20 50 = 1,000) 500 30 34 38 42 46 50 54 58 62 66 70 Stock price 26
Convertible bonds Suppose the bonds have a coupon rate of 8.2%. How much would they be worth? Cashflows* Year 1 2 3 4 10 Cash $82 $82 $82 $82 $1,082 Value if straight debt: $1,000 Value if convertible debt: $1,000 + value of call option * Annual payments, for simplicity 27
Convertible bonds Call option X = $50, S = $32, σ = 35%, r = 6%, T = 10 Black-Scholes value = $10.31 Convertible bond Option value per bond = 20 10.31 = $206.2 Total bond value = 1,000 + 206.2 = $1,206.2 Yield = 5.47%* *Yield = IRR ignoring option value 28