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MIT 15 414 - Financial Management

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Options (2) Class 20 Financial Management, 15.414MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Today Options • Option pricing • Applications: Currency risk and convertible bonds Reading • Brealey and Myers, Chapter 20, 21 2MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 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 3MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Option payoffs (strike = $50) 25 25 20 20 15 15 10 10 5 5 00 -5-5 30 40 50 60 70 30 40 50 60 70 Buy a call Buy a put Stock price Stock price -5 0 5 30 40 50 60 70 -5 0 5 30 40 50 60 70 Sell a call Sell a put -25 -20 -15 -10 Stock price -25 -20 -15 -10 Stock price 4MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Valuation Option pricing 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 5MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 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 6MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Option strategies: Stock + put 30 35 40 45 50 55 60 65 70 30 40 50 60 70 -5 0 5 10 15 20 25 30 40 50 60 70 30 35 40 45 50 55 60 65 70 30 40 50 60 70 Buy stock Buy put Stock + put Stock price Stock price Stock price 7MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Option strategies: Tbill + call 30 35 40 45 50 55 60 65 70 30 40 50 60 70 -5 0 5 10 15 20 25 30 40 50 60 70 30 35 40 45 50 55 60 65 70 30 40 50 60 70 Buy Tbill with FV = 50 Buy call Tbill + call Stock price Stock price Stock price 8MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 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 9MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Black-Scholes Price of a call option C = S × N(d1) – X e-rT N(d2) 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 +σ2T /2) d1 = σ T d2 = d1 σ − T N(⋅) = prob that a standard normal variable is less than d1 or d2 σ = annual standard deviation of the stock return 10MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Cumulative Normal Distribution 0.0 0.1 0.2 0.3 0.4 0.5 N(-2) = 0.023 N(-1) = 0.159 N(0) = 0.500 N(1) = 0.841 N(2) = 0.977 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 11MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 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, σ2003 = 36.1% ln(S/X) +(r +σ2T /2) d1 = = -0.694 σ T d2 = d1 σ − T = -0.842 Call price = S × N(d1) – X e-rT N(d2) = $0.35 12MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Cisco stock price, 1993 – 2003 $90 80 70 60 50 40 30 20 10 0 Aug- Aug- Aug- Aug- Aug- Aug- Aug- Aug- Aug- Aug-93 94 95 96 97 98 99 00 01 02 13MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Cisco returns, 1993 – 2003 40% 30% 20% 10% 0% -10% -20% -30% -40% Aug-93Aug-94Aug-95Aug-96Aug-97Aug-98Aug-99Aug-00Aug-01Aug-02 14MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Cisco option prices Option price $6 5 4 3 2 1 0 15 16 17 18 19 20 21 22 23 24 25 Payoff (intrinsic value) Today's price (2 months) Stock price 15MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Option pricing Factors affecting option prices Call option Put option Stock price (S) Exercise price (X) Time-to-maturity (T) Stock volatility (σ) Interest rate (r) Dividends (D) + – – + + + + + + – – + 16MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Example 2 Call option with X = $25, r = 3% Time to expire Stock price Std. deviation Call option T = 0.25 $18 25 32 18 25 32 30% 30 30 50 50 50 $0.02 1.58 7.26 0.25 2.57 7.75 T = 0.50 18 25 32 18 25 32 30 30 30 50 50 50 0.14 2.29 7.68 0.76 3.67 8.68 1718 MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Option pricing Option price 16 14 12 10 8 6 4 2 0 0 months 1 month 3 months 6 months 9 13172125293337 Stock price 18 41MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 Using Black-Scholes Applications Hedging currency risk Pricing convertible debt 19MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 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? 20MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 s$/₤, Jan 1990 – Sept 2001 1.2 1.35 1.5 1.65 1.8 1.95 2.1 Volatility Full sample: 9.32% After 1992: 8.34% After 2000: 8.33% After 2001: 7.95% J-90 J-91 J-92 J-93 J-94 J-95 J-96 J-97 J-98 J-99 J-00 J-01 21MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 $ revenues as a function of s$/₤ $32 30 28 26 24 22 20 1.30 1.34 1.38 1.42 1.46 1.50 1.54 1.58 1.62 1.66 $26.9 million Exchange rate 22MIT SLOAN SCHOOL OF MANAGEMENT 15.414 Class 20 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 …


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MIT 15 414 - Financial Management

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