Chapter 25 Option Valuation McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved. Key Concepts and Skills • Understand and be able to use Put-Call Parity • Be able to use the Black-Scholes Option Pricing Model • Understand the relationships between option premiums and stock price, exercise price, time to expiration, standard deviation, and the risk-free rate • Understand how the option pricing model can be used to evaluate corporate decisions 25-2 Chapter Outline • Put-Call Parity • The Black-Scholes Option Pricing Model • More about Black-Scholes • Valuation of Equity and Debt in a Leveraged Firm • Options and Corporate Decisions: Some Applications 25-3 Protective Put • Buy the underlying asset and a put option to protect against a decline in the value of the underlying asset • Pay the put premium to limit the downside risk • Similar to paying an insurance premium to protect against potential loss • Trade-off between the amount of protection and the price that you pay for the option 25-4 An Alternative Strategy • You could buy a call option and invest the present value of the exercise price in a risk-free asset • If the value of the asset increases, you can buy it using the call option and your investment • If the value of the asset decreases, you let your option expire and you still have your investment in the risk-free asset 25-5 Comparing the Strategies • Stock + Put – If S < E, exercise put and receive E – If S ≥ E, let put expire and have S • Call + PV(E) – PV(E) will be worth E at expiration of the option – If S < E, let call expire and have investment, E – If S ≥ E, exercise call using the investment and have S Value at Expiration Initial Position S < E S ≥ E Stock + Put E S Call + PV(E) E S 25-6 Put-Call Parity • If the two positions are worth the same at the end, they must cost the same at the beginning • This leads to the put-call parity condition – S + P = C + PV(E) • If this condition does not hold, there is an arbitrage opportunity – Buy the “low” side and sell the “high” side • You can also use this condition to find the value of any of the variables, given the other three 25-7 Example: Finding the Call Price • You have looked in the financial press and found the following information: – Current stock price = $50 – Put price = $1.15 – Exercise price = $45 – Risk-free rate = 5% – Expiration in 1 year • What is the call price? – 50 + 1.15 = C + 45 / (1.05) – C = 8.29 25-8 Continuous Compounding • Continuous compounding is generally used for option valuation • Time value of money equations with continuous compounding – EAR = e q - 1 – PV = FVe -Rt – FV = PVe Rt • Put-call parity with continuous compounding – S + P = C + Ee -Rt 25-9 Example: Continuous Compounding • What is the present value of $100 to be received in three months if the required return is 8%, with continuous compounding? – PV = 100e 08(3/12) = 98.02 • What is the future value of $500 to be received in nine months if the required return is 4%, with continuous compounding? – FV = 500e .04(9/12) = 515.23 25-10 [...]... portfolio 25- 18 Work the Web Example • There are several good options calculators on the Internet • Click on the web surfer to go to ivolatility.com and click on the Basic Calculator under Analysis Services • Price the call option from the earlier example – S = $45; E = $35; R = 4%; t = 5; σ = 2 • You can also choose a stock and value options on a particular stock 25- 19 Figure 25. 1 Insert Figure 25. 1 here 25- 20... American option that does allow for early exercise 25- 16 Table 25. 4 25- 17 Varying Stock Price and Delta • What happens to the value of a call (put) option if the stock price changes, all else equal? • Take the first derivative of the OPM with respect to the stock price and you get delta – For calls: Delta = N(d1) – For puts: Delta = N(d1) - 1 – Delta is often used as the hedge ratio to determine how many options... else equal? • Take the first derivative of the OPM with respect to time and you get theta • Options are often called “wasting” assets, because the value decreases as expiration approaches, even if all else remains the same • Option value = intrinsic value + time premium 25- 22 Figure 25. 2 Insert figure 25. 2 here 25- 23 Example: Time Premiums • What was the time premium for the call and the put in the previous... potential gain 25- 25 Figure 25. 3 Insert figure 25. 3 here 25- 26 Varying the Risk-Free Rate and Rho • What happens to the value of a call (put) as we vary the risk-free rate, all else equal? – The value of a call increases – The value of a put decreases • Take the first derivative of the OPM with respect to the risk-free rate and you get rho • Changes in the risk-free rate have very little impact on options... -(1/.5)ln(.9846) = 031 or 3.1% 25- 11 Black-Scholes Option Pricing Model • The Black-Scholes model was originally developed to price call options • N(d1) and N(d2) are found using the cumulative standard normal distribution tables C = SN ( d1 ) − Ee − Rt N (d 2 ) σ2 S t ln + R + 2 E = d1 σ t d 2 = d1 − σ t 25- 12 Example: OPM • You are looking at a call option with 6 months to expiration... Consider the previous example: – What is the delta for the call option? What does it tell us? • N(d1) = 9767 • The change in option value is approximately equal to delta times the change in stock price – What is the delta for the put option? • N(d1) – 1 = 9767 – 1 = -.0233 – Which option is more sensitive to changes in the stock price? Why? 25- 21 Varying Time to Expiration and Theta • What happens to... range of interest rates 25- 27 Figure 25. 4 Insert figure 25. 4 here 25- 28 Implied Standard Deviations • All of the inputs into the OPM are directly observable, except for the expected standard deviation of returns • The OPM can be used to compute the market’s estimate of future volatility by solving for the standard deviation • This is called the implied standard deviation • Online options calculators are... form solution 25- 29 Work the Web Example • Use the options calculator at www.numa.com to find the implied volatility of a stock of your choice • Click on the web surfer to go to finance.yahoo.com to get the required information • Click on the web surfer to go to numa, enter the information and find the implied volatility 25- 30 Equity as a Call Option • Equity can be viewed as a call option on the firm’s... there is a large expected cash flow from the underlying asset 25- 15 European vs American Options • The Black-Scholes model is strictly for European options • It does not capture the early exercise value that sometimes occurs with a put • If the stock price falls low enough, we would be better off exercising now rather than later • A European option will not allow for early exercise; therefore, the price... call option? 2 2 45 .5 ln + 04 + 2 35 = 1.99 d1 = 2 5 d 2 = 1.99 − 2 5 = 1.85 •Look up N(d1) and N(d2) in Table 25. 3 •N(d1) = (.9761+.9772)/2 = 9767 •N(d2) = (.9671+.9686)/2 = 9679 C = 45(.9767) – 35e-.04(.5)(.9679) C = $10.75 25- 13 Example: OPM in a Spreadsheet • Consider the previous example • Click on the excel icon to see how this problem can be worked in a spreadsheet 25- 14 . option from the earlier example – S = $45; E = $35; R = 4%; t = .5; σ = .2 • You can also choose a stock and value options on a particular stock 25- 19 Figure 25. 1 Insert Figure 25. 1 here 25- 20 . how the option pricing model can be used to evaluate corporate decisions 25- 2 Chapter Outline • Put-Call Parity • The Black-Scholes Option Pricing Model • More about Black-Scholes • Valuation. Chapter 25 Option Valuation McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights