12/21/2012 1 Option Valuation 24 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, time to expiration, standard deviation, and the risk-free rate • Understand how the OPM can be used to evaluate corporate decisions
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12/21/2012
1
Option Valuation
24
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, time to expiration, standard deviation, and the risk-free rate
• Understand how the OPM can be used to evaluate corporate decisions
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Chapter Outline
• Put-Call Parity
• The Black-Scholes Option Pricing Model
• More on Black-Scholes
• Valuation of Equity and Debt in a Leveraged Firm
• Options and Corporate Decisions: Some Applications
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
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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
Comparing the Strategies Value at Expiration
Initial Position S < E S ≥ E
Stock + Put E S
Call + PV(E) E S
• 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
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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
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
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Continuous Compounding
• Continuous compounding is generally used for option valuation
• Time value of money equations with continuous compounding
– EAR = eq - 1
– PV = FVe-Rt
– FV = PVeRt
• Put-call parity with continuous compounding
– S + P = C + Ee-Rt
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
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PCP Example: PCP with Continuous Compounding
• You have found the following information; – Stock price = $60 – Exercise price = $65 – Call price = $3 – Put price = $7 – Expiration is in 6 months
• What is the risk-free rate implied by these prices? – S + P = C + Ee-Rt
– 60 + 7 = 3 + 65e-R(6/12)
– .9846 = e-.5R
– R = -(1/.5)ln(.9846) = .031 or 3.1%
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
t d d
t
t R E
S
d
d N Ee d SN C Rt
s
s
s
- =
+ +
=
- = -
1 2
2
1
2 1
2 ln
) ( ) (
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Example: OPM • You are looking at a call
option with 6 months to expiration and an exercise price of $35. The current stock price is $45 and the risk-free rate is 4%. The standard deviation of underlying asset returns is 20%. What is the value of the call option?
85.15.2.99.1
99.15.2.
5.2
2.04.
35
45ln
2
2
1
=-=
=
++
=
d
d
•Look up N(d1) and N(d2) in Table 24.3
•N(d1) = (.9761+.9772)/2 = .9767
•N(d2) = (.9671+.9686)/2 = .9679
C = 45(.9767) – 35e-.04(.5)(.9679)
C = $10.75
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
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Put Values • The value of a put can be found by finding the
value of the call and then using put-call parity – What is the value of the put in the previous
example? • P = C + Ee-Rt – S
• P = 10.75 + 35e-.04(.5) – 45 = .06
• Note that a put may be worth more if exercised than if sold, while a call is worth more “alive than dead,” unless there is a large expected cash flow from the underlying asset
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 and therefore, the price computed using the model will be too low relative to that of an American option that does allow for early exercise
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Table 24.4
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 we need to hedge a portfolio
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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; s = .2
• You can also choose a stock and value options on a particular stock
• Equity can be viewed as a call option on the firm’s assets whenever the firm carries debt
• The strike price is the cost of making the debt payments
• The underlying asset price is the market value of the firm’s assets
• If the intrinsic value is positive, the firm can exercise the option by paying off the debt
• If the intrinsic value is negative, the firm can let the option expire and turn the firm over to the bondholders
• This concept is useful in valuing certain types of corporate decisions
Valuing Equity and Changes in Assets
• Consider a firm that has a zero-coupon bond that matures in 4 years. The face value is $30 million and the risk-free rate is 6%. The current market value of the firm’s assets is $40 million and the firm’s equity is currently worth $18 million. Suppose the firm is considering a project with an NPV = $500,000.
– What is the implied standard deviation of returns?