8 - 1 Financial options Black-Scholes Option Pricing Model CHAPTER 8 Financial Options and Their Valuation

8 - 2 An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time. What is a financial option?

8 - 3 It does not obligate its owner to take any action. It merely gives the owner the right to buy or sell an asset. What is the single most important characteristic of an option?

8 - 4 Call option: An option to buy a specified number of shares of a security within some future period. Put option: An option to sell a specified number of shares of a security within some future period. Exercise (or strike) price: The price stated in the option contract at which the security can be bought or sold. Option Terminology

8 - 5 Option price: The market price of the option contract. Expiration date: The date the option matures. Exercise value: The value of a call option if it were exercised today = Current stock price - Strike price. Note: The exercise value is zero if the stock price is less than the strike price.

8 - 6 Covered option: A call option written against stock held in an investor’s portfolio. Naked (uncovered) option: An option sold without the stock to back it up. In-the-money call: A call whose exercise price is less than the current price of the underlying stock.

8 - 7 Out-of-the-money call: A call option whose exercise price exceeds the current stock price. LEAPS: Long-term Equity AnticiPation Securities that are similar to conventional options except that they are long-term options with maturities of up to 2 1/2 years.

8 - 8 Exercise price = $25. Stock PriceCall Option Price $25$ Consider the following data:

8 - 9 Create a table which shows (a) stock price, (b) strike price, (c) exercise value, (d) option price, and (e) premium of option price over the exercise value. Price of Strike Exercise Value Stock (a) Price (b) of Option (a) - (b) $25.00$25.00$

Exercise Value Mkt. Price Premium of Option (c) of Option (d) (d) - (c) $ 0.00 $ 3.00 $ Table (Continued)

Call Premium Diagram Stock Price Option value Market price Exercise value

What happens to the premium of the option price over the exercise value as the stock price rises? The premium of the option price over the exercise value declines as the stock price increases. This is due to the declining degree of leverage provided by options as the underlying stock price increases, and the greater loss potential of options at higher option prices.

The stock underlying the call option provides no dividends during the call option’s life. There are no transactions costs for the sale/purchase of either the stock or the option. R RF is known and constant during the option’s life. What are the assumptions of the Black-Scholes Option Pricing Model? (More...)

Security buyers may borrow any fraction of the purchase price at the short-term risk-free rate. No penalty for short selling and sellers receive immediately full cash proceeds at today’s price. Call option can be exercised only on its expiration date. Security trading takes place in continuous time, and stock prices move randomly in continuous time.

V = P[N(d 1 )] - Xe -r RF t [N(d 2 )]. d 1 =.  t d 2 = d 1 -  t. What are the three equations that make up the OPM? ln(P/X) + [r RF + (  2 /2)]t

What is the value of the following call option according to the OPM? Assume: P = $27; X = $25; r RF = 6%; t = 0.5 years:  2 = 0.11 V = $27[N(d 1 )] - $25e -(0.06)(0.5) [N(d 2 )]. ln($27/$25) + [( /2)](0.5) (0.3317)(0.7071) = d 2 = d 1 - (0.3317)(0.7071) = d = = d 1 =

N(d 1 ) = N(0.5736) = = N(d 2 ) = N(0.3391) = = Note: Values obtained from Excel using NORMSDIST function. V = $27(0.7168) - $25e (0.6327) = $ $25( )(0.6327) = $

Current stock price: Call option value increases as the current stock price increases. Exercise price: As the exercise price increases, a call option’s value decreases. What impact do the following para- meters have on a call option’s value?

Option period: As the expiration date is lengthened, a call option’s value increases (more chance of becoming in the money.) Risk-free rate: Call option’s value tends to increase as r RF increases (reduces the PV of the exercise price). Stock return variance: Option value increases with variance of the underlying stock (more chance of becoming in the money).

Put-Call Parity Portfolio 1: Put option, Share of stock, P Portfolio 2: Call option, V PV of exercise price, X

Payoff at Expiration if P<X Portfolio 1: Put: X-P Stock: P Total: X Porfolio 2: Call: 0 Cash: X Total: X

Payoff at Expiration if P  X Portfolio 1: Put: 0 Stock: P Total: P Porfolio 2: Call: P-X Cash: X Total: P

Payoffs are Equal,Values Must Be Equal. Put-Call Parity Relationship Put + Stock = Call + PV of Exercise Price

Real options Decision trees Application of financial options to real options CHAPTER 12 Real Options

Inputs to Black-Scholes Model for Option to Wait X = exercise price = cost to implement project = $70 million. r RF = risk-free rate = 6%. t = time to maturity = 1 year. P = current stock price = Estimated on following slides.  2 = variance of stock return = Estimated on following slides.

Estimate of P For a financial option: P = current price of stock = PV of all of stock’s expected future cash flows. Current price is unaffected by the exercise cost of the option. For a real option: P = PV of all of project’s future expected cash flows. P does not include the project’s cost.

Step 1: Find the PV of future CFs at option’s exercise year. PV at 0Prob Year 1 $45 $ % 40%$30 $ % $15 $37.30 Future Cash Flows Example: $ = $45/1.1 + $45/ $45/ See Ch 12 Mini Case.xls for calculations.

Step 2: Find the expected PV at the current date, Year 0. PV 2004 =PV of Exp. PV 2005 = [(0.3* $111.91) +(0.4*$74.61) +(0.3*$37.3)]/1.1 = $ See Ch 12 Mini Case.xls for calculations. PV Year 0 PV Year 1 $ High $67.82 Average $74.61 Low $37.30

The Input for P in the Black-Scholes Model The input for price is the present value of the project’s expected future cash flows. Based on the previous slides, P = $67.82.

Estimating  2 for the Black-Scholes Model For a financial option,  2 is the variance of the stock’s rate of return. For a real option,  2 is the variance of the project’s rate of return.

Three Ways to Estimate  2 Judgment. The direct approach, using the results from the scenarios. The indirect approach, using the expected distribution of the project’s value.

Estimating  2 with Judgment The typical stock has  2 of about 12%. A project should be riskier than the firm as a whole, since the firm is a portfolio of projects. The company in this example has  2 = 10%, so we might expect the project to have  2 between 12% and 19%.

Estimating  2 with the Direct Approach Use the previous scenario analysis to estimate the return from the present until the option must be exercised. Do this for each scenario Find the variance of these returns, given the probability of each scenario.

Find Returns from the Present until the Option Expires Example: 65.0% = ($ $67.82) / $ See Ch 12 Mini Case.xls for calculations. PV Year 0 PV Year 1 Return $ % High $67.82 Average $ % Low $ %

E(Ret.)=0.3(0.65)+0.4(0.10)+0.3(-0.45) E(Ret.)= 0.10 = 10%.  2 = 0.3( ) ( ) ( ) 2  2 = = 18.2%. Use these scenarios, with their given probabilities, to find the expected return and variance of return.

Estimating  2 with the Indirect Approach From the scenario analysis, we know the project’s expected value and the variance of the project’s expected value at the time the option expires. The questions is: “Given the current value of the project, how risky must its expected return be to generate the observed variance of the project’s value at the time the option expires?”

The Indirect Approach (Cont.) From option pricing for financial options, we know the probability distribution for returns (it is lognormal). This allows us to specify a variance of the rate of return that gives the variance of the project’s value at the time the option expires.

Indirect Estimate of  2 Here is a formula for the variance of a stock’s return, if you know the coefficient of variation of the expected stock price at some time, t, in the future: We can apply this formula to the real option.

From earlier slides, we know the value of the project for each scenario at the expiration date. PV Year 1 $ High Average $74.61 Low $37.30

E(PV)=.3($111.91)+.4($74.61)+.3($37.3) E(PV)= $ Use these scenarios, with their given probabilities, to find the project’s expected PV and  PV.  PV = [.3($ $74.61) 2 +.4($74.61-$74.61) 2 +.3($37.30-$74.61) 2 ] 1/2  PV = $28.90.

Find the project’s expected coefficient of variation, CV PV, at the time the option expires. CV PV = $28.90 /$74.61 = 0.39.

Now use the formula to estimate  2. From our previous scenario analysis, we know the project’s CV, 0.39, at the time it the option expires (t=1 year).

The Estimate of  2 Subjective estimate: 12% to 19%. Direct estimate: 18.2%. Indirect estimate: 14.2% For this example, we chose 14.2%, but we recommend doing sensitivity analysis over a range of  2.

Use the Black-Scholes Model: P = $67.83; X = $70; r RF = 6%; t = 1 year:  2 = V = $67.83[N(d 1 )] - $70e -(0.06)(1) [N(d 2 )]. ln($67.83/$70)+[( /2)](1) (0.142) 0.5 (1).05 = d 2 = d 1 - (0.142) 0.5 (1).05 = d = = d 1 =

N(d 1 ) = N(0.2641) = N(d 2 ) = N( ) = V = $67.83(0.6041) - $70e (0.4551) = $ $70(0.9418)(0.4551) = $ Note: Values of N(d i ) obtained from Excel using NORMSDIST function. See Ch 12 Mini Case.xls for details.

Real Options

Option Theory and Investment Decisions Traditional investment analysis uses the NPV rule to decide on investment projects The concept of net present value uses expected cash flows, discounted at a constant discount rate which is assumed to capture the risk of these cash flows Making investment decisions based on a project’s NPV is equivalent to assuming that the firm’s managers have no ability to respond to uncertainty (no managerial flexibility)

Option Theory and Investment Decisions Example: Suppose that a firm will invest $600m over the next two years ($250m this year and $350m next year) to build a new factory with a life of 20 years Suppose that revenues from this factory will be $80m starting 3 years from now and grow at 8% annually and that the costs to run the factory will be $60m (also staring 3 years from now) and grow at 6% Assume for simplicity that there are no working capital investments, no taxes, no salvage value, and that the firm will use straight-line depreciation starting two years from now

Option Theory and Investment Decisions If the firm’s WACC is 10%, this project’s NPV is -$1.44m so we reject the project The above analysis may fail to capture some aspects of the project The initial investment takes place in stages and can be abandoned at each If the project goes well, perhaps it can be expanded or extended If it goes bad, perhaps it can be scaled back or sold at a floor price Perhaps there is the option to delay building the factory for some time

Option Theory and Investment Decisions Example: Suppose that you are given the choice to put $1 in a toy bank that guarantees $1.05 a year later with certainty This offer is good for only one year Alternatively, interest rates in a real bank are currently at 10% How much is the toy bank offer worth?

Option Theory and Investment Decisions Real options are options that give the right to make decisions on a capital investment project Real options are American options A limitation of traditional investment analysis is that it is static, meaning that it does not do a good job capturing the value of options embedded in a project One can go as far as arguing that the NPV rule systematically undervalues every project

Option Theory and Investment Decisions These options add value to a project and may change a project’s NPV from negative (under traditional analysis) into positive Project’s value = NPV + Value of option

Types of Real Options Option to delay a project (Deferral option) Option to expand or to extend a project Option to abandon a project Option to contract a project Option to scale back a project Switching options Compound options Rainbow options

The Option to Delay a Project If a firm has exclusive rights to a project or a product for a specific period, it can delay taking the project or introducing the product until a later date A project that has a negative NPV today may have a positive NPV at some future date The reason is that expected cash flows and the discount rate may change through time, meaning there are changes in the business environment

The Option to Delay a Project This resembles a call option (deferral option) where The underlying asset is the project The strike price is the investment needed to take the project The life of the option is the period that the firm can delay undertaking the project The value of the option increases with the volatility in the business environment

The Option to Delay a Project NPV of project PV of cash flows from accepting the project Initial investment Loss of rights to the project

The Option to Expand a Project or Take Other Projects Taking a project today may allow a firm to expand the project or consider taking other projects in the future The value of this option may change the NPV of a project These call options are called strategic options Example: Honda, Toyota, GM, Ford are investing in cars with hybrid engines, which, despite current losses, gives them the option to gain technological and production expertise to produce an eco-car profitably in the future

The Option to Abandon a Project A firm may have the option to abandon a project if the cash flows from the project do not match the firm’s expectations If abandoning the project allows the firm to save itself from further losses, then this put option increases the value of the project and makes it more attractive Firms try to build such options in the contracts signed with other parties involved in a project

Other Types of Real Options Switching Real Options are portfolios of call and put options that allow their owner to switch at a fixed cost between two modes of operation Example: the option to shut down a manufacturing plant when demand is low and reopen it when demand picks up (General Motors assembly plants) Compound real options are options on options typically found in phased investment projects

Other Types of Real Options Example: The decision to build a chemical plant may involve three phases: a design phase, an engineering phase and construction At each phase, the firm has the option to stop or defer the project Thus, each phase is an option that depends on the earlier exercise of another option Rainbow options are options driven by multiple sources of uncertainty (e.g., exploration and development)

Value of Real Options The value of real options depends on Expected present value of cash flows from project Exercise price (Investment Outlay) Time to expiration (Time to defer) Uncertainty (volatility) of project’s present value Risk-free rate

Value of Managerial Flexibility Low UncertaintyHigh Uncertainty High Ability to Respond MODERATE FLEXIBILITY VALUE HIGH FLEXIBILITY VALUE Low Ability to Respond LOW FLEXIBILITY VALUE MODERATE FLEXIBILITY VALUE

Decision Trees Decision Tree Analysis is useful for applying binomial pricing methods to (real) options Decision trees are used to analyze projects that involve sequential decisions If the decision made today affects the decision that we are able to make tomorrow, then we should analyze tomorrow’s decision before we act today

Decision Trees Example: Suppose that Delta Airlines is considering offering shuttle service between Champaign and Cincinnati for two years Future demand for this service is uncertain and the company must make a decision today of what type of plane to buy Suppose that there is a 60% chance that demand for this service will be high and 40% chance that it will be low after the service is initiated If demand is high, there is a 75% chance that subsequent demand will also be high

Decision Trees However, if demand is low, the chance of subsequent demand being high drops to 25% The firm has two choices of planes to purchase: A small jet costs $1.5m and can handle large capacity A prop plane costs $0.25m, but it cannot fully handle capacity Operating costs are higher for the jet plane If demand is high in the first year, the firm might consider buying another prop plane before the second year for $0.25m The project’s decision tree looks as follows

Purchase Jet = Purchase Prop = -250 Low (.4) = 300 High (.6) = 1000 High (.6) = 2000 Low (.4) = 500 High (.75) = 2000 High (.25) = 2000 High (.75) = 1500 High (.75) = 1000 High (.25) = 1000 Low (.25) = 300 Low (.75) = 300 Low (.25) = 400 Low (.25) = 500 Low (.75) = 500 Decision node Chance node Decision Tree For Delta Expand (-250) Don’t expand