Option Pricing with Applications to Real Options.ppt

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1、,Financial optionsBlack-Scholes Option Pricing ModelReal optionsDecision treesApplication of financial options to real options,CHAPTER 17Option Pricing with Applications to Real Options,What is a real option?,Real options exist when managers can influence the size and risk of a projects cash flows b

2、y taking different actions during the projects life in response to changing market conditions.Alert managers always look for real options in projects.Smarter managers try to create real options.,An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an ass

3、et at some predetermined price within a specified period of time.,What is a financial option?,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 importantcharacteristic of an option?,Call option: An option to buy a

4、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,Op

5、tion 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.,Covered opti

6、on: A call option written against stock held in an investors 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.,Out-of-the-money call: A call option whose exercise p

7、rice 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.,Exercise price = $25.Stock PriceCall Option Price$25$ 3.00 30 7.50 35 12.00 40 16.50 45 21.00 50

8、25.50,Consider the following data:,Create a table which shows (a) stockprice, (b) strike price, (c) exercisevalue, (d) option price, and (e) premiumof option price over the exercise value.,Price of Strike Exercise ValueStock (a) Price (b) of Option (a) - (b)$25.00$25.00$0.00 30.00 25.00 5.00 35.00 2

9、5.00 10.00 40.00 25.0015.00 45.00 25.0020.00 50.00 25.0025.00,Exercise Value Mkt. Price Premium of Option (c) of Option (d) (d) - (c) $ 0.00 $ 3.00 $ 3.00 5.00 7.50 2.50 10.00 12.00 2.00 15.00 16.50 1.50 20.00 21.00 1.00 25.00 25.50 0.50,Table (Continued),Call Premium Diagram,5 10 15 20 25 30 35 40

10、45 50,Stock Price,Option value,30252015105,Market price,Exercise value,What happens to the premium of the option price over the exercisevalue 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

11、 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 options life.There are no transactions costs for the sale/purchase of either the stock

12、or the option.RRF is known and constant during the options life.,What are the assumptions of theBlack-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

13、 cash proceeds at todays 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 = PN(d1) - Xe -rRFtN(d2).d1 = . td2 = d1 - t.,What are the three equations thatmake up the OPM?,ln(P/X) + rRF

14、 + (2/2)t,What is the value of the following call option according to the OPM?Assume: P = $27; X = $25; rRF = 6%;t = 0.5 years: 2 = 0.11,V = $27N(d1) - $25e-(0.06)(0.5)N(d2). ln($27/$25) + (0.06 + 0.11/2)(0.5)(0.3317)(0.7071) = 0.5736.d2 = d1 - (0.3317)(0.7071) = d1 - 0.2345 = 0.5736 - 0.2345 = 0.33

15、91.,d1 =,N(d1) = N(0.5736) = 0.5000 + 0.2168 = 0.7168.N(d2) = N(0.3391) = 0.5000 + 0.1327 = 0.6327.Note: Values obtained from Excel using NORMSDIST function.V = $27(0.7168) - $25e-0.03(0.6327) = $19.3536 - $25(0.97045)(0.6327) = $4.0036.,Current stock price: Call option value increases as the curren

16、t stock price increases.Exercise price: As the exercise price increases, a call options value decreases.,What impact do the following para-meters have on a call options value?,Option period: As the expiration date is lengthened, a call options value increases (more chance of becoming in the money.)R

17、isk-free rate: Call options value tends to increase as rRF 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).,How are real options different from financial options?,Financial opt

18、ions have an underlying asset that is traded-usually a security like a stock.A real option has an underlying asset that is not a security-for example a project or a growth opportunity, and it isnt traded.,(More.),How are real options different from financial options?,The payoffs for financial option

19、s are specified in the contract.Real options are “found” or created inside of projects. Their payoffs can be varied.,What are some types of real options?,Investment timing optionsGrowth options Expansion of existing product lineNew productsNew geographic markets,Types of real options (Continued),Aba

20、ndonment optionsContractionTemporary suspensionFlexibility options,Five Procedures for ValuingReal Options,1.DCF analysis of expected cash flows, ignoring the option. 2.Qualitative assessment of the real options value.3.Decision tree analysis.4.Standard model for a corresponding financial option.5.F

21、inancial engineering techniques.,Analysis of a Real Option: Basic Project,Initial cost = $70 million, Cost of Capital = 10%, risk-free rate = 6%, cash flows occur for 3 years. AnnualDemand Probability Cash FlowHigh30%$45Average40%$30Low30%$15,Approach 1: DCF Analysis,E(CF)=.3($45)+.4($30)+.3($15) =

22、$30.PV of expected CFs = ($30/1.1) + ($30/1.12) + ($30/1/13) = $74.61 million.Expected NPV = $74.61 - $70 = $4.61 million,Investment Timing Option,If we immediately proceed with the project, its expected NPV is $4.61 million.However, the project is very risky:If demand is high, NPV = $41.91 million.

23、*If demand is low, NPV = -$32.70 million.*_* See Ch 17 Mini Case.xls for calculations.,Investment Timing (Continued),If we wait one year, we will gain additional information regarding demand.If demand is low, we wont implement project. If we wait, the up-front cost and cash flows will stay the same,

24、 except they will be shifted ahead by a year.,Procedure 2: Qualitative Assessment,The value of any real option increases if:the underlying project is very riskythere is a long time before you must exercise the optionThis project is risky and has one year before we must decide, so the option to wait

25、is probably valuable.,Procedure 3: Decision Tree Analysis (Implement only if demand is not low.),Discount the cost of the project at the risk-free rate, since the cost is known. Discount the operating cash flows at the cost of capital. Example: $35.70 = -$70/1.06 + $45/1.12 + $45/1.13 + $45/1.13. Se

26、e Ch 17 Mini Case.xls for calculations.,E(NPV) = 0.3($35.70)+0.4($1.79) + 0.3 ($0)E(NPV) = $11.42.,Use these scenarios, with their given probabilities, to find the projects expected NPV if we wait.,Decision Tree with Option to Wait vs. Original DCF Analysis,Decision tree NPV is higher ($11.42 millio

27、n vs. $4.61).In other words, the option to wait is worth $11.42 million. If we implement project today, we gain $4.61 million but lose the option worth $11.42 million.Therefore, we should wait and decide next year whether to implement project, based on demand.,The Option to Wait Changes Risk,The cas

28、h flows are less risky under the option to wait, since we can avoid the low cash flows. Also, the cost to implement may not be risk-free.Given the change in risk, perhaps we should use different rates to discount the cash flows.But finance theory doesnt tell us how to estimate the right discount rat

29、es, so we normally do sensitivity analysis using a range of different rates.,Procedure 4: Use the existing model of a financial option.,The option to wait resembles a financial call option- we get to “buy” the project for $70 million in one year if value of project in one year is greater than $70 mi

30、llion.This is like a call option with an exercise price of $70 million and an expiration date of one year.,Inputs to Black-Scholes Model for Option to Wait,X = exercise price = cost to implement project = $70 million.rRF = risk-free rate = 6%.t = time to maturity = 1 year.P = current stock price = E

31、stimated 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 stocks expected future cash flows.Current price is unaffected by the exercise cost of the option.For a real option:P = PV of all

32、of projects future expected cash flows.P does not include the projects cost.,Step 1: Find the PV of future CFs at options exercise year.,PV at,2002,Prob.,2003,2004,2005,2006,2003,$45,$45,$45,$111.91,30%,40%,$30,$30,$30,$74.61,30%,$15,$15,$15,$37.30,Future Cash Flows,Example: $111.91 = $45/1.1 + $45/

33、1.12 + $45/1.13.See Ch 17 Mini Case.xls for calculations.,Step 2: Find the expected PV at the current date, 2002.,PV2002=PV of Exp. PV2003 = (0.3* $111.91) +(0.4*$74.61) +(0.3*$37.3)/1.1 = $67.82.See Ch 17 Mini Case.xls for calculations.,PV,2002,PV,2003,$111.91,High,$67.82,Average,$74.61,Low,$37.30,

34、The Input for P in the Black-Scholes Model,The input for price is the present value of the projects expected future cash flows.Based on the previous slides,P = $67.82.,Estimating s2 for the Black-Scholes Model,For a financial option, s2 is the variance of the stocks rate of return.For a real option,

35、 s2 is the variance of the projects rate of return.,Three Ways to Estimate s2,Judgment.The direct approach, using the results from the scenarios.The indirect approach, using the expected distribution of the projects value.,Estimating s2 with Judgment,The typical stock has s2 of about 12%.A project s

36、hould be riskier than the firm as a whole, since the firm is a portfolio of projects.The company in this example has s2 = 10%, so we might expect the project to have s2 between 12% and 19%.,Estimating s2 with the Direct Approach,Use the previous scenario analysis to estimate the return from the pres

37、ent until the option must be exercised. Do this for each scenarioFind the variance of these returns, given the probability of each scenario.,Find Returns from the Present until the Option Expires,Example: 65.0% = ($111.91- $67.82) / $67.82.See Ch 17 Mini Case.xls for calculations.,PV,2002,PV,2003,Re

38、turn,$111.91,65.0%,High,$67.82,Average,$74.61,10.0%,Low,$37.30,-45.0%,E(Ret.)=0.3(0.65)+0.4(0.10)+0.3(-0.45)E(Ret.)= 0.10 = 10%.2 = 0.3(0.65-0.10)2 + 0.4(0.10-0.10)2 + 0.3(-0.45-0.10)22 = 0.182 = 18.2%.,Use these scenarios, with their given probabilities, to find the expected return and variance of

39、return.,Estimating s2 with the Indirect Approach,From the scenario analysis, we know the projects expected value and the variance of the projects 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

40、 the observed variance of the projects 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

41、 of the projects value at the time the option expires.,Indirect Estimate of 2,Here is a formula for the variance of a stocks 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,

42、 we know the value of the project for each scenario at the expiration date.,PV,2003,$111.91,High,Average,$74.61,Low,$37.30,E(PV)=.3($111.91)+.4($74.61)+.3($37.3)E(PV)= $74.61.,Use these scenarios, with their given probabilities, to find the projects expected PV and PV.,PV = .3($111.91-$74.61)2 + .4(

43、$74.61-$74.61)2 + .3($37.30-$74.61)21/2PV = $28.90.,Find the projects expected coefficient of variation, CVPV, at the time the option expires.,CVPV = $28.90 /$74.61 = 0.39.,Now use the formula to estimate 2.,From our previous scenario analysis, we know the projects CV, 0.39, at the time it the optio

44、n 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 s2.,Use the Black-Scholes Model: P = $67.83; X = $70; rRF = 6%;t = 1 year: 2 = 0.142,V = $

45、67.83N(d1) - $70e-(0.06)(1)N(d2). ln($67.83/$70)+(0.06 + 0.142/2)(1)(0.142)0.5 (1).05 = 0.2641.d2 = d1 - (0.142)0.5 (1).05= d1 - 0.3768 = 0.2641 - 0.3768 =- 0.1127.,d1 =,N(d1) = N(0.2641) = 0.6041N(d2) = N(- 0.1127) = 0.4551V = $67.83(0.6041) - $70e-0.06(0.4551) = $40.98 - $70(0.9418)(0.4551) = $10.

46、98.,Note: Values of N(di) obtained from Excel using NORMSDIST function. See Ch 17 Mini Case.xls for details.,Step 5: Use financial engineering techniques.,Although there are many existing models for financial options, sometimes none correspond to the projects real option.In that case, you must use financial engineering techniques, which are covered in later finance courses.Alternatively, you could simply use decision tree analysis.,Other Factors to Consider When Deciding When to Invest,