1、17-1nFinancial optionsnBlack-Scholes Option Pricing ModelnReal optionsnDecision treesnApplication of financial options to real optionsCHAPTER 17Option Pricing with Applications to Real Options17-2What is a real option?nReal options exist when managers can influence the size and risk of a projects ca
2、sh flows by taking different actions during the projects life in response to changing market conditions.nAlert managers always look for real options in projects.nSmarter managers try to create real options.17-3An option is a contract which gives its holder the right,but not the obligation,to buy(or
3、sell)an asset at some predetermined price within a specified period of time.What is a financial option?17-4nIt 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?17-5nCall option:An o
4、ption to buy a specified number of shares of a security within some future period.nPut option:An option to sell a specified number of shares of a security within some future period.nExercise(or strike)price:The price stated in the option contract at which the security can be bought or sold.Option Te
5、rminology17-6nOption price:The market price of the option contract.nExpiration date:The date the option matures.nExercise 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.17-7
6、nCovered option:A call option written against stock held in an investors portfolio.nNaked(uncovered)option:An option sold without the stock to back it up.nIn-the-money call:A call whose exercise price is less than the current price of the underlying stock.17-8nOut-of-the-money call:A call option who
7、se exercise price exceeds the current stock price.nLEAPs: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.17-9Exercise price=$25.Stock PriceCall Option Price$25$3.00 30 7.50 35 12.00 40 16.50
8、 45 21.00 50 25.50Consider the following data:17-10Create 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 25.
9、00 10.00 40.00 25.0015.00 45.00 25.0020.00 50.00 25.0025.0017-11Exercise 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 0.50Table(Continued)17-12Call Premium Diagram5 10 15 20 25 30 35 40 45 50Stock
10、 PriceOption value30252015105Market priceExercise value17-13What happens to the premium of the option price over the exercisevalue as the stock price rises?nThe premium of the option price over the exercise value declines as the stock price increases.nThis is due to the declining degree of leverage
11、provided by options as the underlying stock price increases,and the greater loss potential of options at higher option prices.17-14nThe stock underlying the call option provides no dividends during the call options life.nThere are no transactions costs for the sale/purchase of either the stock or th
12、e option.nRRF is known and constant during the options life.What are the assumptions of theBlack-Scholes Option Pricing Model?(More.)17-15nSecurity buyers may borrow any fraction of the purchase price at the short-term risk-free rate.nNo penalty for short selling and sellers receive immediately full
13、 cash proceeds at todays price.nCall option can be exercised only on its expiration date.n Security trading takes place in continuous time,and stock prices move randomly in continuous time.17-16V =PN(d1)-Xe-rRFtN(d2).d1=.td2=d1-t.What are the three equations thatmake up the OPM?ln(P/X)+rRF+(2/2)t17-
14、17What is the value of the following call option according to the OPM?Assume:P=$27;X=$25;rRF=6%;t=0.5 years:2=0.11V =$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=0.3391.d1=17-18N(d1)=N(0.5736)=0.5000+0.2168 =0.71
15、68.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)=$4.0036.17-19nCurrent stock price:Call option value increases as the current stock price increases.nExercise price:As the exercise price inc
16、reases,a call options value decreases.What impact do the following para-meters have on a call options value?17-20nOption period:As the expiration date is lengthened,a call options value increases(more chance of becoming in the money.)nRisk-free rate:Call options value tends to increase as rRF increa
17、ses(reduces the PV of the exercise price).nStock return variance:Option value increases with variance of the underlying stock(more chance of becoming in the money).17-21How are real options different from financial options?nFinancial options have an underlying asset that is traded-usually a security
18、 like a stock.nA real option has an underlying asset that is not a security-for example a project or a growth opportunity,and it isnt traded.(More.)17-22How are real options different from financial options?nThe payoffs for financial options are specified in the contract.nReal options are“found”or c
19、reated inside of projects.Their payoffs can be varied.17-23What are some types of real options?nInvestment timing optionsnGrowth options lExpansion of existing product linelNew productslNew geographic markets17-24Types of real options(Continued)nAbandonment optionslContractionlTemporary suspensionnF
20、lexibility options17-25Five Procedures for ValuingReal Options1.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.Financial engineering techniques.17-26Analysis o
21、f a Real Option:Basic ProjectnInitial cost=$70 million,Cost of Capital=10%,risk-free rate=6%,cash flows occur for 3 years.AnnualDemand Probability Cash FlowHigh30%$45Average40%$30Low30%$1517-27Approach 1:DCF AnalysisnE(CF)=.3($45)+.4($30)+.3($15)=$30.nPV of expected CFs=($30/1.1)+($30/1.12)+($30/1/1
22、3)=$74.61 million.nExpected NPV=$74.61-$70 =$4.61 million17-28Investment Timing Optionn If we immediately proceed with the project,its expected NPV is$4.61 million.nHowever,the project is very risky:lIf demand is high,NPV=$41.91 million.*lIf demand is low,NPV=-$32.70 million.*_*See Ch 17 Mini Case.x
23、ls for calculations.17-29Investment Timing(Continued)nIf we wait one year,we will gain additional information regarding demand.nIf demand is low,we wont implement project.nIf we wait,the up-front cost and cash flows will stay the same,except they will be shifted ahead by a year.17-30Procedure 2:Qual
24、itative AssessmentnThe value of any real option increases if:lthe underlying project is very riskylthere is a long time before you must exercise the optionnThis project is risky and has one year before we must decide,so the option to wait is probably valuable.17-31Procedure 3:Decision Tree Analysis(
25、Implement only if demand is not low.)CostNPV this2001Prob.2002200320042005Scenarioa-$70$45$45$45$35.7030%$040%-$70$30$30$30$1.7930%$0$0$0$0$0.00Future Cash FlowsDiscount 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.Exa
26、mple:$35.70=-$70/1.06+$45/1.12+$45/1.13+$45/1.13.See Ch 17 Mini Case.xls for calculations.17-32E(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.17-33Decision Tree with Option to Wait vs.Original DCF Ana
27、lysisnDecision tree NPV is higher($11.42 million vs.$4.61).nIn 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.nTherefore,we should wait and decide next year whether to implement project,based on demand.
28、17-34The Option to Wait Changes RisknThe cash 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.nGiven the change in risk,perhaps we should use different rates to discount the cash flows.nBut finance theory doesnt tell
29、 us how to estimate the right discount rates,so we normally do sensitivity analysis using a range of different rates.17-35Procedure 4:Use the existing model of a financial option.nThe option to wait resembles a financial call option-we get to“buy”the project for$70 million in one year if value of pr
30、oject in one year is greater than$70 million.nThis is like a call option with an exercise price of$70 million and an expiration date of one year.17-36Inputs to Black-Scholes Model for Option to WaitnX=exercise price=cost to implement project=$70 million.nrRF=risk-free rate=6%.nt=time to maturity=1 y
31、ear.nP=current stock price=Estimated on following slides.n 2=variance of stock return=Estimated on following slides.17-37Estimate of PnFor a financial option:lP=current price of stock=PV of all of stocks expected future cash flows.lCurrent price is unaffected by the exercise cost of the option.nFor
32、a real option:lP=PV of all of projects future expected cash flows.lP does not include the projects cost.17-38Step 1:Find the PV of future CFs at options exercise year.PV at2002Prob.20032004200520062003$45$45$45$111.9130%40%$30$30$30$74.6130%$15$15$15$37.30Future Cash FlowsExample:$111.91=$45/1.1+$45
33、/1.12+$45/1.13.See Ch 17 Mini Case.xls for calculations.17-39Step 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.PV2002PV2003$111.91High$67.82Average$74.61Low$37.3017-40The Input for P
34、 in the Black-Scholes ModelnThe input for price is the present value of the projects expected future cash flows.nBased on the previous slides,P=$67.82.17-41Estimating 2 for the Black-Scholes ModelnFor a financial option,2 is the variance of the stocks rate of return.nFor a real option,2 is the varia
35、nce of the projects rate of return.17-42Three Ways to Estimate 2nJudgment.nThe direct approach,using the results from the scenarios.nThe indirect approach,using the expected distribution of the projects value.17-43Estimating 2 with JudgmentnThe typical stock has 2 of about 12%.nA project should be r
36、iskier than the firm as a whole,since the firm is a portfolio of projects.nThe company in this example has 2=10%,so we might expect the project to have 2 between 12%and 19%.17-44Estimating 2 with the Direct ApproachnUse the previous scenario analysis to estimate the return from the present until the
37、 option must be exercised.Do this for each scenarionFind the variance of these returns,given the probability of each scenario.17-45Find Returns from the Present until the Option ExpiresExample:65.0%=($111.91-$67.82)/$67.82.See Ch 17 Mini Case.xls for calculations.PV2002PV2003Return$111.9165.0%High$6
38、7.82Average$74.6110.0%Low$37.30-45.0%17-46E(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)2 2=0.182=18.2%.Use these scenarios,with their given probabilities,to find the expected return and variance of return.17-47Estimating 2 with the Indirect
39、 ApproachnFrom the scenario analysis,we know the projects expected value and the variance of the projects expected value at the time the option expires.nThe questions is:“Given the current value of the project,how risky must its expected return be to generate the observed variance of the projects va
40、lue at the time the option expires?”17-48The Indirect Approach(Cont.)nFrom option pricing for financial options,we know the probability distribution for returns(it is lognormal).nThis allows us to specify a variance of the rate of return that gives the variance of the projects value at the time the
41、option expires.17-49Indirect Estimate of 2 nHere 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:t 1CVln22We can apply this formula to the real option.17-50From earlier slides,we know the value of the
42、project for each scenario at the expiration date.PV2003$111.91HighAverage$74.61Low$37.3017-51E(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($74.61-$74.61)2 +.3($37.30-$74.61)21/2
43、 PV=$28.90.17-52Find the projects expected coefficient of variation,CVPV,at the time the option expires.CVPV=$28.90/$74.61=0.39.17-53Now use the formula to estimate 2.nFrom our previous scenario analysis,we know the projects CV,0.39,at the time it the option expires(t=1 year).%2.141 139.0ln2217-54Th
44、e Estimate of 2nSubjective estimate:l12%to 19%.nDirect estimate:l18.2%.nIndirect estimate:l14.2%nFor this example,we chose 14.2%,but we recommend doing sensitivity analysis over a range of 2.17-55Use the Black-Scholes Model:P=$67.83;X=$70;rRF=6%;t=1 year:2=0.142V =$67.83N(d1)-$70e-(0.06)(1)N(d2).ln(
45、$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=17-56N(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)=$10.98.Note:Values of N(di)obtained from Excel using NORMSDIST function.See
46、 Ch 17 Mini Case.xls for details.17-57Step 5:Use financial engineering techniques.nAlthough there are many existing models for financial options,sometimes none correspond to the projects real option.nIn that case,you must use financial engineering techniques,which are covered in later finance course
47、s.nAlternatively,you could simply use decision tree analysis.17-58Other Factors to Consider When Deciding When to InvestnDelaying the project means that cash flows come later rather than sooner.nIt might make sense to proceed today if there are important advantages to being the first competitor to e
48、nter a market.nWaiting may allow you to take advantage of changing conditions.17-59A New Situation:Cost is$75 Million,No Option to WaitCostNPV this2002Prob.200320042005Scenario$45$45$45$36.9130%-$7540%$30$30$30-$0.3930%$15$15$15-$37.70Future Cash FlowsExample:$36.91=-$75+$45/1.1+$45/1.1+$45/1.1.See
49、Ch 17 Mini Case.xls for calculations.17-60Expected NPV of New SituationnE(NPV)=0.3($36.91)+0.4(-$0.39)+0.3(-$37.70)nE(NPV)=-$0.39.nThe project now looks like a loser.17-61Growth Option:You can replicate the original project after it ends in 3 years.nNPV=NPV Original+NPV Replication=-$0.39+-$0.39/(1+
50、0.10)3=-$0.39+-$0.30=-$0.69.nStill a loser,but you would implement Replication only if demand is high.Note:the NPV would be even lower if we separately discounted the$75 million cost of Replication at the risk-free rate.17-62Decision Tree AnalysisNotes:The 2005 CF includes the cost of the project if
