Valuing Real Options in Incomplete Markets

Transcription

1 Valuig Real Optios i Icomplete Markets Bert De Reyck, Zeger Degraeve, ad Jae Gustafsso * Lodo Busiess School, Reget s Park, Lodo NW 4SA, Uited Kigdom bdereyck@lodo.edu, zdegraeve@lodo.edu, gustafsso@lodo.edu Tel. +44 () Helsiki Uiversity of Techology, P.O. Box, 25 HUT, Filad ae.gustafsso@hut.fi Tel () February 24 * Correspodig author. This paper was writte durig the author s visit to Lodo Busiess School from the Systems Aalysis Laboratory of Helsiki Uiversity of Techology. Grats from the Fiish Cultural Foudatio, the Foudatio of Jey ad Atti Wihuri, ad the Foudatio of Emil Aaltoe are gratefully ackowledged. The authors are listed i alphabetical order. i

2 Valuig Real Optios i Icomplete Markets Bert De Reyck, Zeger Degraeve, ad Jae Gustafsso Abstract I this paper, we develop a framework for valuig real optios ad portfolios of real optios i icomplete markets ad show that it is a cosistet geeralizatio of cotiget claims aalysis, which is covetioally used for real optio valuatio i complete markets. The developmet of a framework for icomplete markets is motivated by the difficulty to costruct replicatig portfolios i practice, especially for proects that lead to iovative ew products, which seldom share similarities with existig market-traded assets. The framework relies o (i) a decisio-tree-based mixed asset portfolio selectio model, which is able to capture maagerial flexibility ad relevat opportuity costs, ad (ii) the cocepts of opportuity buyig ad sellig prices, which are exteded from the otios of breakeve buyig ad sellig prices. The use of a portfolio model is ecessary, because i icomplete markets the value of a real optio depeds o the ivestor s preferece model, the available budget, ad o other assets i the portfolio, icludig the real optios they cotai. We demostrate the use of the model through a series of umerical experimets ad compare the results to Capital Asset Pricig Model prices ad Black-Scholes values. Keywords Real Optio Valuatio, Icomplete Markets, Mixed Asset Portfolio Selectio, Capital Budgetig ii

3 2/2/24 :44 AM Itroductio The curret literature o proect valuatio ackowledges that maagemet s flexibility to adapt later decisios to uexpected future developmets ca substatially raise the value of proects (Dixit ad Pidyck 994, Trigeorgis 996, Copelad ad Atikarov 2). This operatioal flexibility shares similarities with fiacial optios, ad cosequetly the term real optio has bee adapted to refer to proect flexibility. For example, possibilities to expad productio whe the markets are up, to abado a proect uder bad market coditios, ad to switch operatios to alterative productio facilities ca be see as optios embedded i a proect. A real optio ca substatially add value to a proect. However, the acquisitio of additioal flexibility may icur a cost as well. Therefore, the ability to aalyze ad value real optios is of crucial importace to achieve a optimal allocatio of resources. I the literature, optios pricig aalysis, or cotiget claims aalysis (CCA; Black ad Scholes 973, Merto 973, Lueberger 998, Hull 999), is typically used to value real optios. This approach is, however, limited by the assumptio that markets are complete, i.e., that the cash flows of each proect ca be replicated usig market-traded securities. Such a replicatio ca be difficult for research ad developmet (R&D) proects that produce iovative ew products, because by ature these products are ofte dissimilar to existig market-traded assets. Ackowledgig this limitatio, researchers have developed methods capable of valuig proects whe proect cash flows caot be replicated (e.g., Smith ad Nau 995; De Reyck, Degraeve, ad Gustafsso 23; De Reyck, Degraeve, ad Vadeborre 23). De Reyck, Degraeve, ad Gustafsso (23), hereafter referred to as DDG, propose a framework for valuig proects i a settig where the ivestor ca ivest i a portfolio of proects as well as i securities i fiacial markets, but where the proect cash flows caot ecessarily be replicated usig securities. I this paper, we exted this framework to proects that iclude real optios ad show how the framework ca be used to value flexibility ad real optios. The methodology is applicable regardless of whether markets are complete or icomplete, ad it ca be used to value sigle real optios as well as portfolios of iteractig real optios. It also supports a large variety of preferece models, from expected utility theory to mea-risk models. We show that this framework is a cosistet geeralizatio of CCA ad demostrate the use of the framework through a series of umerical experimets. The remaider of this paper is structured as follows. Sectio 2 reviews the basic methodology used i real optios valuatio. Sectio 3 presets the mixed asset portfolio selectio model that forms the basis of the valuatio procedure developed i Sectios 4, 5, ad 6. Sectio 7 applies the procedure i umerical experimets, ad Sectio 8 draws coclusios.

4 2/2/24 :44 AM 2 Review of Real Optio Valuatio A optio is the right, but ot the obligatio, to take a actio at a predetermied date i the future, at a predetermied cost, upo the revelatio of specific iformatio. Maagemet s ability to adapt the way i which a proect is carried out ca be see as a collectio of such optios. The literature gives several examples of these real optios; the most commo types are summarized i Table. Table. Some commo real optios. Real optio Descriptio Ucertaity Later decisio Strike price Optio to defer Wait ad make the decisio after the outcome of ucertaity is kow. Ay Ay Immediate cost of the actio take Time-to-build optio Optio to alter operatig scale Optio to abado Growth optio Staged ivestmet. Possibility to start a ew stage of the proect at a cost after the curret stage has fiished. Ability to adust operatig scale at a cost later o. Possibility to termiate the proect ad obtai a salvage value. A ivestmet possibly opeig up ew ivestmet opportuities later o. Outcome of the curret proect stage Ay Startig of a ew stage of the proect Chage of operatig scale (e.g. expad or shrik productio) Ivestmet cost of the ew stage Immediate cost imposed by chage of operatig scale Ay Proect termiatio Salvage value Emergece of a ew proect idea Ivestmet i a ew proect Cost of a ew proect The logic behid real optios ca be described usig decisio trees (Hespos ad Strassma 965, Raiffa 968, Frech 986, Cleme 996), which defie the poits at which decisios ca be take ad the way i which they are related to ufoldig ucertaities (see, e.g., Faulker 996 ad Smith ad Nau 995). Take together, the decisio opportuities i a decisio tree defie the available proect maagemet strategies ad cosequetly the possible cash flow streams that ca be obtaied with the proect. Whe the appropriate risk-adusted discout rate is kow, the et preset value (NPV) method ca be employed to derive the preset value of ay risky cash flow stream, icludig the oes obtaied from decisio trees uder each possible proect maagemet strategy. However, the selectio of a appropriate discout rate is problematic, because the rate depeds o (a) the risk of the proect, which is iflueced by the correlatio of the cash flows with other ivestmets i the portfolio, ad o (b) the opportuity costs imposed by alterative ivestmet opportuities. Several methods for calculatig discout rates have bee proposed, most otably (i) the weighted average cost of capital (WACC; Solomo 963, Brealey ad Myers 2), (ii) expected utility theory (vo Neuma ad Morgester 947), (iii) Capital Asset Pricig Model-based proect valuatio (De 2

5 2/2/24 :44 AM Reyck, Degraeve, ad Vadeborre 23), ad (iv) CCA (see, e.g., Merto 973 ad Lueberger 998). These methods are, however, applicable oly i specific settigs. First, WACC is appropriate oly for average-risk proects i the compay. Secod, expected utility theory, which ca be used to determie a risk-adustmet to the discout rate, values proects i isolatio from other ivestmet opportuities ad hece fails to accout for opportuity costs imposed by other proects ad securities i fiacial markets. Third, a CAPM-based discout rate requires that the CAPM assumptios (Sharpe 964, Liter 965) hold, which may be problematic whe the ivestor is a expected utility maximizer rather tha a meavariace optimizer. Also, the results from DDG idicate that proect valuatios based o the CAPM may ot be accurate, because lumpy proects are ot icluded i the derivatio of the CAPM market equilibrium. Fourth, CCA relies o the assumptio that it is always possible to replicate the cash flows of a proect i each possible state of ature usig market-traded securities. I practice, it ca be difficult to costruct replicatig portfolios for private proects. The limitatios of CCA have raised a eed for a methodology that is applicable to real optio valuatio whe markets are icomplete. Smith ad Nau (995) propose a model for valuig sigle proects which explicitly accouts for security tradig ad maagerial flexibility. However, it does ot provide direct guidelies for valuig real optios embedded i a proect or for icorporatig portfolio effects. Usig a portfolio selectio model, DDG itroduce a valuatio method that is applicable whe the ivestor ca ivest i a portfolio of private proects as well as i securities i fiacial markets. It does ot, however, cosider the existece of real optios. I the followig, we exted this model to capture maagerial flexibility ad show how it ca be used to value real optios i icomplete markets. 3 Portfolio Model Whe markets are complete, real optio values are uiquely defied for each asset. I icomplete markets, however, the values deped o several aspects of the valuatio cotext. The mai factors iclude (i) the ivestor s preferece model, (ii) the available budget, (iii) other assets i the portfolio, which ifluece the aggregate portfolio risk, ad (iv) alterative ivestmet opportuities, which impose opportuity costs. Therefore, it is ecessary to use a portfolio model i the valuatio of real optios i icomplete markets. Followig DDG, we model the valuatio settig as a two-period mixed asset portfolio selectio (MAPS) problem ivolvig two kids of assets: proects ad securities. As opposed to DDG, where the ivestor chooses oly betwee startig ad ot startig proects, the preset model allows proects to be maaged accordig to several proect maagemet strategies. These strategies ca be modeled by associatig a decisio tree with each proect, describig the ivestor s decisios at times ad, ad their relatioship 3

6 2/2/24 :44 AM to the ucertaity about the state at time (see Figure ). Securities ca be bought i ay, possibly fractioal quatities, ad their prices are determied by the CAPM. Actio State Actio 2... AActio AAAA k,, Actio Actio State 2... Actio 2... AActio AAAA k,,2 Actio State l Actio 2... AActio AAAA k,, l Proect k Actio 2... AActio AA Ak + + Time Figure. A geeric decisio tree for proect k i the MAPS model. deotes a collapsed brach. A decisio ode ca be modeled by associatig a zero-oe variable with each actio i the ode ad by requirig that oly oe of them ca be equal to oe at a time (Gustafsso ad Salo 23). I the absece of further ucertaity, optimal actios i the last time period are trivial: a ratioal ivestor always chooses the actio that yields maximum cash flow. Hece, we ca omit decisio variables for time period ad assume that the ivestor chooses the actio with maximum value. 4

7 2/2/24 :44 AM Let there be securities, m proects, ad l states of ature. Proects are flexible i the sese that, for each a,..., A at time. Give the earlier actio proect k, the ivestor ca choose oe of the actios k { k } a k ad the realized state s at time, the ivestor ca the choose oe of the actios a {,..., A } at time (see Figure ). Prices of securities are deoted with k, ak, s k, ak, s S i, i =,...,, ad the amouts of securities i the portfolio by x, i =,...,. The -th security is the risk-free asset. The radom cash flow of security i at time is S i. i The ivestmet cost of proect k whe actio a is selected is deoted by C, ad the correspodig k radom cash flow at time by C. The cash flow of proect k at time, whe (i) the ivestor chooses actio ak {,..., Ak } a k a, k at time, (ii) state s realizes, ad (iii) the ivestor takes actio {,..., A } at time, is deoted by ( ), k, a k, a, s k, a, s k, a, s k k k k k, ak C s a. Sice we kow that the ivestor chooses the actio with maximum value at time, radom variable C ca be expressed by takig maximum k, ak s,..., l. That is, the outcome of C i state s is over the actios at time i each state { } { ( ) } C ( s) = max C s, a a =,..., A. A biary variable z idicates whether actio a k, a k, a k, a, s k, a, s k, a, s k k k k k {,..., Ak } of proect k is selected or ot. To simplify the otatio, let us deote k k a, k k a, k a = a k. Several types of prefereces over risky mixed asset portfolios ca be expressed usig a preferece fuctioal U (cf. DDG). For example, uder expected utility theory a ivestor s prefereces are give by U[X] = E[u(X)], where u is the ivestor s vo Neuma-Morgester utility fuctio. Usig the fuctioal U, a MAPS model maximizig the ivestor s termial wealth ca be formulated as follows: x, z +,, = = = maxu S x C z i i k a k a i k a () subect to i i k, a k, a i= k= a= A k a= S x + C z = B z = k =,..., m k, a { } zk, a, a =,..., Ak k =,..., m x free i =,...,, i (2) (3) where B is the budget. Here, Equatio (2) is the budget costrait ad Equatio (3) esures that oly oe actio ca be selected at each decisio poit. The expressio iside U i () is the ivestor s (radom) termial wealth level. 5

8 2/2/24 :44 AM The mea-risk versio of the MAPS model is expressed slightly differetly. Let ρ be the ivestor s risk measure ad R the risk tolerace, the maximum acceptable level of risk. The mea-risk variat is the obtaied by (i) replacig the preferece fuctioal by the expectatio operator, resultig i the obective fuctio x, z i i + k, a k, a i= k = a= max E S x C z ad (ii) addig the risk costrait for the termial wealth level ρ i i k, a k, a i= k = a= S x + C z R., 4 Aalyzig Real Optio Values The value of a real optio ca be determied by solvig the portfolio selectio problems where the aalyzed proect cotais ad does ot cotai the real optio (see Figure 2). The case without the real optio correspods to the traditioal NPV case, where the proect has oly a sigle strategy, to which the ivestor commits herself for the duratio of the proect. With the real optio, the ivestor may opt to make a decisio later, usig additioal iformatio ad a possible ew strategy give by the real optio. Optio Value Basic Proect (traditioal NPV) Basic Proect with Optio to Defer Proect with Alterative Strategies Real Optio Value Proect with Real Optios Strategy Value Figure 2. Real optio value ad its two compoets. Sice a real optio gives the right to take a actio after the revelatio of iformatio (Howard 996), there are two elemets i a real optio that yield value. First, by providig a additioal strategy for the proect, such as the possibility to expad productio, a real optio yields strategy value. Secod, by allowig the ivestor to choose the proect maagemet strategy later, whe the ucertaity has bee resolved, the real optio yields optio value. Optio value, or the value of waitig, is aalogous to the 6

9 2/2/24 :44 AM expected value of perfect iformatio (EVPI, Raiffa 968). A real optio that does ot yield a ew strategy is a optio to defer, cotaiig oly optio value. The two compoets of real optio value are illustrated i Figure 2. Movig dow the vertical axis implies that the ivestor ca wait util the ucertaity is resolved ad the make a decisio, for example, about startig the proect. Movig horizotally to the right meas that the ivestor is able to implemet additioal strategies. Strategy value ad optio value embedded i a real optio ca be calculated by comparig the settigs o the horizotal axis ad o the vertical axis, respectively. 5 Buyig ad Sellig Prices The value of a real optio ca be defied as the lowest price at which a ratioal ivestor would be willig to sell the real optio if he or she held it, ad as the highest price at which he or she would be willig to buy the real optio if he or she did ot hold it. The cocepts of breakeve sellig ad buyig prices have bee used to calculate aalogous values for proects (Lueberger 998, Smith ad Nau 995). DDG formulate portfolio selectio problems for computig these prices for proects that do ot cotai real optios. Usig the model preseted i Sectio 3, we exted this formulatio to proects with real optios. The resultig optimizatio problems for mea-risk optimizers are give i Table 2. Here, the actio associated with ot startig the proect is idexed with. I Table 2, breakeve sellig ad buyig prices are determied by the amout of moey that makes the ivestor idifferet betwee the situatios where he/she starts ad does ot start the proect. With the breakeve sellig price, this sum is obtaied by first solvig the portfolio selectio problem where the ivestor starts the proect ad the solvig aother optimizatio problem where the ivestor does ot start the proect, but where a amout v, the sellig price, is added to the budget istead. The latter problem is optimized iteratively by varyig s s v util its optimal value matches that of the first portfolio selectio problem. The breakeve buyig price is computed similarly except that the ivestor does ot start the proect i the status quo ad v, the buyig price, is subtracted from the budget i the secod settig. b 7

10 V ' M N P T PR U D F J FH & U O Q S Q & E G I G $ C h _ ` b f bd A B g W Y Y ^ B g a c e c ^ 3 7 X Z Z 2/2/24 :44 AM Table 2. The defiitio of the value of a flexible proect for mea-risk optimizers. Defiitio Status quo Secod settig Breakeve sellig price + v such that W = W s Ivestor ivests i the proect x, z % %% i i k, a k, a i k a + W = max E S x + C z = = = subect to '(' Si xi + Ck, azk, a = B i= k= a= ) * + / //, ρ + - S x + C z., R A k a= i i k, a k, a i= k = a= z = k =,..., m k, a, " " (.) (.2) (.3) (.4) z =, i.e. start the proect (.5) { } zk, a, a =,..., Ak, k =,..., m (.6) x free i =,..., (.7) i Ivestor does ot ivest i the proect JKJ i i k, a k, a x, z L L i= k = a= W = max E S x + C z subect to MM s Si xi + Ck, azk, a = B + v i= k= a= ρ A k a= TT i i k, a k, a i= k = a= k, a S x + C z R z = k =,..., m, (2.) (2.2) (2.3) (2.4) z =, i.e. do ot start the proect (2.5) { } zk, a, a =,..., Ak, k =,..., m (2.6) x free i =,..., (2.7) i Breakeve buyig price + v such that W = W b Ivestor does ot ivest i the proect x, z i i k, a k, a i= k = a= W = max E S x + C z subect to : :: Si xi + Ck, azk, a = B i= k= a= ; < = AA > ρ =? S x + C > R A k a= i i k, a k, a i= k = a= z = k =,..., m k, a, 5 5 (3.) (3.2) (3.3) (3.4) z =, i.e. do ot start the proect (3.5) { } zk, a, a =,..., Ak, k =,..., m (3.6) x free i =,..., (3.7) i Ivestor ivests i the proect x, z ] ]] [ \ i i k, a k, a i= k = a= + W = max E S x + C z subect to b Si xi + Ck, azk, a = B v i= k = a= ρ A k a= ff i i k, a k, a i= k = a= k, a S x + C z R z = k =,..., m, (4.) (4.2) (4.3) (4.4) z =, i.e. start the proect (4.5) { } zk, a, a =,..., Ak, k =,..., m (4.6) x free i =,..., (4.7) i Note that breakeve prices ca be egative, sice they iclude a obligatio to start the proect if it is acquired. A egative breakeve price implies that the ivestor would lose moey by startig the proect. Sice a optio etails the right but ot the obligatio to take a actio, for valuig real optios we eed sellig ad buyig prices that rely o comparig the situatios where the ivestor ca ad caot ivest i the proect (istead of does ad does ot). The lowest price at which the ivestor would be willig to sell a opportuity to start a proect ca be obtaied from the defiitio of the breakeve sellig price by removig the requiremet to start the proect i the status quo, i.e. by removig Equatio (.5) i Table 8

11 2/2/24 :44 AM 2. We defie this price as the opportuity sellig price of the proect. Likewise, the opportuity buyig price of a proect ca be obtaied be removig the startig requiremet i the secod settig, i.e. Equatio (4.5) i Table 2. It is the highest price that the ivestor is willig to pay for a licese to start the proect. Opportuity sellig ad buyig prices have a lower boud of zero, ad they ca be computed by takig a maximum of ad the respective breakeve price, as show by the followig propositio. The proof is give i Appedix A. PROPOSITION. The opportuity sellig (buyig) price of a proect is the maximum of ad the breakeve sellig (buyig) price of the proect. Breakeve sellig ad buyig prices, as well as opportuity sellig ad buyig prices, are ot, i geeral, equal to each other. While this discrepacy is accepted as a geeral property of risk prefereces i expected utility theory (Raiffa 968), it also seems to cotradict the ratioality of these valuatio cocepts. It ca be argued that if the ivestor were willig to sell the proect at a lower price tha at which he/she would be prepared to buy it, the ivestor would create a arbitrage opportuity ad lose a ifiite amout of moey whe aother ivestor repeatedly bought the proect at its sellig price ad sold it back at the buyig price. I a reverse situatio where the ivestor s sellig price for a proect is greater tha the respective buyig price, the ivestor would be irratioal i the sese that he/she would ot take advatage of a arbitrage opportuity if such a opportuity existed where it would be possible to repeatedly buy the proect at the ivestor s buyig price ad sell it at a slightly higher price below the breakeve sellig price. However, these argumets eglect the fact that the breakeve prices are affected by the budget ad that therefore these prices may chage after obtaiig the proect s sellig price ad after payig its buyig price. Ideed, it ca be show that i a sequetial settig where the ivestor first sells the proect (addig the sellig price to the budget) ad the buys the proect back, the ivestor s sellig price ad the respective (sequetial) buyig price are always equal to each other. This observatio is formalized as the followig propositio ad it holds for ay preferece model accommodated by the portfolio models i Sectio 3. The proof is give i Appedix A. Table 3 summarizes the differeces betwee breakeve, opportuity, ad sequetial buyig ad sellig prices. PROPOSITION 2. The ivestor s breakeve sellig (buyig) price ad sequetial buyig (sellig) price are equal to each other. 9

12 2/2/24 :44 AM Table 3. A summary of a ivestor s buyig ad sellig prices. Valuatio cocept Idea Differece from Properties breakeve prices Breakeve buyig ad sellig prices Compariso of settigs where the ivestor does ad does ot - Breakeve buyig price ad sellig price are i geeral Opportuity buyig ad sellig prices Sequetial buyig ad sellig prices ivest i the proect. Compariso of settigs where the ivestor ca ad caot ivest i the proect. A settig where the ivestor first sells (buys) the proect at its breakeve sellig (buyig) price ad the buys (sells) it back at the prevailig breakeve buyig (sellig) price. The latter price is the sequetial buyig (sellig) price. The ivestor is ot obliged to ivest i the proect. The budget used to calculate the sequetial buyig (sellig) price is the origial budget plus the sellig price (mius the buyig price). differet from each other. Lower boud of. Equal to maximum of ad the respective breakeve price. Breakeve sellig (buyig) price ad the sequetial buyig (sellig) price are equal to each other. 6 Valuig Real Optios with Opportuity Buyig ad Sellig Prices The value of a real optio ad a portfolio of real optios ca be determied usig opportuity buyig ad sellig prices ad a portfolio selectio model with three actios at time, (i) Do ot start the proect, (ii) Start the proect without the real optio(s), ad (iii) Start the proect with the real optio(s). Note that whe the proect ivolves real optios other tha the oe(s) beig valued, actio 2 correspods to a variat of the proect that cotais all the real optios except the oe(s) beig valued. This settig ca be exteded to capture proects that have several alterative strategies at time by itroducig a ew actio for each additioal time- strategy. Let us assume that the proect is idexed with ad that the biary decisio variables associated with the actios are z,, z,2, ad z,3, respectively. Whe the ivestor is a mea-risk optimizer, the opportuity sellig ad buyig prices for the real optio(s) ca be obtaied from Table 2 by (i) removig costraits (.5) ad (4.5), implyig that we are usig opportuity prices istead of breakeve prices, ad (ii) replacig costraits (2.5) ad (3.5) with the costrait z,3 =, meaig that the ivestor caot use the valued real optio(s) i the respective settigs. The value of a real optio depeds o whether the proect cotais other real optios or ot. The value obtaied from opportuity sellig ad buyig prices is, i geeral, the real optio s added value, which is determied relative to the situatio without the optio. Note that the proect without the real optio may still cotai other real optios. Whe there are o other real optios i the proect, or whe we remove them from the model before determiig the value of the real optio, we speak of the real optio s isolated value. We defie the respective values for a portfolio of real optios as the oit added value ad oit value (see Figure 3).

13 2/2/24 :44 AM Number of real optios beig valued Sigle real optio Portfolio of real optios Isolated Value Added Value Joit Value Joit Added Value No additioal real optios Additioal real optios Number of other real optios i the proect Figure 3. Differet types of valuatios for real optios. Whe several real optios are available i a proect, isolated real optio values are, i geeral, oadditive; they do ot sum up to the value of the portfolio composed of the same real optios. However, i a sequetial settig where the ivestor buys the real optios oe after the other usig the prevailig buyig price at each time, the obtaied real optio prices do add up to the oit value of the real optio portfolio. These prices are the real optios added values i a sequetial buyig process, where the budget is reduced by the buyig price after each step. We refer to these values as sequetial added values. This sequetial additivity property holds regardless of the order i which the real optios are bought. Idividual real optios ca, however, acquire differet added values depedig o the sequece i which they are bought. These observatios are prove by the followig propositio. The proof is i Appedix A. PROPOSITION 3. The opportuity buyig (sellig) prices of sequetially bought (sold) real optios add up to the opportuity buyig (sellig) price of the portfolio of the real optios regardless of the order i which the real optios are bought (sold). Aother importat property of opportuity sellig ad buyig prices for real optios is that, whe markets are complete, the prices are equal to each other ad give the same result as CCA. The CCA price for the real optio(s) is give by max v CCA, v CCA, v CCA max v CCA, v CCA, {,,2,3 } {,,2 } CCA where v, a idicates the CCA value for actio a of proect, i.e. CCA, a, a i i i= v = C + i S x, where x i is the amout of security i i the replicatig portfolio. The cosistecy property, formalized i Propositio 4, holds for all stadard preferece models captured by the portfolio models i Sectio 3. The proof is i

14 2/2/24 :44 AM Appedix A. PROPOSITION 4. Whe markets are complete, opportuity sellig ad buyig prices for a real optio are equal to its optio pricig value. This property is particularly sigificat, because it idicates that the valuatio procedure usig opportuity sellig ad buyig prices is a geeralizatio of the covetioal real optios valuatio procedure based o CCA. 7 Numerical Experimets I the previous sectios, we have developed a procedure for valuig real optios ad portfolios of real optios i icomplete markets ad showed that this procedure is a cosistet geeralizatio of CCA. I this sectio, we demostrate the valuatio procedure through a series of umerical experimets ad cast light o the followig issues: Q. How are opportuity buyig ad sellig prices for real optios related to CAPM market prices ad Black-Scholes prices (Black ad Scholes 973, Merto 973)? Q2. Whe are opportuity buyig ad sellig prices cosistet with CCA i icomplete markets? Q3. How are opportuity buyig ad sellig prices iflueced by the degree of risk aversio? Q4. How is the value of a real optio related to optio value ad strategy value? Q5. Whe are real optio values additive? 7. Experimetal Setup The experimetal setup cosists of eight equally likely states of ature, three proects, ad two securities costitutig the market. The prices ad cash flows associated with each of the securities ad proects are give i Tables 4 ad 5. The umbers i italic are computed values. The expected rate of retur of the market portfolio is set to 5.33% so that the CAPM price of security 2 is $2, the price used by Smith ad Nau (995). The stadard deviatio of the market portfolio is 35.32%. I the base case, the proects have oly oe possible proect maagemet strategy yieldig the cash flows i Table 5. This case correspods to the traditioal NPV valuatio. 2

15 2/2/24 :44 AM Table 4. Securities. Security 2 Shares issued 5,,,, State $6 $36 State 2 $5 $36 State 3 $4 $2 State 4 $3 $2 State 5 $6 $36 State 6 $5 $36 State 7 $4 $2 State 8 $3 $2 Beta Market price $39.56 $2. Capitalizatio weight 74.79% 25.2% Expected rate of retur 3.76% 2.% St. dev. of rate of retur 28.26% 6.% Table 5. Proects with the basic proect maagemet strategy. Proect A B C Ivestmet cost $8 $ $4 State $5 $4 $8 State 2 $5 $4 $8 State 3 $5 $5 $6 State 4 $5 $ $6 State 5 $5 $7 $8 State 6 $5 $ $8 State 7 $5 $9 $6 State 8 $5 $9 $6 Beta Market price $2.59 $.33 -$4. Expected outcome $ $23.75 $2 St. dev. of outcome $5 $28.26 $6 I Tables 5 8, the market price idicates the CAPM price of each asset if it were traded i fiacial markets, beig ifiitely divisible ad havig a egligible market capitalizatio. Note that proect C is the oe used by Smith ad Nau (995) ad Trigeorgis (996). The risk-free iterest rate is 8%. Table 6 describes several alterative strategies for the proects ad the associated cash flows. We assume that, i additio to the basic strategy, proect A ca be abadoed (idicated by a), yieldig the ivestmet cost back at time, or expaded (deoted by e), or both. The expasio strategy doubles the proect output at a cost of $2, which is paid at time. The abadomet strategy is iheretly uprofitable ad is icluded here oly for the sake of compariso with proects ivolvig real optios 3

16 2/2/24 :44 AM (the abadomet optio will be profitable). Proect B ca be abadoed ad expaded at a cost of $. Proect C ca be expaded at a cost of $2. Table 6. Proects uder alterative proect maagemet strategies. Proect A/a A/e B/a B/e C/e Ivestmet cost $8 $8 $ $ $4 State $8 $8 $ $8 $24 State 2 $8 $8 $ $8 $24 State 3 $8 $8 $ $2 $ State 4 $8 $8 $ $2 $ State 5 $8 -$2 $ $24 $24 State 6 $8 -$2 $ $ $24 State 7 $8 -$2 $ $8 $ State 8 $8 -$2 $ $8 $ Beta Market price -$5.93 -$5.93 -$7.4 $3.7 -$5. Expected outcome $8 $8 $ $47.5 $2 St. dev. of outcome $ $ $ $56.5 $2 The cash flows of the proects havig oe real optio, whe the optimal strategy is chose i each state, are give i Table 7. A asterisk (*) idicates that the proect cotais a real optio, i.e. that the proect has a additioal strategy, as well as the optio to defer the decisio to implemet that strategy. I order to be able to compare the results with Trigeorgis (996), we also examie proect C with a optio to defer (deoted by d). The ivestmet cost whe deferred is.8 $4 = $2.32. Optimizatios i the followig were carried out by usig the discrete time formulatio i Appedix B ad the GAMS software package. Table 7. Proects with oe real optio. Proect A*a A*e B*a B*e C*e C*d Ivestmet cost $8 $8 $ $ $4 $. State $5 $8 $4 $8 $24 $67.68 State 2 $5 $8 $4 $8 $24 $67.68 State 3 $5 $8 $5 $2 $6 $. State 4 $5 $8 $ $2 $6 $. State 5 $8 $5 $7 $24 $24 $67.68 State 6 $8 $5 $ $ $24 $67.68 State 7 $8 $5 $ $9 $6 $. State 8 $8 $5 $ $9 $6 $. Beta Market price $26.48 $26.48 $4. $32.84 $8.22 $25.7 Expected outcome $5 $5 $26.25 $5 $5 $33.84 St. dev. of outcome $35 $65 $25.46 $56.32 $9 $

17 2/2/24 :44 AM 7.2 Sigle Real Optios Let us first examie the case where each proect cotais oe real optio. Whe valuig such a optio, we have to bear i mid that real optios preset i other proects ca ifluece its value, because real optios ca alter the correlatio betwee proects as well as the opportuity costs. I this experimet, we assume that proects A ad B have a expasio optio ad proect C a deferral optio. Table 8 describes the additioal cash flows that the optios i Table 7 yield compared to the basic proects, as well as the associated CAPM market prices. The cash flows for proect C s deferral optio have bee calculated by movig the ivestmet cost of the basic proect to time by icreasig it with the risk-free iterest rate. Observe that the CAPM market prices for optios o proect B satisfy the put-call parity (Merto 973, Hull 999): a portfolio cosistig of a expasio optio, a shorted abadomet optio, ad cash equal to $ /.8 = $92.59 is worth $.33, the price of the uderlyig asset. Table 8. Additioal cash flows from real optios for basic proects. Real Optio a o A e o A a o B e o B e o C d o C State $ $3 $ $4 $6 $ State 2 $ $3 $ $4 $6 $ State 3 $ $3 $ $5 $ $52.32 State 4 $ $3 $ $ $ $52.32 State 5 $3 $ $ $7 $6 $ State 6 $3 $ $ $ $6 $ State 7 $3 $ $ $ $ $52.32 State 8 $3 $ $ $ $ $52.32 Beta Market price $3.89 $3.89 $2.78 $2.5 $22.22 $29.7 Expected outcome $5 $5 $2.5 $26.25 $3 $26.6 St. dev. of outcome $5 $5 $4.33 $25.46 $3 $26.6 Type of optio Put Call Put Call Call Put Price of uderlyig asset $92.59 $92.59 $.33 $.33 $ $ Strike price $8 $2 $ $ $2 $2.32 Volatility 54% 54% 25.38% 25.38% 6% 6% Black-Scholes price $.5 $3.7 $3.7 $22.45 $9.82 $26.22 Table 8 also gives the optio values usig the Black-Scholes call optio formula (Black ad Scholes 973, Lueberger 998, Hull 999). Put optio values have bee obtaied by solvig the Black-Scholes value for the correspodig call optio ad the usig put-call parity. The prices for the uderlyig assets have bee computed from Table 5 by addig the ivestmet cost to the listed market price. Volatility is calculated accordigly. The resultig values differ from the CAPM market prices, because the Black- Scholes formula models ucertaity usig a geometric Browia motio, assumig a logormal 5

18 2/2/24 :44 AM distributio for returs ad a cotiuous-time framework. While these assumptios may be realistic for market-traded assets, they may ot be suitable for private proects, which may have oly a few discrete outcome scearios. Also, sice the CAPM assumes that the assets beig valued are ifiitely divisible, the ivestor s opportuity sellig ad buyig prices are more appropriate measures of real optio value. I the followig, we assume that the ivestor is a mea-variace optimizer for the sake of comparability with the CAPM. Uder this preferece model, the opportuity buyig price ad sellig price for ay proect are the same. The budget is set at $5. The mea-variace ivestor s risk tolerace is defied with respect to a percetage of this budget; e.g. a risk tolerace of 25% implies that the maximum allowed stadard deviatio is $25. Table 9 gives the opportuity buyig ad sellig prices for each real optio as a fuctio of the ivestor s risk tolerace. We ca observe that real optio values coverge towards their CAPM market prices as the ivestor s risk tolerace goes to ifiity, except for proect C, where the optios are priced at their market value mius $4. This discrepacy is due to the fact that without the real optio the ivestor does ot start the proect, ad hece the compariso value is $ rather tha $4, the value of proect C. The real optios o proect C are priced costatly to their CCA values due to the existece of a replicatig portfolio, which ca be costructed from security 2 ad the risk-free asset for all cash flow patters yielded by proect C. Table 9. The real optios isolated opportuity sellig ad buyig prices for a mea-variace ivestor. Real optio value Risk level a o A e o A a o B e o B e o C d o C % $. $. $2.96 $8.85 $8.22 $25.7 2% $6.46 $9.6 $3.4 $7.4 $8.22 $25.7 3% $5.25 $2.8 $3.9 $9.79 $8.22 $25.7 4% $4.85 $2.66 $2.99 $2.34 $8.22 $25.7 5% $4.64 $2.94 $2.94 $2.6 $8.22 $25.7 % $4.25 $3.44 $2.86 $2.8 $8.22 $25.7 % $3.92 $3.84 $2.79 $2.47 $8.22 $25.7 % $3.89 $3.88 $2.78 $2.5 $8.22 $25.7 % $3.89 $3.89 $2.78 $2.5 $8.22 $25.7 Except for very low risk levels, the prices of abadomet optios decrease as the ivestor becomes less risk averse. This results from the fact that these optios have a egative correlatio with the rest of the ivestmet portfolio, implyig that they lower the risk of the portfolio, icreasig the value of the optio for risk averse ivestors. Note that the beta of the abadomet optio o A is zero, eve though it has a egative correlatio with the portfolio (especially with proect A itself); this is explaied by the fact that 6

19 2/2/24 :44 AM beta measures the correlatio of the optio with market securities oly rather tha with the etire mixed asset portfolio. Zero real optio values at a % risk level result from the fact that the portfolio with proect A exceeds the allowed risk limit with or without the real optio. The irregularities at a 2% risk level are caused by the chage i the compositio of the ivestor s security portfolio, which is ecessary to reduce the portfolio risk eough to allow the iclusio of the proect ito the portfolio. These irregularities are caused by the lumpy ature of the ivestmet opportuities, ad could play a maor role whe valuig real-life proects. The real optio values i Table 9 are separated ito strategy ad optio values i Tables ad, respectively. We observe that strategy value is zero for all optios except for the expasio optio o B. This is because the basic proect maagemet strategy is more profitable tha the alterative strategy i all cases except for the expasio strategy for proect B. So except for the expasio optio o proect B, it is the optio compoet of the real optios, i.e. the possibility to make the strategy decisio with more iformatio, that creates value, ot merely the strategy cotaied withi the real optio. I Tables 9 ad, we ca also observe that the optio value for the expasio optio of proect B ad the price of the abadomet optio of proect B coverge to the same value. This is because a expasio optio is essetially oly a deferral optio for a proect usig the expasio strategy; i particular, the optio yields the same additioal cash flows for the multi-strategy proect as the abadomet optio yields for the basic proect. Table. Strategy values for the real optios with respect to a basic proect. Strategy value Risk level a o A e o A a o B e o B e o C d o C % $. $. $. $5.47 $. $. 2% $. $. $. $3.3 $. $. 3% $. $. $. $6.59 $. $. 4% $. $. $. $7.28 $. $. 5% $. $. $. $7.62 $. $. % $. $. $. $8.2 $. $. % $. $. $. $8.68 $. $. % $. $. $. $8.73 $. $. % $. $. $. $8.73 $. $. 7

20 2/2/24 :44 AM Table. Optio values for the real optios with respect to a multi-strategy proect. Optio value Risk level a o A e o A a o B e o B e o C d o C % $. $. $2.96 $3.38 $8.22 $25.7 2% $6.46 $9.6 $3.4 $4.9 $8.22 $25.7 3% $5.25 $2.8 $3.9 $3.2 $8.22 $25.7 4% $4.85 $2.66 $2.99 $3.6 $8.22 $25.7 5% $4.64 $2.94 $2.94 $2.99 $8.22 $25.7 % $4.25 $3.44 $2.86 $2.88 $8.22 $25.7 % $3.92 $3.84 $2.79 $2.79 $8.22 $25.7 % $3.89 $3.88 $2.78 $2.78 $8.22 $25.7 % $3.89 $3.89 $2.78 $2.78 $8.22 $25.7 I summary, we observed that, due to differet uderlyig assumptios, Black-Scholes prices do ot coicide with CAPM values or opportuity buyig ad sellig prices [Q]. Also, if a replicatig portfolio exists for a proect, its real optios are priced to their CCA values at all risk levels [Q2]. I additio, we observed that, i the preset settig, the opportuity prices for egative-beta optios decrease with icreasig risk tolerace, while those with positive beta icrease. However, we also observed that the values of optios with a zero beta could either icrease or decrease depedig o how the optios were correlated with the proect portfolio. Ideed, this suggests that the covergece behavior is determied by the optio s correlatio with the etire mixed asset portfolio rather tha by its correlatio with market securities oly [Q3]. I each case, as log as the value of the basic proect was positive, the values of real optios coverged to their CAPM market prices [Q]. At low risk levels, the lumpy ature of the ivestmet opportuities sometimes resulted i erratic optio values [Q3]. Fially, the results showed that the value of a real optio ca origiate either from strategy value or from optio value [Q4]. For example, the value of the expasio optio o proect C was maily due to its optio compoet. I cotrast, the primary source of value for proect B s optio to expad was the ew expasio strategy, ot the optio compoet. This last poit is importat sice i practice, real optios are sometimes praised for the extra value they create, although it could actually be the strategic compoet cotaied i the real optio that creates value, ot the fact that you have a optio o implemetig that strategy. 7.3 Real Optio Portfolios We will ow cosider the case where each proect icludes all available real optios. The resultig cash flows ad CAPM market prices for the proects are give i colums 2 4 of Table 2. Additioal cash flows from the real optio portfolios with respect to the basic proects are give i colums

21 2/2/24 :44 AM Table 2. Cash flows for proects with two real optios ad for real optio portfolios. Proect Real optio portfolio A*ae B*ae C*de ae o A ae o B de o C Ivestmet cost $8 $ $. N/A N/A N/A State $8 $6 $27.68 $3 $4 $6 State 2 $8 $6 $27.68 $3 $4 $6 State 3 $8 $8 $. $3 $5 $52.32 State 4 $8 $ $. $3 $ $52.32 State 5 $8 $22 $27.68 $3 $7 $6 State 6 $8 $ $27.68 $3 $ $6 State 7 $8 $ $. $3 $ $52.32 State 8 $8 $ $. $3 $ $52.32 Beta Market price $4.37 $26.9 $47.29 $27.78 $24.29 $5.29 Expected outcome $3 $4.25 $63.84 $3 $28.75 $56.6 St. dev. of outcome $5 $42.56 $63.84 $ $23.5 $3.84 Notice that the optio portfolio o A yields a costat cash flow of $3, hece the market price $ Also, observe that (isolated) market prices for the real optios are additive (Table 8). However, this is ot the case i geeral (Trigeorgis 996); additivity stems here from the fact that, i each state, the cash flows of the real optio portfolio are the sum of the cash flows of the idividual optios [Q5]. As a trivial example of oadditivity of market prices, cosider a portfolio of two mutually exclusive expasio optios (e.g., two alterative techologies), oe of which allows the ivestor to double productio at the price of $ ad the other oe does the same at the price of $2. Clearly, the optio with a higher strike price does ot yield additioal value ad hece the market price of the optio portfolio will be equal to the market price of the less expesive optio, eve though both of the optios are valuable i isolatio. Although isolated market prices are geerally o-additive, sequetial added market prices always exhibit additivity because of the liearity of the CAPM. Sequetial prices are obtaied from a sequetial process, where each real optio s additioal cash flows are computed from the situatio that prevails after addig the previous optio. Table 3 shows the opportuity buyig ad sellig prices for the real optio portfolios. Apart from the % risk level, the real optio portfolio o A is priced costatly, because it ca be replicated usig the risk-free asset. The portfolio is priced at $ at a % risk level, because proect B is preferred to A ad a portfolio icludig both proects A ad B violates the risk costrait. The price of the real optio portfolio o C is costat, because of the presece of the twi security. 9

22 2/2/24 :44 AM Table 3. Real optios oit opportuity sellig ad buyig prices for a mea-variace ivestor. Real optio portfolio Risk level ae o A ae o B de o C % $. $22. $ % $27.78 $22.24 $ % $27.78 $23.8 $ % $27.78 $23.5 $ % $27.78 $23.68 $47.29 % $27.78 $23.99 $47.29 % $27.78 $24.26 $47.29 % $27.78 $24.29 $47.29 % $27.78 $24.29 $47.29 As expected, isolated real optio values (Table 9) do ot sum up to the oit values i Table 3. I colums 2 7 of Table 4, we have calculated the real optios added values by comparig the settig where the ivestor has both real optios to the settig where he/she has oly the other, o-valued real optio. However, eve these values, whe coupled with the isolated value of the other real optio from Table 9, do ot exactly add up to the oit values i Table 3, except for proect C. The reaso for this discrepacy is that we have used the budget of $5 to calculate each of the values. To obtai sequetially additive values we have to modify the budget to accommodate the first real optio s buyig or sellig price. For example, whe calculatig the opportuity sellig price of abadomet optio o A at 4% risk level, we would use the budget of $52.66, the origial budget plus the sellig price of the expasio optio (Table 9). The resultig sequetial added values are give i four rightmost colums i Table 4. The added ad sequetial added values for optios o C are the same due to the existece of a replicatig portfolio. Note that values for these optios are ow $4 higher tha the isolated values i Table 9, because the value of the proect with oe real optio is positive ad hece the optios are priced to their market values i Table 8. I summary, real optios isolated opportuity prices, as well as CAPM market prices, are o-additive i geeral. However, both opportuity prices ad CAPM market prices are sequetially additive [Q5]. Table 4. Real optios added opportuity sellig ad buyig prices for a mea-variace ivestor. Added value Sequetial added value Risk level a o A e o A a o B e o B e o C d o C a o A e o A a o B e o B % $. $. $3.25 $9.4 $22.22 $29.7 $. $. $3.25 $9.4 2% $7.7 $.5 $3.39 $9.4 $22.22 $29.7 $8.8 $.32 $4.84 $8.84 3% $5.6 $2.59 $3.8 $2.7 $22.22 $29.7 $5.7 $2.53 $3.39 $2.9 4% $5.7 $2.97 $2.99 $2.56 $22.22 $29.7 $5.2 $2.93 $3.6 $2.5 5% $4.8 $3.7 $2.94 $2.77 $22.22 $29.7 $4.84 $3.4 $3.7 $2.74 % $4.33 $3.54 $2.86 $2.5 $22.22 $29.7 $4.34 $3.53 $2.9 $2.3 % $3.93 $3.85 $2.79 $2.48 $22.22 $29.7 $3.94 $3.86 $2.79 $2.47 % $3.89 $3.89 $2.78 $2.5 $22.22 $29.7 $3.9 $3.89 $2.78 $2.5 % $3.89 $3.89 $2.78 $2.5 $22.22 $29.7 $3.89 $3.89 $2.78 $2.5 2

23 2/2/24 :44 AM 8 Summary ad Coclusios I complete markets, the value of a real optio ca be determied by creatig a replicatig portfolio, which results i a uique value. I icomplete markets, however, a real optio s value depeds o (i) the ivestor s preferece model, implyig that risk eutral, risk averse, ad risk seekig ivestors may value the real optio differetly, (ii) the ivestor s budget, ad (iii) other assets i the portfolio, icludig the real optios they cotai. Therefore, it is ecessary to cosider the ivestor s etire ivestmet portfolio whe valuig real optios i icomplete markets. The valuatio framework preseted i this paper relies o a decisio-tree-based mixed asset portfolio selectio model ad the cocepts of opportuity buyig ad sellig prices, which were exteded from breakeve buyig ad sellig prices. We showed that, whe markets are complete, opportuity buyig ad sellig prices are cosistet with optio pricig values. Also, we showed that these prices are ratioal valuatio measures i the sese that they exhibit (i) sequetial cosistecy, meaig that a real optio s sellig price equals its sequetial buyig price ad vice versa, ad (ii) sequetial additivity, which implies that the sum of values of sequetially bought real optios is equal to the oit value of the portfolio of these real optios. We also highlighted that there are two sources of value i a real optio: optio value ad strategy value, the former of which is aalogous to the expected value of perfect iformatio (Raiffa 968). The latter results from a ew strategy that is cotaied withi the real optio. To examie the sources of value for a real optio, we proposed the examiatio of four types of proects, (i) the basic proect havig oly a sigle strategy, (ii) the multi-strategy proect havig the same strategies as the proect with the real optio(s) but where the strategy is selected at the begiig of the proect, (iii) the basic proect with a optio to defer, ad (iv) the proect with the real optio(s). I our umerical experimets, we compared opportuity buyig ad sellig prices to CAPM market prices ad Black-Scholes prices. Black-Scholes values differ from other valuatios, because the formula assumes geometric Browia motio ad a cotiuous-time framework. I the cotext of private proects, such assumptios may ot be realistic. O the other had, CAPM market prices are typically appropriate oly for market-traded assets that are ifiitely divisible, ad hece they may give biased results if applied to the valuatio of lumpy real optios. The umerical experimets idicate that whe the ivestor is a mea-variace optimizer, real optios 2

24 2/2/24 :44 AM opportuity buyig ad sellig prices coverge towards CAPM market prices as the ivestor s risk tolerace goes to ifiity, provided that the price of the basic proect is o-egative. The results suggest that the prices of real optios with a positive correlatio with the rest of the portfolio grow by risk tolerace, while those with a egative correlatio ca decrease. Hece, CAPM-based real optio valuatio may give biased estimates of real optio value, because beta measures oly correlatio with market-traded securities rather tha with the etire mixed asset portfolio. Some of the umerical results, such as the covergece of opportuity buyig ad sellig prices to CAPM market prices, ca potetially be show to be geeral properties of opportuity sellig ad buyig prices i a mea-variace settig. Also, formal examiatio of the coditios uder which the prices decrease ad icrease by risk tolerace remais as a topic of further research. Refereces Black, F., M. S. Scholes The pricig of optios ad corporate liabilities. Joural of Political Ecoomy Brealey, R., S. Myers. 2. Priciples of Corporate Fiace. McGraw-Hill, New York, NY. Cleme, R. T Makig Hard Decisios A Itroductio to Decisio Aalysis. Duxbury Press, Pacific Grove. Copelad, T., V. Atikarov. 2. Real Optios: A Practitioer s Guide. Texere, New York. De Reyck, B., Z. Degraeve, J. Gustafsso. 23. Proect Valuatio i Mixed Asset Portfolio Selectio. Workig Paper. Lodo Busiess School. De Reyck, B., Z. Degraeve, R. Vadeborre. 23. Proect Optios Valuatio with Net Preset Value ad Decisio Tree Aalysis. Workig Paper. Lodo Busiess School. Dixit, A. K., R. S. Pidyck Ivestmet Uder Ucertaity. Priceto Uiversity Press, Priceto. Frech, S Decisio Theory A Itroductio to the Mathematics of Ratioality. Ellis Horwood Limited, Chichester. Gustafsso, J., A. Salo. 23. Cotiget Portfolio Programmig for the Maagemet of Risky Proects. Workig Paper. Helsiki Uiversity of Techology. Hespos, R. F., P. A. Strassma Stochastic Decisio Trees for the Aalysis of Ivestmet Decisio. Maagemet Sciece () Howard, R. A Optios. I Wise Choices: Decisios, Games ad Negotiatios. Editors: Zeckhauser, R. J., Keeey, R. L. ad Sebeius, J. K. Harvard Busiess School Press, Bosto. Hull, J. C Optios, Futures, ad Other Derivatives. Pretice Hall, New York, NY. Liter, J The Valuatio of Risk Assets ad the Selectio of Risky Ivestmets i Stock Portfolios ad Capital Budgets. Review of Ecoomics ad Statistics 47()