Extended MAD for Real Option Valuation

Transcription

1 Exended MAD for Real Opion Valuaion A Case Sudy of Abandonmen Opion Carol Alexander Xi Chen Charles Ward Absrac This paper exends he markeed asse disclaimer approach for real opion valuaion. In sharp conras o he dominan real opion valuaion ha assumes a sochasic process for an invesmen s capial value, his paper demonsraes he valuaion of a real opion assuming ha cash flow follows a sochasic process. We show ha his mehod is a leas equally effecive and someimes more inuiive. We noe ha, in a discouned cash flow (DCF framework, cerain consrains mus be me, and assuming capial value as a geomeric Brownian moion (GBM is compaible wih simulaneously assuming cash flow as a GBM, We clarify he above argumen wih a simple exbook-sandard case sudy. Key words: real opion, decision making, invesmen opporuniy, geomeric Brownian moion, abandonmen opion, markeed asse disclaimer, change of measure, lease, renal value, marke uiliy, risk olerance, risk aversion Professor of Finance, School of Business, Managemen and Economics, Universiy of Sussex; Phone: +44 ( , c.o.alexander@sussex.ac.uk. PhD Suden in ICMA Cenre, Henley Business School, Universiy of Reading; Phone: +44 ( , x.chen@icmacenre.ac.uk. Professor of Finance, ICMA Cenre, Henley Business School, Universiy of Reading; Phone: +44 ( , c.ward@icmacenre.ac.uk. 1

2 1 Inroducion This paper firs exends he markeed asse disclaimer (MAD approach proposed by Copeland and Anikarov (2001 and show ha his approach can be applied regardless of which assumpion(s abou he gross invesmen value or invesmen cash flows are made. We furher address ha, cerain condiions mus be me when we value a real opion based on a DCF analysis on he base invesmen. We illusrae he exended MAD approach in a numerical case repored by Azevedo-Pereira (2001 in Howell e al. (2001, hereafer referred o he Campeiro case. In paricular, we value a real opion o abandon a real esae lease ha is no associaed wih a marke price. In conras o he original MAD approach which values he abandonmen opion based on assumpions abou he gross lease value evoluion, we ake cash flows (rens as he process ha is driving he uncerainy in he gross lease value. We hen explain how o derive a value for he abandonmen opion ha is consisen wih he value obained under he original MAD approach. Furhermore, when valuing a real opion based on a DCF analysis of he base invesmen, here are several parameers in he valuaion ha we need approximae. An error in he real opion value would be inroduced if hese approximaions are flawed. For insance, irrespecive of wheher i is he invesmen value or cash flow ha is aken as he driving source of uncerainy, we have o change from he physical (P measure for he invesmen valuaion o he risk-neural (Q measure for he real opion valuaion. This means ha we need o derive he risk-neural probabiliy under which he gross invesmen value follows a Q-maringale. The consisen valuaion of he underlying invesmen in he P measure wih he real opion valuaion in he Q measure presens some complexiies which have been ignored by previous papers ha consider he MAD approach. Indeed, Azevedo-Pereira applies a Q measure for he abandonmen opion valuaion ha is no consisen wih he P measure for he lease value.* In he following, Secion 2 briefly describes he sandard MAD approach and hen explains our exension. We show ha he real opion can be valued using his mehod under any assumpion abou he invesmen value or cash flows. In order o numerically illusrae our findings, we inroduce he Campeiro case in Secion 3 and value he real opion consisenly in Secion 4. Secion 5 numerically and inuiively explain he problem involved in oher ways of approximaing (a he risk-adjused discoun rae for cash flows if missing; (b he drif of he gross invesmen value process if assumed o be a GBM; and (c he dividend yield derived from he 2

3 DCF analysis on he base invesmen for he laer valuaion of he aached real opion. Secion 6 concludes. 2 An Exended MAD Approach The original MAD approach comprises he separae valuaion of he base invesmen and he aached real opion; he former employs a convenional DCF analysis, and he laer adops he sandard risk-neural financial opion valuaion echnique. To jusify his approach, CA (2001 argue ha DCF analysis provides he bes esimae of he curren value for he base invesmen when he marke price is unobservable, and ha he aached real opion can be perfecly hedged using he base invesmen, so is valuaion should be carried ou under he risk-neural measure. 1 CA s (2001 approach is limied o heir invesmen cos assumpion, and more imporanly, o he case where daa on he risk-adjused discoun rae for cash flows is known. The DCF analysis esimaes he gross (ne invesmen value a ime 0, and he risk-neural valuaion of he opion requires he evoluion of he gross invesmen value in he risk-neural measure Q. Given ha he poin value of his process under he physical measure P is he discouned sum of all fuure cash flows a ha poin, we would a leas be able o specify he gross invesmen value process under P if we knew he risk-adjused discoun rae for cash flows. However, in pracice, very ofen here are no daa which allow his discoun rae o be obained, and insead, he risk-adjused discoun rae for ne income is known and someimes used as a proxy. 2 We propose an exension of he MAD approach which is more generally applicable. To moivae our ideas we begin wih a brief summary of he original MAD approach which also serves o inroduce some noaion. 2.1 The Original Approach of CA (2001 CA (2001 firs calculae q 0, he gross value of he base invesmen a ime 0 excluding any aached real opion, as he presen value of he sum of all expeced fuure cash flows from 1 The argumen seems widely acceped. For insance, in he asse pricing model by Jagannahan and Wang (1996. Anoher example can be found in enerprise valuaion for mergers and acquisiions (M&A; an enerprise is valued as he porfolio of he asse-in-place and he aached real opions (see Lambrech, 2004.* 2 For insance, in order o value a enerprise, one can download he company-specific weighed average cos of capial (WACC from bloomberg, and use i o discoun he earnings before ineres afer ax (EBIAT; where EBIAT is he ne income and WACC is he risk-adjused discoun rae for ne income. 3

4 he invesmen, each being discouned by is required risk-adjused discoun rae. 3 Hence, in coninuous ime, q 0 = T 0 ( τ E P [x τ ] exp rs x ds dτ, (1 0 where T denoes he invesmen mauriy, x τ is he invesmen cash flow for 0 τ T, and rs x represens he deerminisic risk-adjused discoun rae for he cash flow a ime s, 0 s τ. 4 Regarding he invesmen cos, c 0, CA(2001 assume ha a lump-sum paymen in advance is required. For he real opion valuaion, CA (2001 le q, he gross invesmen value a ime, evolve over ime according o a GBM under he P measure, i.e. dq q = (µ q δ d + σ q dbq, (2 wih boh µ q and σq being deerminisic, δ being he coninuously compounded dividend yield and q 0 given by (1. In his case, he cash flow is generaed from he invesmen as dividends, i.e. x = q δ, wih δ = 1 exp ( δ, (3 where δ is he discreely compounded equivalen of δ. CA (2001 hen change from he P measure o he Q measure, under which he gross invesmen value process (2 becomes dp = (r δ d + σ q p (dbq Q, wih (db q Q = db q + µq δ d, (4 σ q r denoes he deerminisic risk-free rae. For his, hey se δ = EP [x ] E P [q ], (5 where E P [q ] = T ( τ E P [x τ ] exp rs x ds dτ, (6 3 Noe ha, he risk-adjused discoun rae in (1 capures he uncerainy associaed wih only he fuure cash flows and no he invesor s possible abandonmen of hese cash flows. 4 Remark on noaion: we wrie E P [x τ ] := E P 0[x τ ], where E P i [x τ ] denoes he expeced value of x τ a ime i, 0 i τ under he P measure ha is defined for he fuure value of x τ. 4

5 and ( E P [x ] = x 0 exp µ x τ dτ, (7 0 wih µ x being deerminisic. Under his Q measure, hey hen value he real opion. 2.2 The Exended Approach Our firs generalisaion is o replace he lump-sum cos assumpion (c 0 o allow any cos paymen funcion. The oal invesmen cos a ime 0 becomes he sum of all discouned fuure coss, i.e. c 0 = T 0 ( τ E P [k τ ] exp rs k ds dτ, (8 0 where k s denoes he invesmen cos a ime s and rs k is he deerminisic risk-adjused discoun rae for cos a ime s. Noe ha rs k would be a risk-free rae if k s is cerain, and seing k s = 0 for all s > 0 and k 0 = c 0 yields he assumpion in CA s (2001. Le y be he ne income, i.e. y = x k. Hence, he invesmen is allowed o generae ne incomes differen from cash flows, periodically, before is expiry. So he ne invesmen value a ime 0 excluding any aached real opion, is he presen value of he sum of all expeced fuure ne income, i.e. p 0 = T 0 ( τ E P [y τ ] exp rsds y dτ. (9 0 where rs y denoes he deerminisic risk-adjused discoun rae for ne income a ime s. By no arbirage, his value is equal o he gross invesmen value a ime 0 less he oal cos. Hence, when he risk-adjused discoun rae for cash flow is unavailable, we can compue he gross invesmen value as he ne value (9 plus he oal cos (8, insead of using (1, as in CA (2001. Alhough he invesmen value is calculaed under he P measure, for he real opion valuaion, we require a corresponding Q measure under which he gross invesmen value evolves as a maringale over ime. So far we only have (wo alernaive expressions for he invesmen value a ime 0 under he P measure. Firs we model he gross invesmen value a ime under he P measure and hen we find he corresponding Q measure. We can eiher (a assume a cash flow process wih he expecaion (7, and hen he gross invesmen value is a discouned sum of all expeced cash flows in he fuure; or (b follow CA (2001 by assuming a gross invesmen value process which guaranees he expeced cash flows (7. Eiher way, he following relaionship 5

6 beween he gross invesmen value and he cash flow holds, q = x Y, (10 where Y requires knowing he risk-adjused discoun rae for cash flows, i.e. 5 Y = T To see his, we adop assumpion (a, and hence wrie q = T ( τ exp (µ x s rs x ds dτ. (11 ( τ E P [x τ ] exp rs x ds dτ. Now (10 follows from (7. Alernaively, if we use assumpion (b, (10 follows from (3, since by definiion (5 7, δ is equal o he reciprocal of Y. 6 Hence, as long as we know he riskadjused discoun rae for cash flow, we can pin down he gross invesmen value a ime using (10 under any assumpion abou he process for he gross invesmen value, or for he cash flows, no necessarily a GBM. In fac, he gross invesmen value a ime can sill be derived, even when he risk-adjused discoun rae for cash flow is unavailable. For his, noe ha he following condiion mus hold, E P [q ] = E P [p ] + E P [c ], (13 where, E P [p ] = T ( τ T ( τ E P [y τ ] exp rsds y dτ, and E P [c ] = E P [k τ ] exp rs k ds dτ. Therefore, eiher (i Y in (10 akes he alernaive form: Y = T ( τ T ( τ ( τ ] exp (µ x s rs y ds dτ + y,τ [exp rs k ds exp rsds y dτ, 5 Noe ha, many papers se he required risk premium for cash flow, rs x, (or for ne income, rs y equal o he expeced risk premium of he cash flow, µ x s (or of he ne income, µ y s. (See for insance, McDonald and Siegel, 1986.* However, his someimes may no be appropriae. For insance, in enerprise valuaion, he growh rae of EBIAT is no necessarily equal o WACC. 6 Therefore, ( δ = log 1 δ. (12 6

7 ( wih y,τ = kτ 1 x 0 exp µ x τ dτ, and can hence be compued wihou he risk-adjused discoun rae for cash flow; or (ii we ake Y (11 wih he risk-adjused discoun rae for cash 0 flow derived from ha for ne income, i.e. 7 r x = r y (r y rk E P [c ] E P [q ]. (14 Boh alernaives allow us o compue he gross invesmen value a ime when he risk-adjused discoun rae for cash flow is no known bu ha for ne income is known, in conras o CA s approach which can only be applied when he former is available. Now, based on he process for he gross invesmen value (10 in he P measure, we look for an equivalen Q measure. For his, we keep he gross invesmen value a any ime he same under boh measures (according o he law of one price and derive he risk-neural probabiliies under which he expeced gross invesmen value a any ime grows a he risk-free rae over ime because he assumpion is ha his value can be perfecly hedged under he complee marke assumpion.* Thus, ( E Q i [q ] = q i exp (r τ δ τ dτ, 0 i. (15 i Under his Q measure we can hen value he real opion using he sandard risk-neural financial opion valuaion echnique. In he following, we illusrae he exended MAD approach hrough a numerical case sudy. For his, we inroduce he Campeiro case by AP ( The Campeiro Case The Campeiro case concerns an invesor who holds a real esae lease for T years, during which ime he invesor receives regular renal paymens from a enan and has regular mainenance 7 (14 is derived by solving (13 recursively. See Appendix A for furher deails. (14 confirms ha in general, he risk-adjused discoun rae for cash flow canno be used as a proxy for he risk-adjused discoun rae for ne income. An obvious case is when k is consan. Ne income, by definiion, becomes perfecly correlaed wih he cash flow, and he change in cash flow beween ime 1 and 2 is idenical o ha in ne income in he same ime period; however, he relaive changes are differen, and his is he relaive change which deermine he risk-adjused discoun rae. The excepions happen only when he invesmen cos: 1 is close o zero (E P [c ]/E P [q ] 0, or 2 shares he same source of uncerainy wih he ne income (r k r y. In boh cases, r x = r y. Moreover, r x may significanly diverge from r y and ge close o r k for he invesmen whose oal cos is close o is gross value (E P [c ]/E P [q ] 1. Indeed, when he expeced fuure cash flows from he invesmen are only enough o cover he oal cos, he risk involved in he invesmen would be no more han he risk associaed wih he coss, oherwise i would immediaely be abandoned by he holder or unwaned in he marke. 7

8 coss associaed wih he upkeep of he propery. The annual mainenance cos is assumed o be consan, denoed by k, and he uncerainy which drives he lease value and he abandonmen opion value lies only in he amoun of he renal income received during he year. Afer T years he lease expires and he invesor has no furher claim on he propery. A he beginning of each year T. he invesor has a real opion, since he can abandon he lease a no cos, hus saving he annual mainenance cos bu also forgoing he annual ren during years ( + 1,..., T. Clearly he invesor would abandon he lease if all his fuure rens were less han he fuure mainenance coss, oherwise he would make a loss on his lease. We refer o he ren received during year as x, whose process has a drif µ x = % and a volailiy σ x = %. A ime 0, he ren x 0 = Furhermore, he risk-free rae r = 9.531%, and he risk-adjused discoun rae associaed wih he ne income r y = %. 8 In addiion, we call he ren x less he mainenance cos k = 3 he ne income y. In he Campeiro case, he lease is held for 10 years. Noe ha ren and cos are assumed o occur annually in advance so ha T = 9 and we index years using = 0, 1,..., T. 4 Abandonmen Opion Valuaion We assume ha he marke is complee. Now in order o value he opion o abandon he lease, we require a Q measure corresponding o he P measure under which he lease is evaluaed. For he change of measure, we need he gross lease value a ime under he P measure, saring from = 0. Noe ha he risk-adjused discoun rae for ren is unknown. Hence, we compue he gross lease value a ime 0 as he ne value plus he sum of all discouned fuure mainenance coss, i.e. q 0 = p 0 + c 0 = = , (16 where and T T p 0 = E P [y τ ] exp ( r y τ = (x 0 exp (µτ k exp ( r y τ = , τ=0 τ=0 T c 0 = k exp ( rτ = τ=0 8 Alhough, following AP (2001, we employ a discree-ime framework, we remark ha we use coninuousime compounding. The annual ineres raes defined in he Campeiro case are appropriaely re-defined here so ha he effecive annual raes are idenical, for example a 10% annual rae is expressed as 9.531% coninuously compounded. 8

9 To deermine he gross leas value a ime for 0 < T, we firs assume a GBM evoluion of he cash flow: dx x = µ x d + σ x db. (17 Now, we follow he vas majoriy of papers on he MAD approach and presen he above process in discree ime, using he Cox, Ross, and Rubinsein (1979 binomial ree parameerisaion. In he binomial ree for ren, he ime beween successive nodes = 1 since rens and coss occur annually. In each ime period, he underlying can move up by a facor u > 1 or down by anoher facor d < 1, and we denoe he sae a ime by s( = 0, u, d, ud, uu, du,..., uduu ec. In paricular, ( u = exp σ x = 1.38, d = u 1 = 0.725, π x = exp (µx d = 70%, (18 u d where π x denoes he physical ransiion probabiliy of x s( moving up a ime. 9 Hence, we consruc he binomial ree for ren as below, x s(u = x s( u, x s(d = x s( d. Corresponding o his ree, we can build he binomial ree for he gross lease value based on (10 in discree ime, i.e. q s( = x s( Y, wih Y aking he form (11 where he risk-adjused discoun rae for ren is given by ( Noe ha, he value of Y is no limied o he assumpion(s abou he ren or he gross lease value process. Now le us swich o CA s (2001 assumpion (2. In order o build he binomial ree for gross lease value, we need he volailiy of he GBM process (2 o calculae he values of u and d, and ye we only know he volailiy of he ren process. To compue he former from he laer, we use (2 o derive he ren process and hen se he volailiy (which is a funcion of 9 For insance, if he ren moves up a every sep from ime 0 unil he opion expiry year, hen x s(t = x 0u 9 = , where s(t = uuu }{{... uu }. A ime 0, he probabiliy of ren having his value a ime T is πx 9 = 4.035% 9 under he P measure. 10 To build a binomial ree for ne lease value, denoed by p s(, we deduc he oal fuure coss a ime from each value in he gross lease value ree a he same ime. Tha is, p s( = q s( k T exp( r(τ dτ. 9

10 he former equal o he laer. In fac, given (10, no only he ren under (2 also follows a GBM, bu he processes for he ren and for he gross lease value have he same volailiy, i.e. 11 σ q = σx = %. Therefore, regarding he binomial ree for he gross lease value, he values of u and d are idenical o hose in ( We hus consruc he gross lease value ree as follows, q s(u = q s( exp ( δ u, q s(d = q s( exp ( δ d, wih δ given by ( Based on he binomial ree for he gross lease value, we now seek a corresponding Q measure under which he gross lease value fis (15. In discree ime, ha is o derive he risk-neural probabiliy, denoed by π Q, under which he following relaionship holds:14 q s( x s( = E Q s( [ ] qs(+1 exp( r, E Q [ ] ( s( qs(+1 = π Q q s(u + 1 π Q q s(d. where s( + 1 denoes he succeeding saes of s(: s(u and s(d. We solve his equaion for π Q, and hus obain he expression below. π Q = ( qs( x s( exp(r qs(d q s(u q s(d. By bringing in he values in he gross lease value ree, π Q can be quanified. Noe ha, under (17, we can simplify he above formula and presen π Q as a funcion of he risk-adjused discoun rae for ren, π Q = exp(r + rx +1 µ d, (19 u d 11 In Appendix B, we prove he compaibiliy of assumpion (2 and ( Noe ha, he physical ransiion probabiliy of q s( moving up a ime π q, = exp (µq d u d as µ q is ime dependen (see Corresponding o he binomial ree for gross lease value, we can easily consruc a binomial ree for rens using he definiion of ren (3. 14 Noe ha, Azevedo-Pereira made he assumpion (17 bu compued he risk-neural probabiliies wih which he discouned ren (raher han he discouned gross lease value was a maringale. See Secion 5.1 for furher discussion of his imporan poin. π x, 10

11 where π Q is ime dependen, since he risk-adjused discoun rae for ren varies over ime and oher parameers are consans. On he oher hand, when he gross lease value follows a GBM (2, π Q becomes a fixed number: π Q = %. 15 Under hese risk-neural probabiliies, le us now value he abandonmen opion under RNV principle. Irrespecive of wheher we assume (2 or (17, he valuaion process of he abandonmen opion is as follows. We can value i eiher alone or ogeher wih he lease. Inuiively, if he fuure ren is less han he fuure cos, we would abandon he lease, in which case we save he coss bu lose he furher rens; oherwise we do no exercise he opion, hen i would become valueless. Hence, we can presen he pay-off of he opion wih he lease as max{q c, 0} = max{p, 0}, and wihou he lease as max{0, c q } = max{0, p }. Tha is, he opion wih and wihou he lease are analogous o an American call and a pu respecively. To calculae he ne lease value wih he opion a ime 0 under he Q measure, we apply he following backward inducion. { C s( = max y s( + E Q [ ] } s( Cs(+1 exp( r, 0, E Q [ ] ( s( Cs(+1 = π Q C s(u + 1 π Q C s(d, (20 where C s( is he ne value of he lease and he opion in sae s( under he Q measure. This backward inducion sars from he las sep a = 9 wih C s(10 := 0 and goes backwards in ime. 16 The abandonmen opion will have he value P 0 = C 0 p 0. Alernaively, we value he abandonmen opion alone using backward inducion: { P s( = max p s(, E Q [ ] } s( Ps(+1 exp( r, E Q [ ] ( s( Ps(+1 = π Q P s(u + 1 π Q P s(d, 15 Noe ha, under eiher assumpion, he ren process is no a Q-maringale. 16 For insance, C s(9 = max { y s(9, 0 } { [ ] }. I is hen simple o use; C s(8 = max y s(8 + E Q s(8 Cs(9 exp( r, 0 [ ] where E Q s(8 Cs(9 = π Q 8 C s(8u + ( 1 π Q 8 Cs(8d. The res of his backward inducion follows. (21 11

12 where P s( denoes he value of he opion only a sae s( under he Q measure. This backward inducion sars from = 9 wih P s(10 := 0. According o he pu-call pariy (PCP, he above wo valuaion processes are equivalen and hence, under he same assumpion, hey would generae idenical abandonmen opion values (see Appendix C for furher discussion. We presen our calculaion resuls in Table 1 along wih he corresponding assumpions. There is a minor difference beween he opion values under Table 1: A summary of he assumpions (2 and (17 and he corresponding abandonmen opion values. The relaive opion values are in percenage and equal o he abandonmen opion values proporional o he ne lease value (16. Main Assumpion Opion price Relaive opion price (17 x follows a GBM % (2 q follows a GBM % (2 and (17. We believe ha i is only a discreisaion error. Wih more seps in he binomial rees ha we consruc, he wo opion values should be he same. So far we have illusraed he exended MAD approach and valued a real opion wih an assumed sochasic process for he evoluion of eiher he cash flow or he invesmen value. The impression ha praciioners are someimes given is ha only he laer assumpion should be used o value he real opion. In conras, we hold ha eiher assumpion can be appropriae o use in specific insances. In an efficien renal marke (or any marke where he enans (invesors are predominanly ineresed in rens (invesmen cash flows, we migh recommend o value real opions assuming (17. In he condiions where he cash flows are significan and observable, his assumpion may also be preferred. On he oher hand, one would prefer o value he real opion by employing (2 if, referring o a lease (or any oher invesmens, one has no only an abandonmen opion bu oher real opions relaed o he lease (invesmens price such as an opion o sell he lease, o share he ownership of he lease, or o redevelop he lease for oher use. 5 Oher Approximaions of MAD Parameers We hold he relaionship among he invesmen cos and he ne and gross invesmen values (13 and exend he MAD approach so ha any assumpion abou he process of he invesmen value and cash flows can be used o value a real opion. Apar from our approximaion mehod 12

13 of he MAD parameers, π Q, rx, and µ q, here are oher ways o approximae hem which, however, may fail he relaionship (13. Consequenly, he opion value may be biased. For insance, in AP s (2001 sudy of he Campeiro case, π Q was mis-calculaed and hence, i led o a differen and obviously incorrec value of he abandonmen opion. In he following, we firs explain AP s (2001 valuaion of he abandonmen opion. This allows us o illusrae he imporance of changing he measure consisenly in he valuaion. We laer choose differen approximaions of r x and µ q. I will hen be clear ha our approximaions regarding hese parameers are he mos advanced ones. 5.1 A Discussion of AP s (2001 Abandonmen Opion Valuaion In he original sudy of Campeiro case, AP (2001 obained he value for he abandonmen opion. Tha his is differen from our opion value is due o he inconsisen change-ofmeasure in his calculaion. In his subsecion, we exhibi AP s (2001 assumpions and calculaions, so as o illusrae he impac on he real opion valuaion regarding he change-of-measure. Under he P measure, AP (2001 firsly assumed (17 and hen employed he Cox, Ross, and Rubinsein binomial ree parameerisaion o consruc he ree for rens. The values ha he chose for u, d and π x were idenical wih ours (18, so was his calculaion of he ne lease value a ime 0 (16. However, for he opion valuaion, in conras wih π Q (19, he applied a differen risk-neural probabiliy. 17 π Q = exp(r d u d = %, where Q denoes a risk-neural measure which differs from Q. Now he problem is ha he Q measure is inconsisen wih he P measure. To see his, we show ha he gross lease value a ime 0 diverges from q 0 (16, which is agains he law of one price. 18 T q 0 = E Q [x τ ] exp( rτ = (T + 1x 0 = τ=0 We herefore argue ha i is inappropriae o value he abandonmen opion under he Q 17 Wih his ransiion probabiliy, clearly he discouned ren would be a Q -maringale over ime. T 18 We can also calculae he ne lease value a ime 0: p 0 = E Q [y τ ] exp( rτ = , which is also differen from (16. τ=0 13

14 measure. We illusrae his poin by analysing AP s (2001 opion valuaion. He applied (20 wih π Q o he ree of rens and obained he ne lease value wih opion as C Q 0 = He hen claimed he opion value o be C Q 0 p 0 = = Ye his opion value is oo high o be correc, as he abandonmen opion is a deep ou-ofhe-money pu. The opion srike is he oal coss of all ime periods c 0 = (16, whereas he curren underlying price is he gross lease value a ime 0, eiher or ; he laer value is much higher. Inuiively, i can hardly seem plausible ha his opion is worh roughly 20% of he curren underlying price Assuming r x under (17 Is Consan Under (17, in order o value he abandonmen opion, recall ha we apply (14 o calculae he ime-varying risk-adjused discoun rae for ren such ha we obain he process of he gross lease value. A simpler way is o derive a consan risk-adjused discoun rae, denoed by r x, ha ses he gross lease value a ime 0 equal o (16 equal. Hence, (exp( r x T +1 1 δ (exp( r x 1 + δ 1 = 0, = 0. Bu his approximaion is flawed, because for any ime > 0, his equaion would no hold for a fixed value of r x and consequenly, (13 no longer holds. Hence, he abandonmen opion would be mis-priced. Le us calculae he abandonmen opion value so ha we can observe how much difference his approximaion can make. Bringing in he inpus of he Campeiro case, we obain r x = %. The risk-neural probabiliy (16 is hen π Q = % and hence, he value of he abandonmen opion is This value is almos 10% lower han ha calculaed wih he ime-varying r x (14, In fac, r x over-values he rens from = 4 o mauriy, which already have higher expecaions han he shor erm rens ( = 1, 2 and 3, whils he curren ren x 0 remains he same. Hence, if he invesor abandon he lease, he would lose he long erm rens and only obain he shor erm ones. He would herefore end o wai for fuure 19 In fac, he number comprises no only he abandonmen opion value bu also he change in he curren lease value due o AP s (2001 inconsisen change-of-measure from P o Q. This value change can be calculaed as he difference beween he curren gross (ne lease values under he wo measures q 0 q 0 = p 0 p 0 = Noe ha, since he mainenance coss are risk-free under any measure, hey would always have he same values regardless of he measures hey are under. Therefore q 0 q 0 = (p 0 c 0 (p 0 c 0 = p 0 p 0, and we can use eiher he gross or ne lease value o deermine he change in he curren lease value due o he change-of-measure. 14

15 rens o cover he mainenance coss han abandoning he lease early. The abandonmen opion herefore becomes less aracive and consequenly under-valued. 5.3 Assuming µ q under (2 Is Fixed To simplify he calculaion under (2, we may assume a fixed drif, denoed by µ q, insead of a ime-varying µ q (23. However, wih his assumpion, (13 does no hold and herefore he abandonmen opion would be slighly biased. To see his, le us calculae he abandonmen opion value firs. Given (7, we solve for he dividend yield δ and µ q he sysem of non-linear equaions presened as below. δ = δ 1 1 δ 1 exp(µ x µ q. (22 wih δ 0 = x 0 /q 0 and δ 9 = 1. Since he associaed risk-neural probabiliy (19 under (2 applies here, we can hus compue he abandonmen opion value as < This bias on he opion value occurs because he gross lease value is in general over-valued. In paricular, for he same expeced ren a ime, he corresponding dividend yield compued using (22 is lower han ha given by (12. This means he gross lease value ha we calculae here, as he base of he dividend yield, is higher han ha calculaed in he previous valuaion; hence, he invesor would believe in a higher profiabiliy of he lease and herefore be less aemped o abandon he lease, and he abandonmen opion calculaed here is lower han in our previous valuaion. 6 Conclusion In our exended MAD framework, in conras o he lack of enhusiasm in pracice o value real opions using MAD based on an assumed evoluion of cash flows, his paper shows such a valuaion process of a real opion is equivalen and consisen wih he radiional implemenaion of MAD, i.e. valuing a real opion according o an assumed sochasic process of he invesmen values. More precisely, when cash flows and invesmen value share one source of uncerainy, he evoluion of he cash flows (invesmen values would deermine he process of he invesmen values (cash flows. For insance, when he cash flows follow a simple GBM, we show how his leads o a sraighforward derivaion of he implied invesmen value. Based on eiher of hese wo processes, we also demonsrae he valuaion of he real opion aached o he invesmen. 15

16 By valuing he real opion wih various assumpions, his paper demonsraes he link beween he value of he real opion and he valuaion of he underlying invesmen via he changeof-measure. The mos classic and popular operaion in pracice regarding he valuaion of an invesmen is conduced in he DCF framework, whils he paradigm of he real opion valuaion lieraure and pracice is wihin RNV framework. Swiching from he DCF framework o he risk-neural world requires a change-of-measure, mainly he derivaion of he appropriae risk-neural probabiliies. Ye he change-of-measure may be problemaic o achieve in pracice. For insance, before an acquisiion, an invesmen bank may fund he acquirer having calculaed he value of he arge company, produced in he DCF framework (albei heavily adjused or exended. Then he acquirer may simply ake his single value and price is own real opion o acquire wihou running hrough and being consisen wih he assumpions made in he valuaion of he arge company conduced by he invesmen bank. On he oher hand, he danger of using inappropriae approximaion of he parameers in he real opion valuaion is significan and can resul in a mis-esimaed real opion value. A ypical example is he real opion valuaion in he original case sudy by AP (2001, which is examined in Secion 5.1. The mis-esimaion is shown o be oo large o be ignored - AP (2001 valued he abandonmen opion as worh which is roughly a hird of he ne lease value. This is an implausible resul for a deep ou-of-he-money American pu opion and is shown here o be more han weny imes larger han he correc value(s. To conclude, he main key here is o compue he real opion value sricly following he assumpions made in boh he DCF analysis on he base invesmen and he real opion valuaion process, and also o be consisen among assumpions, so ha he model involves less model risks. References J. A. Azevedo-Pereira. Two case sudies on real esae developmen. In S. Howell, A. Sark, D. Newon, D. Paxson, Cavus M., J. Pereira, and K. Pael, ediors, Real Opions: Evaluaing Corporae Invesmen Opporuniies in a Dynamic World. Financial Times, Prenice Hall, T. Copeland and V. Anikarov. Real Opions: A Praciioner s Guide. Texere, New York,

17 J. C. Cox, S. A. Ross, and M. Rubinsein. Opion pricing: A simplified approach. Journal of Financial Economics, 7(3: , S. Howell, A. Sark, D. Newon, D. Paxson, M. Cavus, J. Pereira, and K. Pael. Real Opions: Evaluaing Corporae Invesmen Opporuniies in a Dynamic World. Financial Times, Prenice Hall, R. Jagannahan and Z. Wang. The condiional capm and he cross secion of expeced reurns. Journal of Finance, 51:3 53, B. M. Lambrech. The iming and erms of mergers moivaed by economies of scale. Journal of Financial Economics, 72:41 62, R. McDonald and D. Siegel. The value of waiing o inves. Quarerly Journal of Economics, 101(4: , A Proof of (14 To derive (14, we assume ha he cash flow and he cos happen periodically a he beginning of every ime period (, for 0 T and is very small. By definiion, q = x + E P [q ] exp ( r x. By aking he ime 0 expecaion on boh side and applying he ower propery, we obain E P [q ] = E P [x ] + E P [E P [q ] exp ( r x ] = E P [x ] + E P [q ] exp ( r x. Similarly, E P [p ] = E P [x ] E P [k ] + E P [p ] exp ( r y, E P [c ] = E P [k ] + E P [c ] exp ( r k. Now, given (13, E P [q ] = E P [p ] + E P [c ], we herefore can wrie ( E P [q ] exp ( r x = E P [p ] exp ( r y + EP [c ] exp r k. 17

18 Tha is, ( exp ( r x = exp ( r y (exp + r k exp ( r y E P [c ] E P [q ]. By exracing exp( r y from boh erms on he righ, we obain ( (( exp ( r x = exp ( r y (1 + exp r y rk 1 E P [c ] E P [q ]. Now, we ake he log on boh sides and hence, ( (( r x = r y (1 log + exp r y rk 1 E P [c ] E P [q ]. Divided by, he above equaion becomes, ( ( (( r x = r y 1 log 1 + exp r y rk 1 E P [c ] E P [q ]. When 0, he limi of he second erm on he righ can be calculaed using l Hôpial s rule as below, lim 0 1 log ( log d = lim 0 = lim 0 ( = r y rk ( ( (( 1 + exp r y rk 1 ( 1 + ( exp (( r y rk E P [q ] ( E P r y [c ] rk exp 1 + ( exp (( r y rk E P [c ] E P [q ], d (( r y rk 1 E P [c ] E P [q ] 1 E P [c ] E P [q ] E P [c ] E P [q ] We now bring his resul back o he previous equaion and hus have (14. B The Compaibiliy of Assuming a GBM Evoluion of he Gross Invesmen Value or he Cash flow CA (2001 assume a GBM (2 evoluion of he gross invesmen value. From he relaionship beween he gross invesmen value and he cash flow (10, we can derive he process of he cash 18

19 flow as below, where Y dx = d ( 1 q Y = q Y 1 dq q q Y 1 dy = Y (µ q δ Y x d + σ q Y x db q, = dy /d. Given he expecaion (7 of he fuure cash flows, he drif of he above process is equal o µ x. Hence, we can specify he drif of he process (2, i.e. µ q δ = µ x + Y Y. (23 Alernaively, under he assumpion ha he cash flow follows a GBM (17, he evoluion of he gross invesmen value can be derived from (10, i.e. dq = d (x Y = x Y dx x + x Y dy Y = (µ x + Y q d + σ x q db x. Y Tha is, he gross invesmen value process is also a GBM wih he same drif as in (23: dq q = (µ x + Y d + σ q Y dbq. (24 Also noe ha, he dispersions of he cash flow process and he gross invesmen value process are idenical, i.e. for he same invesmen, σ q = σx, db q = dbx. (25 C The Equivalence of he Opion Valuaion Processes (20 and (21 The opion valuaion processes (20 and (21 are equivalen according o pu-call pariy (PCP: P s( + p s( = C s(. (26 The proof is as follows. We sar wih backward inducion (21 a ime 0. Tha is, P 0 = max {( p 0, E Q [ ] } P s(1 exp( r. 19

20 Adding p 0 o boh sides of he equaion, P 0 + p 0 = max {( p 0 + p 0, E Q [ ] } P s(1 exp( r + p0. (27 We now rewrie he second erm in he maximisaion, E Q [ P s(1 ] exp( r + p0 = E Q [ P s(1 + p s(1 ] exp( r E Q [ p s(1 ] exp( r + p0, where, bringing back he relaionship beween he gross and ne lease value given in Foonoe 10, E Q [ ] ( ] T T p s(1 exp( r + p0 = E [q Q s(1 k e r(τ 1 exp( r + q 0 k exp( rτ τ=1 τ=0. Afer some simple algebra, we rewrie his expression as, E Q [q s(1 k ] ( T exp( rτ exp( r+ q 0 k τ=1 T exp( rτ = τ=0 where, assuming i is q ha follows a Q maringale (??, (q 0 E Q [ q s(1 ] exp( r k, q 0 E Q [ q s(1 ] exp( r = x0. Wih hese decomposiions, we can rearrange (27 as following, { P 0 + p 0 = max 0, E Q [ ] } P s(1 + p s(1 + (x0 k. (28 Now apply he pu-call pariy (26 o (28, we reach o { ( C 0 = max 0, E Q [ ] } C s(1 + x0 k, which is indeed backward inducion (20 a ime 0. This proof applies a any [0, T ]. 20