The Valuation of Multidimensional American Real Options using Computer-based Simulation

Transcription

1 The Valuaion of Mulidimensional American Real Opions using Compuer-based Simulaion Gonzalo Corazar *, Miguel Grave*, Jorge Urzua* Deparameno de Ingeniería Indusrial y de Sisemas, Ponificia Universidad Caólica de Chile, Vicuña Mackenna 486, Saniago, Chile Absrac In his paper we show how a mulidimensional American real opion may be solved using a compuer-based simulaion procedure. We implemen an approach originally proposed for a financial opion and show how i can be used in a much more complex seing. We exend a wellknown naural resource real opion model, originally solved using finie difference mehods, o include a more realisic 3 facor sochasic process for commodiy prices, more in line wih curren research. We show how complexiy may be reduced by adequaely choosing he implemenaion variables. Numerical resuls show ha he procedure may be successfully used for mulidimensional models, noably expanding he applicabiliy of he real opions approach. Scope and purpose Even hough here has been an increasing lieraure on he benefis of using he coningen claim approach o value real asses, limiaions on solving procedures and compuing power have ofen forced academics and praciioners o simplify hese real opion models o a level in which hey loose relevance for real-world decision making. Real opion models presen a higher challenge han heir financial opion counerpars because of wo main reasons: Firs, many real opions have a longer mauriy which makes risk modeling criical and may force considering many risk facors as opposed o he classic Black and Scholes one-facor model. Second, many imes real invesmens have a more complex se of ineracing American opions available, making hem more difficul o value. In recen years new approaches for solving American opions have been proposed which, coupled wih an increasing availabiliy of compuing power, have been successfully applied o solving long-erm financial opions and opening new hopes for increasing he use of his modeling approach for valuing real asses. All Auhors Tel.: ; fax: gcoraza@ing.puc.cl (G Corazar); mgrave@puc.cl (M Grave); jlurzua@ing.puc.cl (J Urzua)

2 . Inroducion Even hough in he las wo decades here has been an increasing lieraure on he benefis of using he coningen claim approach o value real asses, limiaions on solving procedures and compuing power have ofen forced academics and praciioners o simplify hese real opion models o a level in which hey loose relevance for real-world decision making. There are wo main reasons why real opion models may presen a higher challenge han heir financial opion counerpars o be solved. Firs, many real opions have a longer mauriy which makes risk-modeling criical and may force he inclusion of several risk facors, as opposed o only one, like in he classic Black and Scholes-973 sock-opion model. Second, real invesmens many imes have a more complex se of ineracing American-opions available for he decision maker, making hem more difficul o value. Unil some years ago, mos commodiy price models included only one risk-facor and considered consan risk-adjused reurns. These earlier models have several undesirable implicaions, including ha all fuures reurns should be perfecly correlaed and exhibi a similar volailiy, which is no in line wih empirical evidence. In recen years, however, many mulifacor models of commodiy prices have been proposed being much more successful han previous one-facor models in capuring he sochasic behavior of commodiy prices like mean-reversion and a declining volailiy erm-srucure. [Gibson and Schwarz (99), Schwarz (997), Schwarz and Smih (), Casassus and Collin- Dufresne (4), Sorensen () and Corazar and Schwarz (3)]. On he oher hand, he real opions lieraure has also expanded so models ake ino accoun he differen ypes of flexibiliies available o decision makers when managing heir projecs. These flexibiliies include he opion o abandon, o shu down producion, o delay invesmens, o expand capaciy, o reduce coss hrough learning, among many ohers. [Hsu and Schwarz (3), Schwarz (997), Kulailaka (995), Bernardo and Chowdry ()] The higher complexiy of including mulifacor price models ino real opion models wih several flexibiliies has increased he difficuly of solving hem o a poin where he radiional numerical approaches, like he finie difference mehods, are becoming inadequae. This has spanned new research on using some sor of compuer-based simulaion procedure for solving American opions, which coupled wih an increasing availabiliy of compuing power, has been successfully applied o solving mulifacorial financial opions. [Bossaers (989), Tilley (993), Barraquand and Marineau (995), Raymar and Zwecher (997), Broadie and Glasserman (997), Andersen (), Haugh y Kogan ()]. One of he mos promising new approaches in his lieraure is he mehod proposed by Longsaff and Schwarz () (LS) which has been successfully esed for some financial opions of limied complexiy [Senof (4), Moreno and Navas (3), Clemen, Lamberon and Proer ()]. In his paper we show how mulidimensional American real opion models may be solved using a compuer-based simulaion procedure. We exend he Brennan and Schwarz (985) naural resource invesmen one-facor model, originally solved using finie difference mehods, o include a more realisic 3-facor sochasic process for commodiy prices, more in line wih curren research. We implemen he LS procedure showing how o apply i in a much more complex seing and solve his exended real

3 opion model. Numerical resuls show ha he procedure may be successfully used for high-dimensional models, noably expanding he applicabiliy of he real opions approach. The paper is organized as follows. Secion presens he problem o be solved. I describes he classic Brennan and Schwarz (985) real opion model of a naural resource invesmen and how we exend i o include a mulifacor model of commodiy prices. A brief explanaion on he real opions approach for valuing invesmens is also included. Secion 3 presens he proposed soluion. I describes he compuer-based simulaion procedure and how o implemen i so high-dimensional models may be solved. Secion 4 discuses he resuls of he numerical soluion o he original and o he exended Brennan and Schwarz model. Finally, Secion 5 concludes.. The problem. The Real opions approach o valuaion Real Opion Valuaion, or ROV, can be undersood as an adapaion of he heory of financial opions o he valuaion of invesmen projecs. ROV recognizes ha he business environmen is dynamic and uncerain, and ha value can be creaed by idenifying and exercising managerial flexibiliy. Opions are coningen claims on he realizaion of a sochasic even, wih ROV aking a "muli-pah" view of he economy. Given he level of uncerainy, he opimal decision-pah canno be chosen a he ouse. Insead, decisions mus be made sequenially, hopefully wih iniial seps aken in he righ direcion, acively seeking learning opporuniies, and being prepared o swich pahs appropriaely as evens evolve. ROV presens several improvemens over radiional Discoun Cash Flow (DCF) echniques. Firs i includes a beer assessmen of he value of sraegic invesmens and a beer way of communicaing he raionale behind ha value. In mos radiional invesmen valuaions, a base DCF value is calculaed. Then, his base value is "adjused" heurisically o capure a variey of criical phenomena. Ulimaely, he oal esimaed value may be dominaed by he "adjusmen" raher han he "base value." Wih ROV, he enire value of he invesmen is capured rigorously. Concepually, his includes he "base value" and he "opion premium" obained from acively managing he invesmen and appropriaely exercising opions. Second, ROV provides an explici roadmap or opimal policy for achieving he maximum value from a sraegic invesmen. Mos radiional invesmen valuaions consis on a number, and perhaps a se of assumpions underlying ha number. However, he managemen acions required over ime o realize ha value are no clearly idenified. Wih ROV, he value esimae is obained specifically by considering hese managemen acions. As a resul, ROV indicaes precisely which evens are imporan, which milesones o wach for and he necessary acions required o achieve maximum value. There is a broad lieraure on ROV and how o maximize coningen claim value over all available decision sraegies. Among hem, Majd and Pindyck (989) include he effec of he learning curve by considering ha accumulaed producion reduces uni coss, Trigeorgis (993) combines real opions and heir ineracions wih financial flexibiliy, McDonald and Siegel (986) and Majd and Pindyck (987) opimize he invesmen rae, and He and Pindyck (99) and Corazar and Schwarz (993) deermine wo opimal conrol variables. 3

4 The ROV approach has been used o analyze uncerainy on many underlying asses, including exchange raes (Dixi (989)), coss (Pindyck (993)) and commodiies (Ekern (988). Real asses models have included naural resource invesmens, environmenal and new echnology adopion, and sraegic and compeiive opions (Trigeorgis (996), Brennan and Trigeorgis (), and Dixi and Pindyck (994)). In his paper we will exend he classic Brennan and Schwarz (985) model for valuing naural resource invesmens. Oher papers on naural resource invesmens include Paddock, Siegel, and Smih (988), Corazar and Schwarz (997), and Corazar and Casassus (998), Smih and McCardle (998, 999), Lehman (989), and Trigeorgis (99). Recenly real opions analysis is gradually advancing ino he domain of sraegic managemen and economic organizaion. Bernardo and Chowdry () analyze he way in which he organizaion learns from heir invesmen projecs. A relaed model is presened in D ecamps, Marioi and Villeneuve (3). They sudy he choice beween a small and a large projec, where choosing he small projec allows one o re-inves laer in he large projec. Lambrech and Perraudin (3) inroduce incomplee informaion and preempion ino an equilibrium model of firms facing real invesmen decisions. Milersen and Schwarz (4) develop a model o analyze paen-proeced R&D invesmen projecs when here is imperfec compeiion in he developmen and markeing of he resuling produc. Finally, Muro, Nasakkala and Keppo (4) presen a modeling framework for he analysis of invesmens in an oligopolic marke for a homogenous commodiy.. The Brennan and Schwarz (985) Model The valuaion of a copper mine in Brennan and Schwarz (985) (BS85) laid he foundaions for applying opion pricing arbirage argumens o he valuaion of naural resource invesmens. In he model he value maximizing policy under sochasic oupu prices considers he opimal iming of pah dependen, American syle opions o iniiae, emporarily cease or compleely abandon producion. We now briefly describe he BS85 opimizaion problem: Le S() be he spo price of copper, assumed o evolve exogenously according o a one-facor model, as in equaion (): ds = µ d + σ dz S () in which µ is he insananeous reurn, σ is he reurn volailiy and dz is an incremen o a sandard Gauss-Wiener process. I can hen be shown ha he risk-adjused process for commodiy prices can be wrien as: ds = ( ρ C ) d + σ dz () S S wih ρ being he risk-free ineres rae and C being he convenience yield ha accrues o he holder of he commodiy bu no o he holder of is fuures. 4

5 Le V(S,Q,) be he marke value of a copper mine ha produces copper a ime, when he spo copper price is S and he mine has oal reserves of Q and ha is currenly producing (he mine is open). Similarly, le W(S,Q,) be he mine under he same condiions, bu when i is closed. Boh values are no he same, because here is a cos of swiching beween he open and he closed saes, wih k being he cos of closing an open mine and k he cos of opening a closed mine. When open, he mine produces a a consan rae of q, wih a uni cos of A and subjec o he annual income and royaly ax paymen Τ. When closed, he mine has no earnings, bu incurs in a mainenance annual cos of M. In addiion here is an annual propery ax amouning o λ or λ of marke value, depending on wheher he mine is open or closed. Finally, he mine is abandoned when marke value reaches zero. The opimal soluion o his model considers hree criical spo prices S *, S *, S * a which he mine swiches beween closed and abandoned, beween open and closed, and beween closed and open, respecively. The following is he problem o be opimized: + ( ) + + ( ) ( + ) = Max σ S VSS ρs C VS qvq V q S A T ρ λ V { q} σ SS ρ y S ρ λ SW + ( S C) W + W M ( + ) W= (3) (4) Subjec o: V( S, Q=, ) = W( S, Q=, ) = W( S, Q, ) = V( S, Q, ) K ( Q, ) * * (5) (6) * * V( S, Q, ) = max W( S, Q, ) K( Q, ); (7) * * WS ( S, Q, ) VS ( S, Q, ) = if: * W( S, Q, ) K( Q, ) * W( S, Q, ) K( Q, ) (8) W ( S, Q, ) = V ( S, Q, ) S * * S W ( S, Q, ) = S * (9) () Brennan and Schwarz (985) show ha he value of he mine depends on calendar ime only because he coss and he commodiy convenience yield depend on ime, and ha if i can be assumed ha here is a consan inflaion rae, calendar ime simplifies from he model. This simplified, ime-independen model is hen solved using finie difference mehods. 5

6 .3 Exending he Brennan and Schwarz (985) Model Iniial applicaions of he real opions approach were made in he naural resource secor mainly because of heir high irreversible invesmens and he well developed fuures markes characerisic of his secor. Even hough real opion models, like he one we jus described, have been successful in capuring many managerial flexibiliies, in general hey have considered very simple specificaions of he risk processes, which have hindered he applicabiliy of his approach in real-world siuaions. Probably he iniial reason for his simple uncerainy modeling was ha when his approach was developed more han wo decades ago, ha was he sae-of-he ar in commodiy price modeling. In he las wo decades much research has been done o adequaely capure he sochasic process in a more sophisicaed way, bu real opion models have no kep pace wih his commodiy price research, probably due o he added complexiy o obain numerical soluions in a muli-facor seing. Given ha he main goal of his paper is o show how more complex problems may be solved using compuer-based simulaion procedures, in his secion we exend he Brennan and Schwarz (985) model o include a mulifacor specificaion for uncerainy, model which in laer secions will be solved numerically. Commodiy price processes differ on how convenience yield is modeled and on he number of facors used o describe uncerainy. Early models assumed a consan convenience yield and a one-facor Brownian moion. Laer on, mean reversion in spo prices began o be included as a response o he evidence ha volailiy of fuures reurns declines wih mauriy. One-facor mean revering models can be found for example in Laughon and Jacoby (993 and 995), Schwarz (997), Corazar and Schwarz (997). Wih one-facor models, however, all fuures reurns are assumed o be perfecly correlaed which is no consisen wih empirical evidence. To accoun for a more realisic sochasic behavior, wo-facor models, wih mean reversion, were inroduced. Examples are Gibson and Schwarz (99), Schwarz (997) and Schwarz and Smih (). Finally, Corazar and Schwarz (3) propose a hree-facor model for commodiy prices and esimae i using oil fuures. In his paper we use he Corazar and Schwarz (3) (CS3) hree-facor model for commodiy prices, calibraed using copper fuures, and include i in an exension o he Brennan and Schwarz (985) model of a copper mine. We now describe he hree-facor CS3 model. The model has hree sae variables, he commodiy spo price, S, he demeaned convenience yield, y, and he expeced longerm spo price reurn ν. Boh y and ν are mean revering, he firs one o zero and he second one o a long-erm average ν. The auhors show ha he hree facors allow for a greaer flexibiliy in he shape of he fuures price curves. The dynamics of he sae variable are: wih ( ν y) Sd + σ ds = Sdz () dy = κ yd + σ dz ( ) () dν = a ν ν d + σ 3dz3 (3) dzdz = ρd, dzdz3 = ρ3d, dzdz3 = ρ3d (4) 6

7 Defining λ i as he risk premium for each of he hree risk facors, he risk-adjused processes are: ds = ( ν y λ ) Sd + σ Sdz (5) wih dy ( y λ ) d + σ = dz κ (6) * ( ν ν ) λ d σ dν = a( 3) + 3dz3 (7) ( dw )( dw ) = ρ d, ( dw )( dw ) = ρ d y ( dw )( dw ) = ρ d (8) * * * * 3 3 * * 3 3 Corazar and Schwarz (3) calibrae his model for oil prices. To calibrae he model for copper prices we use all copper fuures raded beween 99 and 998 a Nymex, obaining he following parameer values: Parameers Value λ -,3 λ -,39 λ3 -,93 a,379 κ,85 ν -,7 σ,57 σ,96 σ,498 3 ρ,5 ρ,84 3 ρ -,9 3 Table Parameer values of he CS hree-facor price model for copper calibraed using all fuures raded a NYMEX in he period To illusrae he fi of he price model o observed daa, Figure presens a comparison of empirical and model fuures volailiies as a funcion of mauriy. I can be seen ha he price model fis very well he daa. 7

8 Volailiy of Copper Fuures Reurns 99 o 998 3% 5% Volailiy (%) % 5% % 5% Model Volailiy Observed Volailiy T (años) Figure Empirical and model volailiy erm srucure for copper fuures The hree-facor price model can now be used o exend he Brennan and Schwarz (985) real opion model. Wih his specificaion, Equaions (3) and (4) become: σsvss + σvyy + σ3vνν + σσρsvsy + σσ3ρ3vs ν + σσ3ρ3vy ν Max { q} + ( ν y λ) SVS + ( κy λ) Vy + ( a( ν ν) λ3) Vν + q( S a) Tax( S) VT qvq rv = (9) σ SW + σ W + σ W + σ σ ρ SW + σ σ ρ W + σ σ ρ W SS yy 3 νν Sy 3 3 Sν 3 3 yν ( ) ( ) + ν y λ SW + ( κy λ ) W + ( a ν ν λ ) W W M rw = S y 3 ν T () wih he appropriae border condiions. This model, even hough heoreically may be solved using radiional finie difference mehods, may be solve much more easily using he new simulaion mehods shown in he following secions. 3. The Soluion In his secion we sar by a brief explanaion of he Longsaff and Schwarz () (LS) mehod for valuing American opions. For illusraion purpose we use, as our example, a very simple copper mine ha may exrac all available resources insananeously a any momen during he concession period, and a one-facor model for copper prices. Laer, we use he LS mehod o solve a simple copper mine acing as a European opion wih a fixed exercise-ime, bu wih copper prices following a hree-facor model. This problem has a known analyic soluion and helps us o validae our proposiion o use a 8

9 reduced-base implemenaion of he model ha could exend he use of he LS mehod even wih a very high number of risk facors. Finally, we use he LS mehod, wih he proposed reduced-base implemenaion, o solve he exended Brennan and Schwarz (985) model wih prices following hree risk facors. 3. The basic LS mehod In his secion we briefly explain he Longsaff and Schwarz () (LS) approach for he valuaion of American-syle opions. To do his we focus on a simplified copper mine operaion in which he invenory of he mine, Q, may be insananeously exraced (a an infinie exracion rae) once he decision o produce is made. The uni producion-cos is A and he copper spo-price is S which follows a one-facor geomeric Brownian moion: ds S = ( r δ ) d+ σ dz () wih r he risk-free ineres rae and δ he convenience yield. The mehod sars by simulaing a discreizaion of Eq. (): [ ] S = + ( r δ ) S + S σ ε ( ) () wih he ime inerval in years and ε a random variable wih a sandard Normal disribuion. Then, Eq. () is simulaed hrough ime, obaining a price-pah. The process is repeaed N imes, and a price marix S wih N price pahs over a ime horizon T is obained. Like in any American opion valuaion procedure, he opimal exercise decision a any poin in ime is obained as he maximum beween he immediae exercise value and he expeced coninuaion value. Given ha he expeced coninuaion value depends on fuure oucomes, he procedure mus work is way backwards, saring from he end of he ime horizon, T. Then, we sar wih he las price of each pah, T, and, given ha a expiraion he expeced coninuaion value is zero, we compue he opion value in T for he price pah as: CS ( ) = MaxQS ( ( A );) (3) T T One ime-sep backward, a T =, we repea he process for each price pah, bu now we need o esimae he expeced coninuaion value. The LS mehod makes is main conribuion by proposing he use of a leas square regression on a linear combinaion of funcional forms (linear and nonlinear) of he curren values of all model sae variables in order o esimae he expeced coninuaion value. 9

10 Le L j, wih j= o M, be he basis of funcional forms of he sae variable S T used as regressors o explain he realized presen value in rajecory, hen he LS leas square regression is equivalen o solving he following opimizaion problem: Ν M r Min CS ( ) e al( ) T j j S T (4) { a} = j= The opimal coefficiens â are hen used o esimae he expeced coninuaion value GS ˆ ( ) : T M GS ˆ ( ) = al ˆ ( S ) (5) T j j T j= Figure shows he adjusmen of he expeced coninuaion funcion o he realized presen value of all he simulaed pahs (N) Regression Example Coninuaiion Value Simulaions Regression Spo Price Figure Cash flows from simulaions in a facor price process and a sample regression in LSM mehod

11 Then, he opimal decision for each price pah is o choose he maximum beween boh values: he immediae exercise and he expeced coninuaion value. Once we have worked ourselves backwards unil =, we have a final vecor of coninuaion values for each price-pah, which averaged provides us wih an esimaion of is expeced value, which in urn, when compared wih he immediae exercise value gives he opion value a ime =: Opion Value = Max[ Q( S ˆ A); G( S)] (6) 3. A reduced-base implemenaion of he LS mehod in a muli-facor seing As seen on he previous secion, he LS mehod is a very simple way of esimaing he coninuaion value of an American opion based on sandard leas-square regressions on funcional forms of he sae variables. The way hese funcional forms are chosen, however, is no sraighforward and only some general recommendaions are provided which, as is discussed laer in his secion, may prove difficul o implemen in a highdimensional seing. To explain our proposiion we sar by assuming a general mulidimensional model o laer use he hree facor model described earlier as an illusraion. Le s assume ha he dynamics of copper prices is driven by a correlaed sochasic process for he vecor of sae variables x. Then, he expeced coninuaion value funcion is he vecorg ˆ ( x ): M Gˆ x aˆ L x (7) ( ) = ( ) j j j= Longsaff and Schwarz () propose for mulidimensional implemenaions of heir mehod ha he funcional forms include basis funcions from Laguerre, Chebyshev, Gegenbauer, Jacobi polynomials, or even simple powers of he sae variables and heir cross producs. For example, if he sae variables where only wo, X and Y, a simple order-wo expeced coninuaion value funcion would have six regressors, namely: GXY ˆ (, ) = aˆ ˆ ˆ ˆ ˆ ˆ + ax + ay + axy 3 + ax 4 + ay 5 (8) This procedure for finding he basis has he benefi of being simple, bu may presen numerical and performance problems wih mulidimensional models due o is high number of regressors. Figure 3 illusraes how he number of regressors increases wih mulidimensional problems as he order of he base is increased. To conrol for his high number of regressors, many imes mulidimensional problems are esimaed using a low-order base, wih he obvious loss of accuracy.

12 Number of Regressors as a funcion of he Order and Dimension of he problem 6 Number of Regressors Dimension Dimension 3 Dimension 4 Dimension Order Polynomials Figure 3 Number of regressors as a funcion of he dimension of he problem and he order of he base. This procedure for specifying he base, having he advanage of being very simple, does no ake advanage on he srucure of he problem o be solved. Given ha in mos financial opions opimal exercise depends on expeced spo prices and volailiies, we propose using his knowledge on he deerminans of opion value for he base selecion. Then, insead of using funcions of all combinaions of he sae variables, we propose using powers of expeced spo prices which should dramaically reduce model complexiy, while providing an accurae compuaion of opion value. So Eq. (7) becomes under our specificaion: N = + i= i gˆ ( x ) aˆ ae ˆ ( S) (9) N i where E(S) is he expeced spo price under he risk-adjused measure, i.e., he fuures price. Using his reduced-base specificaion we can obain similar valuaion accuracy in a much simpler way, as will be seen in he nex secion. 3.3 Tesing he reduced-base for valuing a hree-facor European Opion. To es our reduced-base proposiion we value an opion ha has a closed-form soluion and compare he analyic soluion o alernaive implemenaions of he LS mehod. The example used is he pricing a ime of he opion o exrac and sale q pounds of copper in T. This is a European opion (insead of an American one), bu we include a hree-facor model for he sochasic process of copper prices. Also, given ha American opions in heir las exercise opporuniy become in pracice a European one, we can compare he expeced coninuaion value funcion a he las exercise dae obained from he LS mehod wih he analyic expression for a European opion wih T- mauriy. Using he hree-facor model shown earlier, we le x = ln( S), hen using Io s lemma o obain he discree-ime form for simulaing he sae variables when :

13 x x σ σ ε y = y + + B (3) κ σ ε υ a υ aν σ 3 ε 3 where B is he Cholesky decomposiion of he insananeous correlaion marix in hree dimensions: B = ρ Ρ ρ 3 Ρ Ρ3 ρ3 ρρ3 Ρ = ρ Ρ = Ρ Ρ = 3 ρ Ρ Ρ 3 (3) ε, ε, ε N(,) 3 The closed-form value for he European opion is: E ln( ) ln( ) ( ) ( ) (,,,, ) S T + Var S T r T rt CS y υ T = q e N( d) KN( d) e (3) where N is he cumulaive normal disribuion, d E ln( ST) + Var ln( ST) ln( K) = Var ln( ST) ; d E ln( ST) ln( K) = Var ln( S ) T (33) E S x y a κ ( T ) a( T ) (ln( T)) = ( e ) / κ+ ν( e ) / λ λ3 + ( ν + λ σ)( T ) κ a λ + (e ) + (e )( aν λ ) / a κ κ ( T ) a( T ) 3 (34) 3

14 σ κ( T ) κ( T ) Var ln( ST) = σ ( T ) + ( T ) ( e ) ( e ) κ + κ κ σ3 at ( ) at ( ) + ( T ) ( e ) ( e ) a + a a ρσσ κ ( T ) ( T ) ( e ) κ κ ρσσ 3 3 at ( ) + ( T ) ( e ) a a ρσ 3 σ3 κ( T ) a( T ) ( a+ κ)( T ) ( T ) ( e ) ( e ) ( e ) aκ + κ a a+ κ (35) Once we have a closed-form soluion o our problem we compare wo differen specificaions for he regression base: firs he original recommendaions and hen our reduced-base implemenaion. We did several numerical implemenaions of LS mehod using Legendre, Laguerre, Hermie, Chebyshev polynomials, all of hem converging o he known analyic soluion in a similar way. We hen implemened our reduced-base proposiion for differen number of regression erms. Figure 4 compares he RMSE of boh approaches as a funcion of he number of regressors. I can be seen ha even hough inroducing more regressors o he base lowers he RMSE for boh approaches, using our reduced-base proposiion requires a much lower number of regressors o achieve a given level of he RMSE. 9% Regression Adjusmen Level o Analyic in Chebyshev Polynomials and Reduced Form Proposed RMSE % 8% 7% 6% 5% 4% 3% Chebyshev Reduced Form % % % Number of Regressors - Increasing order Figure 4 Regression RMSE as a funcion of he number of regressors for Chebyshev Polynomials and he reduced-base form. 4

15 4. Model Implemenaion and Resuls In his secion we show how o implemen he Longsaff and Schwarz () mehod o solve he exended Brennan and Schwarz (985) model. The reduced-base implemenaion proposed earlier in his paper is used o solve his real opion model under prices ha follow he Corazar and Schwarz (3) hree-facor commodiy price model, calibraed for copper. Then, we resric he commodiy price model o mach he one-facor model used in he Brennan and Schwarz (985) paper and compare heir resuls using finie difference mehods wih hose from our simulaion mehod. Finally, we presen our resuls for he real opion model wih he hree-facor price process. 4. Model Implemenaion. In a previous secion we described he exended Brennan and Schwarz (985) model including he opions o abandon a mine, o close an open mine and o open a closed mine. Also we described he compuer-based simulaion approach ha will be used for solving he model. Figure 5 may be useful o undersand he naure of he problem by describing all possible saes during he simulaion. I can be seen ha as ime evolves from o T, he sae variables ha describe he dynamics of copper prices, x = [ S, y, ] υ, evolve following differen pahs. A any poin in ime, and for any value of he price sae variables, he mine may have any amoun of copper reserves beween zero and he iniial reserves Q max. In addiion, he mine a ha poin may be open or closed wih marke values V( x, Q) or W( x, Q), respecively. OPEN CLOSED Sae Variables X X( ) X T ( ) Qmax Reserves Qmin OPEN CLOSED T Figure 5 Sae-space represenaion of he exended Brennan and Schwarz (985) model exended wih a muli-facor price process 5

16 For every sae of he sysem and operaing policy, here is an associaed cash flow for he firm. For example, when he mine is open and he operaing policy is o remain open during years producing q*, he cash flow, CF, is: CF ( S, q *) q * ( S A ) Tax = (36) Recall ha for any price model, he spo price depends on he sae variables x, i.e. S = f( x ). In paricular, for he hree-facor CS model used in his paper, we have: S = f( ) = ' x h x wih ' = [ ] h (37) Also, as noed previously, he mine may be open, closed or abandoned, and may swich from one operaing sae o anoher incurring in fixed coss. Figure 6 summarizes he cash flows of an open mine which will eiher remain open, be closed or abandoned during ime. Figure 7 shows he same informaion, bu for a closed mine. Open Mine Operaing Policy Cash Flow a Value a + Coninue Open CF ( S, q *) V ( x +, Q q ) V (, Q ) x Close K M W (, Q) x + Abandon V = W = Figure 6. Cash flows and Value of an Open Mine as a funcion of Operaing Policy Closed Mine Operaing Policy Cash Flow a Value a + Open CF( S, q*) K V( x +, Q q ) W(, Q) x Coninue Closed M W(, Q) x + Abandon V = W = Figure 7 Cash flows and Value of an Open Mine as a funcion of operaing policy 6

17 As described in a previous secion, afer simulaing all price pahs from ime zero o ime T, he mehod requires making opimal decisions saring a ime T and hen working backwards unil he iniial ime zero is reached. The opimal decision a each poin is aken by maximizing marke value among all available decision alernaives. A ime T, given ha he concession ends, he value of boh he open and he closed mine is zero: V ( x, Q ) W (, Q ) Q = x, T = (38) T Then, a = T here is no ime lef o change he operaing policy so here is no need o esimae an expeced coninuaion value. So he marke values are: ( r+ λ ) V( x, Q) = Max( CF( S, q*);) e T Q (39) T ( r+ λ ) W( x, Q) = Max( CF( S, q*) K ;) e Q (4) T T Then, a = T we mus esimae he expeced coninuaion value. We regress mine value on a linear combinaion of funcional forms of he sae variables L(X), for each invenory level Q: V( X, Q) W( X, Q) = L( X ) a a + e (4) T T T V, Q, T W, Q, T Once he opimal coefficiens are found we can esimae expeced coninuaion values for any mine: Gˆ ˆ ˆ ˆ V, Q, T G W, Q, T = L( X ) av, Q, T a W, Q, T (4) Thus, he expeced coninuaion value a ime = T, as a funcion of he price sae variables x, may be compued. For example, he value of an open mine wih Q unis of resources, condiional on keeping he mine open would be: M gˆ ˆ VQT,, ( x) = avqt,, Lj( x ) (43) j= Given ha we can compue he expeced coninuaion value for all mines (open or closed and wih any amoun of reserves lef), we are now able o obain he opimal operaing decisions by maximizing curren cash flows plus he presen value of expeced coninuaion values. For example, when he mine is open here are hree operaing alernaives available: o coninue open, o close down operaions, or o abandon he mine. Adding curren cash flows o discouned expeced coninuaion values for each of he hree alernaives, he decision maker may choose which is he bes course of acion. Figure 8 shows, for each of he hree alernaives, he expeced presen value (a ime ), he opimal decision should his expeced presen value be he maximum among he alernaives, and he final value a ime using acual realizaions of he price simulaion (insead of expeced values o avoid known biases) a ime +. Figure 9 shows he same informaion, bu if he mine is iniially closed. 7

18 Open Mine Expeced and Realized Value Expeced Value Opimal Decision Realized Value CF( x, *) ˆ q + g VQ, q, ( x ) Coninue Open V( x, Q) = CF( x, q*) + V( x, Q q ) e ( r+ λ ) + + ˆ ( x ) K M gwq,, Close V( x, Q) = K M + W( x, Q) e ( r+ λ ) + Abandon V( x, Q) = Figure 8 Expeced and Realized Value of an Open Mine as a funcion of Operaing Policy Closed Mine Expeced and Realized Value Expeced Value Opimal Decision Realized Value K + CF( x, q*) + gˆ VQ q ( x ),, Open W( x, Q, ) = K + CF( x, q*) + V( x, Q q ) e ( r+ λ ) + + ˆ ( x ) M gwq,, Coninue Closed W( x, Q) = M + W( x, Q) e ( r+ λ ) + Abandon W( x, Q) = Figure 9 Expeced and Realized Value of a Closed Mine as a funcion of Operaing Policy. 8

19 This procedure is repeaed from = T unil =. A = mine values are averaged over all price pahs o provide an iniial esimae of he expeced coninuaion value for he mine: s ( r+ λ ) gˆ VQ, q, = ( x ) = V(, Q q ) e x (44) s = S ( r+ λ ) gˆ WQ,, = ( x ) = W(, Q) e x (45) S = Figures and show he iniial mine values depending on he iniial saus and operaing policy of he mine Open Mine Value Coninue Open V( x, Q) = CF( x, q*) + gˆ ( x ) e VQ, q, = ( r+ λ ) Close V( x, Q) = K M + gˆ ( x ) e WQ,, = ( r+ λ ) Abandon V( x, Q) = Figure Open Mine values as a funcion of he iniial operaion decision. Closed Mine Value Open W( x, Q) = K + CF( x, q*) + gˆ ( x ) e VQ, q, = ( r+ λ ) Coninue Closed W( x, Q) = M + gˆ ( x ) e WQ,, = ( r+ λ ) Abandon W( x, Q) = Figures Closed Mine values as a funcion of he iniial operaion decision. Finally, o deermine he opimal operaing policy, described by he criical values for he sae variables over or under which i is opimal o swich beween mine saes (abandoned, closed and open) he mehod mus find he criical sae variables, x c which equae expeced presen values for differen operaing decisions. 9

20 Figure shows how o find he criical sae variables o close an open mine, o open a closed mine, or o abandon from an open or from a closed mine. Opimal Policy Equilibrium Condiion Open o Closed CF( x, q*) + gˆ ( x ) = K M + gˆ ( x ) c c c VQ, q, WQ,, Closed o Open M + gˆ ( x ) = K + CF( x, q*) + gˆ ( x ) c c c WQ,, VQ, q, Open o Abandon c c CF( x, q*) + gˆ VQ, q, ( x ) = Closed o Abandon c M + gˆ WQ,, ( x ) = Figure Condiions o deermine criical sae variables x c operaing policy for swiching mine 4. Resuls for he original Brennan and Schwarz (985) model. To validae he proposed procedure, in his secion we solve he Brennan and Schwarz (985) real opions model ha was originally solved using radiional finie difference mehods. Recall ha he main difference beween his model and is exension, which will be solved in he following secion, is he price process wih one or hree risk facors, respecively. A simple way of validaing our mehod is o see he one-facor price process as a paricular case of he more general hree-facor process. In his way by resricing cerain parameer values we can perform a beer es on he algorihm by using he same compuer program o solve boh models. Table shows how he Corazar and Schwarz (3) hree facor model may be resriced o behave as he one-facor model used in Brennan and Schwarz (985):

21 Corazar-Schwarz Model Parameers Equivalence o Brennan-Schwarz λ y ( r ) = λ λ 3 a κ ν σ υ δ λ σ σ σ 3 ρ ρ 3 ρ 3 y λ / κ o υ o ν λ / a 3 Table Resricions on he parameers of he Corazar and Schwarz (3) model which induce a one-facor price process similar o Brennan and Schwarz (985). The proposed compuer-based simulaion program was run for 5 price pahs, assuming a concession ha lased for 5 years (he original model assumes an infinie concession), and here are hree opporuniies/year o swich beween operaing saes. Table 3 compares he finie difference values repored in Brennan and Schwarz (985) wih hose obained using he above simulaion procedure. I can be seen ha he simulaion mehod converges very nicely o he known soluion.

22 Spo Price (US$/lb.) Mine Value Finie difference mehod repored in Brennan-Schwarz (985) Mine Value Simulaion mehod Open Closed Open Closed Table 3 Open and Closed Mine Value as a funcion of spo price. Opimal Policy Simulaion Mehod,9,8,7 Criic Price (US$),6,5,4,3,, open close abandon, Reserves Figure 3 Criical prices for opening, closing or abandoning a mine, as a funcion of reserves Our simulaion procedure may also provide he opimal operaing policy. Figure 3 shows he criical prices for abandoning, opening a closed mine, and closing an open mine, as a funcion of reserves. Resuls are very similar o hose repored in Brennan and Schwarz (985), which concludes he validaion of our mehod.

23 4.3 Resuls for he hree-facor exension of he Brennan and Schwarz (985) model. We now repor he soluion o he Brennan and Schwarz (985) model exended o include he Corazar and Schwarz (3) hree-facor commodiy price model. The used parameer values are hose esimaed from all copper fuures prices raded a NYMEX beween 99 and 998 and repored in secion.3. The compuer-based simulaion is run for, price pahs, assuming a 3 year concession horizon, and hree opporuniies o swich operaion mine sae per year. To value he mine for a paricular dae, say January he 4 h, 999, we mus firs deermine he values of he sae variables corresponding o ha dae, which we repor in Table 4. Sae Variable Value S,65 o y,465 o υ,47 o Table 4 Values of he sae variables for January he 4 h, 999 We now run he simulaion and he procedure described in 4. obaining for ha dae a value for he open mine of MMUS$ 6,75, and for he closed mine of MMUS$ 6,68. To explore how mine value changes according o daily variaions in price condiions, we solve for he value of he mine for a 5 year ime span, a each dae beween January 999 and December, 3. Resuls are repored in Figure 4. Daily Values of he Exended Brennan y Schwarz (985) Open Mine MM US$ Daes Figure 4 Daily values of he exended Brennan and Schwarz (985) open mine according o hisorical copper pricing condiions from January 999-December 3 3

24 I is ineresing o noe ha mine value exhibis mean reversion. Even hough i is well known ha copper prices do exhibi mean reversion, given ha a mine produces copper during a long ime span i could be hough ha curren spo prices would no affec oo much mine values, so his value would no display mean reversion. Figure 4 shows his is no he case. To make comparaive saic analysis on how mine value changes wih variaions in he spo price, or in any individual sae variable or parameer value, is raher sraighforward. For example, Figure 5 shows how mine value increases wih copper spo prices. Also i is ineresing o noe ha mine values are convex, as wih all opions, because as value approaches zero he mine increases he probabiliy of abandonmen. Finally, he same figure compares mine value compued wih he real opion model o a simple ne presen value calculaion which does no recognize operaing flexibiliies o abandon or close operaions. I can be seen ha when spo prices are lower, opion values are greaer and hese wo valuaion mehodologies diverge by he mos. By he same oken, when prices are high, flexibiliies are no oo valuable and boh valuaions converge. 5 5 Mine Value Exended Brennan Schwarz (985): y =., v = -. MM US$ Spo Copper Price (US$) NPV Open Mine ROV Closed Mine ROV Figure 5 Mine Value using ROV and NPV as a funcion of spo price We can repea he comparaive saic analysis for any of he sae variables. For example, in Figure 6 we compare Real Opion and radiional NPV values as a funcion of he shor erm price deviaions, y. We assume a raher low iniial spo price of only US$.4. Recall ha our hree-facor price model assumes ha shor-erm price variaions, y, mean rever o zero. Thus, if a any poin in ime y exhibis a high posiive value, fuure prices are expeced o be much lower han curren ones, and given our low iniial spo price assumpion, mine value should basically be explained by is opion value. This can be seen 4

25 in Figure 6 where for large values of y he NPV shows a negaive value, while he ROV value is slighly posiive. Mine Value Exended Brennan Schwarz (985): S =.4, v =.3 5 MM US$ Shor Term Price Deviaions (y) NPV Closed Mine ROV Open Mine ROV Figure 6 Mine Value using ROV and NPV as a funcion of shor-erm price deviaions Comparaive saic analysis can also be performed on opimal policy resuls. For example Figure 7 shows how criical spo prices o open a closed mine depend boh on price condiions, in his case he value of he shor erm price deviaions y, and on he sae of he mine, represened by he reserves lef for exracion. Opimal Opening Prices as a funcion of Shor Term Deviaions (y) Criic Price US$ Mine Reserves MM ons Opening y= -.5 Opening y=.5 Opening y=.5 Figure 7 Criical spo prices for opening a closed mine as a funcion of shor erm price deviaions and mine reserves 5

26 Someimes opimal policy evolves in a non-monoonic way wih he sae of he mine. For example in Figure 8 where as reserves are lower, criical closing prices firs decline o laer sharply increase. Opimal Closing Prices as a funcion of Shor Term Deviaions (y).5 Criic Price US$ Mine Reserves MM ons 5 5 Close y= -.5 Close y=.5 Close y=.5 Figure 8 Criical spo prices for closing an open mine as a funcion of shor erm price deviaions and mine reserves 5. Conclusions Real opions valuaion (ROV) is an emerging paradigm ha provides helpful insighs boh for valuing and for managing real asses. I provides more precise quanificaions on he value of available sraegic and operaional flexibiliies han radiional discouned cash flow echniques. Despie is poenial, he ROV approach has no ye made a srong inroad in corporae decision-making due o several reasons, one of which is he requiremen o keep models oo simple o obain soluions wihin a reasonable amoun of effor. In his paper we show how i is possible o solve very complex mulidimensional American opions resoring o new compuer-based simulaion procedures. We show how o lower complexiy by using a reduced-base implemenaion of he procedure and we validae our proposiion solving a mulidimensional opion wih known analyical soluion. We hen exend a known real opion model proposed by Brennan and Schwarz (985) and solve i using he proposed mehodology. Resuls on differen comparaive saic analysis are provided. This paper makes he case why hese new compuer-based simulaion mehods have he poenial of expanding significanly he use of he ROV approach wihou having o compromise rigorous modeling for solving requiremens. 6

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