A Symmetrical Binomial Lattice Approach, for Modeling Generic One Factor Markov Processes

Transcription

1 A Symmerical Binomial Laice Approach, for Modeling Generic One Facor Markov Processes Carlos de Lamare Basian Pino Universidade Unigranrio Rua da Lapa, 86, 9º andar Rio de Janeiro, , RJ, Brasil Luiz Eduardo Teixeira Brandão IAG Business School - Ponifícia Universidade Caólica - Rio de Janeiro Rua Marques de São Vicene Rio de Janeiro, , RJ, Brasil brandao@iag.puc rio.br Luiz de Magalhães Ozorio Faculdade de Economia Ibmec Av. Presidene Wilson 118 Cenro Rio de Janeiro, , RJ, Brasil lmozorio@ibmecrj.br Absrac In his paper we propose a Symmerical Binomial Laice Approach ha is equivalen o he well known and widely uilized Laice of Cox, Ross & Rubinsein (1979) when modeling Geomeric Brownian Moion ype of processes, bu can be uilized for a wide variey of oher Markov syle sochasic processes. This is due o he highly inuiive consrucion in which firs he expeced value expression of he process is direcly used and he variance is modeled in a symmerical laice, which is added o he firs. We hen demonsrae is applicabiliy wih several Real Opions examples, comparing o he Cox model. Keywords Symmerical Binomial Laice, Markov Processes, Discree Real Opion Modeling. 1 Inroducion: Binomial Approximaion for Markov Processes The mahemaical complexiy associaed wih derivaives and real opions heory derives from he need for a probabilisic soluion for he opimal invesmen decision hroughou he life o an opion. The soluion o his dynamic opimizaion problem, as described by Dixi and Pindyck (1994), is o model he uncerainy of he underlying asse as a sochasic process where he opimum decision value of invesmen is obained by solving a differenial equaion wih he appropriae boundary condiions. In many cases, however, his differenial equaion has no analyical soluion or he simplified assumpions concerning he boundary condiions do no reflec he acual complexiy of he problem. In hese cases, a discree approximaion o he underlying sochasic process can be used in order o obain a soluion ha is compuaionally efficien for he dynamic valuaion problem a hand.

2 One of hese alernaives is he binomial laice, which is a robus, precise and inuiively appealing ool for opion valuaion models. The discree recombinan binomial model developed by Cox, Ross and Rubinsein (1979) o evaluae derivaives is widely acceped as an efficien approximaion o he Black, Scholes and Meron s (1973) model due o is ease of use, flexibiliy and he fac ha i converges weakly o a Geomeric Brownian Moion (GBM) as he ime sep (Δ) decreases. Furhermore, as opposed o he Black, Scholes and Meron model, his approach provides he soluion o he early exercise of American ype opions. The approach used by Cox, e al (1979), where he branch nodes recombine due o he fac ha he upward movemen (u) is he inverse of downward movemen (d), means ha a each sep N, one obains N + 1 node, and no 2 N as in he case of a non recombining ree. The recombinan laice is simple and pracical o implemen in spreadshee such as Excel or even in decision ree programs. In he approach developed by Brandão, Hahn, and Dyer (2005), for example, he payoffs in each branch correspond o cash flows of each sae of he underlying asse. Ofen, however, he uncerainy o be modeled does no behave as a GBM ype of sochasic diffusion process. This occurs when he value of a variable is a funcion of a long erm equilibrium level or mean, as is usually he case of non financial commodiies or ineres raes. Several auhors, such as Bessimbinder, Coughenour, Schwarz (1997, 1988), Laughon and Jacoby (1993) among ohers, sugges ha his ype of variable ofen exhibis auo regressive behavior and poin o he fac ha modeling such variable wih a GBM can exaggerae he range of values depiced and, as a resul, oversae he value of opions wrien on he variable. Bu he Cox, e al. (1979) approach only applies o uncerainies ha can be modeled hrough a GBM process. Alernaively Nelson and Ramaswamy (1990) propose a generic binomial model approach ha can be used o accommodae oher processes han he GBM. Hahn and Dyer (2008) adap i o model a mean reversion process hrough a versaile enough mehodology ha can even be used o model bi variae problems. In his paper we develop an alernaive Symmerical Binomial Laice approach ha is equivalen o he Cox, e al. (1979) approach for GBM modeling, bu is also more generic and able o model oher Markov processes, being more inuiive han he Nelson and Ramaswamy (1990) approach. 2 Cox, Ross & Rubinsein Binomial Model The binomial approach developed by Cox, Ross & Rubinsein (1979) converges weakly o a Geomeric Brownian Moion GBM ype of Markov diffusion process. This is done by maching he firs (Expeced Mean) and Second (Variance) momens of he binomial sep model wih hose of he GBM, which is defined by he differenial equaion:, where: S is he value or price of he uncerain variable, µ is he drif or growh rae of he sochasic process and σ is volailiy parameer. The expeced value expression for his process is: Or considering:, and: Δ = 0 /2Δ, and: Δ If we consider he binomial sep of lengh Δ,

3 p 1-p p 1-p S o S Δ + =S 0 u Figure 1 Geomeric binomial sep Where: u and d are respecively he up and down mulipliers of he laice, and p he probabiliy of an up move along his. In order o mach he firs and second momens of he GBM model, Cox e al. (1979) use hese values:, 1/, and: / 1, or he more usual expression: The resuling expeced value a is: 1, which is dependen on u (and d) and on p, which in urn is a funcion of µ and σ. S o p= µ S Δ+ =S 0 u E[S ]=S 0 (u*p+d*(1-p)) 1-p S Δ- =S 0 d Figure 2 Cox e al (1979) binomial sep As shown, he drif parameer of he GBM modeled wih such a laice is presen only in he probabiliy of he up (and complemenary down) move. The laice values (esimaed wih u and d) only model parly he volailiy of he process, which is also dependen on he values of p. For his reason when esimaing derivaives values, he risk neural approach of he process is done hrough he adjusmen of hese probabiliies. 3 Symmerical Binomial Laice Approach Proposiion The proposiion of his paper is o presen an alernae approach for binomial laice consrucion ha is equivalen o ha of Cox. e al. (1979) when modeling GBM ye applicable o a more wide range of sochasic processes and also valid for derivaives and real opion calculaion. The basic principle is sill o closely mach he firs and second momen of he process o be modeled, bu by using he deerminisic expression of he expeced value (firs momen) direcly in he laice mean value and keeping he laice up and down movemen consrucion o model he volailiy (second momen) of he process.

4 This can be done in he following seps: consider, we assume ha: x = x* + x, where: x is he deerminisic expeced value of he process, so: x = x -1 + (µ-σ 2 /2)Δ, and: x* are he values of an addiive laice, which models an Arihmeic Brownian Moion wih 0 drif, and wih U and D as is addiive (Up and Down) incremens. p x* Δ + = ln(s0 u) = ln(s 0 ) + ln(u) = x* 0 + U x* 0 1-p Figure 3 Addiive binomial sep In his case: x* Δ - = ln(s0 d) = ln(s 0 ) + ln(d) = x* 0 + D,, and: , To represen he whole process of building he addiive sep for he x variable, we can use he following diagram: p=0.5 x* Δ + = + σ x 0 µ x Δ = x 0 + (µ-σ 2 /2)Δ 1-p=0.5 x* Δ - = - σ Symmerical Addiive Laice Analyical Expeced Value Figure 4 Symmerical Laice consrucion seps GBM In his laice he branches are symmerical around he expeced value expression. p=0.5 x Δ + = + σ + x 0 + (µ-σ 2 /2)Δ x o = + x 0 1-p=0.5 Figure 5 Symmerical Laice Nod GBM µ E[x Δ ]= x 0 + (µ-σ 2 /2)Δ x Δ - = - σ + x 0 + (µ-σ 2 /2)Δ To obain he muliplicaive symmerical laice, i is only necessary o calculae a each nod:, and use p = 0.5 hroughou he whole laice. 4 Examples of real opions calculaion wih boh models We will now calculae a real opion using boh Cox, e al. laice and he Symmerical laices approach and show he equivalen resul of boh mehods. We consider a projec wih a presen quarerly Cash Flow of 10 $million. This Cash Flow has a volailiy of σ = 40% and a drif rae (risk adjused) of µ=8%, boh early values. The projec s risk adjused discoun rae is k = 12% and he risk free rae is r = 6%. The projec s Cash Flows

5 are expeced o grow rae µ during 5 years (in 20 quarerly periods) afer which a perpeuiy wihou grow is used. Wih he above premises he Presen Value of he projec is: PV 0 = $ million. To obain he risk free equivalen Presen Value, we use a risk free growh rae µ for he cash flows esimaes as follows: µ = µ - (k r) = 2%. When discouning a he risk free rae r he Cash Flows wih his risk free growh rae, he Presen Value of he projec is: PV 0 = $ million. We firs model a laice wih 20 quarerly periods for he Cash Flows using he Cox, e al. model. In his case, using σ = 40%, µ = 2%, and Δ = 0.25, we obain: , , and Δ Saring wih he end period (20 h quarer) and considering perpeuiy, we hen discoun he laice a he risk free rae r, summing he Cash Flow values of he laice a each nod, and weighing by he probabiliies p and (1 p) above, we end up a ime 0 wih a Presen Value of he projec is: PV 0 = $ million, or an error of 1.3% over he base case value. Now we model he alernae Symmerical Laice hrough he process described earlier above. Again using: σ = 40%, µ = 2%, and Δ = 0.25, we obain: µ - σ 2 /2 = -6% (necessary for x calculaion, and already he risk free adjused drif rae), U = = 0.200, and D = , wih p = 0.5 = (1-p) Again we discoun he risk free rae r he Cash Flows of his Symmerical Laice, considering perpeuiy, from he end period up o he sar, always weighing by 0.5, and now we end up a ime 0 wih a Presen Value of he projec is: PV 0 = $ million, or an error of 0.2% over he base case value. This resul shows a slighly more accurae Symmerical Laice han he radiional Cox, e al. model, bu his canno be generalized since i can be he resul of he parameers of he example used. Bu i poins o he equivalence of boh approaches and he validiy of he Symmerical Laice for GBM modeling. Alhough we are modeling he projecs Cash Flows in he laices, he underlying asse, over which any opion should be exercised, is he Presen Value of he projec iself. We now incorporae in he above projec wo differen Real Opions for he projec a hand. Firs an expansion opion available a any ime during he 5 year laice forecasing in quarerly periods, which is modeled as an American Call Opion ha increases in 90% he Projec Value a he poin of exercise a a cos of 400 $ million. Simulaneously we model an abandonmen opion for he same ime span, which is modeled as an American Pu Opion, in which he projec can be abandoned agains a fixed value (sale of remaining asses) of 350 $ million. These opions are easily modeled wih binomial laices, and in he case of he laices described above, as we modeled direcly he projecs Cash Flows here is no need for alering he consrucion o incorporae hese. When incorporaing he Opions above ino he laices we end up a he sar of hese wih he expanded Presen Values, ha is he radiional Presen Values plus he Opion Value of any flexibiliy incorporaed in he laices. From he Cox, e al. laice model we ge a Real Opion value of: $ million or a 39.9% increase over he base case Presen Value of $ million.

6 Using he Symmerical Laice approach wih he same Real Opions incorporaed we ge a Real Opion value of: $ million or a 40.7% increase over he base case. The above example shows ha he Symmerical Laice approach is similar o ha of Cox, e al. model when modeling GBM ype of uncerain variables, and ha securiies and opions wrien on such a model yield similar resuls, making hus he Symmerical Laice a valid mehod for opions and real opions calculaion. Neverheless he Cox, e al. model is widely used as he main binomial approach for valuing derivaives and opions and anoher model ha does he same hink would no be a significan conribuion unless i brings some sor of new applicaion or improvemen over he radiional one. The main difference of boh is ha wih he Cox, e al. model, he drif rae of he process is no buil in he laice consrucion bu incorporaed ino he ransiion probabiliy of each nod. The symmerical model on he oher hand incorporaes boh he drif and he volailiy in he laice consrucion separaing he wo momens of he sochasic process in wo separae pars of his consrucion. The ransiion probabiliy only weighs symmerically he discouning of he values along he laice. Alhough his approach seems more inuiive for he new laice praciioner, i sill is no subsiue for he Cox, e al. model when i comes o modeling GBM uncerainies. Usually he Presen Value of fuure cash flows is assumed o follow a random walk ype of process, or a GBM, according o Samuelson heorem (apud Copeland & Anikarov, 2003) and his is used o model a decision ree laice based on he Cox, e al. model. Neverheless, frequenly he uncerainy involved wih projec cash flows does no follow a GBM as is he case of commodiies and oher commodiy dependen projecs. Alhough according o Samuelson heorem he presen value of hese cash flows would sill have a random walk behavior, he cash flows hemselves cerainly would no. A number of auhors suppor he general view ha commodiy dependen cash flows, among ohers, generally would follow some ype of auo regressive behavior. In he examples above we direcly modeled he cash flows on he laices o calculae he flexibiliy value. In case hese would no follow a GBM ype of diffusion process, hen he Cox e al modeling would no be fi o esimae he real opion value presen, or migh be overesimaing i grealy. In he following secion we will adap our Symmerical Laice o model a mean reversion process and show ha i can be modified o fi a grea number of similar and generic auoregressive processes. 5 Generic Symmerical Binomial Laice Model The use of binomial laices similar o he classic GBM model of Cox, e al. (1979) o model oher Markov processes has been scany due o he fac ha such models ofen produce ransiion probabiliies greaer han 1 or less han zero when he influence of mean reversion is paricularly srong. Consequenly, discree rinomial and muli nomial rees (Hull, 1999) and Mone Carlo simulaion models have been he primary mehods used o model MR processes. Unforunaely, rinomial rees, such as hose suggesed by Tseng and Lin (2007), Clewlow and Srickland (1999), Hull and Whie (1994 a, 1994 b ) and Hull (1999), require more involved mehodologies for specifying valid branching probabiliies and laice cell sized o ensure convergence of he sochasic process. This requires more sophisicaed programming and resuls in difficuly in applying rinomial rees o a wide range of specific projecs and cases. Mone Carlos simulaion approaches such as he Leas Squares mehod (LS) of Longsaff and Schwarz (2001) are able o accommodae almos any sochasic process, including a combinaion of various processes, hereby eliminaing he so called curse of dimensionaliy

7 and modeling. However, he shorcoming of hese models is in modeling decisions, which can pose problems in he modeling of compound opions, for example. 5.1 Mean Reversion Laice A Mean Revering (MR) sochasic process is a Markov process in which he direcion and inensiy of deviaion are a funcion of he long erm equilibrium level o which he curren price mus rever. The logic behind a Mean Revering Model derives from microeconomics: when prices are depressed (or below heir long erm equilibrium level), he demand for his produc ends o increase while he producion ends o decrease. This is due o he fac ha consumpion of a commodiy, for insance, increases as prices decrease, while low reurns o producers will lead o he decision o pospone invesmen or o close less efficien unis, hereby reducing he supply of he produc. The opposie will occur if prices are high (or above he long erm equilibrium level or mean). As an example, empirical sudies (Pindyck & Rubinfeld, 1991) have shown ha hese microeconomic forces do indeed cause oil prices o exhibi mean revering sochasic behavior. The simples form of MR process is he single facor Ornsein Uhlenbeck process, also called Arihmeic MR process, which is defined by: where x is he naural log of he variable S, η he mean reversion speed, x is he long erm average o which x revers, σ he volailiy of process and dz is he sandard Wiener process. The naural logarihm of he variable is used since in he case of commodiies i is generally assumed ha hese prices have a lognormal disribuion. This is convenien because since S = e x, S canno be negaive. Therefore, he expeced value and variance of he Orsein Uhlenbeck process are given by Dixi and Pindyck (1994), in heir discree model form: Nelson and Ramaswamy MR Laice approach Nelson and Ramaswamy (1990) proposed an approach ha can be used in a wide range of condiions, and which is appropriae for he Ornsein Uhlenbeck process. Their model is a simple binomial sequence of n periods of duraion Δ, wih a ime horizon T: T = n Δ, which hen allows a recombinan binomial ree o be buil. Considering he general form for he differenial equaion of a Markov ype sochasic process given by: dx = μ(x,)d + σ(x,)dz, he Nelson and Ramaswamy model is given by he following equaions: x x x, (up movemen) x x x, (down movemen) x, p 1212 (up probabiliy) x, 1-p (down probabiliy) For he Ornsein Uhlembeck process hese equaions would be: x x x,, and

8 x, Hahn & Dyer (2008) and Basian Pino, Brandão & Hahn (2009) use his approach o model bivariae laices, on which a leas one of he variables follows a Mean Reversion. However, in his model, he probabiliy p can assume values or values greaer han 1. This condiion is remedied by censoring he probabiliies p (and herefore: 1 p), o he range of 0 o 1 in he following manner: p 1 1 x x if p0 and p if p <0, p is censored 1 if p >1, p is censored Symmerical Laice Approach o Mean Reversion The Nelson & Ramaswamy approach is similar o ha of Cox. e al., since he drif of he expeced value of he process is regulaed by he value of p, which in his case varies wih x. This model can be easily adaped o a Symmerical model similar o he one described for GBM modeling. We use he approach suggesed by Hull and Whie (1994 a, 1994 b ) as described in Clewlow and Srickland (1999) and in Hull (1999), for he case of a rinomial ree model of a MR process. Firs, we define a Symmerical Addiive Laice, which models an Ornsein Uhlenbeck arihmeic process wih a long erm mean equal o zero: 0, and iniial value of zero: 0. In his laice he nodes will have a value of. The expeced values of he Ornsein Uhlenbeck x' model are added o he value of he nodes in each period using he real long erm average of he process:, and he real saring value of:. Hence, his Laice of values is used o obain he Laice of a price process S wih lognormal disribuion defined by:, and =. Using he same consrucion as wih he Symmerical GBM Laice, and he parameers of he Nelson & Ramaswami model, we have he following relaionships for he Addiive Symmerical Laice: Δ 2 This las value will also need censoring o he [0,1] range since depending on he values of he expression of can exceed his range. Therefore we will use he following expression: 0, 1, Δ 2 Basian Pino, Brandão & Hahn (2010) use also a sligh variaion of his approach ha does no need censoring and develop he following expression: Δ 2 Δ Bu his model reurns slighly overesimaed values of variance, due o he naure of he Laice consrucion. These auhors use his approach ogeher wih he Symmerical model here proposed and model a swich real opion. Alhough his non censored model is

9 ineresing and applicable, we use he censored probabiliy approach shown above in his paper due o he apparenly higher precision of he model. The x value afer i up movemens, and j down movemens will be: for =(i + j) i, j 0 i j x x x x e i j 0, or: i j i j x, x 1e x e i j i j x The Symmerical Laice consrucion approach is shown here: x 0.5(1+η(- p = max (0, min(1, )(Δ) 1/2 / σ))) x* Δ + = + σ x Δ = +(x 0 )e -ηδ 1-p x* Δ - = - σ x 0 Symmerical Addiive Laice Analyical Expeced Value Figure 6 Symmerical Laice consrucion seps Ornsein Uhlembeck Or: 0.5(1+η(- p = max (0, min(1, )(Δ) 1/2 / σ))) x Δ + = + σ + +(x 0 )e -ηδ E[x Δ ]= +(x 0 )e -ηδ x o = + x 0 1-p Figure 7 Symmerical Laice nod Ornsein Uhlembeck x Δ - = - σ + +(x 0 )e -ηδ We can see a represenaion of a 15 period (wih Δ = 0.25) combinaion of he above consrucion scheme (all in geomeric form) in he laices bellow.

10 2,0 1,5 1,0 0,5 Symmerical Addiive Laice of x* 4,5 4,3 4,1 3,9 3,7 Expeced Value of x' 0,0 3,5-0,5-1,0-1,5-2,0 0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50 2,75 3,00 3,25 3,50 3,75 + 3,3 3,1 2,9 2,7 2,5 0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50 2,75 3,00 3,25 3,50 3, Symmerical Laice for S = Figure 8 Combinaion of Symmerical Laice and Expeced Value showing up and down 95% cer. levels Aenion mus be made o he value of : i is no direcly he naural logarihm of he equilibrium level (mean) of he price modeled :. Applying Io s Lema o he differenial equaion of he geomeric mean reversion model: We end up wih: = The Symmerical Laice for he geomeric MR process, defined by: S e x, is obained by direcly ransforming x (i,j) values in S (i,j). We noe ha in his Symmerical Laice, he adjusmen for risk neuraliy is given in he equaion of expeced value of he process, alering he value of x o: i j x 10-0,00 0,25 0,50 0,75 1,00 1,25 1,50 1,75 2,00 2,25 2,50 2,75 3,00 3,25 3,50 3,75 0 i j i j x x 1e x e i j, Where: λx is he risk premium of he variable x, and λx/η he corresponding normalized risk premium. This adjusmen o ransform a MR process ino a risk neural process is also simpler han ha of he censored model, which requires ha i be done in he ransiion probabiliies along he whole laice. In he following secion we will apply he Symmerical Mean Reversion Laice o he valuaion of a hypoheical real opion, similarly o he example wih a GBM approach seen previously. 5.2 Example of real opion calculaion using Symmerical Mean Reversion Laice We now consider a projec wih he same characerisics as in he GBM example: a presen (=0) quarerly Cash Flow of 10 $ million. This Cash Flow has a volailiy of σ = 40% and a x

11 mean reversion speed parameer of η=1, boh early values. The projec s risk adjused discoun rae is k = 12% and he risk free rae is r = 6%. The projec s Cash Flows are expeced o assume a mean reversion behavior o a long erm equilibrium level of: = 15 $ million for 5 years (in 20 quarerly periods) afer which a perpeuiy wihou grow is used. Since we assume ha he Cash Flows have a mean reversion behavior, he perpeuiy calculaion is differen from a consan grow model (like a GBM). The perpeuiy formula for an Ornsein Uhlembeck process is:, where CF is he las Cash Flow before perpeuiy is assumed and is he value o which CF converges. Wih he above premises he Presen Value of he projec is: PV 0 = $ million. To risk free adjus a Mean Reversion process he normalized risk premium λx/η, is subraced from. We used a normalized risk premium for he Cash Flow process of: λx/η=0.199 and obain a value for he risk free adjused equilibrium level When discouning a he risk free rae r he Cash Flows wih his risk premium, he Presen Value of he projec is now: PV 0 = $ million. Again we firs model a laice wih 20 quarerly periods for he log of he Cash Flows using he Symmerical addiive model. In his case, using σ = 40%, η = 1, 2.403, S 0 = 10 $ million and Δ = 0.25, we obain: 0.200, The ransiion probabiliy p of an up move along he laice is calculaed using he censoring formula give above. The Symmerical Laice for S wihou risk free adjusmen (e.i. wih long erm mean of 15 $ million) and already in geomeric form is shown here bellow, wih indicaion of accumulaed probabiliies a each nod, as well as censored nods (probabiliy of occurrence = 0) CF in $ million Period: Quaer/Year : Probabiliy of occurence beween 1% and 20% : Expeced Value from Symmerical Laice : Probabiliy of occurence beween 20% and 40% : Equilibrium Level S : Probabiliy of occurence above 40% : Censored nod: probabiliy of occurence 0% Figure 9 Symmerical Laice for values of Cash Flow (no risk adjused) Again saring wih he end period (20 h quarer) and considering perpeuiy (wih a mean reversion formula), we hen discoun he laice a he risk free rae r, summing he Cash Flow values of he laice a each nod, and weighing by he probabiliies p and (1 p) calculaed a each nod and censored when needed, we end up a ime 0 wih a Presen Value of he projec is: PV 0 = $ million, or an error of 0.9% over he base case value.

12 Now we incorporae he same real opions calculaed for GBM model: an expansion possibiliy of 90% of he projec value a a cos of 400 $ million and an abandonmen opion for a value of 350 $ million. Using he same backwards discouning procedure as wih he GBM model, bu considering he varying ransiion probabiliies of he MR laice, we end up in = 0, wih an opion value of 29.5 $ million or 6,3% above de base case. Of ineres is o noe ha he opions values wih mean reversion modeling is significanly lower ha when modeling he Cash Flows wih GBM. This is coheren wih heory since mean reversion is an inhibiing facor for opion exercise by is paricular naure. Anoher poin of ineres is ha alhough wih he GBM process he abandonmen opion has a significan value on is own, 85.5 $ million wih he Symmerical Laice, in he MR model is value is close o zero. This is explained by he characerisic of he MR again: if he value of he Cash Flow is significanly low he MR dynamics will correc on average his value inhibiing any abandonmen exercise. The same will happen wih he expansion opion bu o a lesser exen. 5.3 Mean Reversion wih Drif As menioned he binomial approach of Cox e al is limied o GBM ype of derivaives modeling. We propose a Symmerical Laice ha is more generic for oher Markov ype of sochasic processes modeling. Alhough Mean Reversion can be modeled hrough he Nelson & Ramaswani (1990) approach, variaions of his sochasic process canno, or a leas would implicae in limiaions and accommodaions ha could urn he model oo complex o be pracical. We will now model in a Symmerical Laice a Mean Reversion process o which a deerminisic drif rae is added o he long erm equilibrium level. Pindyck (1999) models commodiies prices wih a somewha similar approach using a mean reversion wih an equilibrium level ha evolves quadradically bu in a deerminisic way. There are several wo facor models in he lieraure such as Gibson & Schwarz (1990), Schwarz (1997) model 2, and Schwarz & Smih (2000) among ohers. The model here used can be seen as a paricular case of hese wo facor models, where he equilibrium level does no follow a sochasic process bu has a deerminisic evoluion. Such a model is difficully modeled hrough radiional binomial laices, bu can be very easily incorporaed o he Symmerical approach here proposed. We will only need o include he drif componen in he expeced value calculaion sep of he laice consrucion. This approach is used by Ozorio, Basian Pino & Baidya (2011) and hey name his process as Mean Reversion wih Drif MRM D. Is differenial formulaion is: dx x d cd dz, and expeced value, using he same formulaion as wih he oher E [ S ] expx e ln S (1 e ) 2 2 processes : 0 0 o Therefore we consider a projec wih he same characerisics as in he previous examples: a presen (=0) quarerly Cash Flow of 10 $ million. This Cash Flow has a volailiy of σ = 40% and a mean reversion speed parameer of η=1, boh early values. The projec s risk adjused discoun rae is k = 12% and he risk free rae is r = 6%. The projec s Cash Flows are expeced o assume a mean reversion behavior o a long erm equilibrium level ha is presenly: = 15 $ million bu ha is expeced o grow a a yearly rae of μ = 5% for 20 quarerly periods. Afer his period a perpeuiy wihou grow is assumed. The Symmerical Laice for S before risk free adjusmen and wih long erm mean saring a 15 $ million, and growing a μ = 5% is shown here bellow, wih indicaion of accumulaed

13 probabiliies a each nod, plo, expeced value plo and up and down boundaries for 95% cerainy of S CF in $ million Period: Quaer/Year : Probabiliy of occurence beween 1% and 20% : Expeced Value from Symmerical Laice : Probabiliy of occurence beween 20% and 40% : Equilibrium Level S : Probabiliy of occurence above 40% : Up and Low boundaries of 95% ceriude Figure 10 Symmerical Laice for values of Cash Flow (no risk adjused) For he risk neural ransformaion we can now use he same adjusmen as wih he MRM process. Using he same discouning procedure as wih he previous examples, he Presen Value of he projec is: PV 0 = $ million. When we incorporae he same opions as in he previous examples his we end up in = 0, wih an opion value of 54.5 $ million or 11,2% above de base case. 6 Conclusions In his paper we proposed a Symmerical Laice for Real Opions valuaion ha is more generic han he Cox, e al. (1979) model since i is suiable o a wider number of Markov ypes of sochasic processes. Binomial laices of his ype are no only precise, robus and flexible, bu are inuiive enough for Real Opions eaching and praciioner alike. The approach proposed has an advanage in modeling over he Cox, e al. laice in ha he variance of he process (2 nd momen) is direcly modeled in he symmerical laice herefore leaving he drif of he process (1 s momen) o be modeled independenly. This consrucion scheme may appear more inuiive for he real opion learner besides allowing a more generic parameerizaion of he Markov process ha may appear more suied for he applicaion a hand. We hen apply he Symmerical Laice model o an expansion real opion coupled o an abandonmen opion, and model he projecs cash flows as Geomeric Brownian Moion, Geomeric Mean Reversion and Geomeric Mean Reversion wih Drif where he models long erm mean is subjec o a consan increase rae. The GBM example is compared o he same projec modeled hrough he Cox, e al. approach and we show ha boh models yield similar

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