1 The Choice of Sochasic Process in Real Opion Valuaion Luiz de Magalhães Ozorio Faculdade de Economia Ibmec Av. Presidene Wilson 118 - Cenro - Rio de Janeiro, 20030-020, RJ, Brasil +55 21 9923-1747 lmozorio@ibmecrj.br Carlos de Lamare Basian-Pino UNIGRANRIO - Universidade do Grande Rio Rua da Lapa, 86, 9º andar - Rio de Janeiro, 20021-180, RJ, Brasil +55 21 2531-8804 carbasian@gmail.com Luiz Eduardo Teixeira Brandão IAG Business School - Ponifícia Universidade Caólica - Rio de Janeiro Rua Marques de São Vicene 225 - Rio de Janeiro, 22451-900, RJ, Brasil + 55 21 2138-9304 brandao@iag.puc-rio.br Absrac A main issue in valuaion modeling is he correc choice of he sochasic process ha beer describes he asse price performance. Paricularly, in invesmen projecs ha show a high level of managerial flexibiliy in condiions of uncerainy for which i would be proposed he real opion valuaion models he assumpion of a specific process can have an impac no only on he projec value, bu also on he invesmen rule. This work discusses he choice of sochasic process in real opions valuaion and he main useful ess and heoreical consideraions o give suppor o his ask. Key-words Sochasic Process, Real Opions Valuaion, Geomeric Brownian Moion, Mean Reversion Model, Jump Diffusion Process, Muliple Facor Models.
1 Inroducion The invesmen decisions in socks, financial derivaives and corporae projecs are influenced by uncerainies of differen ypes. One way o deal wih hese uncerainies is o research he sochasic process ha beer describes he random behavior of he asses prices in ime. Typically in he financial derivaives valuaion he Geomeric Brownian Moion (GBM) is assumed as appropriae o describe he behavior of sock prices and sock indexes, as in Black & Scholes (1973) and Cox, Ross & Rubinsein (1979). The GBM is also largely used o describe he uncerainies in corporae projec valuaion by real opions analysis. As a conras, in he valuaion of commodiies and derivaives relaed o hem, i is common o use Mean Reversion Models (MRM) (Gibson & Schwarz, 1990; Dixi & Pindyck, 1994; Schwarz, 1997), assuming ha he commodiies price migh wander randomly in shor erm, bu ha hey end o converge o an equilibrium level in he long run regarding heir marginal cos of producion. Neverheless, commonly i is no so easy o deermine which one GBM or MRM is he more applicable sochasic process. Besides saisical ess, some quesions mus be considered in he sochasic process choice, such as: he economical feaures and he asse lifeime, he difficuly in he parameers calibraion of he seleced sochasic model, he applicabiliy of he chosen process in soluions (analyical or numerical) of he models used o valuaion, among oher facors. This paper discusses he choice of sochasic process in real opion valuaion and he main useful ess and consideraions o give suppor o his ask. The work is srucured as follows: afer his inroducion (i), in (ii) a bibliographical revision of sochasic processes applied o real opion analysis is presened, in (iii) we describe some saisical ess ha can be used o suppor he sochasic process choice, in (iv) we will presen some heoreical consideraions and (v) we conclude. 2 2 Sochasic Process and Real Opions Theory Applicaions We can define sochasic process as variables ha move discreely or coninuously in ime unpredicably or, a leas, parially randomly. Formally, be Ω a se ha represens he randomness, where w Ω denoes a sae of he world and f a funcion which represens a sochasic process. The funcion f depends on x R e w Ω: R Ω R or f(x,w), and i has he following propery: given w Ω, f ( º,w) becomes a funcion of only x. Thus, for differen values of w Ω we ge differen funcions of x. When x represens ime, we can inerpre f(x,w 1 ) and f(x,w 2 ) as wo differen rajecories ha depend on differen saes of he world, as we can see in figure 1: f(x,w) f(x,w 1 ) f(x,w 2 ) Figure 1 Sochasic process rajecories. x
Wih he aim a represening he uncerainies relaed o he invesmens, he choice of he sochasic process is an issue of grea relevance in he asses valuaion modeling. In he case of real opions valuaion in which he uncerainies are sraighforward considered in he fuure cash flow of he asses he relevance is even greaer. A class of sochasic process ha plays an imporan role in financial modeling is Markov Processes. In Markov Processes only he laes observed value is considered o forecas he fuure values, which is consisen wih he Weak Form of Marke Efficien. Among many ypes of Markov Processes, one of he mos popular is Geomeric Brownian Moion, which is he base case used in he modeling of financial opions (Black & Scholes, 1973) and real opion (Brennan & Schwarz, 1985; McDonald & Siegel, 1985, 1986; Paddock, Siegel, & Smih, 1988). GBM is usually defined by he equaion: dx =αxd + σxdz x is he asse price; α is he drif parameer; σ is he volailiy parameer; dz is a Wiener incremen. Among oher pros, he main advanages of GBM are: is mahemaical simpliciy, he small number of parameers o be esimaed and he fac ha i is easy o obain analyical soluions o asse valuaion. In some way, hese characerisics can be considered he main reason o explain is populariy. As a conras, i has as major con he fac ha he prices end o diverge when he ime goes o he infinie, which could creae unrealisic scenarios and i is an undesirable propery in cases of long run asses. In figure 2 we presen a price projecion supposing i follows a GBM. 3 Price 68% Confidence Inerval Forecas Time Figure 2 Price forecas supposing prices follow a GBM. In oher siuaions, when he uncerainies in prices depend on an equilibrium level, such as in case of commodiies and ineres raes, i is debaed if he use of GBM would be appropriae (AL-HARTHY, 2007, GEMAN, 2005, PINDYCK, 2001, 1999, METCALF & HASSET, 1995, SMITH & MCCARDLE, 1998, BRENNAN & SCHWARTZ, 1985, BHATTACHARYA, 1978). In case of commodiies such as oil, copper, sugar and ehanol i is usual o assume
ha he price is driven, a leas parially, by a mean reversion componen, which makes he prices wander randomly in shor erm, bu, in he long run end o converge o he equilibrium level of he prices associaed o he marginal cos of producion. The mos radiional MRM is called Ornsein Uhlenbeck, which is defined by he equaion: dx = η( x x)d + σdz x is he price of commodiy; x is he equilibrium level o which he process revers in he long run; η is he speed of reversion parameer; σ is he volailiy parameer; dz is a Wiener incremen. Alhough MRM is a Markov Process i does no have independen incremens, considering ha he expeced changes of x are a funcion of he difference beween he long run equilibrium level and he las observaion of he process. The sraighforward applicaion of Ornsein- Uhlenbeck model in he prices may generae an inconvenience ha is he appearance of negaive values, which is an undesirable characerisic o price represenaion. An alernaive ha can be used o solve his problem is no direcly applying he Ornsein-Uhlenbeck model in he prices bu in he logarihm of he prices, as in model 1 of Schwaz (1997). In figure 3 we presen a price projecion supposing i follows a MRM. 4 Price 68% Inerval Confidence Mean Forecas Time Figure 3 Price forecas supposing prices follow a MRM. A hird ype of process ha is commonly used in finance is he Poisson Processes ha is also known as Jump Diffusion Processes. Poisson Processes also belong o Markov Processes class and hey are characerized by he occurrence of discree and infrequen jumps in ime. This ype of process is frequenly used in he modeling of rare evens, such as he occurrence of accidens in insurance indusry and he effec on crisis in he oil prices. In he Poisson Process he jump s appearance follows a Poisson disribuion and i may work wih jumps of fixed or variable sizes. As we will see laer, i is common o mix Poisson Process wih oher ypes of processes as GBM and MRM, in order o model he uncerainies in a more realisic way in he real opions valuaion.
As a formal definiion: Be x a Poisson Process, in which all he randomly of he process is concenraed on he appearance of jumps ha have heir sizes deermined by he funcion g(x,). The Poisson Process can be described by he differenial equaion: dx = f(x,)d + g(x,)dq f(x,) and g(x,) are deerminisic known funcions; dq is a Poisson incremen. The ypical parameers of he Poisson Process are: λ which corresponds o he average rae of he jump occurrence for a ime period; λd corresponding o he probabiliy of jump occurrence; 1 λd is he probabiliy of he non-occurrence of he jump; u indicaing he size of he jump; q represening he randomness of he Poisson Process. There are differen ypes of he Poisson Processes. They can be homogeneous, in which he evens are random and he incremens are independen and saionary or non-homogenous, in which he jumps are no saionary. There are also Poisson Processes in which he randomness is observed in he size of he jumps and hey follow a specific probabiliy disribuion. Finally, among he menioned processes, here are he Compensaed Poisson Processes, which are obained by subracing he drif deermining is conversion in a Maringal. In figure 4 i can be seen a Homogeneous Poisson Process wih upward fixed size jumps. 5 N 12 10 8 6 4 2 0,2 0,4 0,6 0,8 1 FIGURA 1 Figure 4 Homogeneous Poisson Process wih upward fixed size jumps. The ask of deermining which is he mos appropriaed process in order o represen he main uncerainies involved in a valuaion is usually no a rivial quesion and, in some cases, analyss realize ha hese uncerainies have elemens of more han one ype of process. As a resul, in order o generae more realisic models, several auhors presened papers in recen decades o propose models of muliple facors ha mix differen ypes of processes. One of he pioneer works ha presened a muliple facors model was Meron (1976) in which is mixed GBM and Poisson Process. The auhor jusified his model o socks, in which he effec of common news in he sock prices would be represened by GBM, while in case of rare even occurrence here would be a Poisson jump. Using a Compensaed Poisson Process, in which he jumps were considered non-sysemaic and using a lognormal disribuion for he size of jumps, Meron
(1976) managed o find a close formula o European call opions. The differenial formula of he model is represened by he equaion: dx/x = [α λk]d + σdz + dq x is he sock price, E[dq]=E[ɸ]λd=kλd; α is he drif parameer; σ is he volailiy parameer; dz is a Wiener incremen; q is a Independen Poisson Process wih non-sysemaic jumps. Anoher work ha presens a Muliple Facors Model wih Jump Diffusion Process is Dias & Rocha (1999), in which he auhors proposed he mix of Poisson Process and MRM o represen he sochasic behavior of oil prices in real opions valuaion, as i can be seen in he equaion: dx/x = [η( x x)d λk] + σdz + dq x is he oil price; dz is a Wiener incremen; η is a speed of reversion parameer; x is he equilibrium level o which he process revers in he long run; k=e[ɸ-1]; dq is a Poisson Process incremen which can assume value zero wih 1-λd of probabiliy and ɸ-1 wih λd of probabiliy. Since k=e[ɸ-1] implies ha E[dx/x]= η( x x)d. In his model, similar o Meron (1976), he common news would cause marginal adjuss in oil prices, while abnormal evens such as crisis, wars and economic booms would cause discree jumps on ime. The uncerainy abou he size and direcion of he jump is represened by ɸ. The jumps can be sysemaic, which do no allow o obain a risk neural porfolio, or non-sysemaic, which allow he use of coningen claims. Oher papers (Gibson & Schwarz, 1990; Schwarz, 1997; Pindyck, 1999; Schwarz & Smih, 2000) are focused on he sochasic behavior of commodiy prices. These works claim ha besides MRM facor price processes of some commodiies may also have a sochasic upward rend facor. In pracical erms, his rend facor would end o increase he equilibrium level o which he process revers in he long run as ime passes. These increases would have addiional moivaions o momenary mismaches of supply and demand (capured by MRM) and hey would be caused by he progressive exhausion of naural resources and incremenal coss relaed o new requiremens of environmenal laws, among oher issues. As a conras, he improvemens in he exploraion and producion echnologies could imply in a downward rend of he commodiy prices. Among oher works ha share he same concep, one ha has a huge 6
populariy is Schwarz & Smih (2000) which proposes a wo sochasic facor model GBM 1 and MRM correlaed and non-observable o describe he behavior of commodiy prices. The sum of hese wo sochasic facor forms he logarihm of he asse price (lns ), as i can be seen in he equaion: ln S S is he spo price of he commodiy; is he facor which represens he changes of he prices in shor erm; is he facor which represens he endency of he prices in he long run. The differenial equaions of he wo sochasic processes are: d d d d dz dzξ.dzχ = ρd. dz is he speedy of reversion parameer of MRM; is he volailiy parameer of he shor run changes in prices; dz is he Wiener incremen of he shor run changes in prices; is he drif parameer of he long run price endency; is he volailiy parameer of he long run price endency; dz is he Wiener incremen of he long run price endency; ρ is he correlaion parameer of he wo facor incremens. In order o esimae he parameers of Schwarz & Smih (2000), he auhors used fuure prices of commodiies and applied he Sae-Space approach combined wih Kalman Filering 2. An ineresing way o summarize and caegorize he sochasic models used in he real opions is presened by Dias (2009), in which he processes are classified in hree levels, as i can be seen in he able 1. 7 1 The model proposes an Arihmeic Brownian Moion (AMB) for he long run endency of he price logarihm, which would be equivalen o a GBM for he prices. 2 The Space-Sae approach is an adequae ool o deal wih sae variables ha are no observable; neverheless i is known ha hose are generaed by a Markov process. When he model is placed in he Space-Sae approach, he Kalman Filer combined wih maximum likelihood esimaors can be used o esimae he parameers of unobservable sae variables, which in he case of Schwarz & Smih (2000) would be he spo price of commodiies.
8 Type of Sochasic Process Model Name References Unpredicable Model Predicable Model More Realisic Models Table 1 - More Usual Sochasic Processes Geomeric Brownian Moion (MGB) Pure Mean Reversion (MRM) Two or Three facor models, and Mean reversion o uncerain long erm mean Mean Reversion wih Jumps Paddock, Siegel & Smih (1988) Dixi & Pindyck (1994), Schwarz (1997, model 1) Gibson & Schwarz (1990), Schwarz (1997, models 2 & 3), Baker, Mayfield & Parsons (1998), Schwarz & Smih (2000) Dias & Rocha (1999, 2001), Aiube, Tio & Baidya (2008) 3 Tess for Deerminaion of Sochasic Processes Some saisical ools can be useful o research which sochasic process would be prevalen in he asse prices and oher ype of uncerainies. One of he mos used approaches in his ask is he Uni Roo Tes, also known as Dickey-Fuller Tes. This es consiss of he analysis of he hypohesis ha he slope (b) of he regression beween he log-reurns and lagged log-reurns of he prices is differen from 1, as shown in he equaion: ln(x ) = a + b ln(x -1 ) + ε Where x is he asse price in he ime. Failure o rejec he null hypohesis would srenghen he idea of he presence of GBM. The criical values of Dickey-Fuller Tes can be seen in he able 2. Significance Level 1% 2.5% 5% 10% Criical Values -3.43-3.12-2.86-2.57 Source: Wooldridge, 2000, p. 580 Table 2 - Asympoic Criical Value of he -es for he Uni Roo wih no rend. In case of auocorrelaion beween he log-reurns of prices and residues of he regression i is recommended he use of Augmened Dickey-Fuller Tes. In his es i should be included sufficien lagged log-reurns so ha he residues become a whie noise. In case log-reurns do no presen saionariy, i is recommended he Dickey-Fuller Tes wih Tendency. This es consiss of applying a regression beween log-reurns and lagged log-reurns including a drif as can be seen in he equaion: ln(x ) = a + b ln(x -1 ) + c + ε Where c is he coefficien of he endency. The criical values of Dickey-Fuller Tes wih Tendency can be seen in he able 3. Significance Level 1% 2.5% 5% 10% Criical Values -3.96-3.66-3.41-3.12 Source: Wooldridge, 2000, p. 583 Table 3 - Asympoic Criical Value of he -es for he Uni Roo wih rend
Generally, i is difficul o rejec he hypohesis ha he process follows a GBM, neverheless i does no mean ha here would be anoher process more suiable o describe he prices behavior. An ineresing resul is when b<1, which would indicae he possibiliy of MRM presence, even in he cases ha GBM has no been rejeced. In order o illusrae his difficuly, we can menion Dixi & Pindyck (1994), in which he auhors expose ha ess made wih 30 and 40 year price series did no allow o rejec he hypohesis ha oil prices would follow GBM. I was necessary o make ess wih 120 year series o manage he rejecion of he uni roo. Oher approach ha can be used o suppor he choice of he sochasic process is o verify if he level of he shocks is persisen, which could be more relevan han he uni roo research. In he auoregressive processes such as MRM he shocks end o dissipae when here is permanen reversion srengh. As a conras, in case of GBM which is no an auoregressive process he shocks in prices are persisen. In order o verify his condiion, Pindyck (1999) proposes a Variance Raio Tes, which consiss of verifying if he log-reurn variance increases proporionally in ime, ha is one of he main imporan hypohesis of GBM. The es measures he level o where he variance converges wih he increase of he lags in he log reurns. The Variance Raio can be measured by he equaion: 1 Var P k P Rk k Var P P 1 The erm Var (.) in he formula represens he variance of he series of he differences beween he log-reurns of prices: P, wih lag of k periods. In case of GBM, i would be expeced ha variance would increase proporionally and linearly o k, which implies ha R k should converge o 1 when k grows. As a conras, in case of MRM he variance is limied o a cerain level even considering he growh of k which implies ha R k should decrease when k grows. In addiion o Dickey-Fuller and Variance Raio Tess, analysis can be made wih Adherence Measures in sample, which compares one sep ahead resuls esimaed by he models and he observaions of he price series correspondens. Among oher measures, 3 approaches ypically used in his analysis are: Pseudo R 2, Mean Quadraic Error (MQE) and Mean Absolue Percenage Error (MAPE). The Pseudo R 2 consiss of he square of he correlaion beween he values of he price series and he forecass one sep ahead, boh relaed o he same period. Larger values (closer o 1) of he Pseudo R 2 indicae a higher adherence of he model. The Pseudo R 2 can be calculaed by he equaion: S, E( S S )) 2 2 Pseudo R ( 1 ( a, b) is he correlaion beween a and b; S is an observaion of he price series; E( S ) S 1 is he esimaed value one sep ahead of he price series. The Mean Quadraic Error is he average of he square of he difference beween he values of he price series and he esimaed prices one sep ahead. Lower values for he MQE indicae a beer predicive abiliy of he model. The MQE can be calculaed by he equaion: MQE Average 2 S E( S S 1) S is an observaion of he price series; 9
10 E ( S S 1) is he esimaed value one sep ahead of he price series. Mean Absolue Percenage Error is he average of he modulus of he difference beween he values of he price series and he esimaed prices one sep ahead, sandardized by he values of he price series. Similarly o MQE, lower values of MAPE indicae a beer predicive abiliy of he model. The MAPE can be calculaed by he equaion: MAPE Average S E( S S 1) S S is an observaion of he price series; E ( S S 1) is he esimaed value one sep ahead of he price series. 4 Theoreical Consideraions abou he Sochasic Process Choice in he Real Opion Valuaion Beyond he saisical ess, he choice of he sochasic process o represen he uncerainies in he real opions valuaion can be suppored by heoreical consideraions referenced in he economic heory. An example would be he assumpion of he equilibrium mechanism in he prices of commodiies which would jusify he use of MRM o represen he behavior of he price of hese asses. In he same vein, he supposiion of he gradual increase in he producion marginal cos and he occurrence of rare evens (such as crisis and wars) would suppor he mix of MRM wih GBM and Poisson Process, respecively, in he search for more realisic models. Anoher relevan issue is he lifeime of he asses. Generally, if he lifeime is relaively shor, furher research o deermine he bes sochasic process could be considered as a maer of minor relevance, allowing he choice of he process o be guided by such issues as he ease in calibraion of parameers and consrucion of he valuaion model. Dixi & Pindyck (1994) show ha in shor periods of ime, price processes of GBM ype are mainly guided by he sochasic shocks, while as ime passes he drif componen becomes more relevan. Thus, as in mos models he randomness is represened by incremens of Wiener ha is, reaed similarly as GBM he search for a more appropriae process o represen he sochasic behavior of prices could be considered an expensive ask, aking ino consideraion he benefis o be obained. As a conras, in cases of long lifeime of he asses, he research o obain a process wih more adherence o he performance of he asse prices could be crucial o is valuaion and he definiion of invesmen rule. Basian-Pino, Brandão & Hahn (2009) show ha in a swich opion valuaion in he sugar-ehanol indusry, he difference in he opion value could change from 20% o 70% in relaion o he base case, when he uncerainies are modeled by MRM and GBM, respecively. Kerr, Marin, Pereira, Kimura & Lima (2009) esimaed he opimum ime o cu rees in he fores producs invesmens considering ha uncerainies could be modeled by MRM and GBM. The auhors concluded ha he criical prices o decide he cu in relaion o he ime, would change subsanially when a differen ype of process is used. In he case sudied by he auhors, he use of MRM could anicipae he exercise decision of he opion when he resuls are compared o GBM. Alhough he mixing of processes can generae more realisic models, his implies in a greaer difficuly in he parameers esimaion. Usually, he muliple facor models require fuure price series of he asses for he parameers esimaion, as in Schwarz (1997) and Schwarz & Smih (2000). In hese works he auhors used he Space-Sae approach mixed wih he Kalman Filer applied in series of fuure prices of commodiies o esimae joinly he parameers of he models. Neverheless, in mos cases of commodiies prices and oher kind of uncerainies, fuure prices are no available, such as ehanol prices and he volume of raffic on a oll road. In
hese cases, despie he advanages of using muliple facor models, he choice of sochasic process migh be influenced by limiaions relaed o he daabase availabiliy. Regarding he qualiy of he available daabases o calibrae he parameers of he models i is imporan o consider he exension and periodiciy of he price series. Generally, i is recommended o use long ime series for esimaing he drif parameers. Taking ino accoun ha he variance of he drif esimaor is proporional o ime, he longer he series he more efficien will be he esimaor. Neverheless, he informaion periodiciy is relevan o calibrae he volailiy parameers. The higher he informaion frequency he beer he esimaor qualiy. Finally, an issue ha should be considered in he choice of he sochasic process is is applicabiliy on close formulas and numerical soluions used in real opions valuaion. Comparing GBM wih oher models, one of is bigges advanages is he small number of parameers o calibrae and he ease of obaining analyical soluions, which are huge incenives o is use. Generally, he use of MRM does no allow he achievemen of analyical soluions o he decision rule, which implies in he use of numerical soluions such as Mone Carlo Simulaion (MCS) and Binomial Laices 3. Usually, i is possible o obain soluions for muliple facor models using MCS and when here is more han one uncerainy o be considered in he analysis. I is imporan o observe ha before 1993 MCS was only used in he soluion of European opions and since hen, wih he developmen of opimizaion mehods pluggable o MCS, i became possible o value American opions (Dias, 2008). 11 5 Conclusions In his work a discussion is raised abou he alernaives of sochasic processes for applying in he real opions valuaion. Besides he main ypes of pure sochasic models, more conemporary models were presened, which mix differen kinds of processes in order o provide a more realisic in he characerizaion of he uncerainies involved in he analysis. In several siuaions mainly in projecs wih long lifeime he choice of sochasic process can be relevan in he real opions valuaion, wih influences on he value and opimal rule of he invesmen. Typically, i is recommended he Dickey-Fuller Tes as a suppor o he choice of sochasic processes; neverheless in mos cases he resuls of he ess are inconclusive. In secion 3 were presened some ools ha can improve Dickey Fuller Tes and oher saisical approaches, in order o obain more conclusive resuls in he analysis. In addiion o saisical ess, some heoreical consideraions based on microeconomics and some resricions caused by he availabiliy of he daabase were considered relevan o define he mos appropriae sochasic model. As furher research in he same field, i is suggesed he sudy of oher ypes of saisical ess for he analysis of adherence of muliple facor models and he analysis of implicaions relaed o he use of hese models on he several kinds of managerial flexibiliies. 6 References Aiube, F. A. L., Tio, E. A. H., & Baidya, T. K. N. (2008). Analysis of Commodiy Prices wih he Paricle Filer. Energy Economics, v. 30, n. 8. Al-Harhy, M. H. (2007). Sochasic Oil Price Models: comparison and impac. The Engineering Economis, 52(3), 269-284. Baker, M., Mayfield, E., & Parsons, J. (1998). Alernaive Models of Uncerain Commodiy Prices for Use wih Modern Asse Pricing Models. The Energy Journal, 19(1), 115-148. 3 Nelson & Ramaswamy (1990) and Basian-Pino, Brandão & Hahn (2010) presen alernaive approaches for binomial laice o he MRM.
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