A Methodology to Forecast Commodity Prices in Vietnam

Transcription

1 Inernaional Journal of Economics and Finance; Vol. 7, No. 5; 2015 ISSN X E-ISSN Published by Canadian Cener of Science and Educaion A Mehodology o Forecas Commodiy Prices in Vienam Thi Ngoc Trang Nguyen 1 & Ngoc Tho Tran 1 1 School of Finance, Universiy of Economics Ho Chi Minh Ciy, Vienam Correspondence: Thi Ngoc Trang Nguyen, School of Finance, Universiy of Economics Ho Chi Minh Ciy, Ho Chi Minh Ciy, Vienam. rangcdn@ueh.edu.vn Received: February 22, 2015 Acceped: March 7, 2015 Online Published: April 25, 2015 doi: /ijef.v7n5p44 URL: hp://dx.doi.org/ /ijef.v7n5p44 Absrac Risk managemen of commodiy prices is an imporan ye challenging ask. Given he complex behaviour of commodiy prices, his creaes he need of using sophisicaed models of commodiy prices dynamics. Obviously, parameer esimaion of such models poses anoher challenge. Previous lieraure has addressed his problem using Markov Chain Mone Carlo, which is compuaionally expensive for parameer esimaion and inference. In his paper we develop an efficien Maximum Likelihood Esimaion procedure based on he characerisic funcion. We hen esimae parameers a sochasic volailiy model wih sochasic drif uilizing he ime-series of rice and coffee prices. We show ha such model produces realisic disribuions of boh commodiy prices. Finally, using he esimaed model parameers we calculae various risk measures such as Value a Risk or Expeced Shorfall. Keywords: risk managemen, commodiies, maximum likelihood 1. Inroducion Commodiies are broadly defined as (physical) producs which can be raded on he marke. Examples of commodiies include energy, meals or agriculural producs such as coffee and rice. These producs are fundamenally imporan o he world economy, which is driven by increasing demand as well as poliical facors including marke regulaion. Buyers and sellers can rade commodiies in he spo marke as well as he derivaives marke. Marke paricipans who undersand he behaviour of commodiy prices are more likely o have more realisic expecaions abou fuure price movemens. Solid undersanding of commodiy markes is also imporan for companies seeking o hedge heir underlying price risk as well as hose invesing in commodiies. Commodiy invesmens differ from socks and bonds in ha hey canno produce a sream of cash flows. Hisorically, commodiy prices increase wih inflaion, which makes commodiies a good hedge agains inflaion. Commodiies offer diversificaion benefis because commodiy reurns end o have low correlaion wih oher asse classes. Over he pas decade, commodiies have become increasingly popular wih many insiuional invesors, as par of heir diversificaion sraegies. Many commodiy markes are characerised by spikes, seasonaliy, mean-reversion and non-consan volailiy. The non-consan volailiy observed in commodiy markes is a moivaion for sochasic volailiy model of commodiy prices. Commodiy prices exhibi significan ime-varying volailiy because elasiciies of shor erm producion and consumpion are low. Volailiy of commodiy prices can reduce he profi of business or invesmen sraegies and negaively affec consumers. Observed variabiliy of commodiy prices elevaes imporance of risk managemen echniques o monior, measure and manage commodiy risk. Effecive risk managemen requires horough undersanding of commodiy price behaviour and sophisicaed modelling of commodiy prices. Demand for robus and effecive ways of risk managemen is also simulaed by he growh of commodiy marke. Managing commodiy risk poses a challenge for boh producers and consumers of commodiies. Because of he naure of heir business, produces hold commodiies in sock. Consequenly, heir operaing cash-flow is affeced by commodiy price changes. As for consumers, hey are in general price-akers and herefore heir ne cash-flow is also affeced by flucuaions of commodiy prices. The remainder of his paper is organized as follows. Secion 2 reviews relaed lieraure. Secion 3 describes he daa and mehodology. Resuls are presened in Secion 4. Finally, Secion 5 concludes he paper. 44

2 Inernaional Journal of Economics and Finance Vol. 7, No. 5; Lieraure Review I is a well know fac ha volailiy of he financial ime-series is ime-varying, which has been already shown by Engle (1982). Moreover, financial ime-series exhibis wha is known as volailiy clusering, meaning ha large (small) moves are followed by large (small) moves. Anoher wo imporan sylized facs of he financial ime-series are heavy ails and non-zero skewness of he reurns disribuion. Sochasic volailiy models have been developed o capure sylized facs of he financial ime-series wih mos noable conribuions from Wiggins (1987), Hull and Whie (1987, 1988), Sco (1987), Sein and Sein (1991) and Heson (1993). The mos pronounced difference beween hese models lies in he specificaion of he volailiy process. Sco (1987), Wiggins (1987) and Sein and Sein (1991) all assume ha he logarihm of he variance follows a mean-revering Ornsein-Uhlenbeck process. In he Hull and Whie model, he variance process follows a geomeric Brownian moion wih he volailiy of volailiy being a linear funcion of he insananeous variance. On he oher hand, he variance process in he Heson model is given by a mean-revering Cox-Ingersoll-Ross process wih he volailiy of volailiy being a linear funcion of he insananeous volailiy. The idea o use sae of ar, ye mahemaically racable sochasic volailiy models o model commodiy prices is no new. In fac, Brooks and Prokopczuk (2013) apply he Heson, Baes, as well as Duffie, Pan, and Singleon (2000) models o commodiy prices. The Baes (1996) model exends he Heson model o accommodae jumps following a compound Poisson process in he sock price process. The SVCJ model of Duffie, Pan, and Singleon (2000) assumes ha boh he price and volailiy jump simulaneously, which is moivaed by he fac ha price jumps are ofen accompanied volailiy jumps. Liu, Chng and Xu (2014) analyse hedge raios of commodiy prices generaed by hese hree models. Benh (2011) applies he Heson model as well as he Barndorff-Nielsen and Shephard model o commodiy prices. Brooks and Prokopczuk (2013) as well as Liu, Chng and Xu (2014) use he Markov Chain Mone Carlo (MCMC) o esimae he parameers of hese models. Solibakke (2014) uses a mulifacor sochasic volailiy model for he carbon prices, also applying MCMC for parameer esimaion and inference. MCMC algorihms are in general compuaionally expensive o achieve a high degree of accuracy. The fac ha MCMC is compuaionally inensive for parameer esimaion and inference is also acknowledged by Solibakke (2014). As an alernaive o MCMC, a Maximum Likelihood mehod can be used o esimae he model parameers. In he opion pricing conex, Trolle and Schwarz propose a model based on he Heah, Jarrow, and Moron (1992) framework. In heir model fuures prices are driven by hree facors while opion prices are driven by wo addiional volailiy facors. The model parameers are esimaes using quasi-maximum likelihood and exended Kalman filer. Yan (2002) proposes a mulivariae sochasic volailiy model ha considers sochasic volailiy, sochasic ineres raes, sochasic convenience yields as well as simulaneous jumps in he spo price and volailiy. In his model, he volailiy follows a square-roo jump-diffusion process whereas he ineres rae follows a square-roo process and he convenience yield follows an Ornsein-Uhlenbeck process. Hikspoors and Jaimungal (2008) assume ha sochasic volailiy in a facor commodiies spo price model follows mean-revering Ornsein-Uhlenbeck process. Li and Linesky (2014) model commodiy prices using subordinae Ornsein-Uhlenbeck process. Their model also feaures sochasic volailiy and mean-revering jumps. In he energy marke conex, Eydeland and Geman (1998) were among he firs o use he Heson model. Barndorff-Nielsen, Benh and Veraar (2013) inroduce volailiy modulaed Lévy-driven Volerra processes for modelling energy prices. Their model allows for jumps as well as sochasic volailiy. They poin ou ha he laer is a key facor in modelling of energy prices. Benh and Vos (2013) propose a mulivariae sochasic volailiy model for energy prices. The model is able o capure sochasic volailiy, price spikes and mean-reversion of prices observed in he energy markes. 3. Daa and Mehodology We obain monhly ime-series of rice (summer-auumn crop) and robusa coffee beans prices from February 2009 unil November 2014 from DaaSream. The ime period considered is limied by he daa avaialbiliy. Boh ime-series are denominaed in Vienam Dong (VND). We choose rice and coffee as hese are by far he wo mos imporan commodiies produced in Vienam. Summary saisics of he daa are presened in Table 1. 45

3 Inernaional Journal of Economics and Finance Vol. 7, No. 5; 2015 Table 1. Summary saisics Mean Variance Skewness Kurosis Minimum Maximum Rice Coffee We conclude ha boh price disribuions are skewed o he lef. The disribuion of rice price is lepokuric, meaning i is peaked and has fa ails, whereas he disribuion of coffee price is playkuric, meaning i is less peaked and has hinner ails. This empirical evidence moivaes he validiy of a sochasic volailiy model is o capure he disribuion of boh rice and coffee prices. Heson model wih sochasic ineres raes has been inroduced in Grzelak and Ooserlee (2011). The dynamics of he model is given by he following se of sochasic differenial equaions: dv ds = r S d + v S dw (1) dr S v ( μ v ) d γ vdw p r ( θ r ) d ηr dw = κ + (2) = λ + (3) where <dw S dw v >=ρ S,v d, <dw S dw r >=ρ S,r d and <dw S dw v >=0. In he above equaion, S denoes he commodiy price, r denoes he drif, v denoes he variance, κ denoes he variance mean-reversion, µ denoes he long-erm variance, γ denoes he volailiy of volailiy, λ denoes he drif mean-reversion, θ denoes he long-erm drif and η denoes he volailiy of drif. ρ S,v is he correlaion beween commodiy price changes and changes in variance while ρ S,r is he correlaion beween commodiy price changes and changes in drif. For he Heson-Hull-Whie model p=0 whereas for he Heson-Cox-Ingersoll-Ross model p=0.5. I is assumed ha he correlaion beween changes in he variance and changes in drif is 0. I is imporan o noe ha he parameers of he variance process in he Heson-Cox-Ingersoll-Ross model have a meaningful economic inerpreaion. For example, he mean-reversion parameer deermines he degree of volailiy clusering. Correlaion beween commodiy price and variance processes affecs he heaviness of he ails. On he oher hand, he peakness of he disribuion depends on he volailiy of volailiy. As shown in Grzelak and Ooserlee (2011), for he above model i is possible o approximae he characerisic funcion by: ( u ) exp( A( u, τ ) + B ( u, τ ) x + C ( u, τ ) r D ( u τ ) v ) ψ = 0 0 +, 0 (4) where x 0 =log(s 0 ) and τ=t-. We refer o Grzelak and Ooserlee (2011) for deails. The probabiliy densiy funcion (pdf) can be obained as follows. Firs, le us use he following inversion formula: ( iω log( s ) ψ ( ω) 1 1 exp Pr ( S > s ) = + dω π Re (5) 2 iω 0 where ψ(ω) is he characerisic funcion. By definiion, he cumulaive disribuion funcion (cdf) is given by: ( s ) = ( S s ) = 1 Pr( S s ) F Pr > (6) Finally, differeniaing F(s) respec o s yields he pdf: f [ ]ω d 1 1 ( s ) = + Re exp( iω log( s ) ψ ( ω) 2 sπ 0 which can be evaluaed numerically using he Gaussian quadraure. Having he pdf, i is also possible o use Maximum Likelihood Esimaor o obain he model parameers. We noe ha he Heson model wih sochasic ineres raes is an affine model. This is because is characerisic funcion is an exponenial of an affine funcion of he sae variables. The class of affine models is very rich and conains many financial models such as Black-Scholes, Meron, Variance Gamma, Heson, Baes or Carr Geman Madan Yor o name bu a few. In fac, he mehodology presened above applies no only o he Heson model wih sochasic ineres raes, bu exends o he whole class of affine models. (7) 46

4 Inernaional Journal of Economics and Finance Vol. 7, No. 5; 2015 The Maximum Likelihood Esimaion procedure oulined above has exensive applicaions in risk managemen. For example, resuling model parameers can be used o projec he disribuion of risk facors over a given ime horizon. Moreover, using numerical inversion of he cdf, i is possible o calculae risk measures such as a Value a Risk (VaR) or Expeced Shorfall (ES). 4. Resuls We apply he mehodology described in he previous secion o esimae he parameers p off he Heson model wih sochasicc ineres raes. We noe ha i akes abou 10 seconds o esimae he model parameers on a lapop wih he i7-3610qm 2.3GHz processor and 16Gbb RAM. Beween he Heson-Hull-WH Whie model and he Heson-Cox-Ingersoll-Ross model he laer is preferred as i has lower Akaike informaion crierion. The esimaed model parameers are presened in Table 2. Table 2. Esimaed model parameers r 0 v 0 Rice-2M Rice Rice+2M Coffee-2M Coffee Coffee+2M κ µ γ λ θ η ρ S,v ρ S,r For boh rice and coffee, he speed of mean-reversion of he variance process is much higher han ha of he drif process. Moreover, he variance process has usually higher volailiy, bu lower long-ermm compared o he drif process. Finally, while boh price processes are negaively correlaed wih he commodiyy price changes, i is he variance process ha exhibis moree negaive correlaion o changes in he commodiy price. To invesigae he robusness of he esimaed model parameers we exended or reduced he lengh of hee sample period by wo monhs and hen reesiamed he model. The resuls of he robusness checks are also presened in Table 2. For example, he firs row of Table 2 repors r he esimaed model parameers for rice when he samplee period is reduced by wo monhs. Similarly, he las row of his able repors he esimaed model parameers for coffee when he sample period is exended by wo monhs. We noe ha he esimaed model parameers are remarkably robus as hey do no change much when exending or reducing he lengh off he sample period. Figure 1. Realized commodiy prices as well l as 5h and 95h perceniless of simulaed d price disribuions 47

5 Inernaional Journal of Economics and Finance Vol. 7, No. 5; 2015 We noe ha in he above experimen we only used 64 monhly observaions ou of 70, because in he nex experimen we would like o perform a simulaion sudy o backes he model. The idea is o forecas 6 monhs forward coffee price a May 2014 and compare he realized coffee price wih he model forecass. These forecass provide informaion abou he likelihood of rising and falling price scenarios occurring. Such informaion is paricularly useful for risk managemen in ha he price risk can be quanified. In Figure 1 we display he realizaions as well as he 5h and 95h perceniles of simulaed rice and coffee price disribuions obained using 100,000 Mone Carlo simulaions. Figure 1 clearly shows ha he realized rice and coffee prices lie beween he 5h and 95h perceniles of simulaed price disribuions. In summary, Heson-Cox-Ingersoll-Ross model produces realisic disribuions of risk facors over a given ime horizon and hus is suiable for risk managemen purposes. Now we urn o calculaion of wo risk measures, namely Value a Risk (VaR) and Expeced Shorfall (ES). The firs one is a saisical measure of downside risk expressed in moneary erms. I is he maximum loss over a arge horizon esimaed wih a given level of confidence. For example, wih 99% confidence, here is a less han 1% chance ha he acual loss will be larger han he VaR. Anoher key par of he VaR definiion is he holding period. The choice of holding period will depend on how he resulan VaR is o be used. VaR is a key measure in risk managemen. A complemenary risk measure o VaR is he ES. I has an advanage over VaR of being a coheren risk measure, because i is no only monoonic, invarian under ranslaion, and homogeneous, bu also sub-addiive. ES is calculaed as an average of losses ha are larger or equal o VaR. 5% VaR esimaed a May 2014 over 6 monhs horizon is 1258 (rice) and 6198 (coffee). I is calculaed as a difference beween he coffee price a May 2014 minus 5h percenile of he inverse cdf of he Heson-Cox-Ingersoll-Ross model, wih parameers given in Table 2, evaluaed a he probabiliy of 5%. On he oher hand, 5% ES esimaed a May 2014 over 6 monhs horizon is 1538 (rice) and 7612 (coffee). I is calculaed as an average of VaR over various percenile levels. For comparison, he loss realized over 6 monhs on a porfolio consising of one uni of rice is 90, whereas he loss realized over 6 monhs on a porfolio consising of one uni of coffee is Conclusion In his paper we address he problem of commodiy risk managemen. Proper commodiy risk managemen is a maer of significan imporance for commodiy invesors as well as firms ha produce and consume commodiies. Complex behaviour of commodiy prices requires sophisicaed models of commodiy prices dynamics. Parameer esimaion of such models is a difficul ask. Previous lieraure has addressed his problem using Markov Chain Mone Carlo, which requires large compuaional effor. This paper conribues o he exising body of lieraure by developing an efficien Maximum Likelihood Esimaion procedure based on he characerisic funcion o esimae parameers of affine models. We use i o esimae parameers of he Heson-Cox-Ingersoll-Ross model uilizing he ime-series of rice and coffee prices. Esimaing he model we find negaive link beween price volailiy and he commodiy price changes. We also find ha he speed of mean-reversion of he variance process is much higher han ha of he drif process. Moreover, we demonsrae ha his model produces realisic disribuions of boh commodiy prices. Finally, we use esimaed model parameers o calculae various risk measures such as Value a Risk or Expeced Shorfall. Acknowledgemens This research was funded by a gran from Universiy of Economics Ho Chi Minh Ciy. References Barndorff-Nielsen, O., Benh, F. E., & Veraar, A. E. D. (2013). Modelling energy spo prices by volailiy modulaed lévy-driven volerra processes. Bernoulli, 19(3), hp://dx.doi.org/ /12-bej476 Baes, D. (1996). Jumps and sochasic volailiy: The exchange rae processes implici in deuschemark opions. Review of Financial Sudies, 9(1), hp://dx.doi.org/ /rfs/ Benh, F. E. (2011). The sochasic volailiy model of barndorff-nielsen and shephard in commodiy markes. Mahemaical Finance, 21(4), hp://dx.doi.org/ /j x Benh, F. E., & Vos, L. (2013). Cross-commodiy spo price modelling wih sochasic volailiy and leverage for energy markes. Advances in Applied Probabiliy, 45(2), hp://dx.doi.org/ /aap/ Brooks, C., & Prokopczuk, M. (2013). The dynamics of commodiy prices. Quaniaive Finance, 13(4), hp://dx.doi.org/ / Duffie, D., Pan, J., & Singleon, K. (2000). Transform analysis and asse pricing for affine jump-diffusions. 48

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