Water price dynamics, water derivatives and general equilibrium modelling

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1 18 th World IMACS / MODSIM Congress, Carns, Australa July Water prce dynamcs, water dervatves and general equlbrum modellng Schreder S. 1 1 School of Mathematcal and Geospatal Scences, RMIT Unversty, Melbourne Emal: Serge.Schreder@rmt.edu.au Abstract: The Councl of Australan Government s (COAG) water reforms of 1994 opened a new era n the rrgated agrculture market. After the separaton of land and water rghts, water became a tradable commodty. In the past, the value of water was sgnfcantly underestmated, whch led to rratonal ways of usng ths valuable natural resource. The water tarffs were not related to the margnal values of crops grown by rrgators, whch caused unsustanable consumpton of water, especally durng the drought years. The major am of the COAG reforms was to enable a proper market value for water and to re-allocate water consumpton to more economcally valuable crops. In , the total water traded n Australa was 1300 GL wth 502 GL of Vctora water traded,.e. about 10% of overall use. Gven the severty of drought condtons throughout Australa n recent years, and the uncertanty of whether ths s a permanent clmate change, water prce and supply s a crtcal ssue for farmers. In ther present stage of development, markets offer farmers lttle nformaton about long-term prces and quanttes, and lmted opportunty for avodng dsastrous water shortfalls or unaffordable water prces beyond the current season. Even for the current season, the present tradng system stll leaves farmers facng unhedgable uncertanty concernng water prces and quanttes. Ths uncertanty appears to be ncreasng. For example, n the Goulburn system, managers am to provde full allocatons for 96 years out of 100. Yet three tmes n the past decade, n , and , allocatons n Goulburn have been below 100%. In response, wth growng concerns over changes n ranfall patterns, there may be a growng bas among rrgators aganst the producton of water-dependent perennal commodtes whch requre secure water allocatons. Moreover, the annual values for these prces and quanttes are not enough for effectve rsk management and farmers are keen to have ths nformaton wth more detaled temporal resoluton, on a fortnghtly or weekly bass. The analyss of hstorc records of water prces over 5 rrgaton seasons n the Goulburn Murray system, suggests that jumps are a dstnctve feature of the water prce seres, and to account for these jumps, the jump dffuson modellng approach s approprate. A Brownan moton plus Compound Posson process s proposed to model the water prce dynamcs. The smulaton result suggests that the stylsed water prce dynamcs can be reproduced by the model. The concept of ntegraton of a modfed computable general equlbrum (CGE) model TERM-H2O, wth a weekly/fortnghtly stochastc model for water prces s ntroduced. As the water market s gradually maturng and water contnues to be a valuable commodty, some new assets exstng on other economc markets can be ntroduced to the water market. Optons are one type of very mportant assets n the fnancal market, whch allow sgnfcant ncreases n the hedgng ablty of ther users. In addton, optons also provde greater prce certanty to ther users and allow them to manage ther fnancal rsks n a more effcent manner. Ths work dscusses and conceptualses the possblty of ntroducng optons nto the water market. Some mportant restrctons for water optons are outlned: a) the opton tradng wll be only ntated after a short ntal perod of water tradng has passed; b) optons are avalable only for a specfed season; c) weekly pool water prces are not mpacted by the actual water allocaton,.e. they are purely stochastc and d) as the water prce tends to drop sharply towards the end of a season, the opton s maturty date s restrcted to much earler than the seasonal endng date. The model of opton prcng for the water market s developed and the potental mpacts on the underlyng water market and market rsks are dscussed. Takng nto account these restrctons and some classcal assumptons from fnancal mathematcs (zero transacton cost, arbtrage opportuntes, no storage costs etc.) a model for European optons for water was developed. The potental mpacts, postve as well as negatve, of the ntroducton of water optons to the market are dscussed. Keywords: Irrgaton water, water tradng, CGE modelng, water optons 3640

2 1. INTRODUCTION Mult-bllons of dollars worth of rrgated lands n Vctora and New South Wales have been severely affected by a seres of droughts, whch were especally serous durng the 2006/07 and 2007/8 seasons, and are expected to contnue well nto the foreseeable future. The agrcultural ndustry n Australa s a very rskprone busness. The major rsk factor s assocated wth the hgh varaton of clmatc condtons whch leads to water abundant years, often assocated wth floods, regularly alternatng wth extreme long-lastng droughts. Clmatologsts have many reasons to beleve that the clmate varablty wll get consderably larger n future. The duraton and severty of droughts wll also ncrease drven by the global change of clmate. The agrculture ndustry of Australa wll face more rsks n future and new measures of rsk management and adaptatons are very topcal at present. The Garnaut Report (2008) ndcates that under no mtgaton measures t s expected a 92% declne n rrgated agrcultural producton n the Basn, affectng dary, frut, vegetables, and grans. The problem of rsk management s an especally mportant problem for regons of rrgated agrculture, such as the Goulburn-Murray rrgaton dstrct n Northern Vctora or the Murray-Murrumbdgee catchment n New South Wales, where securty of water supply depends not only on current clmatc condtons but on many addtonal factors such as the prevous years' precptaton level, current water allocaton, water prce dynamcs and the current water prce. Farmers n such regons tend to grow crops wth hgh margnal value per unt volume of water, very often perennal crops, whose survval depends very much on the level of water supply securty whch makes local ndustry very vulnerable to rsks assocated wth clmatc varatons. A closely related ssue, whch also hghlghts the mportance of rsk management n the agrcultural busness, s the maturng of the Australan water markets. The current system of water tradng n Vctora works as follows: Each farm n an rrgaton regon, for example the Goulburn valley, has a hstorcally determned water enttlement. At the begnnng of each rrgaton season, say September, the water authorty announces what fracton of the enttlement they antcpate wll be avalable for the season. The determnaton of ths fracton depends manly on the level of water reserves. The fracton may be rased durng the season f ranfalls are favourable and lowered f they are not. Throughout each season, farmers can sell ther water rghts to other farmers through a water market whch s organsed by the local water authortes (The Goulburn-Murray Waters n the Goulburn-Murray rrgaton regon). The volume of water traded n Northern Vctora has sgnfcantly ncreased over the last decade. Water tradng s now an mportant manageral problem each rrgator n the regon s faced wth. What s the better choce: to use allocated water to rrgate crops or to sell water allocatons to another market player? If a farmer sells too much water wthout leavng any reserves for rrgaton he faces the rsk of loosng hs crops, especally perennal ones, f precptaton level remans low. If he stores too much enttlements and ranfall level s hgh over the season he looses the possble water revenue. Determnaton of more effcent water tradng strateges for farmers s one of the challengng research problems. Ths objectve has been realzed through the development and mplementaton of TERM-H20, a computable general equlbrum (CGE) model (Dxon et al., 2008). The advantage of applyng CGE models (Dxon et al., 1992) for water tradng predcton has two man aspects. Frstly, all partal equlbrum models are based on a sngle objectve functon (OF) optmsaton. The typcal OFs used are revenue or proft and the assumpton of constant prces of agrcultural commodtes s appled. Ths assumpton s not vald for such large economes as Goulburn- Murray rrgaton dstrct, nor on the macroeconomc scale. In CGE models all commodty prces can be treated as endogenous varables. Secondly, CGE models use as a system nput the nput-output tables provded by the ABS. Smlarly, ther output can be easly represented n the form of ABS data. Ths makes CGE models the most convenent tool for keepng account of tradng water and water used for rrgaton. To keep rrgated water ncluded n the water account, the CGE model should be combned wth a water allocaton model; detals of ths ntegraton process are presented n the accompanyng paper (Fernandes and Schreder, 2009). The TERM-H2O model can predct water prces and quanttes bought and sold, and access the macroeconomc mpacts of water tradng. However, ths model s mplemented wth an annual tme step and s unable to trace the water prce dynamc wthn the rrgaton season. Ths paper provdes a model for predctng a water prces on the weekly bass usng the theory of stochastc processes wth jumps. 2. WATER PRICES AS A STOCHASTIC PROCESS WITH JUMPS 2.1 Water market n Vctora Watermove s a water exchange operated by Goulburn-Murray Water and t acts as an ntermedate to facltate water tradng by provdng market nformaton for people seekng to trade water. It conducts both 3641

3 temporary and permanent tradng throughout Vctora and conducts water exchanges for all water tradng zones. Tradng zones (total of sx n the state) defnes the physcal boundares to, from or wthn whch water may be traded n Vctora. Thus, Watermove determnes a pool prce for each tradng zone where trade can occur. Successful sellers wll receve a prce greater than or equal to ther offer prce and successful buyers wll receve a prce less than or equal to ther offer prce. In most locatons, water rghts and dverson lcences can be traded permanently and temporarly. In some locatons and wth some lmtatons, sales water can be traded temporarly. Sales water s the volume allocatons above water rghts and dverson lcences. A pool prce s calculated by Watermove for all tradng zones where trade can occur. All successful sellers and buyers wthn a tradng zone receve the same pool prce. Sellers are consdered on an ascendng prce bass. The lowest prce seller wthn a tradng zone s the frst seller elgble to trade. Buyers are consdered on a descendng prce bass. The hghest prce buyer wthn a tradng zone s the frst buyer elgble to trade. The pool prce for a tradng zone s greater than or equal to the sell prce offered by successful sellers and less than or equal to the buy prce offered by successful buyers. 2.2 Why jumps? In order to set up a conceptual framework for prcng water prce based optons, t s essental to understand the stylsed features embedded n water prce seres. Thus, before we can make realstc assumptons wth regard to the water prce dynamcs, we need to dentfy these stylsed features and understand the behavour of the water prce dynamcs. For the purpose of ths study, we lmt ourselves to Temporary Water Rght/Dverson Lcence n 1A Greater Goulburn Vctora, as ts pool water prce seres and ther correspondng actual water allocatons are readly avalable. Fgures 1 and 2 show the weekly pool water prces and the actual water allocatons for the rrgaton perods 2005/06 (relatvely wet) and 2006/07 (severely dry). Pool Prces Pool Prces $ Pooled Prce and Water Allocaton (21/07/05-29/06/06) Actual Water Allocaton $100 $90 $80 $70 $60 $50 $40 $30 $20 $10 $0 21/07/2005 4/8/ /08/2005 1/9/ /09/ /09/ /10/ /10/ /11/ /11/2005 8/12/ /12/ /1/ /01/2006 9/2/ /02/2006 9/3/ /03/2006 6/4/ /04/2006 4/5/ /05/2006 1/6/ /06/2006 Dates 29/06/ % 100% 80% 60% 40% 20% 0% Actual Water Allocatons Pool Prces Pool Prces $ Actual Water Allocaton $950 $850 $750 $650 $550 $450 $350 $250 24/08/2006 9/07/ /09/2006 Pooled Prce and Water Allocaton (24/08/06-07/06/07) 10/05/ /10/ /02/ /11/ /11/ /12/2006 1/04/ /01/2007 2/01/ /02/2007 3/01/ /03/ /03/ /04/ /04/ /05/ /05/2007 Dates 7/06/ % 30% 25% 20% 15% 10% 5% 0% Actual Water Allocatons Fgure 1: Pool water prce and water allocaton for the perod 21/07/05 29/06/06. Fgure 2: Pool water prce and water allocaton for the perod 24/08/06 07/06/07. Descrptve statstcs and hstograms were analysed for the prce seres for all fve rrgaton seasons startng from 2002/03. In addton, the Kruskal Walls ANOVA and Medan test was performed to determne whether these fve prce seres are from the same dstrbuton wth the same medan. The analyss of the water prce data dentfed four stylsed features: 1. Weekly pool water prces tend to be at a level specfc to a partcular season; 2. The level of water allocaton has an mpact on the weekly pool water prce levels, but ths mpact s not very strong: prces can move up and down when allocaton level s constant; 3. The weekly pool water prce seres show upward and downward jumps; 4. The pool water prces tend to drop towards the end of the season. To see the frst feature, compare the pool water prce levels n Fgure 1 and Fgure 2. The pool water prces for these two consecutve seasons are very dfferent: the average values dffer by almost tenfold. Ths fndng s also confrmed by the formal Kruskal Walls ANOVA and medan test, whch suggests that these fve pool water prce seres are from dfferent dstrbutons. To see whether water allocaton has any mpact on water prce levels, we check Fgure 1 frst: the weekly pool water prces vary from around $80 to $12 when the water allocaton s between 21% to 100%, wth 100% water allocaton countng for most of the season. From Fgure 2, t can be seen that the weekly pool water prces vary from around $950 to $300 when the water allocaton s between 7% to 27%. Thus, there s a tendency for hgher water prce levels when the water allocaton s low. To confrm that the water prce seres have upward and downward jumps, we look at the followng examples: from Fgure 2, t can be seen that the weekly pool water prce drops from $

4 (21/12/06) to $755 (04/01/07), a 16% drop n prce n one week, and drops further to $ (11/01/07) the followng week (an over 20% drop n prce). Thus, our concluson s that water prce seres contan jumps. The fourth feature that the pool prce tends to drop towards the end of the season s obvous and can be seen from Fgures 1 and 2. Ths could be due to the endng of an rrgaton season n Vctora. The fve water prce seres avalable for analyss show that the water prces tend to have seasonal dfferences, are mpacted to some degree by actual water allocaton (especally when the water allocaton reaches 100% there s a tendency for water prces to follow a fallng trend, wth the excepton of the trend shown n Fgure 1 for the severe drought of 2006/07) and have upward and downward jumps. In addton, the pool water prces tend to drop sharply towards the end of the season. Among the four stylsed features shown n the pool water prce seres, the most mportant feature s the prce jumps n the water prce seres. The other three features, namely dfferent prce levels, mpact of actual water allocaton and tendency to drop n prce towards the end of the season, can be dealt wth separately. However the jump feature s embedded n the pool water prces and can not be assumed away. Ths feature requres specal treatment when we choose a stochastc process to represent the pool water prce dynamcs. Therefore, the stochastc process, whch models water prce dynamcs, must be able to account for these jumps. 2.3 Model for weekly prces of traded water Weekly water prce seres wll be modelled as stochastc processes for rrgaton seasons. Water prces n the southern Murray-Darlng Basn for fve rrgaton seasons, startng at , showed prce jumps. Thus, the jump dffuson modellng approach s approprate. A Brownan moton plus Compound Posson process wth jumps to model the water prce dynamcs s proposed. Ths takes the form: ds(t) = αs(t)dt + σs(t)dw(t) + S(t-)d(Q(t) - βλt) = (α - βλ)s(t)dt + σs(t)dw(t) + S(t-)dQ(t) (1) Here S(t) s the water prce n perod t, W(t) s the Brownan moton and Q(t) s the compound Posson process. The parameter λ s the compound ntensty of the M Posson processes, β = E(Y ) s the average sze of jumps Y. The mean rate of return on the pool water prce s α, and σ s the standard devaton of the Brownan moton process. The soluton has been obtaned by Cu and Schreder (2009) usng methods descrbed n Shreve, (2004). Under the orgnal probablty measure P, the mean rate of return on the pool water prce s α. Y s the random jump sze varable takng values of -1<y 1 < y 2 < <y m, wth probabltes p 1, p 2, p m. Ths assumpton guarantees that although the pool water prce can jump down, t can not jump from a postve to a negatve value or to zero. Ths s smply because Y+1 s always greater than zero based on the above assumpton. The soluton to (1) s N(t) 2 S(t) = S(0)exp{σW(t) + (α - βλ - σ )t} (Y +1) (2) 2 =1 1 Smlarly, under the rsk neutral measure P, the water prce dynamcs can be descrbed as ~ ~ ~ N(t) 2 S(t) = S(0)exp{σ W(t) + (r- βλ - σ )t} (Y +1) (3) 2 =1 1 Fgure 3 shows fve smulated forty weekly pool water prce seres based on the followng specfcatons: S(0) = $100, α = 0.05, β = 0.10, λ = and σ = The szes of fve jumps Y are -0.20, 0.10, 0.15, 0.20 and 0.25, respectvely. They are assumed to occur wth equal probabltes p of

5 5 Smulated 40 Weekly Water Prce Seres Water Prce Week Fgure 3: Fve smulated pool water prce ($/ML) seres; the model parameters for them are specfed above. 3. WATER DERIVATIVES: MODEL FOR EUROPEAN OPTIONS FOR WATER The theory of dervatves s concerned wth explanng the emergence of dervatve markets, the benefts that these markets confer on the economy and the prces at whch dervatve products are traded. As defned by Hull (2006), a dervatve s a fnancal nstrument whose value depends on (or derves from) the values of other, more basc underlyng varables. Very often the varables underlyng dervatves are the prces of traded assets. Typcal dervatves nclude forward contracts, futures and optons. Both forward and futures contracts represent agreements to buy/sell some underlyng asset n the future for a specfed prce. Optons are a type of dervatve that gves the holder the rght, but not the oblgaton, to buy or sell the underlyng asset at gven date (expry or exercse date) for gven (exercse) prce. In ths case optons are called European; f opton holder can exercse t any date before expry date they are called Amercan. In order to develop a conceptual framework to prce optons based on the water prces and account for the observed stylsed features, we specfy four key restrctons: 1. As weekly water prces tend to be at a level specfc to a partcular season, we assume that the opton tradng wll be only ntated after a short ntal perod of water tradng has passed. The reason for ths restrcton s that after a few weeks of water tradng, the opton maker wll have a farly good dea of what the strke prce mght be reasonable for the opton. 2. Due to the sgnfcant water prce dfferences observed between consecutve seasons, we only make optons avalable for a specfed season. Ths mples that we wll not prce optons that have a maturty tme that covers two tradng seasons. 3. We wll assume that the weekly pool water prces are not mpacted by the actual water allocaton. That s, the stochastc process that the water prce follows s ndependent of the water allocaton process. 4. As the water prce tends to drop sharply towards the end of a season, we restrct the opton s maturty date to much earler than the seasonal endng date. By ths restrcton, we can elmnate the mpact of knowng the water prce pattern on the opton prce. Under these four key restrctons, we can set up our conceptual framework for water prce optons. Some classcal assumptons underlyng our opton prcng model should be made: 1. There are no transacton costs, taxes and short sale restrctons; 2. There are no arbtrage opportuntes; 3. The market s effcent and the water prces reflect all avalable nformaton on the market; 4. The rsk free rate s constant wthn the lfe of the opton, and 5. There are no storage costs and convenence yeld. The European call prce wth the pool water prce determned by (2) can be shown as follows: ~ ~ j j ~ ~ ~ j λ (T- t) c(t, x) = exp(- λ (T - t)) E(BSM(T - t, xexp(-βλ(t - t)) (Y +1))) (4) j! j=0 =1 3644

6 Here BSM s the Black-Scholes-Merton European call prce formula (Sreve, 2004). The expectaton operator s there due to the fact that the jump szes are stll random. We omt the proofs of these equatons and refer readers to Cu and Schreder (2009). 4. POTENTIAL IMPACTS ON WATER MARKET The ntroducton of the optons market for the water trades wll have some mpact on the underlyng water market. However, as the actual optons market s yet to be establshed, we can only hypothesse ts potental mpact on the water market. Introducng water optons s lkely to provde farmers wth addtonal ncome, more choces on tradng ther water allocaton and water prce certanty. A well known beneft of opton tradng s the leverage effect: for a fracton of the cost of buyng the underlyng asset, the opton holder can create an exposure smlar to that of physcal ownershp. Thus, farmers can beneft from the leverage effect n the followng way: ncrease ther exposure to the water market wth lmted captal so that they can make addtonal proft from favourable water prce movements whle mnmsng ther potental losses. Ths leverage effect can generate addtonal ncome for farmers who do not want to trade n the physcal water market due to lmted captal resources or other constrants. For nstance, a farmer may not be able to buy hs water due to lmted captal or some other economc reasons, but he can stll buy hs water allocaton n the optons market wth lmted funds to make a proft should water prce move n hs favour. Thus, the leverage effect s especally useful for annual crop growers who can ether buy or sell ther water allocaton dependng on the proftablty of ther agrculture busnesses. In addton, they can defer ther buy or sell decson n the physcal market by holdng only optons. In ths case, they receve a guaranteed prce to sell or buy ther water allocaton and at the same tme have the choce to only trade ther optons to proft wthout actually enterng nto the physcal market. Therefore, the leverage effect of optons tradng s lkely to encourage more farmers to partcpate n both physcal and optons water markets. The optons market s also benefcal to perennal crops growers such as grape and olve tree farmers. Perennal crop farmers have no choce but buy ther water allocaton on a yearly bass to sustan ther agrculture busnesses. The optons market gves them the beneft of hedgng away any adverse water prce movement n advance and at the same tme takng advantage of favourable water prce movements. Ths wll effectvely reduce the cost of ther busness and make ther agrculture busness more proftable. Although the exact mpact of the ntroducton of water optons on the water market volatlty and water prce levels can not be determned at present, farmers are lkely to beneft from water optons. Farmers can use water optons as a tool to manage and secure ther water exposure and proft from postve water prce movement. The opton formula s a very general one and one can calbrate the model to water prce data to determne ts parameters such as jump sze and jump frequency usng the maxmum lkelhood method. Once the model s properly calbrated, t can be used for prcng purposes. In terms of determnng a proper strke prce for an opton, the opton desgner could choose a strke prce that best suts farmers rsk appette, e.g. makng a lower strke prce for a call to ensure the opton has a hgher probablty of beng n the money. In our study, we have suggested that the opton desgner closely examnes the water prces for an ntal perod (a few weeks) of a partcular tradng season to get a good feel for the possble water prce level n order to determne a realstc strke prce. In ths paper, we have set up a conceptual framework for prcng optons for water trades. Although further studes are needed to examne the exact mpact of the ntroducton of water optons on the water market, farmers are lkely to beneft from water optons. Farmers could use water optons as a tool to manage and secure ther water exposure and proft from postve water prce movement. As the demand for water ncreases, t s expected that the need for the development of an optons market for water trades wll emerge. 5. DISCUSSION AND CONCLUSIONS The model of ntra-seasonal water prces s based on the formulae (2) and (3). The ntal water prce for a gven rrgaton season W(0) and values of jump n prces Y wll be estmated by modfed CGE model TERM-H20. The model n the present development s capable of computng ntal water prces whereas calculaton of the values of prce jumps needs further development, whch wll allow t to be mplemented on sub-annual tme steps. Smlarly, the number of these jumps depends on the temporal resoluton of the model under development. Ths approach allows one to translate the stochastc seres of future ranfall, whch s used as an nput for the TERM-H20 model, nto stochastc seres of weekly/fortnghtly water prces. Ths nformaton can be utlzed for assessng the mpacts of dfferent clmatc scenaros to the economy of rrgated regons and to the entre economy of the country. 3645

7 The paper proposes ntroducton of water optons to the market. Ths nnovaton can mpact economy n both postve and negatve ways. The benefts of the ntroducton of water optons to the market can be summarzed as follows: Water prce nsurance for farmers By payng a relatvely small up-front fee (opton prce), farmers are protected aganst adverse water prce movements n the future and stll allow themselves to beneft from potental favourable water prce advances. For water opton holders, no matter how adverse the water prce movement mght be, hs loss s lmted to the amount he pad for hs optons. More choces for farmers Farmers have the choce to partcpate n ether or both of the physcal and optons markets to acheve ther purposes. As optons are cheap compared to the prce of ther underlyng assets, t s deal to hold optons rather than the underlyng assets. By holdng optons, farmers could retan future water prce certanty wthout actually tradng ther water allocaton. Advantages for perennal crops growers The optons market gves them the opportunty to hedge from any adverse changes n water prces n advance and at the same tme takng advantage of favourable water prce movements. Ths wll effectvely reduce the cost of ther busness. The negatve mpacts can be outlned as follows: Informaton gap As farmers generally do not fully understand the rsks nvolved n tradng optons, there s an nformaton gap whch would need to be flled by the government or fnancal ntermedates. Potental water prce dstorton As the water tradng market n Australa s not very large, and water prce s determned by supply and demand, supply and demand could decrease sgnfcantly f a large proporton of farmers use the optons market nstead of the water tradng market. Potentally speculatve actvtes Although optons can be used for hedgng purposes, they can also be used for speculaton and arbtrage. It s lkely that speculatve actvtes wll occur gven the nature of opton tradng. Therefore, there s a need for both government and fnancal nsttutons to set up controls to ensure that tradng actvtes are closely montored. Ths potental problem can also be mnmsed by restrctng the access to the optons market. ACKNOWLEDGMENTS The research work reported n the present paper was mplemented wthn the framework of the ARC Lnkage Project LP Combnng hydrologcal nformaton wth a mult-regonal, computable general equlbrum model and newly newly funded ARC Dscovery Project DP Water dervatves: conceptualsaton, prce modellng and economc mpacts. I thank my co-nvestgators Prof. Peter Dxon and Dr Glyn Wttwer for many frutful dscussons. The model for European water opton prce was developed and coded by Jn Cu as a part of hs Masters thess at School of Mathematcal and Geospatal Scences, RMIT Unversty. I would lke to acknowledge two anonymous revewers and Dr Lynne McArthur who helped me to mprove ths paper. REFERENCES Cu, J. and Schreder, S. (2009), Modellng of prcng and market mpacts for water optons, Journal of Hydrology, 371, Garnaut, R. (2008), The Garnaut Clmate Change Revew, Notes/Garnaut/garnautweb.nsf, Released 30 September Dxon, P.B., Parmenter, B.R., Powell, A.A. and Wlcoxen, P.J. (1992), Notes and problems n appled general equlbrum economcs, Elsever, The Netherlands. Dxon, P.B., Rmmer, T.M. and Wttwer, G. (2008), The benefts of water tradng and the effects of the drought n Australa, Paper presented at 36 th annual Conference of Economsts, Hobart, September. Fernandes M. and Schreder, S. (2009), A new cost mnmsaton resource (water) allocaton model used to smulate the effects of new nfrastructure n the Goulburn Irrgaton System, MODSIM09 proceedngs. Hull, J.C. (2006), Optons, futures and other dervatves, 6 th ed., Pearson Educaton Inc., Upper Saddle Rver, New Jersey. Shreve, S. E. (2004), Stochastc Calculus for Fnance II, Contnuous-Tme Models. Sprng Scence+Busness Meda Inc., USA. 3646