Area Yield Futures and Futures Options: Risk Management and Hedging.

Transcription

1 Area Yield Fuures and Fuures Opions: Risk Managemen and Hedging. Knu K. Aase Norwegian School of Economics and Business Adminisraion 5045 Sandviken - Bergen, Norway Imagine here exis markes for yield fuures conracs as well as ordinary fuures conracs for price. Inuiively one would hink ha a combined use of yield conracs and fuures price conracs ough o provide a reasonable sraegy for securing revenue. In he paper his is made precise - i is shown ha revenue can be locked in by a combined, dynamic use of hese wo markes. The relevan sraegy is characerized: I depends only on observable price informaion in hese wo separae fuures markes, one for quaniy and one for price, no on he specificaion of parameers in uiliy funcions of he agens involved. The abiliy o rade coninuously can no be dispensed wih. These resuls should be of relevance, since markes for crop yield fuures and opions have been esablished. Key words: Area yield opions, fuures, coninuous ime modelling, quaniy and price securing, locking in a cerain revenue, CBOT yield conracs Inroducion This paper develops a sraegy showing how price and quaniy fuures markes can be used ogeher in order o provide a reasonable sraegy for securing revenue. Area yield crop insurance, where he index is based on average yield in a given geographical area, has been offered in India, Brazil, Canada and he USA. Parameric insurance (e.g., rainfall insurance has been proposed in Canada, India and Mexico. A livesock moraliy index has been recenly 1

2 designed o cover herders agains livesock losses in Mongolia (according o Mahul In 1995, he Chicago Board of Trade (CBOT launched is Crop Yield Insurance (CYI fuures and opions conracs in he USA. Relaed challenges exis in he producion of elecriciy, an example being Nord Pool (The Nordic Power Exchange, where financial fuures, forward and opion conracs are raded. Some of hese have characerisics similar o quaniy fuures due o he naure of elecriciy power rading. Price fuures could alernaively be combined wih cerain weaher derivaives for risk managemen purposes. Crop yield insurance conracs are designed o provide a hedge for crop yield risk. For example, CYI fuures users can secure a cerain crop yield several monhs ino he fuure as a emporary subsiue for a laer yieldbased commimen, or hey can alernaively ry o secure he revenue of a given acreage by combining yield conracs wih fuures price conracs. How his can be done is he subjec of he presen paper. In he following we absrac from producion coss, and assume zero local price basis (i.e., local cash price equals fuures price and zero yield basis (i.e., individual farm yield equals index yield. This is o say, we only address marke risk, no idiosyncraic risk. We also ignore asymmeric informaion. Inuiively one would hink ha a combined use of yield conracs and fuures price conracs ough o provide a reasonable sraegy for insuring revenue. In he paper his inuiion is made precise. I is shown ha revenue can be perfecly hedged by a combined, dynamic use of hese wo markes. Moreover, he relevan sraegy is also characerized. This sraegy depends only on observable fuures price informaion in hese wo separae markes. There is a large lieraure on non-marke based risk managemen and insurance of crop yield, using mosly a one period, expeced uiliy framework. Yield conracs have been analyzed from he perspecive of hedging, using a mean variance approach by Vukina, Li and Holhausen Minimizing he variance of revenue was he objecive in Li and Vukina In boh hese papers he yield conracs raded a CBOT are explained, so we need no elaborae on his marke srucure here. A more recen paper is Nayak and Turvey 2000, again using a simple mean-variance model. Two oher papers examine he insurance problem in he expeced uiliy framework (see e.g., Hennessy, Babcock and Hayes 1997; Mahul and Wrigh The focus in his paper is somewha differen from hese invesigaions, in ha our resuls do no depend on any specific assumpions abou uiliy funcions: Our hedging raios can be read sraigh from observed marke prices. However, here are some ineresing connecions o his lieraure which we reurn o below. In a recen paper, Hennessy 2001 discusses he apparen lack of ineres 2

3 in revenue fuures, by idenifying condiions under which revenue fuures are perfec subsiues for price fuures. These condiions hinge upon a nonsochasic relaionship beween producion shock and spo price, no presen in our model. Assuming no ransacion coss, basis risk and correlaion beween yield and price, Mahul and Wrigh 2003 characerize he opimal indemniy payoff ne of he premium for any risk averse agen. Under he same condiions, bu wih he possibiliy of coninuous reselemen, using separae yield and price fuures conracs, we demonsrae how o dynamically replicae his opimal indemniy payoff. This mach of he opimal insurance conrac shows how essenial he possibiliy of dynamic replicaion is, in order o achieve an opimal revenue insurance hrough rade in financial derivaives. Also we can relax he assumpion abou zero correlaion beween yield and price. The paper is organized as follows: In he firs secion we presen he dynamic marke model, and proceed direcly o he main resuls of he paper. In Theorem 1 (and Corollary 1 we show how a farmer can combine a pure yield fuures opion (pure yield fuures conrac wih a pure fuures opion (pure fuures price conrac o insure a revenue similar o wha a farmer could ideally secure if a fuures marke on revenue were o exis. Finally we briefly discuss he possibiliy o exend he analysis o models conaining jumps of unpredicable sizes. This brings us o incomplee financial markes. I urns ou ha our main resuls are sill valid. The final secion concludes. Area Yield Fuures and Opions Inroducion Imagine a counry, or anoher area, secioned ino regions which are uniform in erms of growing condiions for a cerain crop, say corn. In each area here is a quaniy index q, for ime running from 0 o T, where T is he ime of sale and 0 is he ime of sowing. As an example, for he agriculural yield conracs in he USA ha were raded a he CBOT, he values of q were provided by he Unied Saes Deparmen of Agriculure (USDA. One may hink of q as a forecas a each ime of quaniy, measured in bushels per acre, up for sale in his specific region a he final ime T. On his index we assume i is possible o rade fuures, and fuures opions conracs. In order o bring in he quanum uncerainy, we assume ha his index can be modeled as a sochasic process. A farmer in his region may have producion uncerainy ha is well represened by his index, in 3

4 which he relevan number of conracs can be deermined from each farmer s producion area. The idea is ha if he producer can buy opions on his quaniy index or on is corresponding fuures index, he farmer can secure a prespecified quaniy by buying an appropriae number of such coningen claims. This sraegy is of course only 100% efficien if he farmer s yield uncerainy is perfecly represened by he index, an unlikely even, bu a careful selecion of homogeneous regions may make such markes useful for pracical risk managemen purposes. Since agriculural agens are presumably concerned abou revenue in he end, raher han solely abou yield, or abou price, one may hink ha he yield marke may be combined in an appropriae manner wih he fuures marke for crop price o insure a cerain revenue. The condiions under which his can be done is he opic of his paper. A privae insurance marke giving he farmer insurance agains quaniy shorfall is of course difficul o esablish, parly because of he adverse incenives his would creae for he farmers, as he rich lieraure on his opic in agriculural economics journals show. The yield fuures marke may, however, avoid his difficuly, a leas under some presumpions: The agens do no engage in any kind of collecive moral hazard which effecs he yield index, and here is no moral hazard in he consrucion of he index. There is an implici assumpion ha he farmer s acions do no influence he quaniy index o any significan degree. Also he individuals in USDA consrucing he index should have no economic ineres aached o his marke. The model Consider wo fuures markes, one where yield fuures opions are raded, and one where sandard price fuures opions are raded. The quaniy index q( a ime is measured in bushels per acre, and spo price p( a ime is measured in $ per bushel. As earlier explained we absrac from producion coss, and assume zero local price basis and zero yield basis. In his case we can define he revenue R( = q(p(. A yield fuures opion conrac will specify a real funcion g so ha he payoff from a yield fuures opion conrac a he expiry ime T is given by g(q(t bushels per acre, having yield fuures price a ime < T given by F g(q = E Q (g(q(t c. (1 Here we consider an opion on he fuures index as a fuures conrac. 1 To be specific, given is a filered probabiliy space (Ω, F, F, P, where Ω is he se of saes of naure wih generic elemen ω, P is a probabiliy 4

5 measure, he objecive probabiliy, F is he se of evens in Ω given by a σ algebra, F = {F, 0 T } is a filraion saisfying he usual condiions, where F s F if s, F signifying he possible evens ha could happen by ime, or he informaion available by ime. We assume F 0 o be rivial, conaining only evens of probabiliy zero or one, meaning roughly ha here is no informaion available a ime zero, and F T = F, i.e., a ime T all uncerainy is resolved. Here Q appearing in equaion (1 is an equivalen maringale measure, assumed o exis, Q being equivalen o he given probabiliy measure P (i.e., he measures P and Q coincide on he null ses. The symbol E in (1 is he condiional expecaion operaor given he informaion F possibly available by ime. The consan c signifies a conversion facor measured in $ per bushel, so ha he fuures price is measured in $ per acre. For example, for he Iowa Corn Yield Insurance Fuures (icker symbol CA he uni of rading is he Iowa yield esimae imes $100 (e.g., a yield of bushels per acre gives a conrac value of $14, 030. In his secion we se his conversion facor equal o 1 wihou loss of generaliy. Similarly an ordinary fuures opion conrac on price will specify a real funcion h so ha he corresponding payoff from a fuures opion conrac a expiraion is given by h(p(t $ per bushel, and he associaed fuures price a any ime prior o T is given by F h(p = E Q (h(p(t measured in $ per bushel. The linear pricing rule of quaniy fuures implied by he expression (1 is, of course, far from obvious. In addiion o he usual fricions in ordinary fuures markes, like no shor sale possibiliies of he crop, an addiional difficuly arises here, since he index q is no a raded asse. In Aase 2004 his is resolved by considering he quaniy s = pq and idenifying s as a spo price process. Based on he price processes s and p a no-arbirage model is consruced as permied by he financial heory, where s is idenified as he spo price of a leasing conrac of agrarian land for he crop in he paricular region of consideraion. This solves, a leas in heory, he pricing problem of hese conracs. 2 Turning o he dynamics of he wo processes p and q, we assume ha he process q for quaniy and p for price are boh defined on he given probabiliy space as follows: and dq( = µ q (d + σ q (db( (2 dp( = µ p (d + σ p (db(, (3 where B( = (B 1 (, B 2 ( is a sandard wo dimensional Brownian moion, 5

6 σ q ( = (σ q,1 (, σ q,2 ( and σ p ( = (σ p,1 (, σ p,2 ( are adaped volailiy processes saisfying sandard L 2 -ype inegrabiliy and regulariy condiions. Similarly he drif erms µ q ( and µ p ( are adaped sochasic processes saisfying sandard L 1 -ype inegrabiliy condiions. The main resul Consider he produc conracs of he form R gh ( = g(q(h(p(. Our revenue process R( = p(q( would hen follow as a special case, when boh g and h are he ideniy funcion. We wan o invesigae wheher we can lock in a prespecified revenue R gh ( a any ime prior o he expiraion ime T by dynamically rading in he wo separae fuures opions markes described above. To his end imagine firs ha a separae marke for his ype of revenue were available. The fuures price of his conrac we denoe by F g(qh(p, and i mus be given as follows under our assumpions: F g(qh(p = E Q {g(q(t h(p(t }, 0 T. (4 Noice ha his can be wrien E Q {g(q(t h(p(t F g(qh(p } = 0, 0 T, (5 he usual saring poin for analyzing fuures conracs. Equaion (5 implies ha if he fuures price F g(qh(p is agreed upon a ime, hen no money changes hands when he fuures posiion is iniiaed. In order o beer undersand wha follows, le us recall he main feaures of a simple fuures conrac on, say, price. For he holder of one long conrac, he payoff a expiraion is 1 df s = F T F = p T F (6 by he principle of convergence in he fuures marke, where F is he fuures price of one conrac a ime. If an agen holds θ s fuures conracs a ime s in he ime inerval (, T ], he reselemen gain a ime T from his sraegy would similarly be F h(p s θ s df s. (7 Consider a sraegy ha holds Fs g(q pure fuures opions on h(p T, and pure fuures opions on g(q T a each ime s beween and T. The reselemen gain from his sraegy is given by Fs g(q dfs h(p + 6 Fs h(p dfs g(q. (8

7 Using sochasic inegraion by pars, his can be wrien = g(q T h(p T (F g(q F h(p + dfs g(q dfs h(p. (9 Reurning o he basic equaion (5, he saring poin for analyzing fuures conracs, consider he equaliy E Q ( T g(qt h(p T (F g(q F h(p + dfs g(q dfs h(p = 0. (10 If his is rue, i would mean ha he dynamic sraegy given in (8 is equivalen o a fuures conrac on he produc h(p T g(q T wih he associaed fuures price equal o he projecion of he expression (F g(q F h(p + df g(q s df h(p s on he informaion filraion F, i.e., F g(qh(p = E Q ( T g(q F F h(p + df g(q s dfs h(p. (11 We now demonsrae ha he equaliy (10 indeed holds rue under mild condiions. To his end, we will need some echnical condiions, which we relegae o Appendix 1. Assuming hese, we use he following noaion: The fuures price processes F h(p and F g(q can boh be wrien as smooh funcions a(p, and b(q, respecively. Denoe by a p (p, he parial derivaive of he funcion a(p, wih respec o is firs argumen, and similarly for b q (q,. We have he following resul: Theorem 1 Consider he reselemen gain from he sraegy given in (8. This sraegy is equivalen o a fuures price direcly on revenue h(p T g(q T wih associaed fuures price given in (11, which can also be wrien F g(qh(p = F g(q F h(p + E Q = F g(q F h(p + E Q ( ( dfs g(q dfs h(p a p (p s, s(σ p (s σ q (sb q (q s, s ds. (12 In he special case of zero correlaion rae beween yield and price, i.e., σ p (s σ q (s = 0 for all s (, T ], his sraegy is equivalen o a fuures conrac on he produc h(p T g(q T having fuures price a each ime given by F g(qh(p = F g(q F h(p. 7

8 Proof: In order o prove his, according o (5 we have o show ha equaion (10 holds, i.e., E Q ( g(q T h(p T (F g(q F h(p + This would follow if ( E Q Fs g(q dfs h(p + dfs g(q dfs h(p = 0, T. (13 Fs h(p dfs g(q = 0 for any T (14 by equaions (8 and (9. Consider he sandard condiions (22 - (25 of Appendix 1; under hese i is known, essenially by Hølder s inequaliy, ha he sochasic inegrals in (8 boh have zero condiional expecaions under Q, since F g(q and F h(p are boh Q-maringales. Thus we ge he conclusion of he heorem from he expression for he fuures price in equaion (4, he fac ha F g(q and F h(p are boh F -measurable, and from he represenaions (20 and (21 of Appendix 1 for he sochasic processes a(p, and b(q,. The conclusion of he las par follows from he expression (12, since he inegral dfs g(q dfs h(p = 0 in his case. Before we commen on his heorem, we briefly describe he siuaion wih pure fuures conracs only. Consider a sraegy ha holds Fs q fuures conracs on price and Fs p fuures conracs on quaniy a any ime s, where 0 s T, signifying he presen. The reselemen gain from his sraegy is given by F q s df p s + We hen have he following corollary: F p s df q s. (15 Corollary 1 Consider he reselemen gain from he sraegy ha, for any ime s beween he presen ime and he expiraion ime T, holds Fs q fuures conracs on price and Fs p conracs on quaniy, given in (15. This sraegy is equivalen o a fuures price direcly on revenue R T = p T q T wih associaed fuures price given by F R = F q F p + E Q ( a p (p s, s(σ p (s σ q (sb q (q s, s ds. (16 8

9 If σ q (s σ p (s = 0, for all s (, T ] i.e., a zero correlaion rae beween yield and price, hen his sraegy is equivalen o a fuures conrac on revenue R T having fuures price a each ime given by F R := E Q {q(t p(t } = F (q F (p. Proof: Se g(x = x and h(x = x for all real x in Theorem 1. The above resuls show ha here is no need for a specialized fuures marke of, say, revenue R = pq for someone who has access o he wo separae markes for price and yield conracs. One can hen, a leas in principle, achieve exacly he same resuls in erms of risk managemen by simulaneous, dynamic rade in hese wo markes. Since a dynamic sraegy is hen needed, needless o say, we here absrac from ransacions coss. In he siuaion where he correlaion σ p,q (s := σ p (s σ q (s = 0 for all s (, T ], he corresponding fuures price becomes paricularly simple, namely he produc of he corresponding fuures prices F q and F p. Suppose, on he oher hand, ha his correlaion is posiive. The las erm in (16 is accordingly posiive, which raises he fuures price. This seems reasonable due o he increased risk his siuaion represens compared o he one wih a zero covariance funcion: If he harves is poor, he price is also low on he average, boh conribuing o a smaller revenue. If he corresponding correlaion is negaive, which is raher naural of his quaniy, a farmer would ypically no be willing o pay quie as much for his insurance coverage as in he wo oher cases, confirmed by he equaions (12 and (16. Examples and Discussion The resuls of Theorem 1 and Corollary 1 do no depend on any specific assumpions abou uiliy funcions of he agens (excep from some obvious axioms, like agens prefer more o less. The resul is ha a fuures marke for revenue can approximaely be obained hrough he combinaion of he wo markes for yield and price fuures. There exiss a dynamic replicaion sraegy in quaniy fuures and price fuures which, under cerain condiions, is equivalen o a fuures conrac on revenue. Moreover, his sraegy can be obained direcly from fuures price informaion in hese wo separae markes. There are no parameers o esimae, no assumpions abou he relaive risk aversion, or he subjecive ineres rae, or anyhing like ha. Thus his resul could be of pracical ineres. The resuls can perhaps bes be illusraed by an example. Example 1. The sraegy ( F q, F p duplicaes exacly he payoff (F R q T p T from one shor revenue conrac: A he iniiaion ime he fuures prices F q and F p are boh se such ha no money changes hands. 9

10 Insead of he reselemen sraegy in (15 le us consider he very simple sraegy ha sells F p quaniy conracs, priced a F q a ime T, and holds his posiion unil mauriy, and sells F q price conracs, priced a F p a ime T, and holds his posiion ill mauriy as well. The payoff a expiraion for he hypoheical conrac on revenue would be (F R q T p T, for an agen selling one such conrac. On he oher hand, he combined conracs described above would yield he following payoff: (F p p T F q + (F q q T F p, where he firs erm is he payoff of F q shor fuures conracs on price p, and he second erm is he corresponding payoff of F p shor conracs on quaniy q. This laer sum can be seen o be equal o (F R q T p T + (F q q T (F p p T, (17 in he siuaion where F R = F q F p, i.e., when he cross-correlaion rae is zero, so le us for simpliciy consider he case where σ p ( σ q ( = 0 for all. Since F q (F p can be considered as an economic forecas of q T (p T a ime, he remainder erm in (17 should be small of second order (i goes o zero faser han he firs erm in (17 as approaches T, in which case his sraegy may funcion reasonably close o a hypoheical fuures marke for revenue. Of course, his laer ideal marke does no exis, so his simple arrangemen of combining exising markes for quaniy and price separaely may be a reasonable subsiue. The sraegy described in equaion (15 consiues, on he oher hand, a perfec subsiue in his siuaion, as well as in he siuaion when he associaed cross correlaion is differen from zero. Assuming no ransacion coss, basis risk and correlaion beween yield and price, Mahul and Wrigh 2003 characerize he Pareo opimal indemniy payoff ne of he premium for any risk averse agen, and risk neural insurer. I is shown o be (F R q T p T. This argumen requires risk neural pricing, which in our model amouns o equaing he risk adjused probabiliy measure Q and he given one P. As a consequence of his, marke prices are deermined as F R = E (p T E (q T, and he opimal revenue insurance has payoff (E (p T E (q T q T p T. From he relaion in (17 i is seen ha his payoff resuls, bu in addiion here is he remainder erm caused by merely using he sell and hold sraegy. If he dynamic reselemen sraegy (15 is used insead, he correcion erm vanishes, and he opimal payoff is exacly achieved. I is noiceable ha his resul is rue regardless he value of he 10

11 covariance beween price and quaniy, which is also consisen wih sandard (Pareo opimal risk sharing heory, saing ha full insurance is opimal in he siuaion described above. This demonsraes an ineresing connecion o opimal insurance coverage, showing ha he Pareo opimal ne indemniy payoff can be dynamically replicaed by using separae yield and price fuures conracs. In he zero cross correlaion case of he above example, his also gives us he rare opporuniy of finding an expression for he hedging error, he las erm in he expression (17, he deparure from he Pareo opimal conrac, resuling from merely using he sell and hold sraegy when he dynamic replicaion sraegy in (15 is indeed opimal. The rewards from being able o rade coninuously are here brough forward in an explici way. Jump/diffusion uncerainy model In an earlier paper we gave an example of a model for he price of he crop, p, and he quaniy index q, as well as a valuaion model in which marke prices can be found for a large class of relevan financial conracs. In oher words, we give an example how o consruc an equivalen maringale measure Q for yield conracs. Since we have chosen an Iô process framework, his is done by judiciously ransforming o wo price processes (q is no a price process, and hen i is naural o choose a complee model, in which case we have o solve a linear sysem of equaions, and use Girsanov s heorem o esablish a unique marke-price-of-risk process (see Aase Suppose insead ha agriculural yields are exposed o naural disasers in which case i would be naural o include changes more dramaic han coninuous ones in he process dynamics for p and q. Consider he following dynamics dq( = µ q (d + σ q (db( + γ q (, zñ(d, dz (18 R 2 and dp( = µ p (d + σ p (db( + γ p (, zñ(d, dz. (19 R 2 The erm Ñ(d, dz = N(d, dz ν(dzd signifies a compensaed Poisson random measures of an underlying wo dimensional Levy process, independen of he wo dimensional Brownian moion B, and ν(dz is he associaed Levy measure. The idea is ha jumps of random sizes γ i (, z occur a unpredicable ime poins of a Levy process, i = p, q. If a jump happens o ake place a ime, and he underlying jump size of he Levy-process is 11

12 z = (z p, z q, hen he jump size in he quaniy index q is γ q (, z, and in he price process p he corresponding jump size is γ p (, z. Wihou going ino furher echnical deails, we noe he following: This class of models is obviously very general, and can be made o fi well mos observed ime series of daa one can imagine. There is an inegraion by pars formula also for he ype of processes given in he equaions (18 and (19 above. Since his is an essenial par of he proof of Theorem 1, our resuls in Theorem 1 and Corollary 1 can sill be shown o be valid. In he uncorrelaed cases (i in boh resuls we now have in addiion ha R 2 γ p (s, zγ q (s, zν(dz = 0 for all s (, T ]. In he above model we are no able o consruc a unique marke-priceof-risk process as in he aricle referred o earlier, so here will exis many equivalen maringale measures Q (indeed, uncounable many ha will do for pricing purposes, even if no arbirage prevails. All hese measures will coincide on he markeed subspace M L 2 conaining all he random payoffs of he ype ha can be generaed by porfolio formaion of wo differen, correlaed asses wih pricing processes like he one in (18. However, for coningen claims wih componens in he orhogonal complemen M of M (here L 2 = M M, hese componens can no be hedged by he exising financial insrumens. As a consequence we do no have a good pricing heory for his par, and he measures Q will normally no coincide on M. Thus our resuling model is incomplee. Bu his is of no concern o he presen resuls. The agen can sill observe he prices in he wo separae fuures markes, consruc he dynamic sraegy as ime goes, and replicae he payoff of a fuures (or a fuures opion conrac on revenue o he degree ha we have explained above. Thus our resuls are robus o he modelling of uncerainy - more ineresing models han Iô-processes can be used, in principle here are no resricions (oher han echnical ones. Hence marke compleeness is no required for he main resuls o hold. Conclusions We have presened a dynamic model for he analysis of fuures conracs on quaniy and fuures conracs on price in separae markes for such conracs, in order o consruc fuures conracs on revenue. Only marke risk is considered. 12

13 Specifically, we have demonsraed how an agen can lock in a cerain revenue by a combined rade in fuures price and fuures yield conracs, absracing from producion coss. This can be done perfecly if a cerain dynamic sraegy is used, idenified in he paper. This sraegy depends only on fuures prices observed in he wo differen markes for price and yield fuures, and no on he paricular choice of model for he random dynamics. In consequence he resul is independen of a complee marke srucure, and hus fairly robus. The idenified dynamic sraegy is, under cerain condiions, equivalen o a Pareo opimal revenue insurance. Our resuls do no depend upon any specific assumpions abou uiliy funcions, relaive risk aversions, subjecive discoun raes, or oher model parameers. Provided one akes ino accoun ransacions coss in a manner ha is cusomary when hedging derivaives, i is possible o implemen he main resul in pracice. References [1] Aase, Knu K. A pricing Model for Quaniy Conracs. Journal of Risk and Insurance 71, 4: (2004, [2] Goodwin, B. K., and V. H. Smih The economics of crop insurance and disaser aid. Washingon DC. American Enerprise Insiue Press, [3] Hennessy, D. A. Subsiuion beween Revenue Fuures and Price Fuures Conracs: A Noe. The Journal of Fuures Markes, 22, 4(2001: [4] Hennessy, D., B. A. Babcock and D. Hayes. Budgeary and Producer Welfare Effecs of Revenue Insurance Amer. J. of Agr. Econ. 79(Augus 1997: [5] Li, Dong-Feng, and Tomislav Vukina. Effeciveness of dual hedging wih price and yield fuures. Journal of Fuures Markes, 18, 5(1998: [6] Mahul, O. The Financing of Agriculural Producion Risks. Revisiing he role of agriculural insurance., Geneva Associaion Informaion Newsleer 52 (July 2005: 4-9. [7] Mahul, O. and B. D. Wrigh. Designing Opimal Crop Revenue Insurance, Amer. J. Agr. Econ. 85(3(Augus 2003:

14 [8] Nayak, G.N., and C.G. Turvey. The Simulaneous Hedging of Price Risk, Crop Yield Risk and Currency Risk. Canadian Journal of Agriculural Economics 48(2, (2000: [9] Vukina, Tomislav, Dong-Feng Li, and Duncan M. Holhausen. Hedging wih crop yield fuures: A mean-variance analysis. Amer. J. of Agr. Econ. 78, 4(1996: Appendix 1 Here we presen he echnical condiions needed for Theorem 1: The fuures price processes F h(q and F g(y can be boh be wrien as some C 2,1 (R 2 [0, T -funcions a(q, and b(y,, say. Since hey are boh Q-maringales, by Iô s lemma da(q, = a q (q, σ q (d B( (20 db(y, = b y (y, σ y (d B(, (21 where a q (q, means he parial derivaive of he funcion a(q, wih respec o is firs argumen, and similarly for b y (y,, and where B is a sandard wo dimensional Brownian moion wih respec o he measure Q. We will now need he following echnical condiions: We suppose he processes a(q, and b(y, saisfy he following: and (b(y, a q (q, 2 (σ q,1 ( 2 + σ q,2 ( 2 d < a.s. (22 0 ( E (b(y, a q (q, 2 (σ q,1 ( 2 + σ q,2 ( 2 d < (23 0 (a(q, b y (y, 2 (σ y,1 ( 2 + σ y,2 ( 2 d < a.s. (24 0 ( E (a(q, b y (y, 2 (σ y,1 ( 2 + σ y,2 ( 2 d <. (

15 Noes 1 Alernaively i could be an opion conrac requiring an iniial cash paymen, or a hybrid. 2 In pracice he hedging resuling from his use of he differen markes may no be enirely accurae, bu hen one should perhaps have in mind ha he only perfec hedge is found in a Japanese garden. 15