Term Structure of Volatility and Price Jumps in Agricultural Markets - Evidence from Option Data. Steen Koekebakker

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1 Term Structure of Volatlty and Prce Jumps n Agrcultural Markets - Evdence from Opton Data Steen Koekebakker Gudbrand Len E-mal: gudbrand.len@nlf.no Paper prepared for presentaton at the X th EAAE Congress Explorng Dversty n the European Agr-Food System, Zaragoza (Span), 8-3 August Copyrght by Steen Koekebakker,Gudbrand Len. All rghts reserved. Readers may make verbatm copes of ths document for non-commercal purposes by any means, provded that ths copyrght notce appears on all such copes.

2 Term structure of volatlty and prce umps n agrcultural markets - evdence from opton data Steen Koekebakker Agder Unversty College, Department of Economcs and Busness Admnstraton Sevcebox 4, 464 Krstansand, Norway Gudbrand Len (correspondng author) Norwegan Agrcultural Economcs Research Insttute Box 84 Dep, 3 Oslo, Norway Phone: , Fax: E-mal: gudbrand.len@nlf.no Paper submtted to be presented at X th congress, European Assocaton of Agrcultural Economcs (EAAE) 8-3 august, Zaragoza, Span Abstract Emprcal evdence suggests that agrcultural futures prce movements have fat-taled dstrbutons and exhbt sudden and unexpected prce umps. There s also evdence that the volatlty of futures prces contans a term structure dependng on both calendar-tme and tme to maturty. Ths paper extends Bates (99) ump-dffuson opton prcng model by ncludng both seasonal and maturty effects n volatlty. An n-sample ft to market opton prces on wheat futures shows that our model outperforms prevous models consdered n the lterature. A numercal example llustrates the economc sgnfcance of our results for opton valuaton. Keywords: Opton prcng; Futures; Term structure of volatlty; Jump-dffuson; Agrcultural markets An earler verson of ths paper wth the same ttle appeared as Dscusson Paper /9 at Norwegan School of Economcs and Busness Admnstraton, Department of Fnance and Management Scence. We are grateful to Petter Berksund for helpful comments on an earler draft.

3 Term structure of volatlty and prce umps n agrcultural markets - evdence from opton data Abstract Emprcal evdence suggests that agrcultural futures prce movements have fat-taled dstrbutons and exhbt sudden and unexpected prce umps. There s also evdence that the volatlty of futures prces contans a term structure dependng on both calendar-tme and tme to maturty. Ths paper extends Bates (99) ump-dffuson opton prcng model by ncludng both seasonal and maturty effects n volatlty. An n-sample ft to market opton prces on wheat futures shows that our model outperforms prevous models consdered n the lterature. A numercal example llustrates the economc sgnfcance of our results for opton valuaton. Keywords: Opton prcng; Futures; Term structure of volatlty; Jump-dffuson; Agrcultural markets Introducton Black (976) derves a prcng model for European puts and calls on a commodty futures contract, assumng that the futures prce follows a geometrc Brownan moton (GBM). In the lterature on agrcultural futures markets (as n many other markets) however, several emprcal regulartes have been documented, ndcatng that the GBM assumpton may be too smplstc. Research on futures prces has found dstrbutons that are leptokurtc relatve to the normal dstrbutons (e.g. Hudson et al., 987; Hall et al., 989) and the prces often exhbt sudden, unexpected and dscontnuous changes. Jump behavour of ths sort wll typcally occur due to abrupt changes n supply and demand condtons, and naturally t wll affect opton prcng. Hllard and Res (999) used transactons data on soybean futures and futures optons to test Amercan versons of Black's (976) dffuson and Bates' (99) ump-dffuson opton prcng models. Ther results show that Bates' model performs consderably better than Black's model. A number of studes have demonstrated the presence of a term structure of volatlty n agrcultural futures prces. Samuelson (965) stated that the volatlty of futures prce changes per unt of tme ncreases as the tme to maturty decreases. Ths maturty effect s usually referred to as the "Samuelson hypothess". Another vew, the "state varable hypothess" s that the varance of futures prces depends on the dstrbuton of underlyng state varables. For crop commodtes wth annual harvest, seasonalty n the volatlty of futures prces s typcally expected. Emprcal research on the former approach has produced mxed evdence on the maturty effect (Rutledge, 976). Mlonas (986) found strong support for the maturty effect after controllng for the year effect, seasonalty effect and the contract-month effect. Galloway and Kolb (996) concluded that the maturty effect s an mportant source of volatlty n futures prces for commodtes that experence seasonal demand or supply, but not for commodtes where the cost-of-carry model works well. Anderson (985) found support for the maturty effect, but concluded t s secondary to the effect of seasonalty. Anderson also concluded that the prcng of optons on futures contracts should be made for the regular pattern to the volatlty of futures. Bessembnder et al. (996) have reconcled much of the early evdence on the "Samuelson hypothess". They have shown that n markets where spot prce changes nclude a temporary component so nvestors expect some porton of a typcal prce change to revert n the future, the "Samuelson hypothess" wll hold. Mean reverson s more lkely to occur n agrcultural commodty markets than n markets for precous metals or fnancal assets (Bessembnder et al., 995), so we expect to see maturty effects n agrcultural commodty markets.

4 Any regular pattern n the volatlty s nconsstent wth the underlyng assumptons of the Black's (976) and Bates' (99) opton prcng models. Cho and Longstaff (985) appled the formula of Cox and Ross (976) for constant elastcty of varance opton prcng n the presence of seasonal volatlty. They found ths superor to Black's model for prcng optons on soybeans futures. Myers and Hanson (993) present opton-prcng models when tme-varyng volatlty and excess kurtoss n the underlyng futures prce are modelled as a GARCH process. Emprcal results suggest that the GARCH opton-prcng model outperforms the standard Black model. Fackler and Tan (999) proposed a smple one-factor spot prce model wth mean reverson (n the log prce) and seasonal volatlty. They show that futures prces consstent wth ths spot prce model have a volatlty term structure exhbtng both seasonalty and maturty effects. Ther emprcal results ndcate that both phenomena are present n the soybean futures and opton markets. In ths paper we assume that the futures prce follows a ump-dffuson process. The dffuson term ncludes tme dependent volatlty that captures (possbly) both a seasonal and a maturty effect. We derve a futures opton prcng model gven our specfed futures prce dynamcs, and we test our model emprcally usng eleven years of data on Amercan futures opton prces on wheat from Chcago Board of Trade (CBOT). We fnd that our model does a better ob n explanng the opton prces than the models prevously suggested n the lterature. The maturty effect s especally strong n ths market. A numercal example llustrates the economc sgnfcance of our results. Ths paper s organsed as follows: In the next secton we present the model and derve the opton prcng model. Thereafter the data are descrbed and prelmnary evdence on volatlty term structure and ump effects s gven, then the emprcal results are presented. Fnally, we llustrate the economc sgnfcance of volatlty term structure and ump parameters and a numercal example s gven. The paper ends wth a summary and concludng comments. The model We shall present a ump-dffuson model for the futures prce dynamcs and derve an opton prcng model for a European futures opton. Fundamental to the prcng of contngent clams s the dervaton from the real world dstrbuton of the asset prce, to the equvalent "rsk-neutral" dstrbuton, or the equvalent martngale measure (EMM) n modern termnology. The value of a contngent clam s the expected value under the EMM dscounted by the rsk free rate. In the paper by Merton (976), umps are assumed to be symmetrc (zero mean) and nonsystematc. In a stock market model, ths means that umps are of no concern to an nvestor wth a welldversfed portfolo, snce umps on average cancel out. Gven such assumptons of frm specfc ump rsk, parameters concernng the ump part are equal under both the real world probablty measure and the EMM. In our settng, focusng on wheat futures prces, the assumpton of nonsystematc ump rsk may be napproprate. If, for example, bad weather results n a poor harvest, futures prces may ump. However, the occurrence of such an event s lkely to move all the commodty futures prces n the same drecton, and so dversfyng the ump rsk s mpossble. In other words, ump rsk s systematc. To derve the EMM when ump rsk s systematc, we have to make assumptons about the prce of ump rsk. In ths paper we follow Bates (99) closely. Bates assumed frctonless markets, optmally nvested wealth follows a ump-dffuson, and a representatve consumer wth tme-separable power utlty. He then derved the EMM from the real world probablty measure. Under the assumptons on preferences and technology, he showed that ump parameters under the EMM need to be adusted accordng to the preferences of the representatve consumer. In case of rsk neutralty, the ump parameters are equal under both measures. The only dfference between our model and that of Bates s that we mpose tme A full dervaton of the EMM n an equlbrum settng s gven n the appendx n Bates (99). 3

5 dependence n the dffuson term of the GBM. It s well known that the dffuson term s unchanged, gong from one probablty measure to an equvalent probablty measure. Hence, the results n Bates apply to our model as well. We shall set up the model drectly under the EMM. Denote the prce of a futures contract as F(t,T ), where t s today's date and T s the maturty date of the contract. The futures prce s assumed to follow the followng dynamcs under the EMM: df F ( t, T ) ( t, T ) ( t, T ) db() t κdq λκ dt + σ + where B () t s standard Brownan moton under the EMM and κ s the random percentage ump condtonal upon a Posson dstrbuted event, q, occurrng. We assume that ( + κ ) s a lognormal random varable wth mean ( v ) γ and varance () v. Consequently, the expected percentage γ ump sze s E [ κ ] κ e. The frequency of Posson events s λ and q s the Posson counter wth ntensty λ. Note that the ump parameters are ndependent of tme to maturty. Ths means that f a ump occurs, a parallel shft n the term structure of futures prces wll occur. If we observe several futures contracts wth tme to maturty spannng several years nto the future, the ump structure descrbed above may seem nadequate. If, for example, exceptonal bad weather (such as a hurrcane) partly destroys a harvest, then futures prces are lkely to ump. But we would expect contracts wth maturty before the next harvest to experence a greater prce change than contracts wth maturty precedng the next harvest, snce the next harvest s lkely to turn out better than the prevous one. Ths behavour can easly be ncorporated n our model by mposng a term structure on the ump ampltude. Such an extenson s gnored n ths paper snce the maturty of the futures contracts analysed n ths paper never exceed one year. Hence, n our data set, mposng parallel umps may be a satsfactory assumpton. The functon σ ( t,t ) represents the nstantaneous volatlty of the futures prce condtonal on no umps. We want to capture two possble effects n the specfcaton of the volatlty functon; perodc seasonalty and maturty effect. We shall concentrate on the followng canddate l ( t, T ) σ () t σ ( T t) σ () The frst term represents the tme t dependent seasonal volatlty pattern. We model the perodc functon as a truncated Fourer seres σ p () t σ + ( α sn πt + cos πt ) The maturty effect s modelled by negatve exponentals σ δ ( ) ( T t T t e ) Ths model provdes a farly rch volatlty term structure, and as we shall see below, a straghtforward closed-form prcng formula for vanlla European optons can be derved.. Relaton to other models n the commodty lterature Ths model nests several models proposed for commodtes n the lterature. The semnal Black's (976) model s gven by λ δ α. The one-factor model of Schwartz (997), that captures the maturty effect, appears f we set λ α. The ump-dffuson model of 4

6 Bates (99) s δ α. Bates (99) extended wth maturty effect s α, and Bates (99) extended wth seasonal effects s gven by δ.. Valuaton of futures optons Valuaton of both European and Amercan futures optons n ths model are slght generalsatons of the formula gven n Bates (99) and Merton (976). Let n be the number of umps occurrng t, T. Then the soluton to equaton () s n the nterval [ ] F T T n ( T, T ) F( t, T ) exp ( ) ( ) ( ) ( ) κ T t σ s, t ds + σ s, T db s ( + κ ) λ (3) t t The value of a European futures call opton wrtten on the contract ( t,t ) strke prce K and maturty at tme T, s gven by F where r( T t) b( n) ( T t) ( F( t, T ), T ) e ( Pr_ n _ umps) ( F( t, T ) e N( d n ) KN( d n )) c n T T wth where e n ( λt ) b( n) ( T t) ( F( t, T ) e N( d ) KN( d )) λt r( T t) e n n n n! b( n) λκ ( T t) + nγ ( T t) F ln K ( t, T ) + b( n)( T t) d n ω + nv + ω + nv T n d n ω + ω σ ( s, T ) t d nv Put optons can be calculated explctly, or they can be found va the futures opton put-call party. In the emprcal part of ths paper, we use data on Amercan futures optons, consequently some modfcaton of the above model s requred. Bates (99) derves an approxmaton for an Amercan opton n the ump-dffuson framework. Hs approxmaton follows the work of Barone-Ades and Whaley (987) n the standard case where the underlyng asset follows a GBM. We use the same approxmaton as descrbed by Bates (99), replacng the constant volatlty n hs settng wth the tme-dependent volatlty gven by ω above (we name ths model Bates SM later n the paper). ds 3 Prelmnary analyss and data descrpton Weekly data were obtaned for call optons on wheat futures and for the underlyng futures contract traded on the CBOT from January 989 untl December 999. Wheat futures contracts are avalable wth expraton n March, May, July, September, and December. We frst present a smple regresson model to llustrate the term structure of volatlty present n our eleven years sample of futures data. 5

7 3. Term structure effects n futures prce volatlty We ran the followng regresson for each of the fve contracts: V t η + η D + e (4) k k kt t where V t s estmated standard devaton of the log changes of wheat futures prces for month t based on daly data, D kt are seasonal dummy varables for month t: k, February,, k, December, and e t s an error term assumed to follow an AR() process. The regresson model was estmated by Hldreth and Lu (96) grd search method. Table Estmates of seasonalty and maturty coeffcents, March, May, July, September and December wheat futures contracts, t-values are n parentheses March May July September December η.6 (7.3). (.5).7 (.86).9 (.95).3 (.8) η.6 (.). (.5).3 (.4).9 (.).4 (.) η 3.6 (.4).3 (.).35 (.7).3 (.39).4 (.93) η 4.9 (4.99).65 (5.4).54 (.5).6 (.63).3 (.) η 5. (4.6).7 (5.33).67 (.93).5 (.54).35 (.7) η 6. (4.44).8 (.4).7 (3.7).7 (.65).35 (.8) η 7. (4.3). (.).77 (3.47).48 (3.3).4 (.37) η 8. (4.34). (.).3 (.9).73 (5.38).55 (3.3) η 9. (4.65).9 (.6).9 (.4).77 (5.89).73 (4.65) η. (4.83).9 (.).9 (.56).4 (.4).84 (5.79) η. (5.3). (.).9 (.67).5 (.36).96 (7.58) η.3 (3.93).8 (.3).4 (.33).6 (.43).98 (.3) Ad R In Table the results from the regresson are reported n the followng way; January s the constant term, η, February s η + η etc. From the results n Table we see a very pronounced maturty effect, and weak evdence of seasonalty for each contract. Lookng for example at the March contract we see that volatlty starts to rse n December. The volatlty n January, February and March s approxmately sx tmes the volatlty n Aprl. 3 We also see that the volatltes of the remanng months of the March contract are sgnfcantly dfferent from volatlty n January. Note also that the summer months have slghtly hgher volatltes than Aprl and the autumn months. We fnd ths pattern for the other contracts as well. In ths paper we shall nvestgate whether ths term structure effect s prced n the opton market. OLS generally dsplayed autocorrelated resduals. The Hldreth and Lu grd search procedure was employed to yeld consstent parameter estmates. 3 The low t-statstcs n February and March smply mply that the volatltes n those months are ndstngushable from the volatlty n January. 6

8 3. Indcaton of ump behavour from opton prces If wheat futures prces are charactersed solely by determnstc tme-dependent volatlty, they are lognormally dstrbuted. Furthermore, the mpled volatlty from opton prces wll be constant across strke prces. However, f umps are lkely to occur, mpled volatlty wll be skewed. In Fgure we have calculated mpled volatlty from call futures prces at January 8, 995. When backng out mpled volatltes, we used the formula derved by Black (976) adustng for the fact that the optons are of Amercan type usng the approxmaton of Barone-Ades and Whaley (987). Fgure shows no horzontal pattern of mpled volatlty, but an mpled "volatlty smle". A ump dffuson model may produce such a pattern. When futures prces are allowed to ump upwards, out-of-the-money (OTM) call optons have a hgher probablty of endng n-themoney (ITM) than otherwse would be the case, and they wll trade at a hgher prce. Ths n turn creates an upward slopng volatlty pattern for call optons evdent from Fgure. For a call opton ITM, the probablty of a negatve ump wll cause the optons to be worth more than would be the case n a lognormal world. Imp. vol Strke Fgure Implct volatlty patterns from CBOT wheat call optons wth maturty n May 9, 995 at January 8, 995. Impled volatlty for Amercan optons are approxmated as n Barone-Ades and Whaley (987) 3.3 Constructng the data set From the prelmnary analyss above we have seen evdence suggestng that our model, ncludng both umps and tme dependent volatlty, wll capture mportant market characterstcs. We have therefore tested our model on wheat futures opton prces collected from CBOT. The eleven years of data consst of ffty-fve futures contracts. The futures contracts matures n March, May, July, September, and December. At each pont n tme, there are fve contracts traded, meanng that one year s the longest contract an nvestor can enter nto. The optons wrtten on the contracts can be exercsed pror to maturty, hence they are of Amercan type. The last tradng day for the optons s the frst Frday precedng the frst notce day for the underlyng wheat futures contract. The expraton day of a wheat futures opton s on the frst Saturday followng the last day of tradng. We appled several excluson flters to construct the data sample. Frst, our sample starts n 989. We dd not use prces pror to 989 snce market prces then were lkely to be affected by government programs n the Unted States (prce floor of market prces and government-held stocks). Second, only trades on Wednesdays were consdered, yeldng a panel data set wth 7

9 weekly frequency. Weekly samplng s smply a matter of convenence. Daly samplng would place extreme demands on computer memory and tme. Thrd, only settlement (closng) prces were consdered. Fourth, the last sx tradng days of each opton contract were removed to avod the expraton related prce effects (these contracts may nduce lqudty related bases). Ffth, to mtgate the mpact of prce dscreteness on opton valuaton, prce quotes lower than.5 cents/bu were deleted. Sxth, assumng that there s no arbtrage n ths market, opton prces lower or equal to ther ntrnsc values were removed. Three-month Treasury bll yelds were used as a proxy for the rsk free dscount rate. The exogenous varables for each opton n our data set are strke prce, K, futures spot prce, F, today s date, t, the maturty date of the opton contract, T, the maturty date of the futures contract, T, the nstantaneous rsk-free nterest rate, r, observed settlement opton market prce, C t, where s an ndex over transactons (calls of assorted strke prces and maturtes), and t s an ndex over the Wednesdays n the sample. 4 Implct parameter estmaton and n-sample performance 4. Method Besdes the exogenous varables obtaned from the data set, the opton prcng formula requres some parameters as nputs. In the full model the followng parameters need to be estmated: the season and maturty effect-related parameters σ, α,, δ and the ump-related parameters κ,v,λ. There are two man approaches to estmate these parameters; from tme seres analyss of the underlyng asset prce, or by nferrng them from opton prces (Bates, 995). There are two man drawbacks of the former approach. Frst, very long tme seres are necessary to correctly estmate ump parameters, at least f prces ump rarely. Second, parameters obtaned from ths procedure correspond to the actual dstrbuton, and hence the parameters cannot be used n an opton prcng formula, snce the parameters needed for opton prcng are gven under the EMM. The latter approach, to nfer some or all of the dstrbutonal parameters from opton prces condtonal upon postulated models has been used n, e.g., Bates (99, 996, ); Baksh et al. (997); and Hllard and Res (999). Implct parameter estmaton s based on the fact that optons are forward lookng assets and therefore contan nformaton on future dstrbutons. Impled estmaton delvers the parameters under the EMM. We nfer model-specfc parameters from opton prces over an eleven years long tme perod. The model s separately estmated for March, May, July, September and December wheat futures contracts exprng n 989 through 999. In prevous studes, mplct parameters have been nferred from opton prces durng a very short tme nterval, often daly (e.g., Bates (99, 996); Hllard and Res, 999). However, ths method can be appled to data spannng any nterval that has suffcent number of trades (Hllard and Res, 999). Daly recalbratons can fal to pck up longer horzon parameter nstabltes (Bates, ). In ths study, one of the aspects we focus on s the changng volatlty durng the year. Optons wrtten on a specfc contract have only one maturty each year. If we were to use daly data, a model wth tme-dependent volatlty would be ndstngushable from a model wth constant volatlty. Informaton of changng volatlty wll be revealed as the opton prces change durng the course of the year. In other words, we need a long tme span, n order to be able to pck up volatlty term structure effects n ths market. Amercan opton prces, dsturbance term: C t, are assumed to consst of model prces plus a random addtve t ( Ft, K, t, T, T, r, κ, v, λ, σ, α,, ) et C C δ + (5) 8

10 Equaton (5) can be estmated usng non-lnear regresson. The unknown mplct parameters κ, v, λ, σ, α,, δ are estmated by mnmsng the sum of squared errors (SSE) for all opton n the sample gven by SSE T N t [ C C() ] [ e ] t T N t t (6) where s an ndex over transactons (calls of assorted strke prces and maturtes), and t s a tme ndex. The parameters mnmsng (6) were found usng the Quadratc-hll clmbng algorthm n GAUSS. Many alternatve crtera could be used to evaluate performance of opton prcng models. The overall sum of squared errors (SSE) s used as a broad summary measure to determne how well each alternatve opton prcng model fts actual market prces. Assumng normalty of the error term, nested models can be tested usng F-test statstc. 4 Bates (996, ) ponts out that the opton prcng model s poorly dentfed. Ths means that when we mnmse the non-lnear functon (5), qute dfferent parameter values can yeld vrtually dentcal results. As a result of ths, parameter estmates should be nterpreted wth care. 4. Impled parameters and n-sample prcng ft The followng models were estmated (abbrevatons used later n the paper are n parentheses): Black's (976) dffuson (Black76), Bates's (99) ump-dffuson (Bates9), Black's model wth season and maturty effect (Black SM) and Bates wth season and maturty effect (Bates SM). Table shows mplct parameter estmates for March, May, July 5, September and December wheat optons. For the Black SM and Bates SM estmaton was done wth the maturty effects of order,.e., only one parameter for α, and δ, respectvely. 6 As a result of forcng eleven years of data nto one opton prcng model wth constant parameters, the SSE s qute large. However, R values are hgh and vary between.967 and.988 between contracts and models. 4 The F statstc s computed as F[ J n K ] ( SSE SSE ) R U J, where SSE U and SSE R are sum squared SSE n K errors for unrestrcted and restrcted models respectvely, J s number of restrctons, n s number of observatons n the sample, and K s number of parameters n the unrestrcted model. In the nonlnear settng, the F dstrbuton s only approxmate (Greene, 993, p. 336). 5 For July contracts wth the Bates SM model we had a problem n mnmsng functon (6) n one step, so the parameters for ths model were estmated n two steps. In step one all parameters except α and were estmated. The parameters σ and δ from step one were then used as constants n step two. 6 We have also done some estmaton of order for both seasonal parameters and maturty parameters. Generally, usng SSE as the performance crteron there s lttle mprovement from ncludng seasonal and maturty effects of order compared to the more restrctve order seasonal and maturty effects. Estmatons of order for only the seasonal parameters gave almost the same results as estmaton of order for both maturty and seasonal parameters, and are not reported here. However, the results are avalable from the authors upon request. U 9

11 Table Implct parameter estmates for varous models on March, May, July, September and December contracts on wheat n the perod , 3859, 574, 397 and 53 observatons, respectvely. t-values are n parentheses Black76 Black SM Bates9 Bates SM March contracts σ. (54.7).85 (7).5 (3.).8 (955.) γ.4 (5.5).4 (47.9) κ.4.4 ν.9 (54.8).9 (5.4) λ.57 (6.3).59 (45.) δ.85 (47.3) 3.98 (8.6) α -. -(.6) -. -(.) (3.4) -. -(5.8) SSE May contracts σ. (388).5 (897).8 (46).3 (.4) γ.8 (6.4).5 (5.9) κ.9.6 ν.6 (673.8).7 (466.9) λ.4 (.4).6 (8.4) δ.36 (3935).7 (3.3) α -. -(74.) -.3 -(.9) -. -(.3) -.5 -(7.) SSE July contracts σ γ κ ν λ δ. (). (889.7).3 (598.).39 (83.).4 (89.4). (7.5).4..5 (6.5).5 (5.) 6.49 (578.8).5 (93.8). (.9) 4.49 (77.) α -.3 -(6.) -.5 -(5.8) -.8 -(76.7) -. -(6.) SSE September contracts.4 (33.8) 4. (7).8 (9).34 (76.9). (58.).4 (.3)..6.7 (6.8).46 (636.3).56 (6.7).4 (3.7) 7.86 (533.8). (73.) α.4 (444.3) -.5 (4.4).46 (5.3) -.3 (69.8) SSE December contracts.3 (85.3).9 (477.).5 (56.5).3 (4.5). (78.).5 (7.)..5.4 (6.3).35 (4.).65 (44.). (4.4).3 (68.).56 (.7) α. (4.7).5 (5.7) -. -(44.8) -. -(.3) SSE σ γ κ ν λ δ σ γ κ ν λ δ

12 The results provde clear evdence of the mportance of the seasonal and maturty effects; Bates SM performed best for all contracts. Furthermore, the ncluson of seasonal and maturty effects n Black76 sometmes gave approxmately the same and sometmes better ft than Bates9 ump dffuson model. Ths ndcates that the volatlty term structure may be more mportant, n terms of opton prcng, than the possblty of umps. As Hllard and Res (999) found ths analyss also shows that Bates9 performed better than Black76. We have formally tested the models aganst each other usng F-tests. The results gven n Table 3, ndcate that we can reect the other models proposed n the lterature n favour of our model wth both ump and tme dependent volatlty. Table 3 Model specfcaton tests for March, May, July, September and December contracts Null hypothess Restrctons F-value F.95 -crtcal Decson March contracts Bates SM Bates9 δ α Reect H Bates9 Black76 κ ν λ. 8.5 Reect H Black SM Black76 δ α Reect H May contracts Bates SM Bates9 δ α Reect H Bates9 Black76 κ ν λ Reect H Black SM Black76 δ α Reect H July contracts Bates SM Bates9 δ α Reect H Bates9 Black76 κ ν λ Reect H Black SM Black76 δ α Reect H September contracts Bates SM Bates9 δ α Reect H Bates9 Black76 κ ν λ Reect H Black SM Black76 δ α Reect H December contracts Bates SM Bates9 δ α Reect H Bates9 Black76 κ ν λ Reect H Black SM Black76 δ α Reect H 4.3 A closer look at the volatlty term structure From Table we also see that parameters governng the volatlty dynamcs dffer somewhat across contracts. Ths may be explaned partly by the fact that dfferent parameter values may cause qute smlar opton prces, as mentoned above. We have plotted the volatlty term structure for each contract n Fgure, usng the estmated parameters n Table. For each contract, the volatlty term structure spans one year, and ends as the futures contract expres. We see that March, July and September contracts reveal the most profound maturty effect. The December contract combnes hgh summer volatlty and a maturty effect durng autumn. In sum, the December contract seems to be more volatle durng the second half of the year. The July contract shows few sgns of seasonalty at all, but from Table we see that the seasonal parameters are sgnfcantly dfferent. Agan, ths llustrates that the maturty effect has a far bgger mpact on the term structure of volatlty than the seasonal effect.

13 Fgure Estmated term-structure of the volatlty from opton data for March, May, July, September and December futures contracts 4.4 A closer look at the ump parameters As argued elsewhere, mpled volatlty curves reveal the effects of umps on opton prces. As an llustraton of the effect of umps on mpled volatlty, we computed theoretcal opton prces on Amercan calls for dfferent strkes usng parameters from the full model (Bates SM) of the May contract n Table. The futures prce s set to F(t,T) 3, the maturty of the contract T 7 months, and the rsk free rate r.5. We backed out mpled volatlty curves usng 5 strkes (K 4, 7, 3, 33 and 36) for three dfferent opton maturtes (T, 4 and 6 months). The results are gven n Fgure Imp. vol.4.. Tm T4m T6m Strke Fgure 3 Implct volatlty patterns from CBOT wheat call optons where optons contracts have, 4 and 6 months to maturty, respectvely and the underlyng futures contract has 7 months to maturty. Impled volatlty for Amercan optons are approxmated as n Barone-Ades and Whaley (987)

14 We recognse the clear "smle" effect from Fgure, caused by the possblty of both upward and downward umps. It s also evdent that ths "smle" gets more pronounced as opton expraton gets closer. If there s only a short tme to maturty, far OTM optons n a lognormal model wll be worth relatvely lttle, snce an extreme upward prce swngs s very unlkely. In a umpdffuson model, these optons may end up ITM f a ump occurs, and consequently, these optons wll be relatvely more valuable n a ump-dffuson than n a lognormal world. When there s long tme to opton maturty, the ump component plays a less promnent part when t comes to movng futures prces upwards or downwards. In the case of OTM optons say, the dffuson term alone wll be able to move the futures prce so that the opton wll end up ITM. 7 We also note from Fgure 3 that the volatlty curve shfts upwards when opton maturty ncreases. Ths fact s manly caused by the maturty effect captured by the volatlty term structure. 5 A numercal example Fnally, we provde a numercal example showng the economc sgnfcance of our fndngs. Assume that our model specfcaton s correct; that both the volatlty term structure and umps are present n futures prces, and hence our opton prcng formula calculates the true opton prce. What knd of msprcng wll take place f we use the model of Black (976) or Bates (99) prevously suggested n the lterature? We stck to the example above and compute Amercan call opton prces based on parameters from the May contract for dfferent opton maturtes. These prces are compared to Black76 and Bates9 model prces, agan pckng parameters from Table. The results are gven n Table 4. Table 4 Comparson of Amercan wheat futures opton prces usng Black76, Bates9 and Bates SM for dfferent strkes when the underlyng futures contract has 7 months to maturty and the futures prce s set to F(t,T) 3, and the rsk free rate r.5. Parameter estmates for the May contract n Table s used %Dff. K Black76 Bates9 Bates SM Black76 - Bates SM Bates9 - Bates SM T m %.4 % T 7m % 3.9 % % -6.5 % T 4m %. % T 7m % 8.6 % % -.9 % T 6m % -.9 % T 7m % -7. % % -8.9 % Concentratng on the last two columns, we see that Bates SM produce very dfferent opton prces than Black76 and Bates9. We note that the dfference between Bates SM and Black76 s as much as 48% for the nearest OTM call. The general results are as follows: The prces from all 7 In our specal case, there s roughly equal chance for the ump to be ether postve or negatve under the EMM ( κ ). Ths means that as tme to opton expraton ncreases, multple umps wll have a tendency to cancel each other out. Ths wll enforce the flattenng effect on the volatlty smle as tme to expraton ncreases. However, ump effects wll n general be more vsble n terms of mpled volatlty as tme to expraton shortens (see Das and Sundaram (999) for an nvestgaton of term structure effects n a ump-dffuson model). 3

15 three models are more or less the same for ITM calls. Ths s due to the fact that the ntrnsc value domnates the value of an opton when deep ITM, and hence most models would produce qute smlar results. The at-the-money (ATM) prce dfferences are bascally nfluenced by the term structure effect. Both Black76 and Bates9 use an average volatlty for the whole perod as nput. The fact that the volatlty of futures contract ncreases as maturty approaches, means that usng an average value for the volatlty wll produce too hgh opton prces for short maturty optons and too low prces for long maturty optons. We note that the prces from Black76 and Blates9 are n qute good agreement wth each other; however, they dffer qute severely from the Bates SM model. Last, the two alternatve models produce sgnfcantly lower prce for OTM calls than Bates SM. For the Black76 model, ths fact s not surprsng snce OTM calls wll be more valuable n a ump-dffuson world. The results from the Bates9 model deserve some explanaton. We see that the parameters estmated for Bates9 gve a less pronounced smle effect than Bates SM. Ths s because, as the volatlty term structure s restrcted to be flat, the ump parameters wll nfluence both the prces across strkes, and the overall prce level. From the dscusson on mpled volatlty, the ump parameters nfluence both the "smle" and the level of the mpled volatlty curve. 8 In Bates SM, the term structure of volatlty can take care of the level, and the ump parameters can concentrate on "smle" effects. Hence the parameters n Bates9, through the estmaton method, emerge as a compromse of the two effects. The results provded here may be of great mportance n other valuaton contexts. For example, Hllard and Res (999) argue that average based Asan optons are popular n commodty overthe-counter (OTC) markets. They show that Asan opton prces n the Black76 versus Bates9 dffer even more than s the case for European/Amercan optons prces. Our results ndcate, n addton to the ump effect, that Asan opton prces wll dffer qute substantally dependng on where n the lfe of the opton the average s calculated. Especally, the relatve strong maturty effect wll gve very dfferent prces on Asan optons dependng on both the length of averagng perod and how close the averagng perod s to the maturty of the futures contract. 6 Summary and concludng comments In ths paper we have developed an opton prcng model that ncorporates several stylsed facts reported n the lterature on commodty futures prce dynamcs. The volatlty may depend on both calendar-tme and tme to maturty. Furthermore, futures prces are allowed to make sudden dscontnuous umps. We estmated the parameters of the futures prce dynamcs by fttng our model to eleven years of wheat optons data usng non-lnear least squares. Several models suggested n the lterature are nested wthn our model, and they all gave sgnfcantly poorer ft compared wth our more complete model formulaton. In a numercal example we showed that gnorng term structure and ump effects n futures prces may lead to severe ms-prcng of optons. References Anderson, R.W., 985. Some determnants of the volatlty of futures prces. Journal of Futures Markets 5: Ths fact may partly explan the observaton reported n Hllard and Res (999) that parameter values are not stable over tme. In ther estmaton procedure, they calbrate the model each day. Usng ther procedure, Bates9 wll be able to replcate Bates SM on one gven maturty. When ether the opton or futures maturty changes, the parameters n Bates9 must change to capture the volatlty term structure effect. Hence we would expect unstable parameters n the analyss of Hllard and Res (999) f, n fact, there exsts volatlty term structure effects n the underlyng futures data. 4

16 Baksh, G., Cao, C., Chen, Z., 997. Emprcal performance of alternatve opton prcng models. Journal of Fnance 5: Barone-Ades, G., Whaley, R.E., 987. Effcent analytc approxmaton of Amercan opton values. Journal of Fnance 4: 3-3. Bates, D.S., 99. The crash of '87: was t expected? the evdence from optons markets. Journal of Fnance 46: Bates, D.S., 995. Testng opton prcng models. Workng Paper 59, Natonal Bureau of Economc Research. Bates, D.S., 996. Jumps and stochastc volatlty: exchange rate processes mplct n Deutsche mark optons. Revew of Fnancal Studes 9: Bates, D.S.,. Post- 87 crash fears n the SP 5 futures opton markets. Journal of Econometrcs 94: Bessembnder, H., Coughenour, J.F., Segun, P.J., Smoller, M.M., 995. Mean reverson n equlbrum asset prces: evdence from the futures term structure. Journal of Fnance 5: Bessembnder, H., Coughenour, J.F., Segun, P.J., Smoller, M.M., 996. Is there a term structure of futures volatltes? reevaluatng the Samuelson hypothess. Journal of Dervatves Wnter: Black, F., 976. The prcng of commodty contracts. Journal of Fnancal Economcs 3: Cho, J.W., Longstaff, F.A., 985. Prcng optons on agrcultural futures: an applcaton of the constant elastcty of varance opton prcng model. Journal of Futures Markets 5: Cox, J.C., Ross, S.A., 976. The valuaton of optons for alternatve stochastc processes. Journal of Fnancal Economcs 3: Das, S.R., Sundaram, R.K., 999. Of smles and smrks: a term structure perspectve. Journal of Fnancal and Quanttatve Analyss 34: -39. Fackler, P.L., Tan, Y., 999. Volatlty models for commodty markets. Workng Paper, North Carolna Sate Unversty. Galloway, T.M., Kolb, R.W., 996. Futures prces and the maturty effect. Journal of Futures Markets 6: Greene, W., 993. Econometrc Analyss, nd. ed. NJ, Prentce Hall. Hall, J.A., Brorsen, B.W., Irwn, S.H., 989. The dstrbuton of futures prces: a test of the stable paretan and mxture of normal hypothess. Journal of Fnancal and Quanttatve Analyss 4: 5-6. Hldreth, C., Lu, J., 96. Demand relatons wth autocorrelated dsturbances. Techncal Bulletn 76, Mchgan State Unversty Agrcultural Experment Staton. Hllard, J.E., Res, J.A., 999. Jump processes n commodty futures prces and opton prcng. Amercan Journal of Agrcultural Economcs 8: Hudson, M.A., Leuthold, R.M., Sarassoro, G.F., 987. Commodty futures prce changes: recent evdence on wheat, soybeans, and lve cattle. Journal of Futures Markets 7: Merton, R.C., 976. Opton prcng when underlyng stock returns are dscontnuous. Journal of Fnancal Economcs 3: Mlonas, N.T., 986. Prce varablty and the maturty effect n future markets. Journal of Futures Markets 6: Myers, R.J., Hanson, S.D., 993. Prcng commodty optons when underlyng futures prce exhbts tme-varyng volatlty. Amercan Journal of Agrcultural Economcs 75: -3. Rutledge, D.J.S., 976. A note on the varablty of futures prces. Revew of Economcs and Statstcs 58: 8-. Samuelson, P.A., 965. Proof that properly antcpated prces fluctuate randomly. Industral Management Revew 6: Schwartz, E.S., 997. The stochastc behavor of commodty prces: mplcatons for valuaton and hedgng. Journal of Fnance 5: