Evidence of the unspanned stochastic volatility in crude-oil market

Transcription

1 The Academy of Economc Studes The Faculty of Fnance, Insurance, Bankng and Stock Echange Doctoral School of Fnance and Bankng (DOFIN) Dssertaton Paper Evdence of the unspanned stochastc volatlty n crude-ol market Student: Răzvan Danel Tudor Supervsor Professor: PhD. Mosă Altăr Bucharest, July 008

2 Abstract The purpose of ths dssertaton paper s to conduct a comprehensve analyss of unspanned stochastc volatlty n commodty markets wth focus and emprcal evdence on crude-ol market. Usng crude-ol futures and optons on futures data from New York Mercantle Echange (NYMEX) there are presented model-free results that strongly suggest the presence of unspanned stochastc volatlty n the crude-ol market. Sharp ol prces changes eert nfluence on macroeconomc actvty n general and crude-ol ndustry n partcular. The mportance of the results s that they show the etent to whch volatlty rsk s spanned by the futures contracts. The etent to whch crude-ol futures contracts tradng span volatlty wll ndcate f optons on futures are redundant securtes or there s needed a med strategy combnng both types of crude-ol market dervatves (futures and optons) to fully hedge aganst volatlty rsk.

3 1. Introducton Over the last few years, the persstent sharp ol prces changes n both the spot and futures markets have represented perhaps the most strkng challenge to the forecastng abltes of prvate and publc nsttutons worldwde. From the demand sde ncreasng crude-ol prces led to new challenges n hedgng aganst volatlty rsk. Whle volatlty s clearly stochastc, t s not clear to what etent volatlty rsk can be hedged by tradng n the commodtes themselves or, more generally, ther assocated futures contracts, forward or swap contracts, n other words, the etent to whch volatlty s spanned. Estng equlbrum models from commodty markets mply that volatlty rsk s largely spanned by the futures contracts. Manly, they suggest that market volatlty s embedded n nventores whch are the bass for futures prce formaton. Therefore by constructon futures offer a hgh degree of volatlty spannng. The consequence of these models s that they mply that optons on futures are redundant securtes. In spte of ths, the data provded form Bank of Internatonal Settlements BIS strongly suggests that the market for commodty dervatves has ehbted phenomenal growth over the past few years. For echange-traded commodty dervatves, the BIS estmates that the number of outstandng contracts more than doubled from 1.4 mllon n June 003 to 3.1 mllon n June 006. For over-thecounter (OTC) commodty dervatves, the growth has been even stronger wth the BIS estmatng that, over the same perod, the notonal value of outstandng contracts ncreased fve-fold from USD 1.04 trllon to USD 6.39 trllon. Importantly, a large and ncreasng fracton of the commodty dervatves are optons (as opposed to futures, forwards and swaps). Accordng to BIS statstcs, optons now consttute over one-thrd of the number of outstandng echange-traded contracts and almost two-thrds of the notonal value of outstandng OTC contracts. The purpose of ths paper s to show that f, for a gven commodty, volatlty contans mportant unspanned components t cannot be fully hedged and rsk-managed usng only the underlyng nstruments and optons are not redundant securtes. 3

4 The unspanned stochastc volatlty evdence research s conducted n crude-ol market because t s by far the most lqud commodty dervatves market. The data for the analyss was provded by New York Mercantle Echange NYMEX and contans a large set of futures and optons on futures contracts prces. Snce volatlty s not drectly observable I wll use, for dfferent optons maturtes, straddle returns and mpled volatlty of the at-the-money optons straddles as proes for the true volatlty and I wll show the etent to whch futures contracts span volatlty. If volatlty s completely spanned by tradng n futures contracts then the equlbrum models for commodty markets are correct n assumng that commodtes futures prces formaton ncorporates market volatlty. If shown on contrary, t means that optons on futures are not redundant securtes and ther role s to etend the degree of hedgng whch futures contracts tradtonally offer. The reason for choosng these two volatlty proes s that straddle returns are not condtoned on a partcular prcng model. Returns are obtaned from daly optons on futures market prces from NYMEX. Whle usng the mpled volatlty, though t mght be more accurate, t nvolves usng a prcng model. Prevously, ths approach was used to evdence the unspanned stochastc volatlty n fed-ncome market, more specfcally to show the etent to whch tradng of bonds span the term structure of nterest rates. The dssertaton paper s organzed as follows. Secton contans a lterature revew of the models whch treated the stochastc volatlty n commodty and fnancal markets. Secton 3 brefly presents the crude-ol dervatves data used n ths paper and the computatonal aspects behnd data whch was used as nput for the model. In secton 4 the paper contans the model used to evdence of the unspanned stochastc volatlty n crude-ol market. Secton 5 presents model estmaton and analyss. Fnally, n Secton 6 there are to be found the concluson whch can be drawn from ths paper. Secton 7 contans the reference lst and Secton 8 the relevant addtonal nformaton Annees - whch are mentoned n the paper content. 4

5 . Lterature Revew The frst equlbrum models from commodty markets mpled that futures contracts provde nsurance aganst prce volatlty, the level of nventores beng negatvely related to the requred rsk premum of commodty futures. The startng pont of these models was the tradtonal Theory of Storage orgnally proposed by Kaldor (1939). The theory provdes a lnk between the term structure of futures prces and the level of nventores of commodtes. Ths lnk, also known as cost of carry arbtrage, predcts that n order to nduce storage, futures prces and epected spot prces of commodtes have to rse suffcently over tme to compensate nventory holders for the costs assocated wth storage. Developments n ths area were made by Deaton and Laroque (199), Chambers and Baley (1996), Routledge, Sepp and Spatt (000). Ther models predct a lnk between the level of nventores and future spot prce volatlty. Inventores act as buffer stocks whch can be used to absorb shocks to demand and supply, thus dampenng the mpact on spot prces. Deaton and Laroque show that at low nventory levels, the rsk of stock-out (ehauston of nventores) ncreases and epected future spot prce volatlty rses. In an etenson of the Deaton and Laroque model whch ncludes a futures market, RSS show how the shape of the futures curve reflects the state of nventores and sgnals epectatons about future spot prce volatlty. DL (199) and RSS (000) have eplaned the estence of a convenence yeld as arsng from the probablty of a stock-out of nventores. Because they study storage n a rskneutral world, rsk premums are zero by constructon, and futures prces smply reflect epectatons about future spot prces. Another reference model, the model n Ltzenberger and Rabnowtz (1995) and Ng and Prrong (1994), ncorporates the embedded opton n reserves of etractable resource commodtes. Fnally t has smlar mplcatons. The relatonshp between volatlty and the slope of the futures - Ltzenberger and Rabnowtz (1995) showed t for crude ol, and Ng and Prrong (1994), for metals - show that the degree of backwardaton s ndeed postvely related to volatlty, mplyng that volatlty does contan a component that s spanned by the futures contracts. However, whether volatlty also contans mportant unspanned components was not shown. 5

6 Other papers, whch emphasze producton/etracton and nvestment decsons for the formaton of futures prces, nclude those of Casassus, Colln-Dufresne, and Routledge (003), Kogan, Lvdan, and Yaron (005) and Carlson, Khoker, and Ttman (006). In ther paper, Gordon, Hayash and Rouwenhorst (005), analyzed the fundamentals of commodty futures returns and predcted a lnk between the state of nventores, the shape of the futures curve, and epected futures rsk premums. They showed that show that the convenence yeld s a decreasng, non-lnear relatonshp of nventores and also lnked the current spot commodty prce and the current (nearest to maturty) futures prce to the level of nventores, and emprcally documented the nonlnear relatonshp predcted by the estence of the non-negatvty constrant on nventores. In partcular, they showed that low nventory levels for a commodty are assocated wth an nverted ( backwardated ) term structure of futures prces, whle hgh levels of nventores are assocated wth an upward slopng futures curve ( contango ). The estence of unspanned volatlty factors was frst evdenced n fed ncome market. Colln-Dufresne and Goldsten (00) and Hedar and Wu (003) defned unspanned stochastc volatlty as beng those factors drvng Cap and Swapton mpled volatltes that do not drve the term structure of nterest rates. In other words they showed that tradng n underlyng bonds do not span the term structure of nterest rates. There are embedded factors n Cap and Swapton that bonds do not contan and make them more valuable n hedgng aganst nterest rates volatlty rsk That s, n contrast to the predctons of standard short-rate models, bonds do not span the fed ncome market. Usng Colln-Dufresne and Goldsten (00) approach, Trolle and Schwartz (006) etended the problem wth estence of unspanned stochastc volatlty to commodty markets. They developed a tractable model for prcng commodty dervatves n the presence of unspanned stochastc volatlty. The model features correlatons between nnovatons to futures prces and volatlty, quas-analytcal prces of optons on futures and futures curve dynamcs n terms of a low-dmensonal affne state vector. Ther evdence was on crude-ol market due to ts lqudness and showed that n the presence of unspanned stochastc volatlty factors optons are not redundant securtes. The model and the evdence could be etended as well on the other commodty markets. 6

7 Rchter and Sorensen (007) have a work n progress for a stochastc volatlty model n the presence of unspanned volatlty factors for the soybean market. 3. Overvew of the Data As mentoned before, crude-ol market s the most lqud commodty market. The data used n ths paper was delvered by New York Mercantle Echange - NYMEX. It contans large data set of futures and optons on futures prces wth dfferent maturtes and strke prces. The futures data contans daly prces for futures contracts startng wth January 1987 and endng wth May 008. Snce optons on futures prces were avalable for research purposes only for June 00 December 006 perod, I chose to use the futures contracts prces for the same nterval. The NYMEX futures contract trades n unts of 1,000 barrels, and the delvery pont s Cushng, Oklahoma, whch s also accessble to the nternatonal spot markets va ppelnes. The contract provdes for delvery of several grades of domestc and nternatonally traded foregn crude, and serves the dverse needs of the physcal market. The NYMEX symbol for lght-sweet crude-ol s CL. Crude ol futures are lsted nne years forward usng the followng lstng schedule: consecutve months are lsted for the current year and the net fve years; n addton, the June and December contract months are lsted beyond the sth year. Addtonal months wll be added on an annual bass after the December contract epres, so that an addtonal June and December contract would be added nne years forward, and the consecutve months n the sth calendar year wll be flled n. The futures epre on the thrd busness day pror to the 5th calendar day of the month proceedng the delvery month. If the 5th calendar day of the month s a nonbusness day, epraton s on the thrd busness day pror to the busness day proceedng the 5th calendar day. For my purpose I etracted from varous maturtes only futures contracts wth tme-to-maturty 1 Month, 3 Months, 6 Months, 9 Months and 1 Year. The reason s that crude-ol spot prces establshed on spot markets are not avalable. Manly, spot prces are settled on one to one transactons between partners based on current market condtons. Therefore the 1 Month tme-to-maturty futures contract serves as a proy for the crude- 7

8 ol spot prces. The 1 Year tme-to-maturty futures prces are contnuous durng the June 00 December 006 sample so I chose not to nclude them n the analyss. The quarterly maturtes correspond to the tradtonal hedgng strategy of a crude-ol refnng based company. Beng gven the optmal refnng capacty usually the company engages n a rollng futures contract wth quarterly maturtes provdng the company wth the necessary crude ol amount at a certan prce whch can be used for fnancal forecasts. Whle futures offer prce protecton by allowng the holder of a futures contract to lock n a prce level, a major advantage of optons s that the holder of an optons contract s afforded prce protecton, but stll has the ablty to partcpate n favorable market moves. Because the buyer of an optons contract has the optons contract but not the market moves aganst a poston, and a trader holds on to ths opton, the mamum cost s the prce he has already pad for the opton. On the other hand, f the market moves n favor of a poston, the vrtually unlmted proft potental to the buyer of an optons contract s parallel to a futures poston, net of the premum pad for the optons contract. Therefore, protecton from unfavorable market moves s acheved at a known cost, wthout gvng up the ablty to partcpate n favorable market moves. Optons on futures contracts epre three busness days pror to the epraton of the underlyng futures. For the research I chose crude-ol calendar spread optons on futures. The reason s that calendar spread optons are the most traded crude-ol optons dervatves on NYMEX and thus the results of my study wll be more representatve. Also they mply delvery of the underlyng asset as opposed to other dervatves whch are only settled n cash, for eample European Style optons NYMEX symbol LO. Ther NYMEX tradng symbol for calendar spread optons s WA. The contract s smply an optons contract on the prce dfferental between two delvery dates for the same commodty. The prce spread between contract months can be etremely volatle because the energy markets are more senstve to weather and news than any other market. A wdenng of the month-tomonth prce relatonshps can epose market partcpants to severe prce rsk whch could adversely affect the effectveness of a hedge or the value of nventory. The calendar spread optons can allow market partcpants who hedge ther rsk to also take advantage of favorable market moves. 8

9 For the correspondng three maturtes of the futures prces I etracted correspondng calendar spread straddles. One straddle conssts of a call and a put opton wth the same strke. More, I chose the at-the-money straddles snce they are senstve to market volatlty ( Vegas peak for the at-the-money straddles). At-the-money property of an opton means that the opton has the strke prce equal or near to spot prce. Ths computaton s helpful to etract from the whole optons and futures sample the data I need for the evdence of unspanned stochastc volatlty. 4. The Model If equlbrum models are correct and changes n crude ol prces are spanned by the futures contracts then n order to hedge aganst volatlty rsk one may construct a portfolo of futures contracts for ths purpose. One smple alternatve to evdence f the concluson of these models s correct s to smply regress changes n volatlty from crude ol markets on futures contracts prces and see f they fully eplan volatlty changes. But volatlty n crude ol market as well as n other commodty and fnancal markets s stochastc and not drectly observable. Therefore I wll use two reasonable proes for the true and unobservable volatlty atthe-money calendar spread straddles prces and at-the-money calendar spread straddles mpled volatlty. A straddle conssts of a call and a put opton on the same underlyng wth the same strke. When purchasng a straddle the nvestor epects the market to spke n ether drecton. Ths s the case of long at-the-money straddle strategy whch wll be used throughout ths paper as opposte to short at-the-money straddle whch s used when market s epected to be quet (epectng mnor changes n volatlty). Therefore we can say that by purchasng a straddle the nvestor trades volatlty. Straddle profts are unlmted n ether drecton whle losses are lmted to the premum pad for both optons whch form the straddle. The reason for selectng straddles as volatlty proes s the straddle Greeks ndcators. The prce of a near-atm straddle has low senstvty to varatons n the prce 9

10 of the underlyng futures contract (snce deltas are close to zero for ATM straddles) but hgh senstvty to varatons n volatlty (snce Vegas peak for ATM straddles). Delta ( Δ ) and Vega (ν ) for at-the-money straddles: V Δ = S e = e qt * φ( d ), Call qt * φ( d 1 ), Put V qt ν = = S * e φ( d1) τ, for both Call and Put optons. σ The ndcators were derved from the Black-Scholes opton prcng formula where: V Value of the opton; S Stock prce; q Annual dvdend yeld; τ - Tme to maturty (T-t); σ - Volatlty; r Rsk free rate; Φ(d 1 ) - The probablty of eercse under the equvalent eponental martngale probablty measure and the equvalent martngale: φ( ) = y e dy = π y e dy π ln( S / K) + ( r q + σ / ) *τ y = σ τ Bellow graphc ehbts the typcal payoff functon of an at-the-money straddle: 10

11 To avod the non-statonary problem wth straddle and futures prces, whch s common to almost all asset prces I wll use for further analyss straddle returns and futures returns. Straddle returns are computed as follows: Where: r straddle S K ( π call + π put ), S > K + ( π call + π put ), = K S ( π call + π put ), S + ( π call + π put ) < K 0, else S - The spot prce of the underlyng commodty. For calendar spread optons type the underlyng commodty spot prce s the prce dfferental from current market prce and the futures prce of the futures contract whch at the maturty of the opton. K The strke prce; π call, π put - Call and Put opton prces; The futures contracts returns are smply computed ths way: Where: futures _ returns, jmonth = Spot Pr ce FuturesContract Pr ce, jmonth Spot Pr ce - The crude-ol spot prce. In absence of a transparent spot market the spot prce s computed as the prce of the futures contract wth shortest tme to maturty, the contract wth epraton the followng month. FuturesCon tract Pr ce, jmonth - The today observed market prce of the futures contract wth epraton n j months. As mentoned before j = 3, 6, 9 Months. There are three alternatves to evdence the presence of unspanned stochastc volatlty n crude ol market: - Investgate how much of the varaton n the prces of dervatves hghly eposed to stochastc volatlty (so-called straddles ) can be eplaned by varaton n the underlyng futures prces; - Investgate how much of the varaton n mpled volatltes (whch s related to epectatons under the rsk-neutral measure of future volatlty) can be eplaned by varaton n the underlyng futures prces; 11

12 - Investgate how much of the varaton n realzed volatlty, estmated from hgh frequency data, can be eplaned by varaton n the underlyng futures prces. - Investgate how much of the volatlty of the varance swaps can be eplaned by varaton n the underlyng futures prces. Unfortunately hgh-frequency data s avalable only for calendar spread optons. Also varance swaps are qute llqud n the market therefore I wll use only the frst two approaches for evdence. For the approach whch requres the use of straddle mpled volatlty, ths s computed as the average of straddle component put and call optons mpled volatltes. The put and call mpled volatltes are obtaned from put and call formulas of the Black- Scholes model. Brefly, the formulas for call and put optons derved from the Black-Scholes partal dfferental equaton are: V t 1 + σ S V S V + rs S rv rt C( S, T ) = Sφ( d1) Ke φ( d ) = 0 (Black-Scholes PDE) Where: r ( T t) P( S, T ) = Ke φ ( d ) Sφ( d 1) d 1 ln( S / K) + ( r q + σ / ) *τ = σ τ d ln( S / K) + ( r σ / ) * T = = d1 σ σ τ T Φ s the cumulatve dstrbuton functon. Φ(d 1 ) and Φ(d ) are the probabltes of eercse under the equvalent eponental rsk neutral measure and the equvalent rsk neutral probablty measure, respectvely. Beng gven the call and put opton prces, the underlyng futures prce, the current and maturty date, the strke prce and the rsk free rate, the mpled volatlty s computed usng the Newton-Raphson method. It s a root-fndng algorthm that uses the frst few terms of the Taylor seres of a functon f () n the vcnty of a suspected root. Newton's method s also known as Newton's teraton. 1

13 The Taylor seres of f () about the pont + ε s gven by: Keepng terms only to frst order: = 0 1 f ( 0 + ε ) = f ( 0 ) + f '( 0 ) ε + f ''( 0 ) ε +... f + ε ) f ( ) + f '( )ε ( Ths epresson can be used to estmate the amount of offset ε needed to land closer to the root startng from an ntal guess 0. Settng f ( 0 + ε ) = 0 and solvng the above equaton for ε ε 0 gves: ε 0 = f ( 0 f '( 0 ) ) Ths s the frst-order adjustment to the root's poston. By lettng 1 = 0 + ε, calculatng a new ε 1, and so on, the process can be repeated untl t converges to a fed pont (whch s precsely a root) usng: ε 0 = f ( 0 f '( 0 ) ) obtan: Therefore wth a good ntal choce of the root's poston, the algorthm can be appled teratvely to n+ 1 = n f ( n f '( n ) ) Many commodty markets as well as fnancal markets are characterzed by a hgh degree of collnearty between returns. In order to etract the most uncorrelated sources of varaton n a multvarate system I wll use prncpal components analyss (PCA) for the futures returns. Manly, prncpal components analyss objectve s to: - Reduce dmensonalty by takng nto account only the most relevant prncpal components from the whole data set; 13

14 - Avod near multcollnearty ssues for the returns and use n further analyss only uncorrelated components. Also, these components characterze the data and are useful for drawng conclusons; Mathematcal background: The data nput to prncpal component analyss must be statonary. Prncpal component analyss s based on egenvalues and egenvector analyss of V =X XT, the k k symmetrc matr of correlatons between the varables n X. Each prncpal component s a lnear combnaton of these columns, where the weghts are chosen n such way that: - the frst prncpal component eplans the greatest amount of the total varaton n X, the second component eplans the greatest amount of the remanng varaton, and so on; - the prncpal components are uncorrelated to each other; Denotng by W the k k matr of egenvectors of V. Thus: VW=W Λ Where Λ s the k k dagonal matr of egenvalues of V. Then we order the columns of W accordng to the sze of correspondng egenvalue. Thus f W = w ) for,j=1,,k then the m-th column of W, denoted w = w,..., w ), s the k 1 m ( 1m km egenvector correspondng to the egenvalue λm and the column labelng has been chosen so that λ > λ >... > λ 1 k. Therefore the m-th prncpal component of the system s defned by: Where P = w X + w X w m 1 m 1 m X denotes the -th column of X, the standardzed hstorcal nput data on the -th varable n the system. In matr notaton the above defnton becomes: P m = Xw m Each prncpal component s a tme seres of the transformed X varables, and the full T m matr of prncpal components, whch has P m as ts m-th column, may be wrtten as: km X k ( j 14

15 P=XW The procedure leads to uncorrelated components because: P P=W X XW=TW W Λ W s an orthogonal matr, whch means ' 1 W = W and so P P=T Λ. Snce ths s a dagonal matr the columns of P are uncorrelated, and the varance of the m-th prncpal component s λ m (sum of egenvalues). However, the sum of the egenvalues s k, the number of varables n the system. Therefore, the proporton of varaton eplaned by the frst n prncpal components together s: n = 1 Because of the choce of labelng n W the prncpal components have been ordered so that P1 belongs to the frst and largest egenvalue λ 1, P belongs to the frst and largest egenvalue λ, and so on. In a hghly correlated system the frst egenvalue wll be much larger than the others, so the frst prncpal component alone wll eplan a large part of varaton. Snce W ' 1 = W λ /, s equvalent to X=PW, that s: k X = w P + w P w 1 1 k P k Thus each vector of the data nput may be wrtten as a lnear combnaton of the prncpal components. To sum up prncpal component analyss t s a way of dentfyng patterns n data, and epressng the data n such a way as to hghlght ther smlartes and dfferences. Snce patterns n data can be hard to fnd n data of hgh dmenson, where the luury of graphcal representaton s not avalable, prncpal components analyss s a powerful tool to acheve ths. The prncpal components analyss wll be llustrated n the net secton on the hghly correlated crude-ol futures prces returns. 15

16 In order to evdence the presence of unspanned stochastc volatlty n crude-ol market I wll use the frst two approaches mentoned above: nvestgate how much of straddle returns and straddle mpled volatltes varaton can be eplaned by the varaton of the futures returns. The evdence procedure conssts of three steps: a) The frst step s prncpal components analyss of the correlaton matr of daly futures returns. We retan all the prncpal components dentfed n the analyss n an attempt to not eclude from evdence any source of varaton embedded n a component, though that component mght have mnor sgnfcance. b) For each futures contract I regress the daly closest to the at-the-money straddle returns on the futures returns prncpal components. For each futures contract I regress the daly straddle mpled volatltes on futures returns prncpal components. The daly straddle mpled volatlty s related to the average epected (under the rsk-neutral measure) volatlty of the underlyng futures contract over the lfe of the opton. Snce n commodty and fnancal markets the returns dependency s rarely lnear I wll ntroduce n the regresson equaton the squared prncpal components also, n an attempt to take nto account non-lneartes between straddle returns and mpled volatltes and futures returns. Therefore f I take nto account just one prncpal component the regresson equaton to catch non-lnearty wll be: y = α + β + β + ε 1 The coeffcent for the squared prncpal component wll be mportant (as long as t s sgnfcant) just for the sgn ndcatng the convety or concavty of the dependency. Another aspect s takng nto account the cross-products dependences of the straddle returns and mpled volatlty. These dependences reflect changes n the margnal effect of one eplanatory varable gven others. Consderng straddle returns and frst two prncpal components the transformaton can be wrtten as: = α + β + β w + β + β w + β w + ε y By rewrtng the equaton above as: y = α + β w + β w ) + ( β + β + β ) + ε ( w 16

17 17 We can nterpret the ntercept as a functon of w and the slope of as changng wth w and. To sum up, takng nto consderaton both squared components and cross-product between components the regresson equatons for both approaches may be wrtten as: Straddle returns regresson: t y ε β β β β β β β β β α = Impled volatlty regresson: t z ε β β β β β β β β β α = Where:, = 1,,3 - The prncpal components of the futures return data. They numbered accordng to they varaton eplanatory power from prncpal component wth hghest egenvalue to the prncpal component wth smallest egenvalue. y - The straddle returns at maturty. In my case = 3, 6, 9 Months. z - The mpled volatlty of the straddles at maturty. Both regressons wll ndcate the etent to whch volatlty s spanned by the futures contracts. c) Fnally, I wll analyze the prncpal components of the tme seres of resduals from the straddle return regressons and the mpled volatlty regressons. The prncpal components of the resduals are, by constructon, ndependent of those of the futures returns. If there s unspanned stochastc volatlty n the data, there should be at least one sgnfcant eplanatory prncpal component for the varaton due to unspanned factors. If the resduals are smply due to nosy data, there should not be one prncpal component wth hgh eplanatory power among resduals. 5. The Model Estmaton and Analyss Commodty futures prces are characterzed by some mportant propertes: - Commodty futures prces are often backwardated" n that they declne wth tme to delvery,

18 - Spot and futures prces are mean revertng; - Commodty prces are strongly heteroscedastc and prce volatlty s correlated wth the degree of backwardaton; - Unlke fnancal assets, many commodtes have pronounced seasonalty n both prce levels and volatltes. Beng gven S(t) the tme-t crude-ol spot prce and F(t, T ) [P(t, T )] the tme-t prce of a crude-ol futures contract [zero-coupon bond] wth maturty T - t. The futures contract s backwardated f S(t) - P(t, T )F(t, T ) > 0 and strongly backwardated f S(t) - F(t, T ) > 0. For our futures date the results confrm the above backwardaton property: Backwardaton type vs. Maturty of the Futures Contract 3 Months 6 Months 9 Months Backwardaton Degree(%) Strongly Backwardaton Degree(%) Table 1 The smple and strong backwardaton degree As tme to maturty ncreases so does the backwardaton degree. If we take a look on the futures prces graphcal representaton for varous maturtes we see that clearly the market was n contango, although market epectatons derved from the prces of futures contracts for the same maturtes were bearsh. Graph 1 Futures prces 18

19 The strongly heteroscedastcty property of commodtes, whch can be translated as the property of futures prces to have tme dependant functons for mean ( μ (t) ) and varance ( σ ( t) ) poses a serous problem for out further econometrc estmatons. In order to test the valdty of property I wll use Augmented Dckey-Fuller (ADF) test for the 3 Months futures prces. The ADF test s a unt root test whch s carred out by estmatng the followng equaton: Δ y t = α + β t + γ y t 1 + δ 1 Δ y t δ 1 Δ y t p + Ths s the most restrctve form of the test, whch ncludes the ntercept ( α) and the trend ( β ). The null hypothess s that the coeffcent of the level varable (γ ) s 0, whch means the seres s non-statonary, or less than 0 otherwse. I carred out the test for the futures prces wth 3 Month maturty n levels usng only the ntercept. The results were: Null Hypothess: FUTURES_3M has a unt root Eogenous: Constant Lag Length: 0 (Automatc based on SIC, MAXLAG=) ε t t-statstc Prob.* Augmented Dckey-Fuller test statstc Test crtcal values: 1% level % level % level *MacKnnon (1996) one-sded p-values. Table The ADF test results for 3 Months Futures Prces The value of the ADF test s larger than the crtcal values for all levels of confdence meanng that we cannot reject the null hypothess of the futures prces seres beng non-statonary. Therefore I wll further use futures returns nstead of futures prces. Futures returns are computed as shown n 4 th secton as: = futures _ returns, jmonth Spot Pr ce FuturesContract Pr ce, jmonth 19

20 Buldng the futures returns tme seres offer the advantage of statonary. Indeed f we carry out once agan the ADF test for the 3 Months futures returns, the value of the ADF test wll reject the null hypothess. Null Hypothess: FCR_3M has a unt root Eogenous: Constant, Lnear Trend Lag Length: 0 (Automatc based on SIC, MAXLAG=) t-statstc Prob.* Augmented Dckey-Fuller test statstc Test crtcal values: 1% level % level % level *MacKnnon (1996) one-sded p-values. Table 3 The ADF test results for 3 Months Futures Returns The ADF unt root tests for the 6 Months and 9 Months futures returns may be found n the Anne secton of ths paper (Table 13, 14, 15). The graphcal representaton of the crude-ol futures returns for the chosen maturtes show that n crude-ol market futures returns are hghly correlated. Graph Futures returns (3 Months, 6 Months, 9 Months) 0

21 Ths hghly correlaton suggests that for eample the 9 Months futures returns are not only nfluenced by the crude-ol spot prce but also by ntermedate maturtes futures returns. Net I wll present the futures returns correlaton matr. The correlaton coeffcents are closed to 1 ndcatng as well a hgh correlaton degree. FCR_3M FCR_6M FCR_9M FCR_3M FCR_6M FCR_9M Table 4 Futures returns correlaton matr If we eamne the frst column of the model we see that the correlaton degree tends to decrease wth maturty though very slghtly. Snce crude-ol futures returns are eplanatory varables n my classcal lnear regresson model, the hghly correlated returns pose the problem of near multcollnearty. In ths case, t s not possble to estmate all of the betas from the model. In the presence of multcollnearty, t wll be hard to obtan small standard errors. Therefore I wll use futures returns prncpal components analyss as a soluton for the near multcollnearty problem. The startng pont n dentfyng the futures returns prncpal components s the futures returns correlaton matr. Bellow there s presented the egenvalues and egenvectors of the futures returns correlaton matr. The egenvectors are ordered after the correspondng egenvalue, startng wth the hghest. Date: 07/06/08 Tme: 17:16 Sample (adjusted): 6/10/00 10/0/006 Included observatons: 1140 after adjustments Correlaton of FCR_3M FCR_6M FCR_9M Comp 1 Comp Comp 3 Egenvalue Varance Prop Cumulatve Prop

22 Egenvectors: Varable Vector 1 Vector Vector 3 FCR_3M FCR_6M FCR_9M Table 5 The egenvalues and egenvectors of futures returns correlaton matr The frst prncpal component, whch wll be further denoted as PC 1, has the hghest egenvalue, whch s responsble for eplanng 98.76% ( λ / k, where k s the matr dmenson, n my case 3) of the varaton of the future returns. If we look at correspondng egenvector weghts they are qute smlar due to strong correlaton between futures returns. The sgnfcance of the frst prncpal component correspondng egenvector weghts s that an upward shft n the frst prncpal component nduces a downward parallel shft of the futures returns curve. For ths reason frst prncpal component s called the trend component. 1 Graph 3 3M futures return curve reacton to PC1 upward shft.

23 As shown n the theoretcal secton of my paper startng from the egenvectors we can get to the orgnal data applyng the followng formula: X = w P + w P w 1 1 The graph above shows by comparson the 3M futures return curve after nducng an upward shft n the frst component. The downward parallel shft s eplanable due to negatve and smlar weghts of the egenvector. The second prncpal component, whch wll be further denoted as PC, eplans only 1.18% of futures returns varaton. The weghts are ncreasng from - to +. Thus an upward movement of the second prncpal component nduces a change n slope of the futures returns, where short maturtes move down and long maturtes move up. The second prncpal component sgnfcance s that 1.18% of the total futures return varaton s attrbuted to changes n slope. k P k Graph 4 3M futures return curve reacton to PC upward shft. The thrd prncpal component, whch wll be further denoted as PC 3, eplans only 0.03% of the futures returns varaton. The weghts are postve for the short term returns, negatve for the medum term returns and postve for the long term returns. Therefore we can say that the thrd component nfluences the convety of the returns curve. The sgnfcance of the thrd prncpal component s that 0.03% of the total varaton s due to changes n convety. 3

24 Graph 4 3M futures return curve reacton to PC3upward shft. Gven the varance eplaned by each prncpal component I may choose to drop the thrd component and use only the frst two components n the regresson, snce they cumulated eplan 99.97% of the futures returns varaton. I chose not to drop t snce I want to see f changes n convety have sgnfcance n eplanng volatlty n crude-ol markets as well. As I prevously mentoned the man purpose when usng prncpal components analyss was to elmnate the strong correlaton among futures returns. Indeed f we check the correlaton matr of prncpal components we see that correlaton ndces are close to 0 ndcatng that we managed to etract patterns from orgnal data whch move ndependently. PC1 PC PC3 PC PC PC Table 6 Prncpal components correlaton matr I decded to nclude squared prncpal components and prncpal components cross-product n an attempt to take nto account possble non-lnearty between volatlty 4

25 proes and futures returns. Though, ths may lead as well to near multcollnearty ssue. In Anne part of ths paper you may fnd the correlaton matr of prncpal components (Table 16). The correlaton ndces are not hgh, the hghest values are for correlatons between squared prncpal components of the frst two components and cross-product between them ( ). Net, I wll compute the tme seres of volatlty proes I chose. The frst volatlty proy s straddle returns for the same maturty as futures returns (3 Months, 6 Months and 9 Months). Straddle returns were computed usng: r straddle S K ( π call + π put ), S > K + ( π call, = K S ( π call + π put ), S + ( π call + π 0, else + π put put ) ) < K The perod of the sample from whch I etracted the straddles was 10/6/00 1/14/006. When I bult the straddles I looked manly for at-the-money straddles (straddles wth strke prce near or equal to spot prce). The daly frequency of the data was not so hgh, therefore there were days when just only straddle may be computed from the put and call optons avalable. I decded to take t nto account for further evdence mposng though the condton that strke prce dvded by the underlyng asset spot prce to be n the nterval (0.75; 1.5). Where straddle could not be computed due to lack of data or values outsde the (0.75; 1.5) nterval I used for the mssng daly straddle nformaton the prevous avalable straddle returns value. The 3Months straddle returns seres s represented n the bellow graphc. Graph 5 3M straddle returns curve 5

26 Further, I wll carry out the ADF unt root test to see f we can work wth the level seres or there s needed at least on dfference n order to obtan a statonary seres. The output of the ADF test (carred out wth both ntercept and trend ncluded) s: Null Hypothess: WA_3M has a unt root Eogenous: Constant, Lnear Trend Lag Length: (Automatc based on SIC, MAXLAG=) t-statstc Prob.* Augmented Dckey-Fuller test statstc Test crtcal values: 1% level % level % level *MacKnnon (1996) one-sded p-values. Table 7 Calendar spread straddle returns ADF test result The value of the ADF test s lower than test crtcal values. Therefore the null hypothess can be rejected leadng to concluson that calendar spread optons straddle returns for the menton perod are statonary. In Anne part of ths paper there are shown unt root tests results carred out for straddle returns for 6 Months and 9 Months maturtes (Table 17, 18). The second volatlty proy s the straddle mpled volatlty. As mentoned, straddle mpled volatlty s computed as the average of Call and Put optons whch form the straddle mpled volatlty. Straddle IV Call = IV + Put IV The mpled volatlty s derved from Black-Scholes formulas for Call and Put optons usng the optons market prces and the rsk free rate of the US T-Blls wth 3 Months and 6 Months maturtes. For the 9 Months maturty the rsk free rate was not avalable, therefore I computed t as the average of 6 Months and 1 Year rsk free rate. The graphcal representaton of 3M straddle mpled volatlty. 6

27 Graph 6 3M straddle mpled volatlty curve The layout of 3M of straddle mpled volatlty ADF unt root test shows the seres s statonary allowng use t as volatlty proy for our unspanned stochastc volatlty research. Null Hypothess: WA_IV_3M has a unt root Eogenous: Constant, Lnear Trend Lag Length: 4 (Automatc based on SIC, MAXLAG=) t-statstc Prob.* Augmented Dckey-Fuller test statstc Test crtcal values: 1% level % level % level *MacKnnon (1996) one-sded p-values. Table 8 Calendar spread 3M straddle returns ADF test result Net, I wll regress calendar spread (NYMEX symbol WA) straddle returns on prncpal components of the futures returns, squared prncpal components of the returns and cross-products between components. Snce R s the square of the correlaton coeffcent between the values of the dependant varables and correspondng ftted values from the regresson model. Usng straddle returns as a volatlty proy the R wll 7

28 ndcate the etent to whch volatlty s spanned by tradng n the futures contracts, whch nformaton s emphaszed by the prncpal components used as regressors. However, there are some ssues around the R as goodness of ft measure. - If we change the order of the regressors the value wll change; - R wll never fall f we add etra regressors ; Therefore, I wll rely on Adjusted R as goodness of ft measure snce t takes nto account the loss of degrees of freedom assocated wth addng etra varable (squared prncpal components and cross-products between them). R = T 1 1 (1 R ) T k Where, k s the number of degrees of freedom. The layout of the 3M straddle returns regresson on futures returns prncpal components s presented bellow. Dependent Varable: WA_3M Method: Least Squares Date: 06/30/08 Tme: 1:04 Sample (adjusted): 6/10/00 10/0/006 Included observatons: 1140 after adjustments Varable Coeffcent Std. Error t-statstc Prob. C PC PC PC PC1*PC PC*PC PC3*PC PC1*PC PC1*PC PC*PC R-squared Mean dependent var Adjusted R-squared S.D. dependent var

29 S.E. of regresson Akake nfo crteron Sum squared resd Schwarz crteron Log lkelhood F-statstc Durbn-Watson stat Prob(F-statstc) Table 9 Calendar spread 3M straddle returns regresson on prncpal components of futures returns output We notce that the coeffcents for squared thrd prncpal component and product between second and thrd component are not sgnfcant. Though, the thrd prncpal component (the convety nfluence on varance) eplaned only 0.03% of the total futures returns varaton. Lack of sgnfcance for the coeffcent does not nfluence the results. We see that straddle returns have a non-lnear dependency on the futures returns trend component. The values of the coeffcent for PC1 and PC 1 are both postve whch means the straddle returns dependency on trend component takes the shape of an ncreasng conve functon. Snce PC1 s responsble for eplanng 98.76% of the futures return varaton we mght say that an upward movement n PC1 wll lead to a parallel downward shft of the straddle returns. The slope ( PC ) coeffcent s sgnfcant as well. An upward movement n ( PC) wll make straddle returns to decrease for short maturtes and ncrease for long maturtes. Ths change has also a degree of convety ( PC coeffcent s sgnfcant), but snce PC s responsble only wth 1.18% eplanaton for the whole futures returns varaton the convety s slght. Also the margnal nfluence of the components s sgnfcant trend component change on slope component change and trend component change on convety component change). The regresson both R and R are low 0.61 and 0.60, whch ndcates that tradng n futures contracts do not span much of crude-ol prces volatlty embedded n our volatlty proy straddle returns. For commodty and fnancal markets hgh R and R should eceed 0.85, whereas R and be hedged usng only futures contracts. R bellow 0.7 ndcate that volatlty rsk cannot 9

30 One problem whch may appear s the resduals autocorrelaton. A key assumpton for Ordnary Least Squares method s that resduals have the followng property: Snce by cov( u, u ) = 0, j j u we denote the resduals of the regresson estmaton the property assumes that covarance between errors over tme are 0. Havng autocorrelaton among resduals t s not a problem by tself. But snce t s a key assumpton of OLS f t s volated t means that estmated coeffcents may not be sgnfcant. In the Anne part of ths paper there are presented the test performed to evdence and elmnate resduals autocorrelaton (Table 19,0). If we eamne the squared resduals correlogram we see that we have partal autocorrelaton among squared resdual at lag 1 and. We try to model the resduals n order to get rd of the partal autocorrelaton by ntroducng two MA (Movng Average) terms MA (1) and MA (). Runnng the regresson wth second order MA terms leads to a dfferent regresson output. We elmnate resduals autocorrelaton (Durbn-Watson test s close to ). The R and R ncrease (0.76 and 0.75) but the most mportant fact s the sgnfcance of the man coeffcents remans unchanged - PC 1, PC1 -squared, PC * PC. 1 Later on I wll use regresson resduals to evdence the presence of unspanned stochastc volatlty n crude-ol market. Therefore I wll retan the resduals form the orgnal regresson. Runnng the regressons for the 6 Months and 9 Months maturtes ehbts same low values for R and R for 6 Months regresson 0.64 and 0.63 whereas for 9 Months the results are even lower 0.4 and 0.3. The results ndcate that most volatlty rsk cannot be hedged by tradng n the futures contracts. 30

31 Regresson Output Straddle Returns (Volatlty Proy) 3M Straddle Returns 6M Straddle Returns 9M Straddle Returns Futures Returns (Prncpal Components) R^ Adjusted S.E. From R^ Regresson Sum of the Squared Resduals Table 10 R and R from straddle return regressons We notce that the eplanatory power of the futures contracts decreases wth maturty. The standard errors from regresson as well as the sum of the squared resduals ncreases wth maturty meanng that the gap between y yˆ (actual versus ftted of straddle returns) ncreases whle tme-to-maturty ncreases. Now we want to nvestgate how much of the varaton n straddle mpled volatltes (whch s related to epectatons under the rsk-neutral measure of future volatlty) can be eplaned by varaton n the underlyng futures prces. I used the same approach as n straddle returns regressons. The coeffcent sgnfcance t s qute smlar. There s the same partal autocorrelaton problem for the squared resduals. Introducng MA terms n the regresson equaton does not change the sgnfcance of the estmated coeffcents therefore we may assume that the regresson estmaton was successful. In straddle mpled volatlty case the capacty of futures contracts varaton to span crude-ol market volatlty s even lower whch confrms what the concluson from straddle returns regresson that one cannot hedge much aganst volatlty rsk usng only tradng of futures contracts. Impled volatlty regressons output as well as the procedure for elmnatng partal autocorrelaton among resduals are shown n the Anne part of ths paper (Table 1-5). I wll retan the mpled volatlty resduals as well for further evdence. 31

32 Regresson Output R^ Futures Returns (Prncpal Components) Adjusted S.E. From Sum of the Squared R^ Regresson Resduals Straddle Impled Volatlty (Volatlty Proy) 3M Straddle Impled Volatlty 6M Straddle Impled Volatlty 9M Straddle Impled Volatlty Table 11 R and R from straddle mpled volatlty regressons The frst remark s that the hghest eplanatory power s for the shortest maturty (3M) but stll very low. Straddle mpled volatltes R and R evdence that there s a very low etent n whch futures returns can be used to hedge aganst volatlty. Also the eplanatory power of the futures contracts decreases wth maturty. The standard errors from regresson as well as the sum of the squared resduals ncreases wth maturty meanng that the gap between y yˆ (actual versus ftted of straddle returns) ncreases whle tme-to-maturty ncreases. There are both advantages and dsadvantage for usng these approaches to evdence the presence of unspanned stochastc volatlty. Straddle returns: + Straddle returns are not condtoned on a partcular prcng model. They are computed based on NYMEX observed call and put premums, correspondng strke prces. The only assumpton n straddle computaton s the choce of the shortest tme-tomaturty futures contract as a proy for the crude-ol spot prce. V - Straddles have hgh gammas ( = S ϕ ( d 1 ), where V s the opton Sσ T premum Call or Put). Gamma shows how much wll vary the value of the opton at hgh changes n crude-ol spot prce. It ndcates the convety of the opton value. Snce straddles are bult to hedge aganst sgnfcant changes n crude-ol prces they are subject 3

33 to hgh gammas. The assumed sgnfcant varant spot prce s used n computaton of both straddle returns and futures returns. As shown n futures returns prncpal components analyss and n the sgnfcance of estmated coeffcents from the straddle returns regresson straddle returns are conve n futures returns. Though, even f volatlty s completely unspanned by the futures contracts the presence of squared prncpal components (measurng convety of the dependences) may not lead to results close to 0. Ths may be one of the eplanatons for hgher than n straddle mpled volatlty regressons. R and R n straddle returns regressons Straddle mpled volatltes: + If volatlty s completely unspanned by futures contracts result wll be 0 or closed to 0. If we look at the results ths s the case for 9M straddle mpled volatlty regresson. - The results for straddle mpled volatltes are condtoned on the accuracy of the prcng model we use. In our case we condtoned on Black-Scholes model. The thrd and fnal step n my evdence s analyzng of the resduals from the regresson. I retaned the three sets of resduals from each regresson type. Net, I wll etract the prncpal components out of each set of regresson resduals. If there s unspanned stochastc volatlty n the data, there should be large common varaton n the resduals. Usng the prncpal components analyss propertes ths should lead us to a frst prncpal component whch embeds most of the varaton from the resduals. If the resduals are smply due to nosy data, there should not be common varaton n the resduals. The output of prncpal components analyss for the two data sets contanng regresson resduals are presented n the Anne part of ths paper (Table 6, 7). Bellow can be found the synthess of the analyss. 33