Option markets and the stochastic behavior of commodity prices 1

Transcription

1 his is prliminry Wor. ls o no quo. Opion mrs n h sochsic bhior of commoiy prics Gonzlo Corzr Alro rys Ingnirí Inusril y Sisms onifici Unirsi Cólic Chil brury his is prliminry wor bs on h hsis Uilizción l informción los mrcos fuuros y opcions pr l molción socásic los prcios los commoiis: l cso l prólo Rys A. Msr hsis onifici Unirsi Cólic Chil o conc h uhor: mil: gcorzr@ing.puc.cl wb:

2 his is prliminry Wor. ls o no quo..- Inroucion ricing n ris mngmn of commoiy-coningn sss rquirs n qu spcificion n simion of h ris-jus unrlying sochsic commoiy prics. Rcn ffors inclu Gibson R Schwrz E. S. 99 Schwrz E. S. 997 Schwrz E. S. Smih J.E. n Corzr l mong mny ohrs. A shr ribu of ll of hm is hir rlinc only on linr pyou sss fuurs n somims swps for simion purposs. h bnfi of using fuurs prics is h hy r in rlily p mr. On h ohr hn h rwbc of his pproch is h som procss prmrs i.. oliliy my b poorly sim bcus hy o no h srong ffc on fuurs prics. his ppr xplors h us of opion prics in iion o fuurs prics o sim commoiy sochsic prics n iscuss prliminry inc on h bhior of h propos mols for luing opionli sss..- Sochsic Mols n uurs rics. wo-cor Mol n uurs rics W us h wo n h hr-fcor mols scrib in Corzr l : h s ribls r: S y Spo pric Diions from long rm xpc-pric-rurn im h prmrs r: B Expc pric rurn Expc ris-jus pric rurn Mn-rring prmr λ Mr pric of iions from long rm xpc-pric-rurn Voliliy of pric-rurns Voliliy of iions from long rm xpc-pric-rurn Corrlion bwn prics n iions from long rm xpc-pricrurn h ris-jus procss is: * y S Sz S B

3 his is prliminry Wor. ls o no quo. y y λ z z z Unr his mol uur prics r: S y S xp y λ B λ *. hr-cor Mol n uurs rics h s ribls r: S Spo pric y Diions from long rm xpc-pric-rurn Long rm xpc-pric-rurn im h prmrs r: Arg long rm xpc-pric-rurn Mn rring prmr for y Mn rring prmr for λ Mr pric of spo pric ris λ Mr pric of iions from long rm xpc-pric-rurn λ Mr pric of iions from rg long rm xpc-pric-rurn Voliliy of pric-rurns Voliliy of iions from long rm xpc-pric-rurn Voliliy of iions from rg long rm xpc-pric-rurn Corrlion bwn prics n iions from long rm xpc-pricrurn Corrlion bwn prics n long rm xpc-pric-rurn Corrlion bwn iions from long rm xpc-pric-rurn n iions from rg long rm xpc-pric-rurn h ris-jus procss is: S y y λ S Sz y λ z

4 his is prliminry Wor. ls o no quo. z λ z z z z z z uurs prics unr his mol r: y S y S xp λ λ λ.- Europn n Amricn Opion rics In orr o us opion mr pric informion o clibr h procsss w mus h n xprssion for h horicl opion prics unr boh mols. En hough r opions r of h Amricn yp w sr by sing h nlyicl xprssion for h Europn cll c n h Europn pu p opions wrin on fuurs conrc: [] [ ]} { X c r [ ] [ ]} { K p r wih ln X is h unrlying fuurs X is h xrcis pric n is h ccumul rinc on fuurs rurns.

5 his is prliminry Wor. ls o no quo. h bo xprssion hols for boh h wo n h hr-fcor mols prsn in h ls scion rquiring only n jusmn for h lu of h rinc h only prmr which is mol-coningn. or h wo-fcor mol is: [ ] or h hr-fcor mol: r commoiy opions r minly Amricn so w cnno ircly us h bo nlyic xprssions. hus w mus rsor o som numricl procur o incorpor mr opion pric informion. hr r mny procurs o pproxim h lu of n Amricn opion on of h bs-nown ons bing Gs n Shri 985. W us in his ppr rcn impromn propos in Hung l 996 h lus Europn opion lus for srl xrcis

6 his is prliminry Wor. ls o no quo. s n sims h lu of h Amricn pu opion p using Richrson Exrpolion s: 6 7 wih p r rx p r r [ ] [ ] r rx p r r r r r rx p r r r r r r.- Mol Esimion n Rsuls. rocur o sim mol prmrs w xn h procur in Corzr l now incluing opion mr prics. or boh h wo n h hr-fcor mols w follow h sm sps: i.- Compu iniil opion lus ssuming h mr opions r Europn. ii- Esim s ribls n prmr lus which minimiz rrors in mr prics using nlyic xprssions for opion lus.

7 his is prliminry Wor. ls o no quo. iii.- R-compu opion lus s h nlyic xprssion for h Europn opion plus n jusmn using Hung l 996 for Amricn opions i.- Rp sps ii n iii unil rrors conrg. W sim prmr n s ribl lus using s our objci funcion h sum of wigh squr rrors in fuurs n opion prics. By chnging wighs w cn jus h mol o fi br fuurs opions or linr combinion of hm.. D W us ily prics on fuurs n opions on fuurs on Ligh Sw Cru Oil r on YMEX bwn Jnury 997 n Spmbr 999. hr r boh cll n pu opions wih muriis for h nx monhs n 8 n 6 Jun n Dcmbr conrcs for srl xrcis prics.. Rsuls for h wo-fcor n h hr-fcor mol. h following bl prsns lus for h wo n h hr-fcor mols whn w us s only fuurs only opions or boh fuurs n opions. rmr uurs Opions uurs n Opions λ...6 * * ˆ MSE of uurs.6..6 MSE of Opions wo-cor Mol

8 his is prliminry Wor. ls o no quo. rámros uuros Opcions u. Y Opc * ˆ λ λ λ MSE of uurs MSE of Opions.E-5 6.E-5.E hr-cor Mol W cn s h by using only fuurs s ohr pprs h w obin goo fi for fuurs prics bu s r-off w g wors fi on opions. x w nlyz h ffc on fuurs simion of no using opions. h nx figur comprs rrors on fuurs lu n oliliy impli by h fuurs n h opions ss:

9 his is prliminry Wor. ls o no quo. Shor rm monh uurs Errors sim using uurs or Opions D S.E-.E-.E- MSE.E-.E-.E- Min MSE uurs Min MSE Opions 8.E- 6.E-.E-.E-.E % Error Voliliy of uurs wo-cor Mol Long-rm 6.5 yrs uurs Error Esim using uurs or Opions D S.E-.E-.E- MSE.E-.E-.E- Min MSE uurs Min MSE Opions 8.E- 6.E-.E-.E-.E % Error Voliliy of uurs wo-cor Mol W cn s from h bo figurs h minimizing MSE on fuurs s cully rucs h MSE bu h xpns of poorly siming oliliy of fuurs. On h ohr hn if w us opions w cully obining br sims on fuurs oliliy whil rucing h fi on h fuurs ll. his rror on siming h oliliy of fuurs pns on h muriy of fuurs s cn b sn in h following figur:

10 his is prliminry Wor. ls o no quo.. Obsr n sim oliliis wo-fcor Mol.5. Vol.5. Obs. u Opios u Op O Muriy Yrs Obsr n sim oliliis hr-fcor Mol..5 Vol..5 Obs uurs Opions u Op Muriy Yrs 5.- Conclusion wo min conclusions cn b rwn from h bo rsuls. irs whn siming h sochsic procss for commoiy prics opions sms o b non-runn informion n my hlp o quly sim som prmrs of h procss in priculr oliliy. A scon conclusion h my

11 his is prliminry Wor. ls o no quo. b obin is h som jusmns o h pric mol my b xplor o ino ccoun h long-rm mol oliliy sms o b orsim by h mol n h using opions rucs bu os no limin his problm. Rfrncs Corzr G. Schwrz E. Rir. Mr-bs orcss of Commoiy rics using uurs h Annul Confrnc Rl Opions: hory Ms rcic Inrnionl Cmbrig Unirsiy/ROG Cmbrig July 7-8 pp.:: GESKE R. n SHASRI K. 985 Vluion by Approximion: A Comprison of Alrni Opion Vluion chniqus. Journl of inncil n Qunii Anlysis Mrch 985. GIBSO R. n SCHWARZ E. 99 Sochsic Conninc Yil n h ricing of Oil Coningn Clims. h Journl of innc ol. 5 no HUAG J. SUBRAHMAYAM M. y YU G. 996 ricing n Hging Amricn Opions: A Rcursi Ingrion Mho. h Riw of inncil Suis. Vol 9 º : 77- SCHWARZ E. 997 h Sochsic Bhior of Commoiy rics: Implicions for Vluion n Hging. h Journl of innc ol. 5 no SCHWARZ EDUARDO y SMIH JAMES. Shor rm Vriion n Long-rm Dynmic in Commoiis rics. Mngmn Scinc forhcoming