EXAMENSARBETEN I MATEMATIK

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1 EXAMENSARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Opil Csh Mngn v Elin Oxnhil 5 - No 7 MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET, 69 STOCKHOLM

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3 Opil Csh Mngn Elin Oxnhil Exnsr i ik poäng Hndldr: Jn-Erik Björk 5

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5 Asrc W sudy h prol o opiising h low o csh in lrg copny whn h csh rsrv is ssud o ollow jup-diusion procss. This ns h w hv Brownin oion o odl ordinry lucuions nd Poisson procss o odl lrg lucuions, jups, h occur du o rr unprdicl vns such s wrs or nurl dissrs. W driv h soluion o his sochsic opil conrol prol or oh pur diusion odl nd jup-diusion odl. Th soluion will o "ng-ng" yp, i.. hr xiss "swich poin" which i is opil o chng srgy. By nuricl sudis w nlys h swich poin's quliiv propris, s wll s h vlu o ll csh h w k ou ovr i s uncion o h iniil csh lvl. W lso inrpr his prol s h prol o dciding how uch o supply o spciic coodiy w sudy h U.S. crud oil pric in dil sd on n osrvd pric, xiising h ol supplid oun ovr i consrin o xiu producion cpciy.

6 Conns Inroducion. Th Prol. Ojciv. Orgnision o h Thsis Prliinris 4. Sochsic Clculus 4. Th Lplc Trnsor 5. rinisic Opil Conrol 6.4 Sochsic Opil Conrol 6 Th Modl 9. iniion o h Modl 9. Th Opil Conrol Prol. Th Swich Poin.4 Th Vlu uncion.5 Th Lplc Trnsor o h Gnrl Soluion 4 Anlysis o h Pur iusion Modl 5 4. Th Swich Poin 5 4. Th Vlu uncion 6 4. Concviy o h Vlu uncion Nuricl Sudy Th Vlu uncion Th Swich Poin 4.4. Snsiiviy Agins h Brownin Risk 5 Anlysis o h Jup-iusion Modl 5 5. A Gnrl Equion or h Swich Poin 5 5. Nuricl Sudy: Th Pric o Crud Oil Esiion o h Prrs Th Swich Poin 5.. Th Vlu uncion 4

7 5. Conclusions 5 Rrncs 7

8 Inroducion. Th Prol Igin h w r ishrn; inrsd in drining how uch slon w cn ish ro lk wihou hrning h survivl o h. W cn ssu h h nur o slons in h lk volvs in i ccording o diusion procss. Bu sois, rr vns cn occur h cn kill ny slons on poin in i,.g. nurl dissrs or wrs. I is suil o odl hs vns y Poisson procss wih sll innsiy r. This is iporn, s hs vns dinily hv n ipc on how ny slons w cn ish wihou hrning hir survivl. Th prol o inding how ny slons h w cn ish is n opil conrol prol, in which w wn o xiis h ol nur o slons h w cn ish, nd in h s i w us k sur h hy surviv. This prol cn rnsld ino ny ohr xpls, nd hy r ll quivln in hicl sing. W lis so xpls hr, sing h ky hicl propris o ch o h:. Th nur o slons in lk Conrol uncion: How ny w cully ish up. Opiision incniv: Mxiis h ol oun o slons ishd ovr i. Conlic: I w ish oo uch, h survivls o slons in h lk is hrnd. Consrin: Bound on xiu ishing cpciy.. Csh rsrv o lrg copny Conrol uncion: Tking csh ou or.g. dividnds nd invsns. Opiision incniv: Mxiis h ol oun o csh h w k ou ovr i. Conlic: Risk or liquidion i csh lvl cos oo sll. Consrin: Bound on xiu oun h w cn k ou on poin in i.

9 . Prics o rw rils Conrol uncion: How uch o supply. Opiision incniv: Mxiis h ol supplid oun ovr i. Conlic: I pric cos oo low, i is no proil o produc. Consrin: Bound on supplirs' producion cpciy. This Thsis will inly concrnd wih xpls nd, u h prol cn sily rnsld ino xpl s wll s ohr ons. No h h xpls r nlysd undr urhr ssupions, which w will prsn lr. By sudying hs xpls, w s h h in prol is o ind h spciic conrol uncion h opiiss h opiision incniv nd sisis h consrin, nd in h s i no ks h risk oulind in h conlic. So wh do w xpc h soluion o his prol o? I w look h scond xpl, h opiision incniv lls us o xiis h ol oun o csh h w k ou, so h nswr should o k ou csh h xiu possil spd. On h ohr hnd, h conlic lls us o sop i h csh is low so lvl, so h hr will no risk or liquidion. Th in diiculy is hus o ind his "swich poin" or which i is opil o sop h pyns. W r lso inrsd in drining h vlu o ll dividnds pid ou ovr i or givn csh lvl, which w opiisd.. Ojciv Our nlysis is ing sd on h ricl y Jnlnc nd Shiryv [], in which his prol is nlysd or pur diusion odl. Bu his ricl is qui horic nd no so ccssil or non-hicins. Thror, w will k prs o hs rsuls xplici nd lso do so nuricl sudis, which will srv s n iporn vriicion o h rsuls nd in h s i giv us h possiiliy o visulis h rsuls in vrious plos. This is possil s on gs n nlyic soluion h is rlivly sy o dl wih.

10 urhr, w will look h prol s dscrid in h ricl y Blhj [], h prsns h jup-diusion odl, h opil conrol prol nd solvs i or so xpls o h disriuion o h jup's siz. W will look h cs wih n xponnil disriuion, s w will s h i is h sis on o dl wih. Bu in [] no conclusions r drwn concrning quliiv propris o h rsuls. Our ojciv is o nlys h quliiv propris y conducing so nuricl invsigions. Th diiculy wih his odl is h i is oo diicul o driv n nlyic soluion in his cs, so w us rly on nuricl xpls.. Orgnision o h Thsis SECTION : In scion w will provid h prrquisis ncssry o undrsnd h hicl orulion o h prol, i.. sic diniions ro sochsic clculus nd so inorion on h Lplc rnsor, h w will us o solv h opiision prol. W will lso s up h gnrl opil conrol prol in oh h drinisic nd sochsic cs. SECTION : In Scion w will prsn h jup-diusion odl w will nlys. W prsn hods o oin h swich poin nd h vlu uncion. W lso drin h Lplc rnsor or h gnrl soluion xplicily. SECTION 4: Scion 4 will concrnd wih nlysing h pur diusion odl. In his cs i is qui sy o drin h swich poin nd h vlu uncion. W show h h vlu uncion is lwys concv in his cs, which is no gnrlly ru whn w work wih h jup-diusion odl. W lso sudy h nur o h soluion y nuricl sudis. SECTION 5: In Scion 5 w will nlys h jup-diusion odl whn h siz o h jups is xponnilly disriud. W driv gnrl quion or h swich poin. Thn w look h crud oil pric nd sis prrs sd on his d. Ar h w do so nuricl sudis o nlys wh hppns whn w llow on or or jups o occur.

11 Prliinris. Sochsic Clculus.. EINITION: A -lgr on I is no py; nd I A hn A c ; nd Ω ; U i i Ω is collcion o suss on Ω such h Ii I A, A,... hn A nd A. i.. EINITION: Ω,, P is proiliy spc i Ω is h s o ll possil oucos, is -lgr nd P dnos h proiliy uncion... EINITION: A collcion } o -lgrs on { > Ω is ilrion i s s...4 EINITION: A sochsic procss x is collcion o rndo vrils { x, T} { x α, T, α Ω} dind on proiliy spc Ω,, P...5 EINITION: A sochsic procss } is sndrd Winr procss or Brownin oion i w ; { w > w } hs sionry indpndn incrns, i.. { > w w hs h s disriuion s s w h ws h, s T such h h, s h T nd i T such h <... < n nd n w hv h w w,..., w w r indpndn rndo vrils; >, w ~ N, ; nd } hs coninuous spl phs. { w > n n 4

12 ..6 REMARK: W hv h dw ~ N, d s dw ε d whr ε is rndo drwing ro sndrdisd norl disriuion...7 EINITION: A sochsic procss } is Poisson procss i i sisis h irs { N > nd scond propris o h sndrd Winr procss, nd >, N ~ Poi λ, whr Poi dnos h Poisson disriuion. or xpl, i hs h ollowing propris clld h Poisson Posuls: Th nur o vns occurring in nonovrlpping i inrvls r indpndn; Th proiliy srucur is i-invrin; Th proiliy o xcly on vn in n ininisil inrvl is pproxily proporionl o is lngh; nd Th proiliy o inding or hn on vn in n ininisil inrvl is sllr hn h proiliy o inding on.. Th Lplc Trnsor.. EINITION: Th uncion y y dind on [, longs o L [, i y d <.. EINITION: Th Lplc rnsor o uncion y L [, is uncion Ly dind s L y y d Th Lplc rnsor sisis svrl nic propris. or xpl, L y L y y, so on cn us Lplc rnsors o rnsor dirnil quions ino lgric quions. 5

13 . rinisic Opil Conrol.. EINITION: ux is clld uiliy uncion i u nd i u is sricly concv coninuously dirnil uncion, such h h driviv ro o. x u x dcrss sricly.. EXAMPLE: A coon xpl o uiliy uncion is u x x... THE PROBLEM: Th drinisic opil conrol prol is s up s ollows: L ux uiliy uncion nd ρ > discoun cor. Th prol is o ind h uncion ux h xiiss Thror, w l ρ u x& d whr rsourcs. J A Mx ρ u x& d dx x & nd w hv h condiion x & d A, which xprsss h ol disposl o d..4 THE SOLUTION: Th soluion o his prol is ccording o Eulr givn y h dirnil quion: ρ u x& C whr C is consn..4 Sochsic Opil Conrol.4. THE PROBLEM: Th sochsic opil conrol prol is s up s ollows: W dno y h consupion nd y x } h lvl o cpil i h is govrnd y h c { ollowing procss: 6

14 dx c d x dw whr w } is sndrd Winr procss nd h x in ron o i ns h w hv { goric procss. > is h voliliy consn o h Winr procss. Th prol is o ind h uncion c h xiiss h xpcd vlu E u c ρ d W lso hv h condiion h h procss sops i x τ or so τ. W no h h soluion o his prol is no givn in dvnc. As gnrl rrnc w rr o h ccoun o sochsic opiision s prsnd in chpr - in h ook y Kin- Schwrz..4. THE SOLUTION: I h soluion is no givn in dvnc, how dos on co ou soluion o his prol? W sr y dining J A ; Mx E u c ρ d whn h iniil ss vlu is A, i.. x A. To drin n xprssion o J A ;, w considr sll inrvl in i [, δ ] in which c s c or ll s δ. Th vlu i δ is pproxily qul o δ J A ; i h vlu o h ss i is J A ;, sinc δ J A ; is consud in his i inrvl, givn h h consupion is xiisd. This givs h h proi is qul o u c δ ρ A J A; δ s w g wh w consu in h i inrvl [, δ ] nd w surc h discound vlu o his. This proi is pproxily qul o whr J x ; J A c δ A dw ; δ w δ is rndo vril such h E δ nd Vr w δ. w δ δ Using hs propris nd vluing in Tylor xpnsion yilds h xpcd proi: so w should choos h uncion E[ J x ; ] E[ J A c δ A dw ; ] δ A J A ; c J A ; δ J A; δ c so h w xiis u c c J A; u c J A ; δ o δ 7

15 whr h c h u is sricly concv iplis h h prol hs uniqu soluion. 8

16 Th Modl In his scion, w us h inrprion o h prol o opiising h low o csh in lrg copny.. iniion o h Modl L h csh hld i zro, µ h consn growh r, > h Brownin voliliy r, λ h innsiy o h Poisson procss N } nd s rndo vril { > dnoing h siz o h Poisson jup. s cn discr or coninuous. W k h ollowing ssupions: Th liquidion vlu o h ir is zro, i.. h ir gos ino nkrupcy h irs i is csh rsrvs cos ngiv; Th ir hs no ccss o xrnl inncing; Th ir hs no possiiliy o invs is csh rsrvs in h sock rk or in riskr ss. Undr h ssupions nd diniions ov, h ir s csh rsrvs volv ccording o h procss: d µ d dw s dn dl whr L } dnos h ol oun o csh kn ou up o i. L is ssud o { > non-ngiv nd righ-coninuous. In dciding h srgy L i w only hv ccss o h inorion givn h spciic i, i.. is dpd o ilrion }. L { > L is lso ssud o oundd y so-clld chnologicl consrin, i.. L < K < or so consn on poin in i. K. This is h ound on h xiu oun o csh h on cn k ou 9

17 .. REMARK: In rliy, h consns µ,, λ nd h disriuion or s in h odl ov should sid ro piricl d using sisicl siion chniqus. This ns h whn on insrs vlus o hs prrs ino h rsuls drivd in his Thsis on gs n inxc odl h lwys involvs so kind o rror, which in urn cn sid y osrving h signl/nois rio in h d nd nlysing h rror o h siion hod on uss. This rror should lwys nlysd or on srs using odls h r inxc nd h xcnss o h rsuls only dpnds on how good h siions o h prrs r. Th ruh lis in h piricl d, no in h hicl odls. Ths r jus pproxiions o h d nd should lwys hndld wih cr. W should lso sy h Brownin oion in gnrl is good or prdicion, u d wih ching hisoricl d s i is sochsic procss.. Th Opil Conrol Prol Rcll ro h inroducion h w wnd o xiis h ol oun o csh h w k ou. Th xpcd vlu o his is V τ ρ E dl whr τ in{ : } is h on o nkrupcy nd ρ > is h discoun cor. Th xpcion is kn condiiond on h iniil rsrv. V is clld h vlu uncion or h opil rurn uncion. By xiising h xpcd vlu lik his, w xpc h opil policy o king ou csh o sohing lik his: whn h csh lvl is lrg nough, i is opil o k ou h oun K. And i h csh lvl is sll, i is opil o k ou nohing. So h srgy isl is no so xciing. Th in prol is o ind wh w cll h swich poin, low which w shll k ou nohing. Th xisnc o such swich poin nd h soluion o his opil conrol prol is givn y h ollowing proposiion:.. PROPOSITION: Thr is swich poin such h Whn, w hv h L nd V nd r givn y h ollowing quions:

18 ρ V µ V V Whn >, w hv h W s h whn, V λ E V V, s [ V s V ] V L K nd V V., h rginl vlu o king ou csh qul h rginl vlu o rining csh. Th vlu uncion is rlcd his poin; h soluion is o "ng-ng" yp.. Th Swich Poin How dos on ind h swich poin ollows: irs, w ind h uncion h solvs? According o h ov, w should procd s µ ρ λ E [ s s ] Thn h swich poin is h uniqu poin in h s { :, > } h iniiss... REMARK: Th quion h w hv o solv is pril ingro-dirnil quion or non-zro λ, involving h wo unknown uncions nd h rquncy uncion or h jup siz. Th quion is only possil o solv nlyiclly in w css, s w will s lr. An inrsing osrvion is o copr his quion wih ons oind whn pricing opions whn h undrlying sock ollows jup-diusion procss. I h opion cn rlisd nyi o h holdrs choic, nd i h holdr lso cn choos how uch o h invsd vlu h h/sh prr o rlis, w s h on is inrsd in inding h poin in i whn i is opil o rlis h opion nd how uch i is opil o rlis h i. Th soluion will lso in his cs o "ng-ng" yp, i.. givn poin in i i is opil o ihr rlis s uch s you cn or o rlis nohing dpnding on h currn vlu o h undrlying sock.

19 .4 Th Vlu uncion Th vlu uncion s uncion o h iniil csh lvl is:, V V, <.4. REMARK: On osrvion w cn k idily is h V or ll, i.. V is linr in his rgion. So whn nlysing concviy/convxiy on only nds o considr h cs whn <. W lso s h h slop o h vlu uncion is lwys on r h swich poin, indpndn o ny o h prrs..5 Th Lplc Trnsor o h Gnrl Soluion To solv h prol, w will drin h Lplc rnsor o h gnrl soluion. This is s r s on cn sy hings ou h gnrl soluion wihou king ny urhr ssupions on h rquncy uncion or h jup siz..5. PROPOSITION: I h uncion is in L [, nd i sisis h quion µ ρ λ E [ s s ] hn hs h Lplc rnsor L whr s is rndo vril. µ ρ λ λ E s [ s ] PROO: As, w hv h L L nd L L. I w y s ψ s dno h rquncy uncion or s, h quion o solv cos

20 µ ρ λ whr h Lplc rnsor o h ingrl is λ s ψ s ds s ψ s dsd ψ s s dds s ψ s s d s d ds s s s ψ s s dds ψ s dds ψ s dds s L s ψ s ds L s E s [ ] W now k h Lplc rnsor o h whol quion, nd oin: [ L ] µ L ρ λ L λ L E s s [ ] L L [ ] s µ ρ λ λ E [ ] µ ρ λ λ E s [ s ] s.5. REMARK: W s h in gnrl i will vry diicul o ind h invrs Lplc rnsor nd hry n nlyicl xprssion or h uncion. In c, h only css w cn nlys nlyiclly r h ons whn ihr λ whn w hv pur diusion odl or whn s hs n xponnil disriuion, s hs r h only css h ks oh h noinor nd h dnoinor o h Lplc rnsor lgric. s.5. REMARK: Th uncion E s [ ] pprs s ilir nd vry usul ool in hicl sisics..5.4 EXAMPLE: Considr h cs whn s is discr sochsic vril, dnoing jups h occur du o inrs r dcisions. To prcis, w l s wih proiliy ½ nd s wih proiliy ½. Thn w hv h rnsor or h uncion cos s Es [ ] nd h Lplc

21 L λ λ ρ µ W s h in his cs, h invrs Lplc rnsor is ipossil o ind wihou h us o dvncd rsidu chniqus. 4

22 4 Anlysis o h Pur iusion Modl Throughou his scion, w ssu h hr r no Poisson jup prsn in h odl, i.. λ. This cs ws sudid y Jnlnc nd Shiryv in 995 []. Blow, w shll k so o hir rsuls xplici nd do so nuricl invsigions. 4. Th Swich Poin 4.. PROPOSITION: Th uncion is sinh whr µ nd ρ µ PROO: Th Lplc rnsor o h uncion cos L ρ µ so w hv h sinh i w l µ nd ρ µ. 4.. PROPOSITION: Assuing h, w hv h h swich poin is > ln whr µ nd ρ µ PROO: By king drivivs w oin: 5

23 ] cosh sinh [ ] cosh sinh [ ] cosh sinh [ Ling givs h cosh sinh ] [ ] [ ] [ ] [ ] [ ] [ ln ln s w hv h. < W hv o nlys whn, which is whn s sn y rwriing s > > ] [ ] [ 4. Th Vlu uncion 4.. PROPOSITION: Th vlu uncion is givn y < V ln, ln ln, sinh whr µ nd ρ µ PROO: Insring h swich poin ino w oin 6

24 ] cosh sinh [ ] [ ] [ ] [ so w hv h sinh nd sinh ] [ ] [ 4 4. Concviy o h Vlu uncion W rcll ro rlir osrvions h V or ll. Th ollowing proposiion provids rous rsul h no holds in h cs wih non-zro λ, s w will s in h nx scion. 7

25 4.. PROPOSITION: or ll < V <, i.. h uncion V is sricly concv in his inrvl. PROO: ] cosh sinh [ V ] [ ] [ < i < < < which ws ssud. 4.4 Nuricl Sudy Th ojciv wih his scion is o nlys h rsuls oind in h pur diusion odl or vrious vlus o h prrs. W sr y osrving h coponns o h vlu uncion nd w xpc i o concv up o h swich poin nd linr r h. Thn w will sudy h swich poin nd how i dpnds on h prrs. W xpc i o incrsing wih nd dcrsing wih ρ. Bu how dos i dpnd on µ? Our clculions will show h his dpndnc looks lik h disriuion, n inrsing c h y hv conoic iplicions. inlly w will sudy whn h ir is snsiiv owrds h Brownin risk inducd y non-zro. W will show h whn is sll h ir is snsiiv up o h swich poin, nd whn incrss ov so swich poin, h ir is snsiiv up o poin low h swich poin h will convrg qui s o zro. 8

26 4.4. Th Vlu uncion 9 9 I II V V III IV V V IGURE: In his igur, h wo coponns o h vlu uncion V r plod. Th vlu uncion ollows h uncion h srs zro, nd whn i rchs h swich poin i swichs o ollow h linr lin. W s h h vlu uncion is indd concv up o h swich poin, nd r h i cos linr i.. ihr concv or convx. Th prr vlus r s ollows: I: µ /, /, ρ / II: µ /, /, ρ / 5 III: µ /, / 5, ρ / IV: µ /, /, ρ / 9

27 4.4. Th Swich Poin W will nx sudy wh hppns whn w l on o h prrs r, nd w ix h rs o h h s vlus s ov. 7 I 7 II 7 III IGURE: Th uncion. O cours w wr xpcing h swich poin o n incrsing uncion o, s lrgr ns grr risk nd hror h ir nds o rin or csh gins possil liquidion. On h ohr hnd, i convrgs owrds zro, hr is no nd o rin ny csh s h odl gs drinisic, i.. h - risk is liind. This is kind o srng soluion, s i is opil o k ou vryhing, wi unil h csh grows wih µ nd hn k vryhing ou onc gin. Th prr vlus r s ollows: I: µ /, ρ /

28 II: µ /, ρ / 5 III: µ /, ρ /.5 I.5 II ρ 5 ρ 5.5 III ρ IGURE: Th uncion ρ ρ. W s h h swich poin srs dclining rpidly whn ρ incrss i w sr zro, nd w s h ρ us hld suicinly sll or h swich poin o no convrg o zro. So h swich poin is vry snsiiv o chngs in h discoun cor! Th spcil hing wih h discoun cor is h i is n xrnl prr coprd o h ohr ons h r givn ro d gnrd inrnlly wihin h ir. Th discoun cor dcids how proil i is o rin csh coprd o consuing nd his igur illusrs how h rlionship wih discoun cor looks lik. Th prr vlus r s ollows: I: µ /, /

29 II: µ /, / 5 III: µ /, /.5.5 I II µ 5 µ 5.5 III µ IGURE: Th uncion µ µ. W s h his uncion hs shp siilr o disriuion; i srs zro, i is concv up o xiu poin nd hn convx r h. This is lss inuiiv hn h ohr plos. Th xisnc o xiu poin cn xplind y h ollowing conlic: Th c h h ir gos ino ruin qui s cus o h > givs h lrgr vlu o µ is dsirl. On h ohr hnd, i µ is oo lrg hn h ir dos no nd o rin ny csh s h lrg µ provids gurn h csh will low in s. W s h whn µ convrgs owrds zro; h swich poin convrgs owrds zro. I is hus opil o no rin ny csh whn µ is zro s h -risk cn ring i down. Th prr vlus r s ollows: I: /, ρ /

30 II: /, ρ / 5 III: / 5, ρ / 4.4. Snsiiviy Agins h Brownin Risk inlly, w discuss vriion, which prhps is i niv. Th snsiiviy is sudid y osrving h sign o h driviv o h vlu uncion wih rspc o. Whn >, his driviv is indpndn o nd hus i hs h s sign or ll i is posiiv. Thn h ir is no snsiiv o h Brownin risk. Whn, h driviv o h vlu uncion wih rspc o is uncion o, so or ixd vlu o hr is in h pur diusion odl on poin or which h sign o h driviv chngs. This ns h or sllr hn his poin h driviv is ngiv nd h ir is snsiiv o h Brownin risk nd or lrgr hn his poin h ir is no snsiiv. W no h hr is jus on such poin, s indicd y h ollowing igur: 5 V IGURE: Th uncion h srs in is h swich poin s uncion o. This is plod s his is h uppr ound on or which h plo is vlid. Th ohr uncion is oind y ling h driviv o h vlu uncion wih rspc o qul o zro nd king ou s uncion o, o s which vlus o h h sign o his driviv chngs. W s h or sllr hn h inrscion wn hs wo uncions, w

31 hv h h sign chngs h swich poin ; h ir is hus snsiiv up o h swich poin. In his igur h inrscion poin wn h wo uncions is pprox or lrgr hn h, h igur indics h hr is poin lss hn or which h sign chngs. This ns h vn i h csh gos low h swich poin h ir dos no nd o snsiiv or h Browninn risk. Th prr vlus r: µ /, ρ /. nd 4

32 5 Anlysis o h Jup-iusion Modl Throughou his scion w nlys h jup-diusion odl s oulind in scion. W ssu h s is xponnilly disriud wih prr, which ns h h rquncy uncion or s is s s ψ s such h nd ds Th xpcd siz o h jup is E s /. This odl is nlysd o so xn in []. W will irs driv n quion h givs us uniqu swich poin. Thn w will sudy nuricl xpl, driv prrs ro givn d, solv h swich poin quion nd drin h vlu uncion. W hn nlys so quliiv propris o h swich poin nd h vlu uncion h w g. 5. A Gnrl Equion or h Swich Poin This scion culins in Thor 5..4 h givs gnrl quion or h swich poin. Bu h hor nds so prprion. Using h rquncy uncion ov, w hv h E [ s s ] s ψ s ds s s ds s s ds So h Lplc rnsor o h uncion cos L [ µ ρ λ] λ [ µ ρ] λ µ ρ λ Th polynoil 5

33 λ ρ µ hs hr disinc rl roos or h prr vlus h w r inrsd in, which cn chckd y lgric copuions h discriinn is posiiv. W do i nuriclly or ll s o prrs h w will us. W hv λ ρ µ ρ λ ρ µ µ nd y using h sndrd chniqu or inding roos o cuic polynoils, w hv provd h ollowing proposiion: 5.. PROPOSITION: Th roos o h polynoil quion ρ λ ρ µ µ r cos π θ k p k or } k {,, whr µ, λ ρ µ c, ρ d,, c p, 7 c d q, / rccos p q θ W hv h h Lplc rnsor cos L K whr K, nd will drind y pril dcoposiion. ETERMINING K, AN : Th quliy ov is quivln o K K 6

34 K K K K K K K nd nd ] [ Suing up, w hv h 5.. PROPOSITION: Pril dcoposiion yilds h K inlly, king h invrs Lplc rnsor yilds K nd w hv provd h ollowing proposiion: 5.. PROPOSITION: Assu h is xponnilly disriud wih prr. L s, nd h roos o Proposiion 5.. nd l nd givn y Proposiion 5... Thn 7

35 irniing h uncion o Proposiion 5.. wic nd ling h scond driviv qul o zro yilds h ollowing hor: 5..4 THEOREM: Assu h s is xponnilly disriud wih prr. L, nd h roos o Proposiion 5.. nd l nd givn y Proposiion 5... Thn h uniqu poin h iniiss is givn y providd.w cll his h swich poin quion. 5. Nuricl Sudy: Th Pric o Crud Oil W will nlys h soluion in dil or h U.S. crud oil pric. Th odl w us suis h pric qui wll, spcilly s hr r ig jups prsn in h pric d, which cn odlld y Poisson procss. W will s h h sll lucuions look vry uch lik Brownin spl ph, which lso shows h his odl is vry suil o us. Bu on us hv ccss o lrg oun o d; ohrwis h odl will no i. Los o d lso nsur r prr siions. No h his pproch suis prics o rw rils in gnrl, no only crud oil. Bor h susqun nlysis, w insr n inroducory discussion. Assu h h supplirs on h rk h w r considring only supply on priculr coodiy which w low will ssu o crud oil, h h rk ss h pric o his coodiy nd h w cn osrv how his pric volvs in i. Thn h supplirs will go ou o usinss i hy do no k proi. This ns h h pric or h coodiy will nvr go low so posiiv non-zro hrshold, low which i is no proil o supply his coodiy. Th supplirs y copnis o dirn siz h hv dirn invsn cpciy, so h hrshold is so vrg lvl or ll copnis conriuing o h pric indx or his coodiy. Whn h pric rchs h hrshold, h proi o 8

36 supplying h coodiy is qul o zro. This hrshold is h swich poin, s h supplirs should sop hir producion i h pric gos low h poin. W s h h conrol uncion is h oun o h coodiy supplid, nd h opiision incniv is o xiis h ol oun supplid ovr i. O cours, his is sipliicion o rliy, u I hink h i is or clos o rliy hn o jus ssu h h log-pric ollows.g. jup-diusion procss whn odlling pric indx. 5.. Esiion o h Prrs Th ollowing igur is visulision o h d s h w will us: $6 $5 $4 Pric in US$ $ $ $ $ Yr 5... IGURE: Th U.S. crud oil pric wn Jnury 986 nd Mrch 5. W s h i is suil o ssu h h pric ollows jup-diusion procss d µ d dw s dn dl 9

37 Th prrs h w will work wih r h ollowing no h hs r pproxi vlus: λ / 4 / wo Poisson vns occur in h lps o 4 onhs, ρ., µ.7 oind y rgrssion on h d s r king ou h Poisson jups. Ar norlising h iniil vlu o, w s h /. In h originl unis,. 9, which is h dviion ro siuld norl disriuion, nd w choos h vlu h is our odiid d s oind y xprining wih dirn s nd ploing gins d, which is pprox..9. Blow, w plo h dviions ro h d s oghr wih siuld rndo nurs h hv norl disriuion wih xpcion zro nd sndrd dviion.9: $8 $7 $6 $5 Vlu in US$ $4 $ $ $ $ Sord Osrvions Rndo 5... IGURE: Th dviions in h d s nd rndo dviions ro norl disriuion wih xpcion zro nd dviion.9. This shows in so sns how wll h odl is suid or h d. W s h h d hs lrgr proiliy or lrg vlus hn h norl disriuion hs, u h siilriy is qui good! Hr, on hs h possiiliy o choos h is good wih sll jups, or good wih lrg jups. W hv chosn o k h gos s good s possil wih oh sll nd lrg jups. Ar norlision,.9 /

38 5.. Th Swich Poin irs, w clcul h swich poin wih ll h prrs s sid o.677 norlisd nd.845 in h originl unis $. This is visulisd oghr wih h d low. $6 $5 $4 US$ $ $ $ $ Yr Swich Poin 5... IGURE: Th originl d nd h clculd swich poin. W s h hisoriclly, h pric hs nvr lln low h swich poin. Nx, our inrs lis in nlysing h quliiv propris o h uncion, λ, λ. W hn λ, h proiliy o lrg upwrd jup is zro. I λ is conrolll, h conrollr y wn o incrs λ lil, o g posiiv proiliy o lrg upwrd jup h y ring lrg incrs in vlu o h conrollr. I is lik winning on lory cus o h sll proiliy h h vn occurs u h gr vlu h n upwrd jup y ring. Bu o l o ply in his lory, h conrollr us incrs λ ro zro nd hry lso incrs h proiliy o downwrd jup. So his is no lory whr you only cn win sohing you cn lso los lrg oun. This conlic ns h or sll λ, h uncion λ λ cn incrsing or dcrsing. W will s h h ohr risk prr,, dcids whhr h uncion λ λ is incrsing or dcrsing or sll

39 λ. W will s h whn is suicinly sll, h xpcd proi o n upwrd jup is grr hn h xpcd loss o downwrd jup or sll λ nd hus h uncion λ λ will incrsing or sll λ. And whn grows, hr xiss hrshold or which h opposi cos ru, i.. h uncion λ λ will dcrsing or sll λ. Bu wh hppns whn λ? Thn h risk o lrg downwrd jup cos so ig so h i is dngrous o kp nyhing nd hus h uncion λ λ us convrg o zro, nd h convrgnc spd us dcrsing in. So w s h i h uncion λ λ is incrsing or sll λ, hr xiss λ or which h uncion ins glol xi u. And i h uncion λ λ is dcrsing or sll λ hn i is dcrsing or ll λ. All hs propris o h uncion λ λ r illusrd y h igur low: 4,5 4,5 Swich Poin,5,5,5,5,,5,,5,,5,4,45,5,55,6,65,7,75,8,85,9,95 Ld Sig,9 Sig, IGURE: Th uncion λ λ s nuricl soluion o h swich poin quion wih norlisd prrs s sid in scion 5.., xcp or nd h w prscri wo vlus ccording o h igur. λ h is r,

40 I is lso n inrsing cs o l r nd look h uncion or so ixd vlus o λ. W will s h whn λ is sll, hr is gr uncriny in h odl whhr h λ -vn will occur or no. This will g iv h uncion spcil shp; i will lucu widly up nd down. W will lso s h or non-zro λ, h uncion nd no onoon nyor. or lrgr λ, h rsul will or rous s on knows h h λ -vn proly will occur, nd cn l o c hrr. Th igur low indics h h uncion cully ss o onoon or suicinly lrg λ. Anohr qusion o pos is: os h lii li, λ indicd y h ollowing igur: xis? I ss o h wy, s 5 4,5 4,5 Swich Poin,5,5,5,,,,4,5,6,7,8,9,,,,4,5,6,7,8,9 Sig Ld/ Ld, 5... IGURE: Th uncion or wo vlus o λ, s nuricl soluion o h swich poin quion wih norlisd prrs s sid in scion 5... This igur visuliss h uncriny inducd y sll λ, which cuss lrg lucuions in h uncion.

41 W s in h igur ov h h lii li, λ ss o xis nd qul o vry sll posiiv oun. And his is qui nurl, s h supplirs should lwys, i no h pric is vry sll posiiv lvl, produc hir xiu cpciy whn ; i λ is sll hn h vn is no likly o occur, so i is r o supply down o vry sll pric nd rn h ony vn i h pric gs down. And i λ is lrg, h risk inducs h i lso is r o supply down o vry sll pric, s h highly prol vn cn ring h pric down s wll. 5.. Th Vlu uncion 8 V 6 4,5,,5,,5,,5,4,45,5,55,6,65,7,75,8,85,9,95 Ld/ Ld, 5... IGURE: Th uncion V or h norlisd prrs o scion 5.. or wo prscrid vlus o λ. Blow, w nlys h vlu uncion quliivly. W s h whn h iniil csh lvl is sll, w r vry snsiiv o chngs in λ. or sll λ, h possiiliy o lrg upwrd jup ssigns lrg vlu whn is sll, 4

42 rlcd in h pk in h igur. or sll, h proi o h vn o winning in h lory hs gr posiiv ipc on h vlu. Bu whn incrss, w r lss vulnrl o h c o lrg jup so h ipc o lrg jup dcrss nd so h vlu uncion dcrss. And hn h swich poin cos nd r h h vlu uncion is linr. W s h h sllr λ, h grr is h pk. W wr xpcing h h lrgr λ, h or dngrous h lory cos s hn w cn los lrg oun. O cours, whn λ grows, w g or nd or rid o lrg downwrd jup nd his is rlcd in h vlu uncion, which dcrss whn λ incrss. A non-zro λ ss o provid h xisnc o such pk, i.. h xisnc o poin such h V. This is consisn wih h rsul h w oind in h prvious scion; h h vlu uncion lwys is concv whn λ. Thn, i is ovious h no poin wih V xiss. Whn λ is non-zro, h propry V cn usd o ind h or which h vlu uncion ins his locl xiu. Whn λ ws zro, i ws d hing o hv n iniil csh lvl low h swich poin, s h vlu uncion ws concv. Bu in h cs o non-zro nd qui sll λ, w s h hving n iniil csh lvl h is low h swich poin u no oo clos o zro is good hing. W s h n " opil iniil csh lvl" is h on h xiiss h vlu uncion nd no is driviv. 5. Conclusions W will hr suris h rsuls h w ound y h nuricl invsigion ov. W showd nuriclly h whn h Poisson innsiy r is sll, w r lindd y h proi h lrg upwrd jup y ring nd dos no cr ou h qully lrg proiliy o downwrd jup. Bu whn h Poisson innsiy r incrsd, h ocus urnd owrds h risk o downwrd jup nd h qully lrg proiliy o n upwrd jup did no r nyor. W lso ound h ollowing: 5

43 THE SWITCH POINT Sll Brownin voliliy r: h swich poin s uncion o h Poisson innsiy r is incrsing, ins glol xiu nd hn dcrss owrds zro. Lrg Brownin voliliy r: I is dcrsing ll h i owrds zro. Whn Poisson innsiy r is zro, h swich poin s uncion o h Brownin voliliy r is onoon incrsing. or non-zro Poisson innsiy r, h swich poin s uncion o h Brownin voliliy r nd no onoon nyor. Espcilly in h cs o sll Poisson innsiy r, whn hr is gr uncriny whhr h Poisson vn will occur or no in givn i inrvl. In his cs i lucus widly up nd down. I cos onoon whn h Poisson innsiy r is suicinly lrg. THE VALUE UNCTION Whn Poisson innsiy r is zro, w sw h i ws d hing o hv n iniil csh lvl low h swich poin, s h vlu uncion is concv. Bu in h cs o non-zro nd qui sll Poisson innsiy r, w showd h hving n iniil csh lvl low h swich poin is good hing, s h possiiliy o n upwrd jup givs lrg vlu ing in his inrvl i no iniil csh lvl is vry clos o zro. W sw h n "opil iniil csh lvl" is h on which h irs driviv o h vlu uncion is qul o zro. By inrpring h prol s prol o dciding how uch o supply o spciic coodiy sd on n osrvd pric, w ound whn sudying h crud oil pric h h pric hd hisoriclly nvr n low h swich poin h w oind y clculions. 6

44 Rrncs [] Blhj, M. 4, Opiizing ividnd Pyns whn Csh Rsrvs ollow Copound Procss, Sochsic innc 4. [] Jnlnc, P. nd A. N. Shiryv 995, Opiizion o h low o dividnds, Russin Mhicl Survys, 5, [] Mikosch, T. 999, Elnry Sochsic Clculus wih innc in Viw, World Sciniic. [4] Con, R. nd Tnkov, P. 4, inncil Modlling Wih Jup Procsss, Chpn & Hll [5] hlgrn, M. 4, So Applicions o Vriionl Mhods in Mhicl innc nd Nurics, Lund Insiu o Tchnology 7