The Asset Pricing When the Interest Rates Are Differentiable Stochastic Processes

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1 T A Picin Wn In A Diffnibl Socic Poc Gnndy dd Bluin S Uniiy 4 F. Soin., in, 5, Blu Tl , Fx E-mil: mdd@fpm.bu.unibl.by Abc Ti pp conid poblm of picin fo c wn o-m in poc do no moin popy. In i c pic cn b dmind lo by ibl om of no obbl. In m im fom pcicl poin of iw, mmicl xpion fo pic i ccpbl fo picipn of m if i includ only obbl ibl. Tfo pocdu of liminion fom i mmicl xpion of ll no obbl componn of co of ibl ould b dlopd. In ocic poblm i i umd o limin no obbl indx by in condiionl xpcion. Suc ppoc i ud in i pp. I i uppod in poc i diffnibl bu i mmicl dii of om od i diffuion poc. In i c lu of i poc fuu im dpnd on lu of poc nd i dii pn im. I mn i dpndnc on poc p. I i did xpion fo dminion pic und condiion. In i lion o uul fomul fo pic om mulipli i ddd dpnd on ocic popi of mmicl dii of in poc. Exnion of Vic modl on diffnibl poc i inoducd. T compion bond pic fo i xnion wi bond pic of ndd Vic modl i md. T pln of pp i follow. In Inoducion poblm ubiuion i md nd ibl dmind. In Scion no bi condiion fo muli-fco modl of m ucu i in. T quion fo pic nl muli-fco modl i did in Scion 3. In nx cion i i own o limin no obbl componn of ibl. Scion 5 conin nlyi of diffnibl o-m in poc. Spcil c of in poc wi on dii i in in dil in Scion 6. Equion fo pic wn o-m in poc i diffnibl i did in Scion 7. In Scion 8 xnion of Vic modl i conidd.

2 . Inoducion In muli-fco modl i i uppod wold, on wic pic Р dpnd, i dmind by l qunii, wic cn b iou quod m indx, nd lo conncd wi m no obbl ibl. L' din co compod of qunii, ou, w m М, nd М i compl numb of m ibl, on wic pic dpnd, bu m numb fom ibl, wic quod obd in m. In on-fco modl of m ucu of in М m, i.. co i nfomd in inl ibl nd i inl ibl of i o-m inn in. Fom pcicl poin of iw mmicl xpion fo pic i ccpbl fo picipn of m, if i includ only obbl ibl. Tfo pocdu of xcpion fom i mmicl xpion of ll no obbl componn of co of ibl ould b dlopd. Somin imil plc nd in on-fco modl of m ucu of in. T pic of dicoun bond wi d of muiy Т momn of im i dmind i nominl lu, dicound in nionmn djud o i o momn, i.. on n inl of im [, Т]. How lu of o-m in i nown only momn, nd i lu fuu momn of im up o momn Т unnown. T fomul obind in condiion fo pic of bond nd ccpbl fo picipn of m i imply condiionl mmicl xpcion on pobbiliy mu djud by i of dicound nominl lu on no obbl fuu lu of o-m in {, < T} und condiion. A u of objci pobbiliy mu, nlly pin, m occu only nominl lu i wid in im by fco, i dpndn fom m pic of i Vic, 977. Tu in i c cully xcpion of lu no obbl ibl fom mmicl xpion mn by xpcion of dicound nominl lu of bond on no obbl ibl. Ti pocdu cn b ccpd lo in muli-fco modl und dminion of pic of finncil. Only in i c no obbl ibl will b no only fuu lu of ll of ibl, bu lo pn lu o m ibl, wic no obd momn of im. Suc pocdu cully i pojcion of funcion of pic in in compl pc of ibl on ubpc of obbl ibl. T opo of uc pojcin in c of onfco modl i clculion of condiionl mmicl xpcion. In i n fu w ll nm compl pic mmicl xpion fo pic in in compl pc of ibl, nd n pic i pojcion on ubpc of obbl ibl. L' um, co cn o im ifi o followin ocic diffnil quion d µ, d, dw, w µ, i М-co of dif of ibl,, i М q-mix m olilii, nd dw co of incmn of q-dimnionl ndd Win poc wi muully indpndn componn. L' um fu, compl pic cn b pnd dminiic funcion fom, nd Т, nd i funcion i diffnibl in pc o ll ibl ncy numb of im. W inf um, n do no py ny inmdi

3 pymn, nd ll pymn md in muiy d Т. Tn funcion of compl pic Р,, Т Р Т will b by fomul Io o cn duin im ccodin o ocic diffnil quion dр Т Р Т µ Т d Р Т Т dw, in wic cl fco of dif µ Т nd q-co-ow olilii Т bif dinion of followin xpion upcip in bc mn µ Т nd Т cciz n wi muiy d Т, upcip Т wiou bc will mn npoiion: T T T T P P P T µ µ,,,, T P 3 T T P,. T P Fo dfinin w ll noic P T i М-co-ow nd P T i М М-mix; A c of mix А.. T o Abi Condiion fo uli-fco odl of Tm Sucu In bi oy of picin uul wy of diin of quion fo dminion of pic i quimn of bnc of bi oppounii in m, w n dd diffin only by muiy d Т j, j n. Acully no bi condiion i lo pil diffnil quion fo n compl T pic. Fu fo biy ind of P j T, µ j T j, w ll wi ccodinly Р j, j j µ,, j n.. To obin no bi condiion in i m fi on compo if pofolio of nd n qui yild of uc pofolio in ccucy mu b qul o if in. L' um om ino compod pofolio of, inin momn of im in n wi muiy d Т j lu of iz V j uppo V j >, if ino puc, on wic in d Т j will ci ppopi pymn, nd V j <, if ino iu obliion, i.. in d Т j will b oblid o py ncy co. Tn compl lu of i pofolio momn of im will b qul V j j P,, T j j j j j j P, w j din numb of wi muiy d Т j conind in pofolio momn of im. T incmn of lu of uc pofolio fo im inl, d will b qul

4 d j j dp j j V j dp P j j. Uin quion dmin incmn of pic wi muiy d Т j w ci ocic diffnil quion fo lu of pofolio d j V j dp j j j j V j j P j j µ d V dw. 4 Suc pofolio will b if, if ocic componn of i quion will b qul o j zo, i.. V. Fo fu onin i i connin o inoduc n- j j co-ow V V, V,, V, М-mix P/ wi lmn P/ j lnp j /, j,, nd q-mix P/, wi ow j, j, dmind by lion 3. Tn condiion pofolio will b if i quliy V. I cn b conidd quion fo dminion of co V of inmn in if pofolio. Fom i i iibl, ncy condiion of xinc of if pofolio i inquliy: n min{n P, n, } ρ <. W do no conid of cou uninin ion wn co of inmn i zo. So l ncy condiion of xinc of if pofolio ρ < plc. L' pn mix in bloc fom in c, wn ρ < q, 5 w pn nondnd bloc of iz ρ ρ. I in ou condiion lwy cn b cd by numin of. O bloc dmind in ppopi wy. Accodin o i pnion w ll wi down in bloc fom co of inmn in pofolio V V V. Tn condiion of xinc of if pofolio V fom V V, V V. 6 Fom i follow bwn V nd V ould b followin lin dpndnc V V, 7 nd co V ifi quion V, 8 wic fmily of no zo dciion mix i dnd. Tu iy of if pofolio i dmind by iy of dciion V of

5 quion 8, c of wic uniquly dmin ppopi V by fomul 7, nd lo wol co V. In c, wn ρ q, in mix 5 bloc nd bn, cond quion in 6 nd quion 8 lo bn, nd V cn b con ny no zo ρ-co. T no bi condiion qui, yild of ny if pofolio in ccucy ould b qul o if in, wic lu momn of im w ll din by ymbol. A ul, if in i on of componn of co of ibl. Fom lion 4 nd 6 follow, quion of dynmic of lu of if pofolio loo li d V j j j µ d V µ V µ d V µ µ d. 9 H w in fo biy of cod ud bloc ucu of co µ Т µ µ µ T Т µ µ Т. In od o w no bi ould b ifid condiion j d V j µ d d V j d. j If o n co-column nd wi idnicl componn in c of m wi dimnion ccodinly ρ nd ρ n l xpion in cn b win down j V j Tn fom 9 nd quliy will follow j d V V d V d. V [µ µ ], wic ould b cid ou fo ny if pofolio, i.. fo ny no zo co V fom iy bn dmind by quion 8. Ti impli y componn of co µ µ ould b qul o zo. Ti quimn i quiln o fc fo ny j, ppopi componn V j of co V, quliy plc j µ d µ j µ j µ j d, w j din ow of mix wi numb j. Fom follow, no bi condiion i cid ou, if mix

6 µ ρ µ ρ µ ρ µ ρ n ρ. In un, i i quiln o fi lin of mix i lin combinion o ρ of lin. T lmn of mix dmind by quliy 3, nd j i - componn of ow j. Fom w obin no bi condiion in finl fom ρ µ j j λ,, j. 3 H λ,, nlly pin, cn b ny funcion no dpndn fom j. I i ncy o noic, numb of ummnd in i p 3 i qul ρ q, i.. no ncily coincid wi numb q n indpndn ocic componn of quion, ρ i n of mix 5, compod fom ow j, dmind by quliy 3. Fom bo nlyi i do no follow of ny commndion concnin fom of funcion λ,, fo i i conidd, y ould b in fom ny o on nd m xnl fco, funcion of dif nd oliliy in quion. I i ccpd o nm funcion λ, m pic of i conncd wi influnc of unciny, poducd ocic componn wi numb. 3. Equion fo A Pic Gnl uli-fco odl T quliy 3 cn b conidd pil diffnil quion fo pic wi muiy d Т j if xplici fom funcion µ Т nd Т fom quliy 3 will b ubiud in i. L' n fo compcn of cod ρ-co-column λ, λ, λ, λ ρ, Т. Tn quliy 3 cn b win down fo ρ q T P P T T P, λ,. T P T µ,,, Р 4 T quion 4 i quion fo dfiniion of pic wi muiy d Т in nl mn fo muli-fco modl. To quion 4 i i ncy o dd boundy condiion Р, Т, Т ΨТ, wic dmin pymn in muiy d xcuion of conc nd flc ipuld bfo condiion of conc. Unfounly, oluion of quion 4 in xplici fom fo nl c cn no b win down. I i poibl o p only bou oluion in n nlyicl fom fo om pcil c. L' conid wo of m. T oliliy mix, do no dpnd on, co funcion µ, nd, λ, lin in pc o. In i c funcion by lion

7 , Т, δ, µ, β α,, λ, η ξ, 5 w α, β, δ, η, ξ co nd mix of ppopi iz, wic in nl c cn dpnd on im. W bli, if in i dmind by om combinion of componn of co o i on of componn. Tfo w ll n lo -co-ow uc а. If i on of componn of co n componn zo wi on xcpion: componn wi numb, wic if in in co, i qul o uni. Tn dciion of quion 4 i funcion Р,, Т ΨТ ехр{а, Т В, Т}, 6 w cl funcion A, Т nd -co-ow B, Т found fom followin diffnil quion o din dii on im А В η β ½ В δ В Т, 7 В а В ξ α, 8 wi boundy condiion AТ, Т nd BТ, Т. ix funcion, Т, nd co funcion µ, nd, λ, lin funcion in pc o, i.., Т, δ D γ D, µ, β α,, λ, η ξ, 9 w α, β, δ, γ, η, ξ co nd mix of ppopi iz, wic nlly cn dpnd on im. T ymbol D din nfomion of co in dionl mix, fo xmpl γ γ γ γ γ, γ D γ γ. In conidd c funcion 6 lo i dciion of quion 4, ow in i c cl funcion A, Т nd М-co B, Т found fom followin diffnil quion А Вη β ½ Вδ В Т, В а Вξ α ½ Bγ D В D, wi fom boundy condiion AТ, Т nd BТ, Т. T c widly nown fo dicoun bond pyin uni in d of muiy Ψ Т fom pp, w dod m ucu fo on-fco modl wi conn fco, wn М q,, funcion A nd B in cl, nd pm α, β, δ, γ, η, ξ nfom o conn. T xplici fom of funcion A, Т nd B, Т, wn γ и ξ, w obind in fmou pp Vic 977; in un,

8 oluion fo c δ и η i found in widly nown pp Cox, Inol nd o 985; wn ll ix pm non zo dciion i bou in dd nd Cox 996. T dild compi nlyi of funcion A, Т nd B, Т, nd lo pobbiliy popi of poc of o-m in dmind by modl, conin in Ili. T pocdu of cpion of oluion of quion 4 in conidd c i ducd quion icci fo co B in c fi ould b old. Unfounly, in n nlyicl fom i cn b old only in cl c fo conn pm. Fo mo complx iuion i i ncy o find oluion of icci quion by numicl mod. T funcion A, if B i nown, i dmind by impl inion, ow cn ul in inl wic no clculd xplicily. Blow w ll conid on mo c, wn dciion of quion 4 cn b found in clod fom fo pciclly impon c. 4. Eliminion of no Obbl Componn of S Vibl T oluion 6, i found by dcibd wy, dmin compl pic. I would b ccpbl fo picipn of m, if ll componn of co obbl. If om componn of i co no obd in m li w dmind m numb М m l componn of co, i i ncy o limin m fom oluion. Fo i w mu ill o find diibuion of pobbilii of componn nd o clcul condiionl mmicl xpcion of oluion 6 on componn fixd obbl ibl condiion. Ti will i fomul fo pic ccpbl fo picipn of m. L' conid i pocdu fo on impon c of noml diibuion of co of ibl. I mn, dniy of pobbilii of co fom f, d Σ / π ехр{ ½ Е Т Σ Е}, 3 w Е E i co of xpcion of ibl, nd Σ Σ mix of i coinc. Fo conninc of ubqun diin w ll pli co on wo p: obbl G т Т nd no obbl Н т т М Т. Accodin o i w ll pn in bloc fom co Е nd mix Σ: Е E E, Σ Σ Σ Σ Σ, 4 w Е nd Е mmicl xpcion obbl nd no obbl componn pcily, Σ nd Σ ccodin o i mix of coinc pcily, nd Σ nd Σ mix muul coinc of co obbl nd no obbl componn. Tn dniy of pobbilii of ibl cn b win down in fom fg, H, xp G E H E π T Σ Σ Σ d Σ Σ Σ Σ Σ G E H E. 5

9 Ti dniy of pobbilii fo ou pupo i connin o pn in fom of poduc uncondiionl fo G nd condiionl fo H fixd G dnii fg fh G xp xp T G E Σ G E π m d Σ T H E Σ Σ Σ Σ H E π d Σ Σ Σ Σ, 6 w fo biy dinion i ud E E Σ Σ G E. 7 In conidd c i i poibl o wi down compl pic 6 G РG, Н,, Т ΨТ ехр A, T B B, 8 H w co B i pnd in bloc fom ccodin o pliin of co of ibl ino wo p G nd H. ow fo diin of fomul fo pic, wic could b ud in m, i i ncy o clcul condiionl xpcion of funcion РG, Н,, Т on diibuion of co H fixd co G. Tn w ll РG,, Т ΨТ ехр{а, Т В G} Е G, {ехрв H} ΨТ ехр{а,т В E Σ Σ E ½ В Σ Σ Σ Σ В Т } ехр{в B Σ Σ G }. 9 A l, if obbl nd no obbl m indx iiclly indpndn mon ml in ou c i mn, Σ nd Σ, n w ll obin РG,, Т ΨТ ехр{а, Т В G} ехр{в E ½ В Σ В Т }. 3 T no obbl pm in i fomul dmin l mulipli. By i mulipli obind fomul diff fom fomul of m pic nown fom liu,. L' mind, co G i compod by obbl m pm, i.. G m T. Tu in i fomul fo pic ud i funcion dmind by ccpd modl wn i uppod wi noml diibuion of ibl, o ob-

10 bl m pm,, m. So in modl of oluion of m indx i cn b ud in l condiion. 5. Diffnibl Poc of So-Tm In In owlmin numb of c ocic modl of dynmic of o-m in bd on poc wi indpndn incmn diffuion poc, wic coninuou non diffnibl o poc nd dcibd by quion of fom, wn co of ibl condiion dn in inl ibl. A m im, mpiicl idnc p, l poc of in no lwy o popi. Ti poblm w dicud in mny pp fom diffn poiion, nully quid nw wy of conucion of modl of dynmic of o-m in. W ll um modl offd in dd. T id of i modl i bd on umpion, poc of o-m in diffnibl М im, nd i М - mmicl dii i diffuion poc ifyin o ppopi ocic diffnil quion. In conidd c o obin quion fo dminion of pic dmiin oluion in xplici fom, w ll conid lin ocic diffnil quion of od М in pc o wi oliliy i non dpndn fom nd coninuou dmind fco, i.. d М а М М d а d bd dw, 3 o coninuou mmicl dii, М, diffnil d d, nd mmicl dii of od М ocic diffnil d М а М М d а d bd dw, 3 L' noic, quion 3 bcom omonou odiny diffnil quion fo dmind funcion, wic М dii d d d d I i poibl o pn nl oluion of quion u,,, 34 ou lu of poc nd i mmicl dii, М, iniil momn of im, nd lo om pil oluion u,, ppopi o pcil of iniil condiion:, j, fo ll j. L u um now, lu {, М } ndom ibl. Tn funcion dmind 34 will coninuou in qu mn mmicl dii up o od М inclui nd will b uniqu oluion of omonou ocic quion 3 wi ndom iniil condiion {, М }. T dciion of ocic diffnil quion 3 wi zo iniil condiion i dmind by fomul Øndl, 998

11 u, dw,, 35 w fo ny fixd funcion u, of ibl,, i oluion of omonou diffnil quion 33 wi iniil condiion u,, u,,, u М,, u М,. Tu, if o oluion 35 of quion 3 wi zo iniil condiion {, М } nd o dd o i dciion 34 omonou quion 33, n obind um will i oluion of quion 3 wi iniil condiion {, М }. Fo u of i oluion und diin of quion of dminion of pic, dmi diin of fomul in n xplici fom, i i mo connin o wi oluion of quion 33 in o fom. L' dmin М-co of ibl of o-m in follow, d d, М. 36 In dinion infiniiml incmn of fi М componn of co will b dmind follow d d, М, 37 bu und umpion l componn М М ifi o ocic diffnil quion 33, fomul 33, llow o wi ind of quion 3 followin ym of М diffnil quion of fi od in diffnil d d, d М М d, 38 d М а М М d а d bd dw. T oluion of i ym of quion i connin o pn in mix fom. Fo i pupo w ll wi 4 d d d d d dw. b If now fo compcn of cod of buly xpion o inoduc mix dinion, i quion will b win down

12 w dinion ud d α d β d δ dw, 39, β, δ, 4 b α. 4 I mn quion 39 i ppopi o quion in µ, α β,, δ, ow mon componn only on ocic bio. T oluion of ym 39 wi iniil condiion in ind fom i {, М} 4 U, U, β d U, γ dw, 43 w U, i fundmnl mix of oluion of omonou ym of odiny diffnil quion α din mmicl dii of co in pc. L' pn om uful popi of fundmnl mix of oluion U, α U,, U, U, α. U, U, U, fo ny,,. U, U,, U, I, I idniy mix. d U, ехр α d

13 Bid if mix α α i indpndn on, n mix U, dpnd only on on ibl diffnc bu no on wo ibl, i.. in i c U, U, U I, U UU, U U, αu Uα. L' dd now o oluion 43. A wll i w ncy o xpc, bcu of liniy of quion 3 i oluion i nomlly diibud ocic poc wi condiionl fixd xpcion nd coinc mix Е{ } U, Vа{ } U, β d, 44 T T U, δ δ U, d. 45 A pcicl pplicion on n in uully in poc fo oclld dy im xi, wn on nou l inl of im of poc oluion i xpcion nd i coinc no ndnci o unlimid inc. In i c inl in quliy 44 nd 45 ould xi. Fo i i i ncy fo y mix U,. Tu dpndnc fom i lo nd limiin niion w uncondiionl xpcion nd coinc mix if y xi Е{} U, β d, 46 T T Vа{} U, δ δ U, d. 47 Tfo, poblm of olin of quion 39 un o of findin of fundmnl mix of oluion U,. In nl c i i impoibl o find i mix. How impl oluion did in c, wn mix 4 i indpndn fom, i cofficin in quion 3 conn. T followin fomul fo U, i fi in i c: w U, U Vе Λ V, 48 е Λ, V. 49 H {, М} inlu of mix α, i.. oo of quion dα I. T fomul win down fo mo impl c, wn ll inlu iou. L' noic, fo xinc of dy im fo poc in i c, i.. fo xinc of inl in 46 47, i i ncy, ll inlu of mix α i w ni, o d ni l p in c of complx numb.

14 6. Exmpl. In Poc H On Dii L poc of if in mmicl dii, wic follow o diffuion poc d а d а d b d dw, 5 cciic polynomil а а, wic oo, 4!. 5 In mix fom quion 5 fom d d d d b dw. 5 Fom 5 i follow, fo xinc of dy poc of if in i i ncy, fco а nd а in quion 5 w ni. T mix V nd V in 48 fom V, V, 53 nd fundmnl mix of oluion i did in fom U. 54 Soluion of quion 5 in fom 43 i did b 55 u u u u u dw

15 und iniil condiion nd momn of im. A follow fom 44 45, fi wo m in 55 fom condiionl mmicl xpcion Е. T coinc mix V indpndn on nd xpd in fom V } V{ }, Co{ }, Co{ } V{ 56 d. T xplici fom of lmn of mix 56 by followin qulii V{ } 4, V{ } 4, Co{, }. L' noic, ll lmn of coinc mix 55 nd o zo, w ncy o xpc, in i c fixd nd unciny of dipp. T fomul fo uncondiionl xpcion nd coinc mix in c of ni oo nd cn b did, if in xpion fo cciic o p o limi. Tu if o ino ccoun oo nd conncd wi cofficin of quion 5 by lion а, а, а 4а, nd in i c on umpion а <, а <, n fo uncondiionl xpcion nd coinc mix of co of ibl w ll ci followin xpion Е b, V. 57

16 Tu, in dy im in i indpndn fom mmicl dii m coinc qul o zo xpcion b/ nd inc / а, wil mmicl dii of in zo xpcion nd inc / а. In m of fomul 9 E b/, E, Σ / а, Σ / а, Σ. If obbl m indx i only in nd i no obbl n in xpion 9 G, H. 7. Equion fo A Pic wn So-Tm In Poc i Diffnibl In c of diffnibl o-m in i bio o im no o popi, fo i i nul o conid, "compl" pic dpnd no only on in, bu lo fom i mmicl dii, i.. fom ll componn of co. Fomlly i co ifi quion 39, wic i pcil c of quion. Tfo quion fo dfiniion of pic in dcibd by quion, i lid fo in dcibd by quion 39 oo. Only min o find ou, ow fu of quion 39 ffc on fom of pic, if i pic cn b find xplici xpion. L' dd o conidion of quion 4 fo c, wn poc of cn of if o-m in i dcibd by quion 3 nd ibl o-m in nd i mmicl dii, М, wic compo co. Fom lion 39-4 follow, in i c q, nd М- co funcion of dif µ, nd mix of olilii i dn in М-co, in quion will fom µ, b, 58,,, Т,. Som implificion of quion 4 follow fom. Ty dmind by pcific fom 58 of co nd mix, P T µ T T T P b P P,

17 P T T,, P T, 59 P T, λ, λ, P T. In i p of quliy 59 ll ibl nd funcion cl, includin m pic of i λ,. Subiuin xpion 59 in quion 4, w ci i in followin fom w ll mind, ccodin o ou dinion P T T P T P b λ, P T Р Т. 6 To conid poblm of oluion of quion 6 concly i i ncy funcion λ, w dmind in xplici fom. A wll li in 5 w ll um, i i lin in pc o co, i.., λ, η ξ, ξ ξ ξ ξ. Tu in conidd c ll umpion 5 ld, xpion 6 cn b conidd oluion of quion 6. Tn quion nd fo funcion A, Т nd B, Т В В В М followin fom А η bв М ½ В М, 6 В ξ а В М, 6 В ξ а В М В, М. 63 Tu compl funcion of pic of n ci n ffin ucu in pc o nd i dmind by fomul Р,, Т ехр { A, T B } xp B, T, 64 w ibl, М, no obd. Tfo i min o compu xpcion 64 in pc o no obbl ibl. Ti ul in o finl fomul of yp 3. In c conidd only fi componn of ibl co i obd, o-m in, nd o ibl i mmicl dii. Bcu cofficin in quion 6 63 conn funcion A, Т nd B, Т will dpnd on inl umn, m o muiy T, i.. A, Т A, nd B, Т B. And lo A da d, B db d. Fu conid followin pcil c. Aum m pic of i i conn nd i indpndn on o-m

18 in nd i mmicl dii. Tn ξ nd quion 6 63 o wi i iniil condiion compo ym B а В М, B, B B а В М, B, М. 65 T mix of ym 65 i compl coincid wi mix α dmind by 4. Tfo i un ou quion 65 nd 39 dmind fo m mix α. Tn fundmnl mix of oluion bd on m inlu fo c in quion umd diffn nd ni o ni l p fo c of complx inlu. So funcion B wll dmind nd followin popi: y qul o zo nd nd o om limi lu. T limi lu B y found fom 65 bcu of B. So B /, B /, М. ow will coninu nlyi of xmpl of piou cion, w М. T oluion of ym 65 B, B, 66 w nd oo of cciic polynomil of ym 65 dmind by xpion 5. T funcion A i compud inl A [ η b B / B ]d. 67 Subiuion 66 nd 67 in fomul 57 wi B B nd B B will i finl fomul fo bond pic in i xmpl. 8. Exnion of Vic odl I i inin o now ow only fomul fo pic c conidd diff fom ppopi fomul in uul nlyi. Fo i xmpl will conid Vic modl of o-m in nd i modificion fo ppoc conidd. Vic 977 uppod o-m in follow poc wi ocic diffnil quion d θ d dw, 68 w >, θ >, >. Ti poc i obd nd dicoun bond pic im wi i dmind by fomul w Р V, ехр { B } A, 69 V V

19 A V [ λ θ BV / BV ]d. BV /. 7 H λ i conn m pic of i. A i i nown poc 68 lu of dy xpcion E[ ] θ nd dy inc V[ ] /. ow w will conuc poc 5 would b quiln o poc 68 in uc n i will m dy xpcion E[ ] θ nd m dy inc V[ ] /. Fo i i i ufficin,, b θ. Ti ul in o quion d [ θ ] d dw. 7 T quion 7 cn b conidd xnion of Vic modl on diffnibl poc of o-m in. Fo poc 7 oo of cciic polynomil ½ 4, ½ 4, nd funcion B nd B compud by fomul В В 4 4,. 7 T compl nlyi fo ny lu of pm i ou id of fmwo ou nlyi fo w will uppo only i pm mll lu mpiicl ul confim i, fo xmpl Pon nd Sun 994, i.. w uppo < ¼. To obin nlyicl ul fo compion wi Vic modl w will conid ppoximion of funcion B nd B will b bd on mlln of pm. o i ppoximion 4 O. Tfo О, О, Tn funcion B nd B will b В В О, 3 3 О. 73 Compion of funcion wi 7 i В В V О В V ε, 74

20 В В V 3 О В V ε, w funcion ε nd ε cn b conidd om djumn funcion ow diffnc funcion B nd B fom funcion В V. T djumn funcion nonni nd uc ε, ε,. Ty icd fom bo by lu mxε < О, mxε < О. 75 Tu bond pic in Vic modl i dmind by xpion Р V, ехр{в V [ λ θв V ½ В V ]d}, 76 w В V < i dmind in 7 wil in modifid modl conidd i pic i dmind by fomul Р, Р V, 77 ехр ε B B d V 4 θ λ ε ε. T l mulipli flc ffc on bond pic of xnion of Vic modl o o-m in poc no moin poc nd diffnibl on im. I ould b wiin fi m in xponn will b no n nil ol nd min conibuion in diffnc fom diion fomul will b i wo l m. Bcu y cn diffn in ffc of modificion ould b cful iniion. fnc Cox, J., Inoll J., o S A oy of m ucu of in. Economic. Vol. 53,. P Ili,. G.. T compi nlyi of m ucu modl of ffin yild cl. Poc. of - Inn. AFI Sympoium. Tomø. P dd G.A.. T Poc wi Dpndn Incmn micl odl of In Poc, Poc. of - Inn. AFI Sympoium. Tomø. P dd, G. A., Cox S. H T m pic of i fo ffin in m ucu. Poc. of 6- Inn. AFI Sympoium. umb. Р Øеndаl, B Socic Diffnil Equion. An Inoducion wi Applicion. Blin: Spin-Vl. Pon,. D. nd T.-S. Sun Exploiin Condiionl Dniy in Eimin Tm Sucu: An Applicion o Cox, Inoll nd o odl. Jounl of Finnc, 49, Vič, O An quilibium ccizion of m ucu. Jounl of Finncil Economic. Vol. 5. P

A Simple Method for Determining the Manoeuvring Indices K and T from Zigzag Trial Data

A Simple Method for Determining the Manoeuvring Indices K and T from Zigzag Trial Data Rind 8-- Wbsi: wwwshimoionsnl Ro 67, Jun 97, Dlf Univsiy of chnoloy, Shi Hydomchnics Lbooy, Mklw, 68 CD Dlf, h Nhlnds A Siml Mhod fo Dminin h Mnouvin Indics K nd fom Ziz il D JMJ Jouné Dlf Univsiy of chnoloy