Pricing derivatives if trading in the underlying is limited: Froot and Stein, and Merton 98
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1 Pcg d f dg h udlyg lmd: oo d d Mo 98 A o of ou uggd pplco of h modl oo d dc h pcg of d o h o (co-ffcly) dl. hy d pl dffl quo fo h lu k would g o opo o h co dd ly (.g. du o lck of lqudy o ho-llg co) u fo whch ol mk pc ll. h do follow: h k um h h ol mk pc of h udlyg odd (.g. llqud h of commo ock) follow gomc Bow moo (logoml dffuo poc) wh df (ou pcd u) μ d olly σ : d d A h pc of h udd ol (d co wh h umd pcg modl fo dl ) h opo pc couou d wc dffl fuco of m d h udlyg pc o Io lmm 3 c ppld o h poc d uqu kg of pco g po fo h pcd chg of h opo lu : pg 73 : Rk Mgm Cpl Budgg d Cpl ucu Polcy fo cl Iuo: A Igd Appoch Kh A. oo d Jmy C. h Joul of cl coomc 998 o h h opo lu wc dffl fuco of udlyg pc d m (d hc Io lmm pplcl) dd cocluo h do fo dd g Ro C. Mo 77 O h Pcg of Cog Clm d h Modgl Mll hom Joul of cl coomc Vol p If dg h udlyg o pol Mo gu Applco of Opo Pcg hoy: 5 Y L Amc coomc Rw Vol. 88 Iu 3 Ju 998 p h o c ll df d lu fuco f h udlyg w dl d hdgg gy h ppom h fuco y dymclly llocg fud w pofolo of dl ky o ck h udlyg d h kl. o h hdgg gy h ppom h d lu fuco h wgh of h h ckg pofolo cho o h h c of h ou ckg o.. h dffc w h ou u of h ckg pofolo d h udl mmzd. h mpl h h ou ckg o ucold wh y dd ou u Mo h how h ook Couou m c h d do of h ook h poof c foud o pg 396 wh h co o hom 5.3. A l poof g low pg 5-6. I follow h h ckg o lo ucold wh h mk pofolo (o y pcd fco) d hfo Mo o mk wh oly ymc k pcd ( ud h oo d pcg modl umpo fo dd ud h h l po h log d fc g h fo dl) h pc of h d wll h m f h udlyg w dd Mo 98 pg 333. o ddol foml uo why h h c o h wh couou lcg h poly duo of h udlyg u o y ou pod h m h poly duo of h ckg pofolo u mu h ckg o. h udlyg c hc plcd wh h ckg pofolo mu ddol dom pym copodg o h ckg o h d of ch ou pod whch h l cou oly o-ymc k d do o qu ch ouly h gg of h pod h pcd lu of zo d olly uch h h c of h ckg pofolo plu h c of h pym o mch h ckg o qul h udl udlyg c. Ad wh plco gy fo h udlyg foud h m h codo fo h pplco of Io lmm m wll h c wh ol pc. 3 h ppd A. o Io Lmm.
2 I. I u oo h oo d hudl fo w mll pou : 4 II. M R p Wh R copodg o h Ru mu of l k o fo h k 5 co g h coc of h u of h udl compo of h w m wh h u of h g pofolo of udl co h coc of h w m M u wh h pcd mk fco h kl h mk pc of k d h pcd u qud y h k fo mll m o h w udl. 6 Accodgly h ou hudl M fo opo wh pc c w follow (wh ow g pd ul wh couou compoudg d wh u y): p p co M R co p M R p h chg of h opo lu d hc h opo u fuco of h ou u of h udlyg ock : 7 Ad hfo oh h coc of h opo wh h mk d coc wh h udl p of h g pofolo ( d p ) c pd fuco of h pc coc pm of h udlyg ock ( d p ): M M M R p 4 o o o h do of h hudl h Oco 4 po h log d h fc g h o h h h fo ucg fcg co (k f ). 5 h d o pg h pyoff fuco of h k fuu o-ochc m oppouy f dducg fcg co uly fuco fo dl h Oco d om 4 po o h oo- modl h log d h fc g h (o h h pou po R w llld K ). 6 A h mk pc of h ock co wh h pcg modl fo dd M d h k would o py h ol mk pc fo h o pfcly dl ock (ul p ) u would qu dcou (o wllg o py pmum dpdg o h g of p ) oo d pg 73 (o h h c of h udl compo do o pp h fo mll m h mpc o pofolo c du o h d-lo c codd o glgl dff o h pol coc mpc). 7 g f.
3 3 Ad wh quo II. o c w h hudl fuco of h (mmum) pcd ock u h k would qu f hd o hold h ock dcly: h f (qud) ou pcd u of h opo fom h k ppc. Mulplyg wh h opo lu g h qud ou pcd chg of h opo lu: III. Ad h pcd chg of h lu of h opo lo g y I. f h opo pcd fly fom h k ppc quo I. d III. qul: 8 IV. lghly g IV.: d wh : uh gg g quo 7 oo d : Rgg g: Ad wh ow lo pd ulzd fgu (d hc mulpld wh ): 8 h m pcpl fo h l do of h opo lu dffl quo y Blck d chol ( h ppd A. d h l po h log o h do) ppld h h pcd chg of h opo lu dd wh Io lmm compd o h pcd chg of h lu qud y h o whl Blck-chol h qud pcd chg h mk qud pcd chg dmd wh h CAPM h ddol compo cludd y h k o comp fo h mpc of h k pofolo udd k.
4 ow dd h whol quo y : h lly h dffl quo g y Mo fo uop opo o ock pyg co ddd yld. 9 Hc h copodg opo pcg fomul c ppld o clcul f pc fom h k ppc y g h co couou ddd yld qul o. h fomul fo ock pyg co ddd yld ld o h m ul dcoug h ock pc wh h ddd yld d h pplyg h copodg Blck-chol- Mo fomul fo uop opo o o-ddd pyg ock wh h dcoud ock pc (d lghly modfd olly mu).. y g h udlyg po pc fo h opo luo o: A u o fo h qulc h f dcoug h pc of h ddd pyg ock wh h c ddd yld (d djug h olly) h poly duo of h ock pc wll h m h poly duo fo h pc of (ohw qul) oddd-pyg ock wh cu pc qul o h dcoud pc. Pcg d o udd f pc o cououly ol Mo og fo pcg opo o udl wh pc oly ol h m h opo w d py d d o h ppoch wh ol pc dcd o fooo. A h c wh ol pc f h udlyg o dd pofolo of dd c ud o ck h udlyg u cloly pol. h u of h udlyg ou l h dffc of h ckg pofolo u d h 9 Ro C. Mo hoy of Rol Opo Pcg h Bll Joul of coomc d Mgm cc Vol. 4 o. (pg 973)pp quo 44 d ooo 6 p.7. h opo pcg fomul g h mpl h h ddd kl ou pd y fml mll m l dug h lf of h opo. A plc do fo h pcl c c foud Joh C. Hull Opo uu & Oh D h do ( h followg fd o Hull p. 37). Hull fuh how h h pu cll py lo hold fo opo o ock pyg co ddd yld d d h luo fomul fo pu o ddd pyg ock (Hull p.37/373 d chcl o 6 hp://www-.om.uooo.c/~hull/chclo/chclo6.pdf). o h d of h pl d of h opo pc wh pc o h cu m l Mo u h pl d wh pc o m ll po whch ld o h dff g of h pl d Mo 73. Hull p. 37. h olly mu h o djud wh low ock pc g pc chg lly lg (Hull p.343 f. o h h djum llld h ppomo oducd h mo gl co wh ddd o cly g pd wh co yld how fo h pcl c wh co ou ddd yld lk h h ppomo c). Id. hg h m poly duo c h pd hg h m lu y of h wold. uh cll h h ddol u compo qud y h k oo d c pd dcou (o pol pmum c of g coc of h poo wh h g pofolo of udl ) o h ock pc. g f. 6 d h fc g h. 4
5 5 ckg o (wh h ckg o dfd h dffc w ou ckg pofolo u d ou udlyg u). Wh pc g oly ol u d py d of h opo dug h lf of h opo h lu of h ckg pofolo cou h m of h lu of h udlyg. h ou ckg o uol d cumul o ol ckg o h com oly l opo py. A h mom h u lu of h udlyg g ld d h m jump fom h lu of h ckg pofolo o h u lu. h m h h udlyg pc c modlld pcl c of md-jump dffuo poc mo dl fuh low. o d h pop of h ckg pofolo mod fooo o h h ou u of ckg pofolo cog of poo ll dl wh pofolo wgh w g y: w Wh h ou u of h udl llld h ou ckg o dfd follow: w h pcd ou ckg o hfo: w w d h c of h ou ckg o: j j j w w w h ckg pofolo uppod o ck h udlyg u cloly pol.. h c of h ckg o mmzd whch g h followg of codo: j j j w w Mo 98 p. 335
6 w j j j I.. wh h ckg c mmzd h coc of h ckg pofolo wh y dl qul h coc of h udl udlyg wh h dl. I follow fo h coc of h ckg o of h opml ckg pofolo wh y dl : w j j j hfo h ckg o ucold wh h ckg pofolo d hc: O wh ulzd ou c d: d d d Mo um h h ckg o omlly dud. 3 uh h plco pofolo lcd cououly d ll oll d pcd u co o h olly d pcd lu of h ckg o co. 4 h ou ckg o c hfo pd y ochc dffl quo fo l X (wh y g lu X ): dx Xd X uch h =dx/x h ou ckg o d wh z g W poc. Mo um g lu of X =. X h c pd h lu p ool doll of pofolo whch = co of log poo h ckg pofolo fcd wh ho poo wh qul ool z h udl. Mo f o X cumul popool ckg o. A joly oml 5 d f o ddd umd h poc umd y Mo fo h ckg pofolo of dd c w follow: 3 h poc fo h ckg o Mo 98 p. 33 wh h ochc compo co W poc. 4 h follow fom Mo 98 pg 33 wh y h ch po m h wgh h plcg pofolo cho o mmz h ckg o c d h umpo h h ou coc m cludg h o hdgd d ll dd y co o m. 5 Mo do o plcly h umpo how ckg o d udlyg omlly dud d ucold d h um of wo ucold oml l lo oml o o u h h hold h h u of h dd d h udd mu joly oml. 6
7 d d Wh z g W poc (h lly quo 4 Mo 98 h djud fo zo ddd). h ou u of h ckg pofolo hc: d d h udd follow ml poc umpo Mo 98 g h djud fo h c of zo ddd: dv Vd V wh z g W poc. Hc h ou u of h udd (whch uol) qul: dv V d Wh h poc fo how o h lu of h opo py d c (omwh fomlly) w follow: 6 Wh h pod ddd o fml mll m l wh lgh d umd fom ll. Ad copodgly V d X c pd : z V V d: z X X h ou ckg o h dffc of h ou u of h plco pofolo d h udl : dx X X d X X d dv d V dv V V 6 h co o h ock pc f pod log h d h ppd A. o Io Lmm. 7
8 8 A Mo gu f h CAPM (o oh qulum modl h mpl h h mk pc oly ymc k) hold d h ckg o ucold wh y dl d hc h mk pofolo (o pcd fco).. h ckg o puly oymc h h ou pcd u of udl d plco pofolo qul d h pcd ou ckg o zo. uug X h quo o fo X : X X A X = : X Ad hc: X Ad wh d d d h mplf o: X Bcu V = : V V X O:
9 V. V Wh h lzd cumul ckg o. h copod o pcl c of Mo Md Jump-Dffuo modl 7 wh h udlyg jumpg cly o m h jump compo X g logomlly dud h pcd jump z g zo d h jump occug. A Mo how ud h codo h lu of opo = h dd Blck chol lu cp h h p u m c o ud : 8 VI. V Wh h c of L(X ) whch = qul o d c fom: X X X o h: X Ad hc h c mu fo V o ud fo pcg = V X X A l po m (>) o co u cu h c of h loghm of X do o chg dug h lf of h opo h ckg o o ol fo py. o clcul ulzd p u m c o mu d dd h co y h mg m o py (llld τ y Mo).. h c p u m fo V o ud h V uch h wh τ ppochg zo h c of h cououly compoudd u of V ll muy V do o ppoch zo would h c whou h jump u ppoch. 9 7 Opo Pcg Wh Udlyg ock Ru A Dcouou Ro C. Mo Joul of cl coomc Vol p pg 8 of h ouc wh h gl dcpo of h jump oducd oo logou o h oo h (V=X) d fuh Mo plo o quo 8 o pg 35 d o h h pcl c wh h codo g o ppl. 8 fooo 3. pg 36 Mo 76 d pg 335 Mo Mo 98 pg
10 Comg oo d d Mo 98 I ummy c h cly o jump wll occu dug h lf of h opo d X logoml wh pcd lu of o h duo of h fl ock pc wll cly qul h duo h fl ock pc would h f h w o jump u f h ock (ulzd) c w h um of h c of h loghm of h cumul popool ckg o ddd y m o py d h ckg pofolo (ulzd) c whch c fom quo V d VI. uh kow h h jump wll hpp h d of h lf of h opo. o h o h fl ock pc fom h ppc y po m fo py h h m poly duo f h ock w dd d would follow logoml dffuo poc wh h mod o-popool--m c. I follow h y po m h pc of opo o h ock h poly duo of h ou chg of h opo pc d hc h ou u duo of h opo dcl o ho of opo o dd u ohw qul ock wh c qul o h o-popool--m c. Hc h k hould d h hudl fo h pcd u of opo o llqud wh uol pc logouly o h c wh ol pc. A mod h f co h k pmum qud y h k o h mk qud pcd u c d ddd pd co hc mdg h pcg fomul o dju fo h ddol pmum c ghfowdly do y dcoug h udlyg l po pc wh h (d djug ol olly d cumul ckg o o flc h lowd pc) 3. Rcll h h k wll chg pmum cdg h mk qud u oly fo h udl p of h opo k. h compo of h ou k pmum o ddd o h mk qum hfo fuco of h coc of h udl p h ou ckg o wh h k g pofolo of udl. A mod o h c of h loghm of h cumul popool ckg o do o dc wh m dug h lf of h opo o fomo ou h cumul ckg o g ld d hc fom h ppc of h k h h m duo opo py o m h m of luo. o h m o lo h djum o h udlyg po pc o comp fo h couo of h ckg o o h k ggg pofolo of udl k y co o m wh pc uol whch qu dcoug h ock pc fo luo l m lo wh h ou ddd (h ddol pmum chgd y h k o comp fo h mpc o udl pofolo k) mulpld wh h l m o py h f. h um wll followg Mo h md of h fd o o-popool--m c g Mo 76 p. 36 f. 3. I oly = qul o h ock ou (ulzd) c od o. h logou o Hull og fo h m of ddd pg 4 d h fc g h. A m l h = wh h cu pc of h udlyg qul o h pc of h ckg pofolo h udlyg pc uol h mpc of ll ckg o lzo g ld oly opo py whou dco of lzo h my ul o m fo h cu po m. 3 f. d h fc g h fo h olly djum
11 Appd: A.: Io Lmm Plmy: ylo polyoml: f ; k k f k! k ylo (h ylo polyoml wh ppoch fy): 4 f k k '' k ' f f f f k! ylo l l: 5 o fuco of wo l h ylo o cod od ou h po (wh h co pl d g h f d wh pc o of h f d of wh pc o ) 6. I g ppom lu of f oh d chg y mll mou. o h pl d of wh pc o mpl holdg co l oo o mphz h : Io poc:. 7 d.. d fuco of d. d wh z followg W poc o h: : z 4 fo mpl hp://.wkpd.og/wk/ylo_ 5.g. hp://.wkpd.og/wk/ylo_#ylo l_l) 6.g. hp://.wkpd.og/wk/pl_d#hgh_od_pl_d 7 hp://.wkpd.og/wk/pl_d#oo
12 ( dd oml l d h lm fo ppochg zo) Dc m o: z o Do of Io Lmm: 8 p : pply ylo l l o fuco of m d l h follow Io poc: wh d : p : uc h gg of h m pod codd ( ) o g h pc chg: 9 A.I. I ody clculu ypclly m of hgh od (poduc cog l wh po lg h o) would god cu wh d ppoch zo h m would ppoch zo f. How h h o complly h c fo. o h look h dc m o of h Io poc: A.II. Ad clcul :.5 8 h followg follow Hull Opo uu & Oh D 9 h do (fd o Hull h followg) p wh om ddol plo fo whch fc g. 9 h h g quo 4.A.3 of h do of Io Lmm Hull c lo foud h ld pck o chp 4 p. 5. ll hp://www-.om.uooo.c/~hull/ofodld/d.hml
13 3 h co m whch mulpl of. p 3: dm h ppom lm of ppoch zo: 3 Wh ppoch zo.5 ppoch cu d 5. ppoch zo f h. h pcd lu of cu zo d h c of d hc: o h uc h pcd lu fom o g h ochc compo of : Wh ppoch zo h ochc compo wll ppoch zo codly f h h ochc compo whch ( quo A.II) d hc codd glgl compd o h m d god. hfo c plcd wh quo A.I: p 4: uu fo : z z uu fo z : hp://
14 4 A oh ummd coffc of d o hgh pow h hy g oh god (lk oh uch m h ylo- ppomo) o h mplf o: p 5: uu fo dl : z z p 6: Rgg: z p 7: kg lm ( ppochg zo) g Io Lmm: d d A poc fo ock pc: 3 d d o dc m: A.III: z Wh u c (pcd lu d olly) g p.. fgu d h u of y. h poc kow gomc Bow moo. If h ock pc follow h poc h ock ou u (dc u (fml) mll l) : d d. 3.g. Hull p
15 Apply Io Lmm o fuco of d : A.IV.: d d Apply Io lmm o L(): 3 L o gl f follow Io poc h poc fo =L() : d d Ad mo pcfclly f follow gomc Bow moo o h h poc fo d : d d Hc: d d.. d d ~ d d (.. d log-omlly dud mg h loghm omlly dud). d hc h followg hold fo h ock pc f ho pod of m: d d d.. wh h ou u d/ g dfd o d h u of h ock pd of u wh couou compoudg dug m l d Hull p
16 6 h pcd lu of log-omlly dud l w who loghm omlly dud wh m m d dd do o : 34 o m w o h: d d A foml po fo h ock pc f pod log h d: Wh d d d d ud h umpo of followg gomc Bow moo d co o c d foml 35 po fo h dom l f log pod h d fo mpl pod cog wo (fml mll) pod.. wo pod wh h lgh d of ch ppochg zo wh h d dcg h um of uch pod of lgh d h wll h pd: d d d d d d d d d d Copodgly wh g h po m f uch pod wh lgh d h ppoch zo h ock pc : d Hc h pcd u L L wh couou compoudg o h ol pod : 33 o h dfo of h of u wh couou compoudg d lohp o dc u ud cy.g. Hull Opo uu & Oh D 9 h do pg 8. o dcuo of h lohp w uc ou u d h copodg dom u wh couou compoudg d ock pc Hull co 5.3 (pg 35). 34 Hull chcl o o Opo uu & Oh D (9 h d). 35 foml po would u ochc gl o.
17 7 d d h c of h u wh couou compoudg : d Ad h g po w = lo h lgh of h ol pod mud y.. d - o h h pcd u wh couou compoudg c w : d L L d h c of L L : d Ad wh h u ch u-pod g omlly dud ( follow gomc Bow moo) h u wh couou compoudg o h ol pod um of dpd oml l d hc o g fo h ulzd u o h ol pod: L L ~ A.: CAPM-d do of h Blck-chol dffl quo: 36 A ock pc follow h poc pcfd o (quo A.III.): z o h chg of h pc of d h pc of whch fuco of d mll l o g wh Io lmm o (quo A.IV.): z O (omwh clo o B ogl oo): 37 A.V. 36 h Blck d Myo chol h Pcg of Opo d Copo Ll Joul of Polcl coomy Vol. 8 Iu p Blck/chol dd y Δ o g ulzd ou u
18 8 Blck/chol um h CAPM hold fo ou u. 38 D h opo : W h ou opo u fuco of h ock u : Coc of opo u wh mk u: M M co co Hc h opo fuco of h udlyg ock : If CAPM hold h ou pcd u of h opo mu qul: M M Wh h ou u of h ock. h pcd chg h opo pc hfo: A.VI.: h pcd olu chg ul lo fom kg h pcd lu of oh d of A.IV.: A.VII.: g A.VI. d A.VII. qul ucg o oh d ddg y d mplfyg g h Blck-chol dffl quo: Blck/chol pg 646 d fooo o pg Blck/chol quo 7 pg 643
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