or Imov r.amam.sais. hor. Probabiliy and Mah. Sais. Vip. 77, 27 No. 77, 28, Pags 57 69 S 94-99747-9 Aricl lcronically publishd on January 4, 29 ON PRICING CONINGEN CLAIMS IN A WO INERES RAES JUMP-DIFFUSION MODEL VIA MARKE COMPLEIONS UDC 57.927.25+57.928.5 S. KANE AND A. MELNIKOV Absrac. his papr dals wih h problm of hdging coningn claims in h framwork of a wo facors jump-diffusion modl wih diffrn crdi and dposi ras. h uppr and lowr hdging prics ar drivd for Europan opions by mans of auxiliary complions of h iniial mark.. Inroducion In wll-known financial mark modls on considrs a uniqu inrs ra for boh dposi and crdi s for insanc h books by Ellio and Kopp [], Karazas and Shrv [2]. In raliy, h crdi ra is always highr han h dposi ra. Such a mark consrain brings nw difficulis in h problm of hdging coningn claims s Brgman [4], Korn [3], Bar [3], and also Cvianic and Karazas [8], Cvianić [6, 7], Föllmr and Kramkov [], Karazas and Shrv [2], Cvianić, Pham, and ouzi [9], Sonr and ouzi [8] rgarding ohr mark consrains. In conras wih compl marks, hr is no symmry bwn sllr and buyr posiions in h cas of a mark wih consrains. h fair pric of h drivaiv scuriy opion is spli o h uppr and lowr prics. Hnc, h problm of hdging a givn coningn claim is o find hs prics. W considr h problm in a jump-diffusion sing and driv h formulas for h abov prics in rms of paramrs of h iniial modl. W giv an xnsion of h mhodology of complions in a wo inrs ras jumpdiffusion financial mark and show how our rsuls ar applid in h Black Schols modl s Korn [3] and in h Mron modl Mron [7]. 2. Dscripion of h modl and auxiliary rsuls L {Ω, F, F=F, P} b a sandard sochasic basis. Suppos hr ar wo risky asss S i, i =, 2, whos prics ar dscribd by h quaions 2. ds i = Si µ i d + σ i dw ν i dπ, i =, 2. Hr W is a sandard Winr procss and Π is a Poisson procss wih posiiv innsiy λ. h filraion F is gnrad by h indpndn procsss W and Π, µ i R, σ i >, ν i <. 2 Mahmaics Subjc Classificaion. Primary 6H3, 62P5, 9B28; Scondary 6J75, 6G44, 9B3. Ky words and phrass. Consraind mark, complion, hdging and pricing, jump-diffusion, diffrn inrs ras. h papr was suppord by h discovry gran NSERC #26855. 57 c 29 Amrican Mahmaical Sociy Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
58 S. KANE AND A. MELNIKOV 2.2 W also assum ha hr ar a dposi accoun B and a crdi accoun B 2 saisfying db i = B i r i d, i =, 2. Dno by B,B 2,S,S 2 h mark dscribd by h abov asss. Any non-ngaiv F -masurabl random variabl f is calld a coningn claim wih h mauriy im. In h B,B 2,S,S 2 -mark, a porfolio π =β,β 2,γ,γ 2 isanf -prdicabl procss, whr w dno rspcivly by β i and γ i h numbr of unis of h i h bond and i h sock in h walh. h valu of h porfolio π is givn by 2.3 V = β B + β 2 B 2 + γ S + γ 2 S 2 a.s. A porfolio π is calld slf-financing SF if i has h following propry: 2.4 dv = β db + β2 db2 + γ ds + γ2 ds2 a.s. Such a porfolio will b calld admissibl if V a.s. for all. h s of admissibl porfolios wih iniial capial x is dnod by Ax. h sllr has h obligaion o dlivr h claim f a mauriy, and in rurn h rcivs an iniial amoun x. h amoun x will grow o X x f. h buyr is borrowing h iniial amoun y, y<, which grows o Y y f a mauriy, h rcivs h claim f and pays his db Y. h sllr and h buyr posiions can b idnifid wih h walh procss X andhdb procss Y rspcivly. Morovr, h procsss X and Y ar h capials of slf-financing and admissibl porfolios. Undr h abov condiions and 2. 2.2 h walh procss X and h db procss Y hav h form [ dx = X α α 2 + r d α α 2 r 2 d + α ds + α 2 ds 2 ] 2.5, 2.6 dy = Y S [ α α 2 + r 2 d α α 2 r d + α ds S Hr α i = γs i i /X i rsp. γs i i /Y i, i =, 2, is h proporion of cash invsd on h i h sock in h walh procss rsp. db procss, and a + =max{,a}, a = min{,a}. No ha hroughou h papr α will b also calld a sragy. In his papr, a porfolio π wih iniial capial x is calld a hdg for h sllr if h corrsponding walh procss saisfis X x,π f P-a.s. Similarly a porfolio π is a hdg for h buyr if h db procss is such ha Y y,π f P-a.s. For h sllr, w say ha a hdg π is minimal if X π X π P-a.s., for all and for any ohr hdg π. For h buyr, a hdg π is minimal if Y π Y π P-a.s., for all and foranyohrhdgπ. L us considr h spcial cas whr h financial mark has h sam dposi and crdi ras: r = r 2 = r, and hnc, B = B 2 = B. In h framwork of such a B,S,S 2 -mark, h capial rsp. db gnrad by an admissibl porfolio procss π := β,γ,γ 2 is dscribd as follows: 2.7 2.8 dx X X = β B + γ S + γ 2 S 2 a.s., = dy [ = α α 2 rd+ α ds Y S + α 2 ds 2 ] S 2. + α 2 S 2 ds 2 S 2 ]. Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
PRICING CONINGEN CLAIMS 59 If σ ν 2 σ 2 ν, hn h paramrs µ r ν 2 µ 2 r ν φ = σ 2.9 ν 2 σ 2 ν, µ r σ 2 µ 2 r σ ψ = σ 2 ν σ ν 2 λ dfin s Mlnikov al. [5] a dnsiy Z of a uniqu maringal masur P in h B,S,S 2 -mark as a sochasic xponn } 2. Z = E N =xp {φw φ2 2 +λλ +lnλ ln λπ, whr N = φw + ψπ λ. Undr such a masur, h givn Poisson procss Π has innsiy λ = λ + ψ, and W = W φ is a Winr procss. W considr coningn claims of h form f := fs. I is wll known ha h fair pric of h call opion S K+ in his B,S,S 2 - mark is givn by h formula 2. E r S K + = λ λ n C BS [S ν n ν λ ],r,σ,k, n= = C r, whr E is h xpcd valu undr h risk nural masur, and C BS x, r, σ, K, =xφd K r Φd 2, d = lnx/k+ r + σ 2 /2 σ, d 2 = d σ, Φx = x 2 /2 d. 2π S Aas [], Bardhan and Chao [2], Colwll and Ellio [5], Mrcurio and Runggaldir [6], Mlnikov al. [5]. W driv h wll-known pric of a pu opion from h call-pu pariy. h nx lmma sudis h monoonic propris of C r s 2. as a funcion of r. Lmma 2.. If h following inqualiis ar fulfilld: 2. λ /, or 2.2 λ / and ν, or 2.3 λ /,ν and ν λ +ν λ hn ρ C := C/ is posiiv. If h nx inqualiis ar saisfid: 2. λ /, or 2.2 λ / and ν, or 2.3 λ /, ν and hn ρ P := P/ is ngaiv. Φd 2 Φd 2,, +ν λ Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
6 S. KANE AND A. MELNIKOV h proof of his lmma is providd in h Appndix. L us now urn o h B,B 2,S,S 2 -mark. 3. Main rsuls and pricing formulas o sudy h hdging problm in h framwork of B,B 2,S,S 2 -mark w dfin a variy of B,S,S 2 orb d,s,s 2 -marks wih h inrs ras r = r d = r + d, whr d =d is a prdicabl procss such ha d [,r 2 r ]. Considr firs h posiion of a sllr. From his/hr viwpoin, h invsor wishs o find h minimal iniial amoun possibl o invs in gnraing a walh procss maching a las f. Such a pric is providd by h iniial capial of h minimal hdg if i xiss agains h claim f. In h B,B 2,S,S 2 -mark, h uppr hdging pric or sllr pric will b givn by h following Samn s Korn [3] for h Black Schols modl. Samn 3.. L d =d b a prdicabl procss wih valus in h inrval [,r 2 r ]. Assum ha α d := α,α 2, h opimal hdging sragy agains h claim f B d,s,s 2 -mark, saisfis h condiion in h 3. r 2 r d α α 2 + d α α 2 + =. hn C r d rsp. P r d, h iniial pric of h minimal hdg in B d,s,s 2 agains f,isqualoc + rsp. P +, h iniial pric of h minimal hdging sragy in B,B 2,S,S 2. Namly C r d = C + rsp. P r d = P +. Bfor giving h proof of his samn, w show ha h s of soluions of h quaion 3. is non-mpy a las for h Europan pu and call opions. Exampl. L f =S K+ ; h Europan call pric 2. can b xprssd as follows: C r d =S ν n ν λ λ n λ Φd K rd λ n λ Φd 2. For any im h sllr borrows mony: α α 2 <. In h s rm of C r d h cofficin of S is h numbr of unis of sock ndd, h 2 nd rm, always non-posiiv assum i is ngaiv, is invsd in a bank accoun. aking h lar ino accoun α α 2 + = in rlaion 3. yilds o r 2 r d α α 2 =. Sinc α α 2 >, w driv r 2 r d =. Hncr 2 r = d,andhpair r 2 r,α r 2 r saisfis h rlaion 3.. Proof of Samn 3.. L us firs show ha undr rlaion 3., h minimal hdging sragy α inhb d,s,s 2 -mark is a hdging sragy in h B,B 2,S,S 2 - mark. L C r d b h iniial capial associad o ha hdg in h B d,s,s 2 -mark. Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
PRICING CONINGEN CLAIMS 6 If α saisfis 3., hn w rwri h lar quaion as follows: r 2 r d α α 2 + d α α 2 + = r 2 α α 2 r + d α α 2 + d α α 2 + = r 2 α α 2 r d α α 2 r α α 2 =. + Hnc, and dx α,d X α,d r d α α 2 = r α α 2 + r 2 α α 2 = r d α α 2 d + α ds S + α 2 ds2 S 2 = r α α 2 + r 2 α α 2 d + α ds S + α 2 ds2 S 2 = dxα X α. hrfor, h walh procsss X α,d C r d and X α C r d on h B d,s,s 2 - and B,B 2,S,S 2 -marks, rspcivly, coincid, and in paricular X α,d C r d=xα C r d=fs. Hnc, h minimal hdg α in h B d,s,s 2 -mark agains f is a hdg in h B,B 2,S,S 2 -mark whn rlaion 3. holds. W now show ha h abov sragy α wih iniial capial C r d is minimal among h hdgs agains fs inhb,b 2,S,S 2 -mark. For ha purpos, w will show ha in h B,B 2,S,S 2 -mark, h iniial capial of an arbirary sragy α a hdgingf is grar han or qual o h iniial capial of h minimal hdg α C r d: C r d := E d, [ fs rd ] x, whr E d, is h xpcd valu undr h maringal masur P d, s rlaion 2. in h B d,s,s 2 -mark, and x rprsns h iniial capial of α a, an arbirary sragy in h B,B 2,S,S 2 mark. L X αa b h walh procss gnrad by α a in h B,B 2,S,S 2 -mark. W will prov ha E d, [ f ] rd E d, [ X ] αa x. rd Considr h discound walh procss X := X αa rd ; hn by using Iô s formula w obain d X = r d rd X αa d + rd dx αa = rd X αa [ a ] dx α r d d X αa [ + = rd X αa α,a α 2,a r α,a α 2,a r 2 r d d + α,a ds S + α 2,a No ha from h consrucion of h maringal masur s Mlnikov al. [5], = W d φ is a P d, -Winr procss, and Π λ isap d, -maringal. hrfor W d, ds 2 S 2 ]. Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
62 S. KANE AND A. MELNIKOV w can rwri h dynamics of h socks as follows: ds S = r d d + σ dw d, ν dπ λ, ds 2 S 2 = r d d + σ 2 dw d, ν 2 dπ λ, and [ d X + = rd X αa α,a α 2,a r α,a α 2,a r 2 α,a α 2,a r d d + α,a σ dw d, ν dπ λ + α 2,a σ 2 dw d, ν 2 dπ λ ]. Sinc r d = r + d, i follows ha [ d X = X αa rd α,a 3.2 + α,a α 2,a σ + α 2,a r r 2 d α,a α 2,a d σ 2 dw d, α,a ν + α 2,a ν 2 dπ λ Now, noic ha Xu αa u α,a rd u αu 2,a r r 2 d αu,a αu 2,a du, and X αa u rd u α,a u σ + αu 2,a σ 2 dw d, u α,a u ν + α 2,a u ν 2 dπ u λ u is a P d, -local maringal. Wihou loss of gnraliy w assum h lar is a P d, - maringal. Whnc upon firs ingraing h rlaion 3.2 and hn aking h P d, - xpcaion, w obain, for all in [,], 3.3 E d, [ X ]=E d, [X αa rd ] x. h sragy α a is a hdg for f and yilds o X αard = X f rd [ ] [ ], hncforh C r d = E d, f rd E d, X αa rd x, whr x is h iniial capial of an arbirary hdg for f in h B,B 2,S,S 2 -mark. Furhr, providd rlaion 3. is fulfilld, C r d is an iniial pric of a hdg for f in h lar mark. hrfor, C r d = C +,whrc + is h iniial capial of h minimal hdg in h B,B 2,S,S 2 -mark. h proof is similar for h pu cas; hnc P r d = P +. Scondly, w sudy h posiion of a buyr in h following samn. Samn 3.2. L d =d b a prdicabl procss in [,r 2 r ], and assum ha α d, h minimal hdging sragy agains f in h B d,s,s 2 -mark saisfis h quaion 3.4 r 2 r d α α 2 + + d α α 2 =. hn h sragy α d is a hdg agains f in B,B 2,S,S 2 ; ]. Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
PRICING CONINGEN CLAIMS 63 2 furhrmor, α d provids h minimal hdg agains f in h B,B 2,S,S 2 - mark. In ordr o proof Samn 3.2, w firs sa h following lmma. Lmma 3.3. h minimal hdging sragy for a sllr agains f in B d,s,s 2 is h minimal hdging sragy for a buyr agains f inhsammark. Proof. In h unconsraind B d,s,s 2 -mark, h sochasic diffrnial quaions of h db and walh procss gnrad by a sragy α saisfy s 2.8 dx α X α = dy α Y α. If α d is a hdg for h sllr agains f in B d,s,s 2, hn X α d,x y = x as h iniial pric for h db procss yilds o Y α d,y = X α d,x = f. = f.now,aking Hncforh, α d is a hdg agains f in B d,s,s 2 s hdg for a buyr. Proof of Samn 3.2. L α, h minimal hdg agains f in h B d,s,s 2 -mark, wih iniial db C r d, saisfy h rlaion 3.4. W rwri h rlaion 3.4 as follows: r 2 r d α α 2 + + d α α 2 = r 2 α α 2 + r d α α 2 r α α 2 =, r d α α 2 = r 2 α α 2 + r α α 2. Dno by Y d and Y h db procsss gnrad by α in h B d,s,s 2 - and B,B 2,S,S 2 -marks, rspcivly; hn dy d Y d From h abov qualiy, aking C r d f in h B,B 2,S,S 2 -mark. = dy. Y as iniial pric yilds o α bing a hdg agains o prov h minimaliy of h abov sragy, w considr an arbirary sragy α a wih iniial db procss y, ly αa b h db procss gnrad by α a, and dno Ỹ αa = rd Y αa.usingiô s formula w driv d Ỹ α a = Y αa r d d + rd 3.5 dỹ αa αa dy, Y αa [ + = rd Y αa α,a α 2,a r 2 α,a α 2,a Noic ha from h drif rm α,a α 2,a = r α,a α 2,a r d d + α,a σ dw d, ν dπ λ + α 2,a σ 2 dw d, ν 2 dπ λ ]. + r 2 α,a α 2,a r α,a α 2,a r d + r α,a α 2,a 2 r d r α,a α 2,a r d. Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
64 S. KANE AND A. MELNIKOV Hnc, rdu Yu αa α,a 3.6 u α 2,a + u r 2 r d αu,a αu 2,a r r d du. Furhr, w can assum wihou loss of gnraliy ha rdu Yu αa α,a u σ dwu d, ν dπ u λ u + αu 2,a σ 2 dwu d, ν 2 dπ u λ u is a P d, -maringal. Consqunly 3.7 dỹ αa u rdu Yu αa αu,a σ dw d, u ν dπ u λ u + αu 2,a σ 2 dwu d, ν 2 dπ u λ u. Sinc α a is a hdg, i follows ha Y αa f [, and w driv ] [ ] C d = E d, rd f E d, Ỹ αa, from h rlaions 3.5, 3.6, and 3.7 E d, [ Ỹ αa ] = E d, [ Y αa rd ] y. Hnc C d y for any arbirary sragy wih iniial db y. Sinchpairα, C d is a hdg agains f in h B,B 2,S,S 2 -mark, i provids h minimal hdg. h proof holds for boh pu and call opions. L us giv an approximaion of h arbirag-fr prics of h claim f =S K+. h ky ingrdin of h mhod rlis on h following. aking h suprmum rsp. infimum ovr h auxiliary marks of h acual prics, w find som naural approximaions for h uppr and lowr hdging prics of h claim, and hnc w approxima h arbirag-fr inrval of prics by aking [ ] inf C d [,r 2 r r d, sup C r d. ] d [,r 2 r ] Exploiing h call-pu pariy, a similar mhod is usd for f =K S +. W hav considrd auxiliary marks of h form B d,s,s 2 wih consan d and hrfor wih consan inrs ras r d = r + d. o formula our pricing rsuls w inroduc h following condiions drivd from Lmma 2.: I ν λ / ; II ν and λ / and III ν and λ / and Φd 2 ; +ν λ ν λ Φd +ν λ 2. Undr hs condiions, combining Lmma 2., Samn 3., and Samn 3.2, w arriv a h following horm. Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
PRICING CONINGEN CLAIMS 65 horm 3.4. If condiion I or II or III is fulfilld, hn h following pricing formulas hold: sup C r d sup C r d C r 2, d [,r 2 r ] d [,r 2 r ],d is cons. sup P r d sup P r d P r, d [,r 3.8 r ] d [,r 2 r ],d is cons. inf d [,r 2 r r ] d inf d [,r 2 r r ],d is cons. d C r, inf d [,r 2 r r ] d inf d [,r 2 r r ],d is cons. d P r 2. Corollary 3.5 S Korn [3]. h Black Schols modl saisfis 3.8 sinc condiions I, II and III ar fulfilld: and Φd 2. ν =, ν λ = Corollary 3.6. hpurjumpcashmronmodlsaisfis3.8 sinc condiion I holds: ν λ =. 4. Appndix Proof of Lmma 2.. a h cas of a call opion. For convninc w will us a diffrn rprsnaion of C/ dpnding on whhr λ / is posiiv or ngaiv. Diffrniaing 2. yilds o C = λ λ n λ An n 4. + K r λ n λ Φd 2 n if λ, n and 4.2 C = λ ν n ν λ Kr n + K r λ n n λ n λ Bn λ n λ Φd 2 n +Φd 2 n λ Φd 2 n if λ, whr An andbn hav h following xprssions: An =S ν n+ ν λ Φd n +Φd n K r Φd 2 n +Φd 2 n, Bn =S ν n ν λ Φd n +Φd n K r Φd 2 n + Φd 2 n ; Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
66 S. KANE AND A. MELNIKOV w dno σ by σ, and d 2 n = ln [S/K]+nln ν + ν λ + r σ 2 /2 σ d n =d 2 n+σ. On can asily show ha An andbn. W only giv h proof ha An, sinc a similar mhod can b usd o show ha Bn. W hav An =S ν n+ ν λ Φd n+φd n K r Φd 2 n+φd 2 n, whr d i n +=d i n+lnν/σ, i =, 2, and An = S ν d n+ d2 n+ ν λ x2 /2 dx K r n+ x2 /2 dx 2π d n = S ν ln ν n+ ν λ σ 2π = ln ν σ 2π = ln ν σ 2π x+d n 2 /2 dx ln ν K r σ d 2 n, x+d 2n 2 /2 dx S ν n+ ν λ x+d n 2 /2 K r x+d 2n 2 /2 dx S ν n+ ν λ x+d n 2 /2 xσ ν dx. If ln ν is ngaiv hn xσ ν is also ngaiv. Hnc, An is posiiv. Similarly, if ln ν is posiiv hn xσ ν is also posiiv. Hnc, An is always posiiv. Consqunly: For λ /, from h sign of An and quaion 4. w obain C/ >. 2 Similarly for λ /, sinc Bn and from quaion 4.2, w only nd o find h sign of 4.3 ν λ Kr n Exprssion 4.3 can b ransformd as λ n 4.4 Φd 2 n K r n λ n λ Φd 2 n +Φd 2 n +ν λ + K r λ n λ Φd 2 n. n ν λ Φd 2n +. Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
PRICING CONINGEN CLAIMS 67 o guaran h posiiviy of h abov xprssion, i is sufficin o prov ha X = Φd 2 n +ν λ ν λ Φd 2n + is posiiv. L us now considr wo cass ν orν and no ha from h xprssion of d 2 n h following always holds: ν Φd 2 n +Φd 2 n. a If ν, hn from h prvious rlaion, Φ is a non-dcrasing funcion of n and + ν λ / > ; hrfor Hnc, if Φd 2 +ν λ Φd 2 ν λ <X. λ ν +ν λ hn X>and C/ >. b If ν, hn Φ is a non-incrasing funcion of n, i.., Φd 2 n + Φd 2 n, and Φd 2 n +<X. hrfor, X is non-ngaiv and C/ >. b h cas of a pu opion. For h pu opion, ρ is givn by h following: 4.5 P = λ n λ n, λ S ν n+ ν λ Φd n +Φd n K r Φd 2 n +Φd 2 n K r λ n λ Φd 2 n n if λ, and 4.6 P = λ ν λ n λ S ν n ν λ Φd n +Φd n n ν λ λ n Kr n K r λ n λ Φd 2 n n K r Φd 2 n + Φd 2 n λ Φd 2 n + Φd 2 n if λ. If λ /, hn from h sign of An wg P/ <. Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
68 S. KANE AND A. MELNIKOV 2 Now if λ / >, sinc Bn andν >, h firs rm of P/ is ngaiv. W only nd o drmin h sign of ν λ λ n Kr λ Φd 2 n +Φd 2 n n 4.7 K r λ n λ Φd 2 n. n As in h call cas, w no ha ν Φd 2 n +Φd 2 n and disinguish wo cass ν andν. Again w considr h problm of finding h sign of 4.8 ν λ Φd2 n +Φd 2 n Φd 2 n. a For ν, w rwri h xprssion 4.8 as Y = ν λ Ψd2 n Ψd 2 n + Ψd 2 n 4.9 = Ψd 2 n ν λ + ν λ Ψd 2n +, whr Φ = Ψ. h abov xprssion is ngaiv if an uppr bound of Y is ngaiv. Bu, for h sam rason as in h call cas hr Ψ is an incrasing funcion of n, Hnc for or Y < Ψd 2 ν λ Ψd 2 = Φd 2 > + ν λ λ ν +ν λ Φd 2 < +ν λ P/ is ngaiv. b For ν wnohaφd 2 n rsp. Ψ d 2 n is a non-dcrasing rsp. non-incrasing funcion of n and Y Ψd 2 n. Hnc Y is ngaiv and so is P/. Bibliography. K. K. Aas, Coningn claim valuaion whn h scuriy pric is a combinaion of an Iô procss and a random poin procss, Soch. Procss. Appl. 28 988, 85 22. MR952829 89k:95 2. J. Bardhan and X. Chao, Pricing opions on scuriis wih disconinuous rurns, Soch. Procss. Appl. 48 993, 23 37. MR2377 94g:9 3. Y. Bar, Opion hdging in h binomial modl wih diffring inrs ras, UspkhiMah. Nauk 53 998, no. 5, 227 228; English ransl. in Russian Mah. Survys 53 998, 84 85. MR699 4. Y. Brgman, Opion Pricing wih Diffrn Inrs Ras for Borrowing and for Lnding, Working Papr Univrsiy of California, vol. 9, Brlky, 98. 5. D. Colwll and R. Ellio, Disconinuous ass prics and non-aainabl coningn claims and corpora policy, Mah. Financ 3 993, 295 38.. Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us
PRICING CONINGEN CLAIMS 69 6. J. Cvianic, Opimal rading undr consrains, Lcurs Nos in Mahmaics, vol. 656, Springr-Vrlag, Brlin, 997, pp. 23 9. MR4782 7. J. Cvianic, hory of porfolio opimizaion in marks wih fricions, Handbooks in Mah. Financ: Opion Pricing, Inrs Ras and Risk Managmn E. Jouini and M. Musila, ds., Cambridg Univrsiy Prss, 2. MR848547 22d:94 8. J. Cvianic and I. Karazas, Hdging coningn claims wih consraind porfolio, h Annals of Applid Probabiliy 33 993, 652 68. MR23369 95c:922 9. J. Cvianic, H. Pham, and N. ouzi, Supr-rplicaion in sochasic volailiy modls undr porfolio consrains, J. of Appl. Probabiliy 36 999, 523 545. MR724796 2a:948. R. Ellio and P. E. Kopp, Mahmaics of Financial Marks, Springr-Vrlag, Brlin, 998. MR298795 25g:9. H. Föllmr and D. O. Kramkov, Opional dcomposiions undr consrains, Probabiliy hory and Rlad Filds 9 997, 25. MR46997 98j:665 2. I. Karazas and S. Shrv, Mhods of Mahmaical Financ, Springr-Vrlag, Nw York, 998. MR64352 2:976 3. R. Korn, Coningn claim valuaion in a mark wih diffrn inrs ras, Mahmaical Mhods of Opraions Rsarch 42 995, 255 274. MR358829 4. R. Kruchnko and A. V. Mlnikov, Quanil hdging for a jump-diffusion financial mark, rnds in Mahmaics M. Kohlmann, d., Birkhäusr-Vrlag, Basl/Swizrland, 2, pp. 25 229. MR882833 5. A. V. Mlnikov, M. Nchav, and S. Volkov, Mahmaics of Financial Obligaions, Amr. Mah. Soc,, Providnc, 22. MR9876 23f:955 6. F. Mrcurio and W. Runggaldir, Opion pricing for jump-diffusions: approximaions and hir inrpraion, Mah. Financ 3 993, 9 2. 7. R.C.Mron,Coninuous-im Financ, Basil-Blackwll, Oxford, 99. 8. H. Sonr and N. ouzi, Suprrplicaion undr gamma consrains, Journal on Conrol and Opimizaion 39 2, 73 96. MR7899 22h:968 Offic of h Suprinndan of Financial Insiuions, orono, M5H39, Canada E-mail addrss: slly.kan@osfi-bsif.gc.ca Dparmn of Mahmaical and Saisical Scincs, Univrsiy of Albra, Edmonon, 6G2G, Canada E-mail addrss: mlnikov@ualbra.ca Rcivd 3/NOV/26 Originally publishd in English Licns or copyrigh rsricions may apply o rdisribuion; s hp://www.ams.org/journal-rms-of-us