ON PRICING CONTINGENT CLAIMS IN A TWO INTEREST RATES JUMP-DIFFUSION MODEL VIA MARKET COMPLETIONS

Transcription

1 or Imov r.amam.sais. hor. Probabiliy and Mah. Sais. Vip. 77, 27 No. 77, 28, Pags S Aricl lcronically publishd on January 4, 29 ON PRICING CONINGEN CLAIMS IN A WO INERES RAES JUMP-DIFFUSION MODEL VIA MARKE COMPLEIONS UDC 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 # c 29 Amrican Mahmaical Sociy Licns or copyrigh rsricions may apply o rdisribuion; s hp://

2 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 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: 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://

3 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://

4 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://

5 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://

6 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 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 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://

7 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 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://

8 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://

9 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://

10 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://

11 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://

12 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 , MR k:95 2. J. Bardhan and X. Chao, Pricing opions on scuriis wih disconinuous rurns, Soch. Procss. Appl , MR g:9 3. Y. Bar, Opion hdging in h binomial modl wih diffring inrs ras, UspkhiMah. Nauk , no. 5, ; English ransl. in Russian Mah. Survys , MR Y. Brgman, Opion Pricing wih Diffrn Inrs Ras for Borrowing and for Lnding, Working Papr Univrsiy of California, vol. 9, Brlky, D. Colwll and R. Ellio, Disconinuous ass prics and non-aainabl coningn claims and corpora policy, Mah. Financ 3 993, Licns or copyrigh rsricions may apply o rdisribuion; s hp://

13 PRICING CONINGEN CLAIMS J. Cvianic, Opimal rading undr consrains, Lcurs Nos in Mahmaics, vol. 656, Springr-Vrlag, Brlin, 997, pp MR 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. MR d:94 8. J. Cvianic and I. Karazas, Hdging coningn claims wih consraind porfolio, h Annals of Applid Probabiliy , MR c: J. Cvianic, H. Pham, and N. ouzi, Supr-rplicaion in sochasic volailiy modls undr porfolio consrains, J. of Appl. Probabiliy , MR a:948. R. Ellio and P. E. Kopp, Mahmaics of Financial Marks, Springr-Vrlag, Brlin, 998. MR g:9. H. Föllmr and D. O. Kramkov, Opional dcomposiions undr consrains, Probabiliy hory and Rlad Filds 9 997, 25. MR j: I. Karazas and S. Shrv, Mhods of Mahmaical Financ, Springr-Vrlag, Nw York, 998. MR : R. Korn, Coningn claim valuaion in a mark wih diffrn inrs ras, Mahmaical Mhods of Opraions Rsarch , MR 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 MR A. V. Mlnikov, M. Nchav, and S. Volkov, Mahmaics of Financial Obligaions, Amr. Mah. Soc,, Providnc, 22. MR f: F. Mrcurio and W. Runggaldir, Opion pricing for jump-diffusions: approximaions and hir inrpraion, Mah. Financ 3 993, R.C.Mron,Coninuous-im Financ, Basil-Blackwll, Oxford, H. Sonr and N. ouzi, Suprrplicaion undr gamma consrains, Journal on Conrol and Opimizaion 39 2, MR h:968 Offic of h Suprinndan of Financial Insiuions, orono, M5H39, Canada addrss: slly.kan@osfi-bsif.gc.ca Dparmn of Mahmaical and Saisical Scincs, Univrsiy of Albra, Edmonon, 6G2G, Canada addrss: mlnikov@ualbra.ca Rcivd 3/NOV/26 Originally publishd in English Licns or copyrigh rsricions may apply o rdisribuion; s hp://