HEDGING OPTIONS FOR A LARGE INVESTOR AND FORWARD BACKWARD SDE S BY JAKSA ˇ CVITANIC. Columbia University and Purdue University

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1 he Annals of Applie Probabiliy 996, Vol. 6, No., EDGING OPIONS FOR A LARGE INVESOR AND FORWARD BACKWARD SDE S BY JAKSA ˇ CVIANIC AND JIN MA Columbia Universiy an Purue Universiy In he classical coninuous-ime financial marke moel, sock prices have been unersoo as soluions o linear sochasic ifferenial equaions, an an imporan problem o solve is he problem of heging opions Ž funcions of he sock price values a he expiraion ae.. In his paper we consier he heging problem no only wih a price moel ha is nonlinear, bu also wih coefficiens of he price equaions ha can epen on he porfolio sraegy an he wealh process of he heger. In mahemaical erminology, he problem ranslaes o solving a forwarbackwar sochasic ifferenial equaion wih he forwar iffusion par being egenerae. We show ha, uner reasonable coniions, he four sep scheme of Ma, Proer an Yong for solving forwarbackwar SDE s sill works in his case, an we exen he classical resuls of heging coningen claims o his new moel. Inclue in he examples is he case of he sock volailiy increase cause by overpricing he opion, as well as he case of ifferen ineres raes for borrowing an lening.. Inroucion an summary. In he usual coninuous-ime moel of a sock marke, going back o Meron 9, a sock price process P is moele as a soluion of a linear sochasic ifferenial equaion, wih given rif an noise Ž volailiy. coefficiens. An assumpion ha has long been viewe as sanar is ha he invesor is small in he sense ha hisher financial saus an raing sraegy shoul no affec he moel of he marke prices. herefore, in he classical moel he coefficiens of he price equaions are inepenen of he wealh an porfolio processes of he invesor. In his paper we consier he case in which he influence of he invesor s financial behavior is no a priori known o be irrelevan an he price moel is no necessarily linear. In oher wors, we assume ha he rif an he volailiy erms can boh be nonlinear in he price process an also epen on he wealh process X an he porfolio process of he invesor. Such a moel is useful when he invesor is no-oo-small ; we call himher large in he sequel, since he small invesor assumpion is obviously remove. Naural Receive December 994; revise January 996. Research suppore in par by NSF Gran DMS an Army Research Office Gran DAA Research suppore in par by NSF Gran DMS AMS 99 subjec classificaions. Primary 90A09, 6030; seconary 90A, 93A0. Key wors an phrases. Forwarbackwar sochasic ifferenial equaions, coningen claims, heging sraegy, large invesor. 370

2 EDGING OPIONS FOR LARGE INVESORS 37 examples inclue a marke wih ifferen ineres raes for borrowing an lening Ž wih a small invesor. an moels in which volailiy of he prices can change ue o he srange behavior of he large invesor. We suy a heging problem for his invesor on a finie ime horizon 0, : given an iniial sock price p PŽ 0. an a esire erminal wealh value gž PŽ.. Ž which we call an opion, as a special case of coningen claims., he invesor wans o fin a porfolio process an an iniial wealh x XŽ 0., such ha he corresponing wealh process saisfies XŽ. gž PŽ... Moreover, heshe wans o fin he heging porfolio process ha goes wih he smalles iniial wealh possible. ha smalles iniial wealh x is hen he upper boun for he price of he opion gž PŽ..; ha is, no one shoul be willing o pay more han x a ime 0 for he opion worh gž PŽ.. a ime. ŽFor more on opion pricing heory, refer o he famous Black an Scholes paper ; see also,, 3, 4 for he maringale heory.. In mahemaical erminology, he problem ranslaes o fining a soluion Ž P, X,., wih minimal XŽ 0., of a forwarbackwar sochasic ifferenial equaion Ž FBSDE., wih P being he forwar an X he backwar componen. We use he four sep scheme of Ma, Proer an Yong 8 o solve he FBSDE Žwe refer he reaer o ha paper for more references on he relaively new noion an heory of FBSDE s.. isorically, i was he special case of FBSDE s, calle backwar sochasic ifferenial equaions Ž BSDE s., ha was firs evelope in a mahemaical conex by Paroux an Peng 0, an inepenenly by Duffie an Epsein 6 in finance; see also an. In our conex, he erm pure backwar case will mean hose cases in which he esire erminal value oes no epen on he wealh or porfolio process, eiher because he price process oes no, or because he value is equal o an a priori given F -measurable ranom variable which may be of a more general form han gž PŽ... A problem similar o he one of his paper, bu only in he pure backwar case, wih he volailiy of socks being inepenen of he invesor s policy an using mehos an assumpions compleely ifferen from ours, was suie by El Karoui, Peng an Quenez 8 an Cvianic 3. he forwarbackwar case as a moel in finance is use by Duffie, Ma an Yong 7 for a ifferen problem concerning he erm srucure of ineres raes. For relae work on he ineracions beween heging sraegies an marke prices, in he Ž equilibrium. conex of several agens, see he very ineresing papers by Plaen an Schweizer 3 an Grossman 0 an references herein. We shoul noe here ha in he presen moel, he price equaion is no longer linear. So, unlike he classical case, i is alreay quesionable how o keep he price processes P Ž., i,...,, from becoming negaive Ž i in such a case he marke will be esroye, by common sense.. One seemingly simple way o rea his, say wih, is o a a local-ime erm o he price equaion, so ha he process P becomes iffusion reflece a zero. he isavanage of oing his is ha in such a moel here woul be an opporuniy for arbirage, ha is, a chance o make a posiive amoun of

3 37 J. CVIANIC AND J. MA money ou of zero iniial wealh, wih a posiive probabiliy. his has been commonly viewe as an unesirable propery of a financial marke. For more on his maer, refer o 5, an he example in 5. Anoher meho, which is he one we shall employ in his paper, is o pose some coniions on he coefficiens of he price equaion so ha he soluion PŽ., whenever i exiss, will always say insie he region Ž x,..., x. x i 0, i 4,...,. In oher wors, he bounary is naural. I urns ou ha one of he coniions is ha he forwar equaion Ž price equaion. will have o be egenerae; ha is, he volailiy funcion will vanish on he bounary. On he oher han, however, his egeneracy will cause echnical ifficulies for us o obain he unique aape soluion for he FBSDE, because he resul in 8 canno be applie irecly. We show ha, uner cerain coniions, such a conflic can be resolve in a saisfacory way, by exening he resuls of 8. Some oher possible mehos ha migh lea o posiive bu weaker resuls, such as replacing he pricewealh pair by he log-pricewealh pair or using a suiable change of probabiliy measure, will also be iscusse. In his paper we will only consier he problem of heging an opion. Namely, he erminal value of he wealh is specifie as gž PŽ.., where PŽ. is he value of socks a he expiraion ae an g is some smooh funcion of linear growh. he complee resoluion o he heging problem for general coningen claims Ž i.e., arbirary erminal coniions for he wealh. will require furher evelopmen in he heory of FBSDE s an will be suie separaely. he paper is organize as follows. We escribe he moel an give efiniions an some preliminary resuls in Secion. In Secion 3 we suy he corresponing FBSDE s using he four sep scheme an prove he amissibiliy of he soluion as a heging sraegy. Secion 4 is evoe o wo comparison heorems. he firs leas o he uniqueness of he FBSDE an implies ha he soluion o he FBSDE will inee prouce he opimal heging sraegy; ha is, he corresponing wealh process has he smalles iniial enowmen x XŽ 0. among hose ha o he heging. he secon heorem shows ha he opimal sraegy is monoone wih respec o he erminal value; he higher he opion value a he expiraion ae, he higher he premium. In Secion 5 we give examples ha moivae our moel, incluing he case in which here is an increase in he sock s volailiy if he opion is overprice an an example of a marke wih ifferen ineres raes for borrowing an lening. he former leas o a phenomenon unknown in he classical case: he heging is guaranee if one sells he opion a he fair price Ž e.g., using he BlackScholes formula.; however, if one sells he opion for more han ha, hen one may no be able o o he heging because of he corresponing change of he volailiy in he marke. Finally, we give an example Ž in he Appenix. showing ha he classical comparison heorem for a backwar SDE nee no hol in he presen forwarbackwar case, supporing our argumen in Secion 4.

4 EDGING OPIONS FOR LARGE INVESORS 373. Problem formulaion. Le us consier a marke M in which asses are rae coninuously. One is calle a bank accoun which is riskless an he ohers are calle socks, which are assume o be risky. We consier an invesor in his marke an, conrary o he usual small invesor hypohesis, we assume ha boh his invesor s wealh an sraegy, once expose, migh influence he prices of he financial insrumens. More precisely, we assume ha he price of he bank accoun evolves accoring o he ifferenial equaion Ž.. P P r, X Ž.,, 0, 0 0 P0Ž 0., an ha he price of he socks evolves accoring o he sochasic ifferenial equaion, for 0, Ž.. P b, P Ž., X Ž., i i Ý i j j Ž, P Ž., X Ž.,. W Ž., P Ž 0. p 0, i,...,, i i where 0 is he mauriy ae or uraion an X is he wealh process, while Ž,...,. is he porfolio process of he invesor an W Ž W,..., W. is a -imensional sanar Brownian moion efine on a complee probabiliy space Ž, F, P., wih he filraion F 4 0, which is he W P-augmenaion of he naural filraion F WŽ s.: 0 s 4 generae by W. We require now Ž an specify laer. ha he funcions b an are such ha he soluion P i s are posiive processes. Now le us suppose ha he invesor will sar wih an iniial enowmen x 0 an ry o allocae his wealh ino he bank an socks accoring o a cerain sraegy a each ime 0,. We efine each process i o be he amoun of money ha he invesor pus ino he ih sock; hus he amoun invese in he bank will be X Ý Ž. i i. Furhermore, if we allow he invesor o consume a cerain amoun of money a each ime an enoe he cumulaive consumpion up o ime by CŽ., hen C is a nonecreasing, F 4-aape process, CŽ I is inuiive ha he change of he wealh in a small ime incremen, h. can be escribe approximaely by Ž.3. X Ž h. X X Ý P h P i i i Ý Ž i i. i Pi P0 P Ž h. P CŽ h. CŽ his amouns o saying ha he wealh process saisfies he sochasic if- j

5 374 J. CVIANIC AND J. MA ferenial equaion Ž.4. X Ý X P P C where Ž.5. i i i Ý i 0 i Pi P0 ½ i Ý bi Ž, P Ž., X Ž.,. P i i Ý i j Ž. j 5 j, P, X, W X Ý i i P0 r Ž, X Ž., P. C P0 ˆb Ž, P Ž., X Ž.,. ˆ Ž, P Ž., X Ž.,. W CŽ., X Ž 0. x 0, i Ý i ž / Ý i i p i ˆb Ž, p, x,. x r Ž, x,. b Ž, p, x,., Ý i ˆ Ž, p, x,. Ž, p, x,., j,...,, j i p i i j for, p, x, 0,. In his paper we shall use he follow ing noaion hroughou: we enoe he posiive orhen by Ž x,..., x. 4 ² : x i 0, i,..., ; he inner prouc in by, ; he norm in by an ha of, he space of all marices, by an he ranspose of a marix A Žresp. a vecor x. by A Žresp. x.. We also enoe o be he vecor,..., an efine a Ž iagonal. marix-value funcion : by x x 0.6 x, x x,..., x.. Ž x I is obvious ha x x for any x, an whenever x, Ž x. is inverible an Ž x. is of he same form as Ž x. wih x,..., x being replace by x,..., x. We can hen rewrie he funcions ˆb an ˆ in Ž.5. as Ž.7. ˆ ² : bž, p, x,. xr Ž, x,., b Ž, p, x,. r Ž, x,., ² :, p, x,,, p, x,, ˆ i

6 where EDGING OPIONS FOR LARGE INVESORS 375 ž / b b b Ž, p, x,. Ž p. bž, p, x,.,..., Ž, p, x,., p p Ž.8. i j Ž, p, x,. Ž p. Ž, p, x,. Ž, p, x,.. ½ 5 p i i, j o be consisen wih he classical moel, we henceforh call b he appreciaion rae an he volailiy marix of he sock marke. We now give more precise efiniions of he quaniies appearing in Ž.4.. DEFINIION.. Ž i. A porfolio process Ž.; 0 4 is a real-value, progressively F 4-measurable process, such ha E 0. Ž ii. A consumpion process C CŽ.; 0 4 is a real-value, F 4 - aape process, wih nonecreasing an RCLL Žrigh-coninuous wih lef limis. pahs, such ha CŽ 0. 0 an CŽ., a.s. P. Ž iii. For a given porfolioconsumpion pair Ž, C., he price process wih he iniial value p 0 an he wealh process wih iniial capial x 0 are he soluions o he SDE s Ž.. an Ž.4., respecively, which will ofen be enoe by P P p, x,, C an X X p, x,, C, whenever he epenence of he soluion on p, x,, C nees o be specifie. We will make use of he following saning assumpions: Ž A. he funcion b, : 0, an g: are wice coninuously iffereniable. he funcions b an, ogeher wih heir firs orer parial erivaives in p, x an are boune, uniformly in, p, x,. Furher, we assume ha parial erivaives of b an in p saisfy b ½ i j p p 5 Ž, p, x,. k k Ž.9. sup p k, pk, i, j, k,...,. A he funcion saisfies, p, x, 0 for all, p, x, wih p, an here exiss a posiive consan 0, such ha Ž.0. a Ž, p, x,. I for all Ž, p, x,., where a Ž.. Ž A3. he funcion r is wice coninuously iffereniable an such ha he following coniions are saisfie: a For, x, 0,, 0 rž, x,. K, for some consan K 0. Ž b. he parial erivaives of r in x an, enoe by a generic funcion

7 376 J. CVIANIC AND J. MA, saisfy Ž. x,,. lim sup x, x,. REMARK.. Ž. We noe ha Assumpions Ž A. an Ž A. obviously conain hose cases in which bž, p, x,. Ž p. b Ž, x,. an Ž, p, x,. Ž p. Ž, x,., where b an are boune, coninuously iffereniable funcions wih boune firs orer parial erivaives, an is posiive efinie an boune away from zero, as we ofen see in he classical moel. In paricular, our seing will conain he BlackScholes moel as a special case. Coniion Ž A3. is somewha resricive, which is largely ue o he generaliy of our seing. I also conains he classical case when r r. Moreover, as we shall see in examples in Secion 5, he meho escribe below someimes works even if he assumpions are far from being saisfie. he bouneness of he funcions b an imply ha b an will vanish on he se, he bounary of. In oher wors, has o be egenerae on. his requiremen is o guaranee ha he sock prices say posiive all he ime so ha he marke is no esroye, as we shall prove in he following lemma. LEMMA.3. Suppose ha Ž A. an Ž A. hol. hen for any porfolioconsumpion pair Ž, C. an iniial wealh x, he price process P saisfies P 0, i,..., for all 0, i, almos surely, provie he iniial prices p,..., p are posiive. PROOF. Le Ž, C. an he iniial values Ž p, x. be given. Le Ž P, X. enoe he soluions o he forwar SDE s. an.4. By efiniion of b an, we can rewrie Ž.. in he form Ž.. Pi pi Pi Ž s. bi Ž s, P Ž s., X Ž s., Ž s.. s 0 Ž. i s, P s, X s, s W s, where is he ih row vecor of he marix. Denoing b i i b Ž, PŽ., XŽ.,. an Ž, PŽ., XŽ., Ž.. i, an recalling heir bouneness, we see ha he processes Pi can be wrien as sochasic exponenials Žsee, e.g., 6, 4 or 5., ½ 5 i i i i i 0 P p exp b Ž s. Ž s. s Ž s. W Ž s., i,..., ; hence he conclusion follows. Lemma.3 now enables us o give he following efiniion of he amissible porfolioconsumpion sraegy.

8 EDGING OPIONS FOR LARGE INVESORS 377 DEFINIION.4. For a given x 0, a pair of porfolioconsumpion processes Ž, C. is calle amissible Ž wih respec o x. if, for any p 0, he corresponing price process P an wealh process X saisfy P i 0, i,..., i.e., P an X 0, 0,, a.s. P. For each x, we enoe he se of all amissible porfolioconsumpion pairs by AŽ x.. We claim ha AŽ x. for all x. o see his, we firs noe ha for any x 0, p an Ž, C., we can solve a pair of Ž forwar. SDE s for P an X. By Lemma., we know ha i is always rue ha P, a.s. P. herefore, we nee only show ha for each x 0 here exiss a pair Ž, C., p, x,, such ha for all p, X C 0, a.s. P. his can be one by choosing 0 an C 0. By he efiniion of ˆb an ˆ Ž.5., we see ha he wealh process will saisfy X x X Ž s. r s, X Ž s., 0 s, 0 whence X x exp ržs, XŽ s., 0. s4 0 0 for all. In oher wors, he rivial pair Ž 0, 0. AŽ x.. o conclue his secion, we give he following efiniion. DEFINIION.5. An opion is an F-measurable ranom variable B gž PŽ.., where g is a real funcion. he heging price of he opion is efine by Ž.3. hž B. inf Ž B., where 4 Ž.4. Ž B. x : Ž, C. AŽ x., s.. X x,, C Ž. B a.s.. 3. Forwar backwar SDE s. In his secion we suy he FBSDE s ha will play an imporan role in our fuure iscussions. Consier he FBSDE given by Ž 3.. where Ž 3.. P p b s, P Ž s., X Ž s., Ž s. s 0 Ž s, P Ž s., X Ž s., Ž s.. W Ž s., 0 ˆ X g P Ž. b s, P Ž s., X Ž., Ž s. s ˆ Ž s, P Ž s., X Ž s., Ž s.. W Ž s. CŽ. CŽ., ˆ ² : ² : b, p, x, xr, x,, b, p, x, r, x,, ˆ, p, x,,, p, x,. We firs give he efiniion of an aape soluion o he FBSDE 3..

9 378 J. CVIANIC AND J. MA DEFINIION 3.. A quaruple Ž P, X,, C. is calle an aape soluion o he FBSDE Ž 3.., if he following hol: Ž i. P, X an are F 4 -aape, square inegrable processes. Ž ii. C is an F 4-aape, RCLL, nonecreasing process, such ha CŽ 0. 0 an CŽ.. In wha follows we shall only consier he FBSDE Ž 3.. wih C 0, namely, he FBSDE Ž 3.3. P p b s, P Ž s., X Ž s., Ž s. s 0 Ž s, P Ž s., X Ž s., Ž s.. W Ž s., 0 ˆ X g P Ž. b s, P Ž s., X Ž s., Ž s. s ˆ Ž s, P Ž s., X Ž s., Ž s.. W Ž s.. he exisence of an aape soluion o such an FBSDE will lea o he nonempyness of he se Ž gž PŽ... of Ž.4., an in fac, o hž gž PŽ... XŽ 0., where X is he backwar componen of he soluion o FBSDE Ž We shall also assume ha he funcion g saisfies eiher one of he following wo coniions: A4 a he funcion g is boune, C an nonnegaive. Is parial erivaives up o secon orer are all boune. Ž b. he funcion g is nonnegaive an lim gž p. p. Moreover, g has boune, coninuous parial erivaives up o hir orer an here exis consans K, M 0 such ha Ž. p g p p K g p, sup p g p p M. p Furher, we assume ha he parial erivaives of ½ 5 i j i j Ž, p, x,. x k in x an saisfy 3.4 sup x x, i, j, k,...,. REMARK 3.. Clearly, he firs inequaliy in Ž A4.Ž b. hols for any g ha behaves like a polynomial for p large, bu he secon coniion resrics i o one ha has a mos quaraic growh. he coniion Ž 3.4. can be calle a compaibiliy coniion o compensae for he unbouneness of g. We noe ha i also conains he classical moels as special cases. An example of a funcion saisfying Ž A., Ž A. an Ž A4.Ž b. coul be Ž, p, x,. Ž Ž p arcan x.. wih saisfying Ž A..

10 EDGING OPIONS FOR LARGE INVESORS 379 In orer o prove he exisence an uniqueness of he aape soluion o Ž 3.3., we follow he four sep scheme esigne in 8. he main iea of he four sep scheme is base on he well-known FeynmanKac formula, combine wih a ecoupling proceure for he FBSDE. We shoul noe ha in he presen case, he funcion is egenerae on he bounary of, an he funcion g is allowe o be unboune coniion Ž A4.Ž b.. hus he resul in 8 oes no apply irecly. owever, by using he special srucure of he funcions ˆb an ˆ an some ransformaions, we show ha he four sep scheme will remain vali in he presen case. For convenience of presenaion, we shall iscuss he exisence par firs in his secion an efer he proof of uniqueness o he nex secion, as a corollary of our comparison heorem. Le us firs review he four sep scheme 8. Four sep scheme. Le us firs keep in min ha he soluion o Ž 3.3., whenever i exiss, will saisfy P, for all, a.s. P, hanks o Lemma.3. So we migh as well resric ourselves o he region Ž, p, x,. 0, E wihou furher specificaion an we procee as follows. SEP. In orer o mach iffusion erms, fin a smooh mapping z: 0, so ha Ž 3.5. q, p, x, zž, p, x, q. ˆ, p, x, zž, p, x, q. 0 Ž, p, x, q. E. In our case, 3.5 becomes q, p, x, zž, p, x, q. Ž 3.6. z Ž, p, x, q. Ž p., p, x, zž, p, x, q. 0, hence z, p, x, q p q since 0 by Ž A. an is a iagonal, nonsingular marix. One shoul noe ha in he presen case we solve for he funcion zž,,,. irecly, which makes our soluion more explici han ha in 8. SEP. Wih he inenion of seing zž, p,,. Ž p. p p, solve he quasilinear parabolic equaion for Ž, p.: 0 r Ž, p,, Ž p. p. p p4 Ž 3.7. ² Ž p. p: Ž p. b, p,, Ž p., ˆb, p,, Ž p., Ž, p. g Ž p., p. In our case, by an easy compuaion using 3., we have ha Ž p. Ž p. ² Ž p. Ž p. p: ˆb, p,, Ž p. r,, Ž p. b, p,, p r,, p p,.

11 380 J. CVIANIC AND J. MA hus 3.7 becomes Ž 3.8. Ž. 4 0 r, p,, p p p p Ž p. Ž p. ² p, : r,, Ž p., Ž, p. g Ž p., p. Ž 3.9. SEP 3. Seing Ž p. Ž p. b Ž, p. b, p, Ž, p., Ž p. Ž, p.,, p, p,, p, p, p, solve he forwar SDE Ž 3.0. P p b Ž s, P Ž s.. s Ž s, P Ž s.. W Ž s.. SEP 4. Seing 0 0 Ž 3.. X Ž, P Ž.., Ž P. Ž, P Ž.., show ha P, X, is he unique aape soluion o 3.. Before we procee any furher, le us give a lemma which shows ha if Ž. P, X, is a soluion o he FBSDE 3.3, hen he pair, 0 A X 0. LEMMA 3.3. Suppose ha Ž A. Ž A3. hol an le Ž P, X,. be an aape soluion o Ž 3.3., wih gž p. 0 for all p 0. hen he pair Ž, 0. is an amissible heging sraegy wih respec o XŽ 0., in he sense of Definiion.4. PROOF. Le Ž P, X,. be an aape soluion o Ž he process P will say insie for all 0,, a.s. P, hanks o Lemma.3. Since he P-square inegrabiliy of is alreay conaine in he efiniion of he aape soluion o FBSDE, i remains o show ha he wealh process X 0 for all 0. o his en, le us efine, for he given processes Ž P, X,., a Ž ranom. funcion Ž 3.. f Ž, x, z. r Ž, X Ž.,. x z, Ž, P Ž., X Ž.,. 4 an consier he linear backwar SDE Ž. Ž. ; b, P, X, r, X,, Ž. Ž. ² : 3.3 x g P f s, x s, z s s z s, W s. p

12 EDGING OPIONS FOR LARGE INVESORS 38 Comparing wih 3. an 3.3, we see ha he pair x, z efine by Ž 3.4. x X, z, P, X,, Ž. 0,, is an aape soluion o Ž On he oher han, noe ha he funcion f is linear in x an z wih boune erivaives by assumpions Ž A. an Ž A., an ha fž, 0, We see ha f is a sanar generaor for he BSDE Ž 3.3., in he erminology of 8. herefore, a comparison heorem for he classical Ž linear. BSDE s Žsee 8. leas o X x 0, a.s. P, whenever gž PŽ.. 0, a.s. P, proving he lemma. Our main resul in his secion is he following heorem. EOREM 3.4. Suppose ha he saning assumpions Ž A. Ž A3. an A4 b hol. hen for any given p, he FBSDE Ž 3.3. amis a unique aape soluion Ž P, X,., given by Ž 3.. wih being he soluion of Ž PROOF. We follow he four sep scheme menione above. Sep is obvious. For Sep, we claim he following asserion which migh be of ineres in is own righ: here exiss a unique classical soluion Ž,. o he PDE Ž 3.8., efine on, p 0,, which enjoys he following properies: i g is uniformly boune for, p 0,. Ž ii. he parial erivaives of saisfy, for some consan K 0, Ž 3.5. Ž p. Ž, p. K Ž p., p p p sup p, p K., p 0, o prove he asserion, le us firs consier he funcion ˆ g. I is obvious ha ˆ, ˆ g an ˆ g, an ˆ p p p p p p p p p saisfies he PDE ½ ž ž // ž / 5 ž ž // ² : p p p p ˆ ˆ ˆ ˆ p p p p p p 0 r, p, g p, p g p g 3.6 r, ˆ g p, p ˆ g p p, ˆ g ˆ g, o simplify noaion, le us se Ž 3.7. ˆ Ž, p. 0, p. Ž p. Ž, p, x,., p, x g Ž p., Ž p. g Ž p., Ž p. r Ž, p, x,. r, x g Ž p., Ž p. g Ž p..

13 38 J. CVIANIC AND J. MA hus, noing ha for p, q, we have rž, p, x,.² p, q: ² rž, p, x,., Ž p. q :, Ž 3.6. becomes ½ ž / 5 ; ˆ ˆ ˆ ˆ p p p 0 r, p,, p ž p/ p ž p/ Ž 3.8. r, p, ˆ, Ž p. ˆ, Ž p. ˆr, p, ˆ, Ž p. ˆ, where Ž 3.9. ˆ Ž, p. 0, p, ˆr Ž, p, x,. r Ž, p, x,. g p pž p. 4 ² : r Ž, p, x,., Ž p. g Ž p. r Ž, p, x,. Ž x g Ž p... Nex, we efine a change of variable by seing L p an Ž,. ˆ Ž, L., where L an L are efine by LŽ,...,. Ž e,..., e.,, L p,..., p log p,..., log p, p. hen i is easily checke ha ˆ Ž,. Ž, e., Ž 3.0. Ž,. Ž L. ˆ Ž, L., p ˆ ˆ p p p Ž,. Ž L. Ž, L. Ž L. Ž, L.. p ž / Plugging Ž 3.0. ino Ž 3.8. an oing some compuaion, we obain a quasilin ear parabolic PDE for : ½ 5 ž / Ž. ž ž / / 0 r, L,, L L ž / ; ž / Ž 3.. r,,, ½ 0 0 ž / 5 b,,,, ˆb,,, 0ž / ; 0ž /, where r, L,,, ˆr, L,,, Ž,. g Ž e.,, Ž 3.. 0Ž,, x,. Ž L. Ž, L, x,., b0,, x, r, L, x, iag 0 0,, x,, ˆb Ž,, x,. ˆr Ž, L, x,.. 0

14 EDGING OPIONS FOR LARGE INVESORS 383 ere ˆr is efine by Ž 3.9., an for a marix A we enoe by iag A he vecor compose of he iagonal elemens of A. Now by using coniion Ž A. noe Ž.0., one sees ha here exiss a consan 0 such ha a Ž,, x,. I 0 for all Ž,, x, ,. Also, by efiniion Ž 3.7., we see ha for all i, j, k,...,, an Ž,, x,., i hols Ž suppressing he variables. ha Ž 3.3. Ž a 0. i j a i j a i j g k e e p x p k k k k ai j g g l k l Ý k l l l pk pl pl e e e. Noe ha by Ž A4.Ž b. we have ha for each k, Ž q p. k k e Ž L. gž L. KŽ gž L.. an 0 gž L. x gž L.. hus coniions Ž A., Ž.9. an Ž A4.Ž b. imply ha a i j sup, L, x g Ž L., Ž L. g L e k p, p Ž,, x,. k ai j g sup Ž, L, x g Ž L., Ž L. g Ž L.. e k p x p a i j K sup ½ Ž, L, x g Ž L., Ž L. g pž L.. Ž,, x,. x Ž x g Ž L.. 5, Ž,, x,. k ai j g sup Ž, L, x g Ž L., Ž L. g Ž L.. e k p p a i j K sup, L, x g Ž L., Ž L. g Ž L ½ Ž p.. Ž,, x,. k Ž x g Ž L... 5 Ž,, x,. k k ha is, he funcion a0 has boune firs orer parial erivaives in, uniformly in Ž,, x,.. Noe ha he firs orer parial erivaives of a0 in x an are he same as hose of a Ž wih corresponing change of variables., an we conclue ha a0 is uniformly Lipschiz in, x an. o o a similar analysis for b an ˆb, we noe ha for any k,..., 0 0

15 384 J. CVIANIC AND J. MA an,, x,, i hols by efiniion of r ha r r g r g g k l k l Ý k l k x pk l l pk pl pl e e e e, ² r Ž, L, x,., Ž L. g pž L.: k g r Ž, L, x,. Ý e l Ž L. p k l l g g l k l r Ž, L, x,. Ý e e k l e, p p p l l k l k r Ž, L, x,. Ž x g Ž L.. g r Ž, L, x,. Ž x g Ž L.. r Ž, L, x,. e k p an ha he funcion k 4 ½ 5 r Ž, L, x,. g p pž L. r a0ž,, x,. Ž L. g p pž L. is uniformly boune an Lipschiz in, x an by coniion Ž A4.Ž b.. hus we obain by a similar argumen as before he uniform bouneness an Lipschiz propery of he funcions b an ˆ 0 b0 as well. herefore, by heorem 4.5 in 8 or by applying he resuls in 7 irecly, we conclue ha he, PDE 3.8 has a unique classical soluion in C for any Ž 0,.. Furhermore,, ogeher wih is firs an secon parial erivaives in, is uniformly boune hroughou 0,. If we go back o he original variable an noe he relaions in Ž 3.0., we obain ha he funcion variables ˆ is uniformly boune an is parial erivaives saisfy sup Ž p. ˆ Ž, p., sup Ž p. ˆ Ž, p.. p Ž, p. Ž, p. his, ogeher wih he efiniion of ˆ an coniion Ž A4.Ž b., leas o he esimaes Ž 3.5.; he asserion, an hence Sep, is prove. As for Sep 3, we noe ha p an p p hemselves are unboune an will blow up when p0. Consequenly, b an are only locally Lipschiz an no even efine a p 0. herefore a lile bi more careful consieraion is neee here, an we procee as follows. Firs we observe ha he local Lipschiz propery of b an is enough for us o show he exisence an uniqueness of he local soluion of Ž 3.0. insie Ž wih he possibiliy of explosion.. owever, by a similar argumen as ha in Lemma.3, one can linearize Ž 3.0. an use Assumpion Ž A. o show ha whenever he solu ion exiss, i will neiher leave nor exploe before. Ž In fac he soluion is a square-inegrable process.. ence Sep 3 is complee. Finally, Sep 4 is rivial, an noing ha he square inegrabiliy is he irec p p k

16 EDGING OPIONS FOR LARGE INVESORS 385 consequence of efiniion Ž 3.., propery Ž 3.5. an he square inegrabiliy of he process PŽ., he exisence is prove. he uniqueness of he aape soluion will be prove in Corollary 4.. he proof of he heorem is herefore complee. We noe from he proof of heorem 3.4 ha Coniion Ž A4.Ž b. is only use o guaranee he bouneness of he parial erivaives of a 0, which is unnecessary if g is boune, because in such a case he inermeiae soluion ˆ is no neee. For he same reason, coniion Ž A3. can be relaxe o he following. A3 he funcion r saisfies all he coniions of A3 excep ha. is replace by Ž 3.4. lim sup Ž x. Ž, x,.. x, Furhermore, he esimaes 3.5 of he soluion o he PDE 3.8 can be improve o p p Ž, p. Ž, p. 3.5 sup p, p ; sup p, p. In oher wors, we have he following corollary. COROLLARY 3.5. Uner Assumpions Ž A., Ž A., Ž A3. an Ž A4.Ž a., heorem 3.4 remains vali. Furhermore, he classical soluion of Ž 3.8. saisfies he esimae Ž Discussion. A seemingly simpler way of proving he exisence an uniqueness of FBSDE Ž 3.3. can be carrie ou in he following way if some even sronger coniions are saisfie by he coefficiens b an. We skech he iea here, because i migh be useful for some oher applicaions. Le us suppose ha Ž A. Ž A3. hol an suppose ha an he funcion g is boune an C for simpliciy. Le us also assume ha he funcions b an an heir parial erivaives in x an, enoe by a generic, saisfy he coniions Ž A5. lim sup Ž, p, x,. uniformly in Ž, p, x., p an he parial erivaives of b an in p, enoe by a generic, saisfy Ž A6. lim sup Ž, p, x,..

17 386 J. CVIANIC AND J. MA Žhese coniions are obviously saisfie in he classical wealhpolicy inepenen moels.. By Lemma.3, we know ha any soluion o Ž 3.3. mus saisfy P 0 for all 0,, a.s. P. herefore, we can efine a process log P for 0,. A simple compuaion using Io s ˆ formula will show ha Ž, X,. saisfies he FBSDE Ž 3.6. where 0 Ž 0. b s, Ž s., X Ž s., Ž s. s 0 s, Ž s., X Ž s., Ž s. W Ž s., ˆ X g Ž Ž.. b Ž s, Ž s., X Ž., Ž s.. s ˆ Ž s, Ž s., X Ž s., Ž s.. W Ž s., bž, e, x,. Ž, e, x,. bž,, x,., e e Ž, e, x,. Ž,, x,., e ˆ ˆ bž,, x,. bž, e, x,., ˆ Ž,, x,. ˆ Ž, e, x,.. Obviously, he exisence an uniqueness of he aape soluion o he FBSDE Ž 3.3. is equivalen o ha of Ž Since log PŽ., we will call Ž 3.3. he pricewealh equaion an Ž 3.6. he log-pricewealh equaion. One can herefore suy eiher one of hem, whichever is easier. I is fairly easy o check ha uner coniions Ž A. Ž A3., ogeher wih Ž A5. an Ž A6., he funcions b an are boune wih boune firs orer parial erivaives in p, x an, while he funcions ˆb an ˆ are of linear growh in x an, wih boune firs orer parial erivaives. herefore by heorem 4.5 in 7, he FBSDE Ž 3.6. has a unique aape soluion; herefore so oes Ž 3.3. Ž wih.. Comparing he above resul o heorem 3.4 or Corollary 3.5, we see ha applying he resul of 8 irecly, say, o he log-pricewealh equaion is no always he easies way, because one woul have o use coniions Ž A5. an Ž A6., which is obviously unnecessary, as we see in he heorem an is corollary. One of he main reasons for his o happen is ha in our seing, he coefficiens ˆb an ˆ are explicily relae o b, an r, an he corresponing quasilinear parabolic PDE is simplifie rasically so ha he exisence an uniqueness of classical soluions can be prove wih fewer resricions on he coefficiens. Such a phenomenon can also be parially explaine from a finance poin of view. In fac, by he sanar financial

18 EDGING OPIONS FOR LARGE INVESORS 387 mahemaics ool of he change of probabiliy measure, we can in a sense replace he appreciaion rae b by he ineres rae r, as in he classical case, which may faciliae some analysis. Such an equivalen probabiliy measure correspons o he classical equivalen maringale measure. owever, we shoul poin ou here ha his equivalen measure is now boh wealh an policy epenen; herefore one canno use i o simplify he FBSDE a priori o erive he aape soluion if he Brownian moion is no allowe o change. 4. Comparison heorems an main resuls. In his secion we shall prove wo comparison heorems. As corollaries of he firs comparison heorem, we prove he uniqueness par of heorem 3.4 an ha he unique aape soluion o he FBSDE Ž 3.3. is an opimal sraegy among all amissible ones. he secon comparison heorem shows ha he opimal sraegy is monoone in he opion value a he expiraion ae; namely, he higher he value of he opion, he higher he premium he buyer has o pay. We shoul noe ha ue o he special feaure of he FBSDE, hese comparison heorems are much weaker han hose of he pure backwar case. herefore, some new phenomena ha are ifferen from hose in classical heory migh be worh suying. Le us firs noe ha, in he forwarbackwar case, one canno easily jump o a conclusion like X Ž. g Ž P Ž.. g Ž P Ž.. X Ž. from he assumpion ha g Ž p. g Ž p., p, because in he presen siuaion, he price process is policywealh epenen an P Ž. an P Ž. are ifferen if g an g are so. hus, unlike he classical Ž pure backwar. case, no simple comparison can be mae for he processes X an X, excep for 0 Ž see heorem 4.4 an he counerexample in he Appenix.. We noneheless have he following resuls, which will be enough for our purpose. Noe ha in wha follows when we say coniion Ž A4. hols, we mean eiher Ž A4.Ž a. or Ž A4.Ž b. hols. EOREM 4. Ž Comparison heorem.. Suppose ha Ž A. Ž A4.Ž a. or Ž b. hol. Le iniial prices p be given an le Ž, C. be any amissible pair such ha he corresponing pricewealh process Ž P, Y. saisfies YŽ. gž PŽ.., a.s. hen Y Ž, PŽ.., where is he soluion o Ž In paricular, YŽ 0. Ž 0, p. XŽ 0., where X is he soluion o he FBSDE Ž 3.3. saring from p consruce by he four sep scheme. PROOF. We only consier he case when coniion Ž A4.Ž b. hols, since he oher case, when Ž A4.Ž a. hols an g is boune, is much easier an can be prove in a similar way. he meho we use is similar in spiri o he meho of linearizing he backwar equaion use in 8, excep we have o be more careful in orer o fin generaors which will be Lipschiz. Le Ž P, Y,, C. be given such ha Ž, C. AŽYŽ 0.. an YŽ. gž PŽ.., a.s. We firs efine

19 388 J. CVIANIC AND J. MA a change of probabiliy measure as follows: le Ž Ž.. 0, P, Y, b Ž, P Ž., Y Ž.,. r Ž, X Ž.,., ½ 5 Ž 4.. Z0 exp 0Ž s. W Ž s. 0Ž s. s, 0 0 P 0 Z 0Ž., P so ha he process W W Ž s s is a Brownian moion on he new probabiliy space Ž, F, P. 0. Furhermore, he pricewealh FBSDE becomes noe he efiniions of ˆb, an in Ž.7. an Ž.8. ˆ P p P Ž s. r s, Y Ž s., Ž s. s s, P Ž s., Y Ž s., Ž s. W Ž s., Ž 4.. Y g P Ž. r s, Y Ž., Ž s. Y Ž s. s 0 ² : s, s, P s, Y s, s W s CŽ. CŽ.. Noe ha in he presen case he PDE Ž 3.8. is egenerae, an he funcion g is no boune, so he soluion o Ž 3.8. an is parial erivaives will blow up as p approaches an infiniy. herefore he usual esimaes such as hose in 8 will no work, an some more careful consieraion will be neee. o overcome his ifficuly, we procee as follows. Firs we apply Io s ˆ formula o he process gž P. from o o ge ½ ² : g P g P Ž. g P Ž s., r s, Y Ž s., Ž s. P Ž s. Ž 4.3. p Ž. Ž. 45 r s, P s, Y s, s g p p P s s p 0 ² : g P Ž s., s, P Ž s., Y Ž s., Ž s. W Ž s.. hen we efine a process Yˆ Y gž PŽ.., which obviously saisfies he Ž backwar. SDE ½ ² : Yˆ Yˆ r Ž s, Y Ž s., Ž s.. Y Ž s. g Ž P Ž s.., P Ž s. Ž 4.4. Ž. Ž. 45 r s, P s, Y s, s g p p P s s p 0 ² : Ž s. P Ž s. g P Ž s., s, P Ž s., Y Ž s., Ž s. W Ž s. CŽ. CŽ.. p

20 EDGING OPIONS FOR LARGE INVESORS 389 We now use he noaion ˆ g as ha in he proof of heorem 3.4. hen i suffices o show ha Yˆ ˆ Ž, P. for all 0,, a.s. P 0. o his en, le us enoe Y ˆ Ž, PŽ.., Ž P. ˆ Ž, P. g Ž P. p p an Yˆ Y Ž., Ž. Y. Applying Io s ˆ formula o he pro- cess Ž., we obain Y ½ Yˆ Ž. r Ž s, Y Ž s., Ž s.. Y Ž 4.5. p p ; Y Ž s. g Ž P Ž s.. ˆ Ž s, P Ž s.., P Ž s. ½ ˆ Ž. Ž. ˆ p pž s, P Ž s.. g p pž P Ž s.. 55 s s s, P s r s, P s, Y s, s ˆ p p Ž s. Ž P Ž s.. g Ž P Ž s.. Ž s, P Ž s.., ; s, P Ž s., Y Ž s., Ž s. W Ž s. 0 0 ˆ ² Ž. : Y A s s s, s, P s, Y s, s W s CŽ. CŽ., where he process A in he las erm above is efine in he obvious way. We now recall ha he funcion ˆ saisfies PDE Ž 3.6., recall he efiniion of an also noe ha ² p, q: ² Ž p. q, : an ha ˆ Ž, P. gž P. Y Ž.. We can easily rewrie A as Y AŽ s. r s, Y Ž s., Ž s. Y Ž s. r s, Y Ž s. Ž s., Ž s. Y Ž s. Ž s. Ž 4.6. where ; ˆ ½ Ž Ž.. Ž. 5 r Ž s, Y Ž s., Ž s.. Ž s. r s, ˆ Ž s, P Ž s.., Ž s. Ž s., r s, P s, Y s, s, s, s, P s s, P s IŽ s. IŽ s. I3Ž s., ž p p / Ž, p. Ž p. Ž, p. g Ž p., ˆp p Ž, p, x,,, q. Ž, p, q g Ž p.,. Ž, p, x,., an I s are efine in he obvious way. Now noicing ha i I Ž s. r s, Y Ž s., Ž s. Y Ž s. r s, Y Ž s. Ž s., Ž s. Y Ž s. Ž s. Y Y r s, Y Ž s. Ž s., Ž s. r s, Y Ž s. Ž s., Ž s. Y Y Y Y

21 390 J. CVIANIC AND J. MA Y Ž s. Ž s. Y ½ 4 x 5 0 xžy Ž s. Y Ž s.. r s, x, Ž s. x Y Ž s. r Ž s, Y Ž s. Y Ž s., Ž s. Ž s.. 0 ² : s s s, s, Y Y Ž s. Ž s., Ž s. Y ; we have from coniion Ž A3. ha boh an are aape processes an are uniformly boune in Ž,.. Similarly, by coniions Ž A., Ž A3. an Ž A4.Ž b., we see ha he process Ž, P. is uniformly boune an ha here exis uniformly boune, aape processes, 3 an, 3 such ha ; r Ž s, ˆ Ž s, P Ž s.., Ž s.. Ž s. r Ž s, ˆ Ž s, P Ž s.., Ž s.. Ž s., ; I Ž s. r Ž s, Y Ž s., Ž s.. Ž s. r s, ˆ Ž s, P Ž s.., Ž s. Ž s., Y ² : ² : s s s, s, I s s s s, s. 3 3 Y 3 herefore, leing Ý 3, Ý 3, we obain ha i i i i ² : A,, Y where an are boh aape, uniformly boune processes. In oher wors, we have Ž 4.7. ˆ ² : 4 Y s s s, s s Y Y 0 ² : s, s, P s, Y s, s W s CŽ. CŽ.. he remainer of he proof is similar o ha in he comparison heorem for backwar SDE s Žsee 8.. We inclue i for he sake of compleeness. Define anoher change of probabiliy measure similar o Ž 4.. by seing, P Ž., Y Ž., Ž., ½ Z exp s W s s s, P Z. P 0

22 EDGING OPIONS FOR LARGE INVESORS 39 hen i is clear ha Z is a P0-maringale by virue of he bouneness of. ence W W Ž s. 0 0 s is a P-Brownian moion. Moreover, we have ž / Y ž / 0 0 exp Ž s. s Ž. exp Ž s. s Ž 4.8. ž / ž s / s 0 ² : exp u u s, s, P s, Y s, s W s exp Ž u. u CŽ s.. 0 By efiniion of an he esimae Ž 3.5., we see ha is obviously square inegrable Ž noe also ha is square inegrable by is amissibiliy.. Recalling ha is boune, we see ha he firs erm in he righ-han sie of Ž 4.8. is a maringale. aking coniional expecaions on boh sies above, we obain ha Ž 4.9. exp Ž s. s E exp Ž s. s Ž. ½ ž / ž / 0 0 s exp Ž u. u CŽ s. F5 0 Y ž / for all 0,, a.s. P. hus Ž. YŽ. gž PŽ.. Y 0 implies ha 0, 0, Y, a.s. P, whence a.s. P0 an finally a.s. P. he proof is complee. he main resuls of his paper are hen conaine in he following corollaries. COROLLARY 4. Ž Uniqueness of FBSDE.. Suppose ha coniions Ž A. Ž A4.Ž a. or Ž b. hol. Le Ž P, Y,. be an aape soluion o he FBSDE Ž hen i mus be he same as he one consruce from he four sep scheme. In oher wors, he FBSDE Ž 3.3. has a unique aape soluion an i can be consruce via Ž 3.0. an Ž 3... PROOF. Noe ha in his case C 0 an Ž. YŽ. gž PŽ.. 0. he asserion follows from Ž COROLLARY 4.3 Ž he opimal sraegy.. Uner Assumpions Ž A. Ž A4.Ž a. or Ž b., we have hž gž PŽ... XŽ 0., where P, X are he firs wo componens of he aape soluion o he FBSDE Ž Furhermore, he opimal heging sraegy is given by Ž, 0., where is he hir componen of he aape soluion of FBSDE Ž PROOF. his is a irec consequence of heorem 4. an Corollary 4..

23 39 J. CVIANIC AND J. MA We noe ha he uniqueness oes no hol if we allow nonzero consumpion in FBSDE Ž An inuiively obvious example is ha one can firs fin a porfolio sraegy o hege gžpž.. by solving Ž 3.3. an hen fin anoher porfolio sraegy o hege gž PŽ.. k by solving Ž 3.3. wih gž PŽ.. being replace by gž PŽ.. k, k 0 an consume k ollars a ime. As menione before, we canno always use he mehos for comparison of backwar SDE s in our case. In paricular, he following comparison heorem is prove using comparison resuls for eerminisic PDE s. EOREM 4.4 Ž Monooniciy in erminal coniion.. Suppose ha Ž A. Ž i i A4 a or b hol, an le P, X, i., i,, be he unique aape soluions o 3.3, wih he same iniial prices p bu ifferen erminal i i Ž i. coniions X g P, i,, respecively. If g, g all saisfy he coniion A4 an g p g p for all p, hen i hols ha X Ž 0. X Ž 0.. PROOF. By he exisence an uniqueness of he aape soluion o he FBSDE 3.3, we know ha X an X mus have he form X Ž, P Ž.., X Ž, P Ž.., where an are he classical soluions o he PDE Ž 3.8. wih erminal coniions g an g, respecively. We claim ha he inequaliy Ž, p., p mus hol for all, p 0,. o see his, le us make he change of variable p L again, where he mapping L is efine in he proof of heorem 3.4, an efine u i Ž,. i, L. I follows from he proof of heorem 3.4 ha u an u saisfy he PDE Ž 4.0. respecively, where 4 ² : 0 u r,, u, u u b,, u, u, u ur, u, u, už 0,. g i Ž e.,, 0 Ž,, x,. Ž L. Ž, L, x,., b Ž,, x,. r Ž, x,. iag Ž,, x,., 0 r Ž, x,. r Ž, x,.. Noe ha by a sanar argumen Žcf. 7. one can show ha if we enoe u i Ž,. R o be he soluion o he iniial bounary value problem 0 u r Ž,, u, u. u Ž ² : b,, u, u, u ur, u, u, i u B R Ž,. g Ž e., R, už 0,. g i Ž e., B R, 0

24 EDGING OPIONS FOR LARGE INVESORS i i i,, respecively, where BR ; R, hen ur u uniformly on compacs, i,, as R. herefore, we nee only show ha u Ž,. R u Ž,. for all Ž,. 0, R BR an R 0. For any 0, consier he PDE, Ž ² : u r,, u, u u b,, u, u, u ur, u, u, 0 u B R Ž,. g Ž e., R, už 0,. g Ž e., B R, an enoe is soluion by u R,. I is no har o check, using a sanar Ž. echnique cf., e.g., 9, ha ur, converges o u R, uniformly in 0,. Nex, we efine a funcion Ž ˆ. ˆ ² Ž ˆ. ˆ: F,, x, q, q r,, x, q q b,, q, q, q xr, x, q. 0 Clearly F is coninuously iffereniable in all variables, an u an u R, R saisfy u R, F Ž,, u R,, Ž u R,., Ž u R,.., u R RŽ,, u R, Ž u R., Ž u R.., u Ž,. u Ž,., Ž,. 0, B 04 B. R, R R herefore by heorem II.6 of 9, we have ur, ur in B R. By leing 0 an hen R, we obain ha u, u, for all, 0,, whence Ž,. Ž,.. In paricular, we have X Ž 0. Ž 0, p. Ž0, p. X Ž 0., proving he heorem. REMARK 4.5. From, p, p we canno conclue ha X X for all, since in general here is no comparison beween Ž, P. an Ž, P. Ž see he example in he Appenix.. 5. Examples. In his secion we provie some examples which moivae our moel. We noe ha in some of he examples he saning assumpions of his paper are no acually saisfie, bu because of he special feaures of hese moels, we can verify, by using some exising resuls, ha our meho will also erive he righ answer o hese problems. herefore i woul be ineresing o evelop our mehoology furher, o a wier class of problems wih more general coefficiens, alhough, echnically, i will be much more challenging. EXAMPLE 5. Differen ineres raes for borrowing an lening. Suppose we wan o moel a marke in which here are ifferen ineres raes for

25 394 J. CVIANIC AND J. MA borrowing R an lening r, 0 r R. In., we le bž, P Ž., X Ž.,. bp Ž., Ž, P Ž., X. P for some real b an 0. Recall ha is he amoun ha he invesor pus in sock. In., le Ž 5.. r Ž, X Ž.,. r Ž. X Ž.4 R Ž. X Ž.4. Also le gž p. Ž p q. ; ha is, we wan o hege a European call opion wih exercise price q 0. We guess now Ž an verify laer. ha he heging porfolio will always borrow Žas is well known for a European call in he sanar moel. an, herefore, we shall work wih rž, XŽ.,. R. hen Ž 3.8. becomes 0 p p p R pp, wih Ž, p. Ž p q.. his is nohing else bu a BlackScholes PDE for he price of a European opion on a sock wih volailiy an wih he marke s riskless ineres rae equal o R. I has an explici soluion Žsee, e.g., 4., which is easily seen o saisfy pp an, herefore, also saisfies Ž 3.8. wih he funcion r given by Ž 5... Now, by Ž 3.., we have X Ž, P. an P Ž, PŽ.., C p 0. We recover herefore he re- suls of Example 9.5 of 4. REMARK 5.. In he previous example he smoohness assumpions in Ž A3. an Ž A4. on r an g are, in fac, no saisfie. Neverheless, he resuls hol again, see 4 for rigorous proofs which o no require assumpions like Ž A3., Ž A4., bu require some concaviy assumpions which are saisfie in he example. Of course, regaring assumpion Ž A3., we were lucky, since he isconinuiy in he funcion r isappears for he soluion of Ž his also makes he PDE linear an allows weakening of he assumpions. he message is ha one can ry o consruc a heging sraegy using he proceure of his paper, even if no all of he assumpions are saisfie, an see wha happens. REMARK 5.3 Ž Inepenence of rif.. he PDE Ž 3.8. oes no epen on he rif funcion b, an neiher oes he price XŽ 0.. his is a familiar fac from he sanar BlackScholes worl, vali even in his general moel, where he rif can be influence by he porfolio sraegy. herefore, i is of no ineres o look a examples in which b akes ifferen forms. EXAMPLE 5.4 Ž Large invesor.. We inicae here one of he possible moels in our framework, no inclue in he sanar heory. Suppose ha our invesor is really an imporan one, so ha, if heshe invess oo much in bons, he governmen Ž or he marke. ecies o ecrease he bon ineres rae. For example, we can assume ha rž, x,. is a ecreasing funcion of x, for x large.

26 EDGING OPIONS FOR LARGE INVESORS 395 EXAMPLE 5.5 Ž Several agensequilibrium moel.. In Plaen an Schweizer 3 an SDE for he sock price is obaine from equilibrium consieraions; boh is rif an volailiy coefficiens epen on he heging sraegy of he agens in he marke in a raher complex fashion. As he auhors menion, I is no clear a all how one shoul compue opion prices in an economy where agens sraegies affec he unerlying sock price process. Our resuls provie he price ha woul enable he seller o hege agains all he risk, ha is, he upper boun for he price. EXAMPLE 5.6 Ž Cheaing oes no always pay for he large invesor.. Suppose ha, up o ime 0, we have he sanar BlackScholes moel; ha is, he ineres rae r is consan an he volailiy funcion is given by Ž, p, x. p, 0 Žhe rif funcion oes no maer, as noe in Remark hen, a ime 0, he large invesor sells he opion worh gž PŽ.. a ime, for he price of Ž0, PŽ 0.., where is a pricing funcion. ypically, Ž, P. Ž, PŽ.., where is he BlackScholes pricing funcion, given by he soluion o Ž 3.8., wih r an as above. aving a sric inequaliy means ha he large invesor is rying o chea, ha is, o sell he opion for more han is worh, he BlackScholes price. Suppose ha he invesor, le us call himher he seller, fins buyers for he opion a his price. his woul creae insabiliies in he marke an arbirage opporuniies if he volailiy of he sock were o remain he same. Le us assume ha he effec is fel as a change in he volailiy, so ha he funcion Ž, p, x. is no longer equal o p. A naural example woul be Ž, p, x. p fž Ž, p. Ž, p.., wih f increasing. Also assume ha Ž, p, x. remains equal o p if Ž,. Ž,.; ha is, if here is no cheaing aempe a any ime, we remain in he BlackScholes worl an herefore heging is possible if one sars wih iniial capial equal o he BlackScholes price. If, on he conrary, he opion sells for more, hen he volailiy increases an he minimal heging price changes. For example, for a European call opion he iniial value of he soluion o he PDE Ž 3.8. increases wih Ž a leas for consan.. herefore, he cheaing price Ži.e., some price greaer han he BlackScholes price. migh be smaller han he minimal heging price in he marke wih he increase sock volailiy, an heging migh no be possible, in which case cheaing oes no pay. On he oher han, here are cases for which heging is possible an cheaing woul pay. From he poin of view of he marke, his moel can inicae how he volailiy has o change if he opion is overprice, in orer for he seller no o make an arbirage profi. For example, in a simple moel in which Ž,. Ž,. an Ž, p, x. pž., i is easy o calculae Ž for sanar European call opions. wha woul have o be in orer ha here be no arbirage profi or, equivalenly, in orer o have Ž0, PŽ 0.. equal o he BlackScholes price of a sock wih volailiy. If is less han he criical value, cheaing pays, an if i is larger, cheaing oes no pay. ere we have a phenomenon unknown in he classical moels: selling he opion for is fair, BlackScholes price ensures ha heging is possible, bu selling for more han ha price oes no guaranee heging.

27 396 J. CVIANIC AND J. MA In general, if he seller of he opion has an iea of how much cheaing will affec he volailiy of he sock hen heshe shoul also have an iea of how much heshe can safely chea, by solving he PDE Ž 3.8. for ifferen s Ž assuming, of course, ha here are always buyers willing o buy he opion.. Anoher example woul be he case wih Ž, p, x. p fž x Ž, p.., where, again, f is increasing an fž z. p for z 0. Assuming ha he seller will always reinves he profis raher han consume, X Ž, P. coul be hough of as a measure of arbirage profi a ime. I is clear ha he BlackScholes pricing funcion is a soluion o Ž 3.8., even wih his moifie volailiy funcion Ž, p, x.. herefore, if Ž, p, x. is such ha assumpions Ž A. Ž A4. are saisfie, hen he BlackScholes price Ž 0. is he smalles price ha sill guaranees successful heging Ž by Corollary owever, i is no clear ha heging is guaranee if he invesor sells he opion for more han he minimal heging price Ž 0.; since here is no consumpion, we will have X Ž, P. for small, so ha he volailiy possibly increases. If he increase is significan compare o he price of he opion, here migh be no heging porfolio. For example, one coul have Ž, p, x. p arcanž x Ž, p.. 4. Wih gž p. p we have Ž, p. p. Suppose, for example, ha he seller sells for he price of p an invess his amoun in he marke, buying a leas one whole share of he sock. If heshe oes no consume, hen he volailiy will always be greaer han. APPENDIX We now give an example showing ha in he forwarbackwar case g g oes no necessarily imply ha X X Ž., where X an X are he corresponing Ž backwar componens. of he FBSDE Ž 3.3. wih erminal coniions g an g, respecively. Inee, he reverse can happen wih posiive probabiliy. Le us assume ha an r 0. Le he price equaion be P Ž A.. P P W Ž., P Ž 0. p; X hence he wealh equaion is see.4 X W Ž., X Ž. g Ž P Ž... Ž X. We firs assume ha gž p. g Ž p. p Ž i.e., one wans o hege he price iself.. he corresponing PDE Ž 3.8. is now of he form A. 0 p p p,, p g p p. Le us enoe he soluion of Ž A.. by. Since g Ž p. p saisfies Ž A4., we easily see ha Ž, p. p is he unique classical soluion o Ž A... hus, by Ž 3.., X P Ž.. Moreover, from Ž A.., X p expw 4.

28 EDGING OPIONS FOR LARGE INVESORS 397 We now le g Ž p. p an enoe by he classical soluion o Ž A.., wih he erminal coniion being replace by g Ž p.. Clearly, Ž, p. p. herefore, X P. Moreover, from Ž A.., X 4 W Ž. p exp W. herefore, X X pe e, which can be boh posiive or negaive wih posiive probabiliy for any 0. Acknowlegmens. he auhors woul like o hank Professors Jiongmin Yong of Fuan Universiy, China, an Marin Schweizer of echnische Universia Berlin, Germany, for helpful iscussions. In paricular, Marin Schweizer an he wo anonymous referees poine ou ha our original, longer proof of Lemma.3 is no neee. hanks are also ue one of he referees for many useful suggesions, in paricular, for simplifying some of he proofs by using resuls from 8. REFERENCES ANSEL, J. P. an SRICKER, C. Ž Lois e maringale, ensies e ecomposiion e Follmer-Schweizer. Ann. Insi.. Poincare BLACK, F. an SCOLES, M. Ž he pricing of opions an corporae liabiliies. J. Poli. Economy CVIANIC, J. Ž Nonlinear financial markes: heging an porfolio opimizaion. Proceeings of he Isaac Newon Insiue for Mahemaical Sciences. o appear. 4 CVIANIC, J. an KARAZAS, I. Ž eging coningen claims wih consraine porfolios. Ann. Appl. Probab DELBAEN, F. an SCACERMAYER, W. Ž A general version of he funamenal heorem of asse pricing. Mah. Ann DUFFIE, D. an EPSEIN, L. G. Ž 99.. Sochasic ifferenial uiliy. Economerica Ž Appenix wih Skiaas, C.. 7 DUFFIE, D., MA, J. an YONG, J. Ž Black s console rae conjecure. Ann. Appl. Probab EL KAROUI, N., PENG, S. an QUENEZ, M. C. Ž Backwar sochasic ifferenial equaions in finance an opimizaion. Preprin. 9 FRIEDMAN, A. Ž Parial Differenial Equaions of Parabolic ype. Prenice-all, Englewoo Cliffs, NJ. 0 GROSSMAN, S. J. Ž An analysis of he implicaions for sock an fuures price volailiy of program raing an ynamic heging sraegies. Journal of Business ARRISON, J. M. an KREPS, D. M. Ž Maringales an arbirage in muliperio securiy markes. J. Econom. heory ARRISON, J. M. an PLISKA, S. R. Ž 98.. Maringales an sochasic inegrals in he heory of coninuous raing. Sochasic Process. Appl ARRISON, J. M. an PLISKA, S. R. Ž A sochasic calculus moel of coninuous ime raing: complee markes. Sochasic Process. Appl KARAZAS, I. Ž Opimizaion problems in he heory of coninuous raing. SIAM J. Conrol Opim KARAZAS, I., LEOCZKY, J. P. an SREVE, S. E. Ž 99.. Equilibrium moels wih singular asse prices. Mah. Finance 9. 6 KARAZAS, I. an SREVE, S. E. Ž 99.. Brownian Moion an Sochasic Calculus, n e. Springer, New York. 7 LADYZENSKAJA, O. A., SOLONNIKOV, V. A. an URAL CEVA, N. N. Ž Linear an Quasilinear Equaions of Parabolic ype. Amer. Mah. Soc., Provience, RI. 8 MA, J., PROER, P. an YONG, J. Ž Solving forwarbackwar sochasic ifferenial equaions explicilya four sep scheme. Probab. heory Relae Fiels