BS3-9 MAFI.cls Ocober 4, 22 5:28 Mahemaical Finance, Vol. 3, No. January 23), 35 5 EFFICIEN COMPUAION OF HEDGING PORFOLIOS FOR OPIONS WIH DISCONINUOUS PAYOFFS JAK SA CVIANIĆ Deparmen of Mahemaics, Universiy of Souhern California JIN MA Deparmen of Mahemaics, Purdue Universiy JIANFENG ZHANG School of Mahemaics, Universiy of Minnesoa We consider he problem of compuing hedging porfolios for opions ha may have disconinuous payoffs, in he framework of diffusion models in which he number of facors may be larger han he number of Brownian moions driving he model. Exending he work of Fournié e al. 999), as well as Ma and Zhang 2), using inegraion by pars of Malliavin calculus, we find wo represenaions of he hedging porfolio in erms of expeced values of random variables ha do no involve differeniaing he payoff funcion. Once his has been accomplished, he hedging porfolio can be compued by simple Mone Carlo. We find he heoreical bound for he error of he wo mehods. We also perform numerical experimens in order o compare hese mehods o wo exising mehods, and find ha no mehod is clearly superior o ohers. KEY WORDS: hedging opions, Malliavin calculus, Mone Carlo mehods. INRODUCION Unil quie recenly he mehod mos widely used in pracice for evaluaing hedging porfolios of opions in sandard diffusion models was based on he fac ha he opimal number of shares o be held is ypically obained by differeniaing he opion price wih respec o he underlying facors: namely, one would compue he price for some iniial value of a facor X, increase i by a small amoun x, find he price for he perurbed facor, compue he difference, and divide by x. his division usually makes he mehod much more compuaionally expensive han finding he opion price. ypically, if one uses Mone Carlo wih n seps, he error goes from he order of n /2 o he order of n /4 or n /3 see Boyle, Broadie, and Glasserman 997). On he oher hand, in special cases one can use so-called direc esimaes ha avoid resimulaion for he perurbed he auhors hank P. Glasserman for useful commens. he firs auhor s research was suppored in par by NSF gran DMS--99549. Work by he second and hird auhors was suppored in par by NSF gran 99772. Address correspondence o Jak sa Cvianić, Deparmen of Mahemaics, USC, 42 W Downey Way, MC 3, Los Angeles 989-3; e-mail: cvianic@mah.usc.edu. C 23 Blackwell Publishing Inc., 35 Main S., Malden, MA 248, USA, and 8 Cowley Road, Oxford, OX4 JF, UK. 35
BS3-9 MAFI.cls Ocober 4, 22 5:28 36 J. CVIANIĆ, J. MA, AND J. ZHANG iniial condiions see Boyle e al. for references). In recen papers by Fournié e al. 999, 2), his was generalized o finie-dimensional Markovian models, and i was shown ha he hedging porfolio can be calculaed as an expeced value of a funcional involving he gradien of he opion payoff. his brings down he order of he error o n /2 if Mone Carlo is used. Moreover, using Malliavin calculus, Fournié e al. showed ha his expression can be ransformed o avoid he need for compuing he gradien of he payoff, which is quie useful because ypical opion payoffs are no everywhere differeniable. Ma and Zhang 2) exended hese represenaions o models in which he porfolio process migh ener he drif of he wealh process in a nonlinear fashion. he models considered in his paper have he following wo main feaures: i) he number of facors may be larger han he number of Brownian moions, and ii) he payoff is disconinuous. A ypical model wih feaure i) is one in which he underlying sock is driven by one Brownian moion, bu he ineres rae and volailiy are also diffusion processes driven by he same Brownian moion. he prooypical example of feaure ii) is he digial opion, which will be he focus of our resuls. We shall consider wo numerical mehods for compuing he hedging porfolios. he firs mehod is based on inegraion by pars in Malliavin calculus, a echnique used in he aforemenioned papers. his leads o a new represenaion of he hedging porfolio ha is no covered by any exising resul. Since such a represenaion by naure does no involve differeniaing he payoff funcion, i can hen be compued by direc Mone Carlo. he second mehod is as follows: We firs arificially increase he number of Brownian moions o mach he number of facors by perurbing he volailiy marix o a nonsingular square) one wih an appropriae number of addiional columns, indexed by. We hen use he resuls from he aforemenioned papers o obain a represenaion of he hedging porfolios in he arificial markes. Finally, we show ha as hese porfolios converge o he hedging porfolio in he original marke model. We hen apply hese mehods o opions wih disconinuous payoffs, and confirm by numerical experimens ha hese procedures provide feasible algorihms for compuing hedging porfolios. We compare our wo mehods, called MandM mehods, respecively, o oher mehods ha are applicable o models wih disconinuous payoffs and wih he number of Brownian moions being smaller han he number of facors. hese include he sandard finie difference dela mehod, or mehod, and he rerieval of volailiy mehod of Cvianic, Goukasian, and Zapaero 2), which we call he RVM mehod. We show ha, wih he appropriae choice of numerical parameers, he M mehod has he smalles sandard error, bu no much smaller han he mehod and he RVM mehod. However, i seems ha he M mehod is no as sensiive wih respec o he choice of as he mehod is wih respec o he size of he perurbaion, or as much as he RVM mehod is wih respec o is free parameers. Moreover, in he example we have only a wo-dimensional facor process. he compuaional ime of he mehod increases linearly wih he number of facors, since i has o compue approximae derivaives of he opion price wih respec o all he facors. On he oher hand, a disadvanage of he M, M, and RVM mehods is ha hey do no provide he sensiiviies of he opion price o individual parameers hey only provide he value of he hedging porfolio. Somewha surprisingly, he M mehod has he larges error, for he same amoun of processing ime. In fac, he M mehod requires he smalles number of ime seps and simulaion pahs, bu i requires a lo of ime for compuing he Skorohod inegral. However, we do no have o worry abou he choice of any small parameers for he M mehod, oher han he ime sep size.
BS3-9 MAFI.cls Ocober 4, 22 5:28 EFFICIEN COMPUAION OF HEDGING PORFOLIOS 37 In he special framework of models for LIBOR raes, Glasserman and Zhao 999) addressed similar issues, bu hey focused on he compuaion of Greeks, and no on he hedging porfolio, using mehods differen from ours. he conclusions hey derived from heir numerical experimens seem o be consisen wih ours. he res of he paper is organized as follows. Secion 2 describes he model and he problem. Secion 3 gives he new represenaion formulas for hedging porfolios. Secion 4 compues he Skorohod inegral involved in a represenaion. Secion 5 presens he second, approximaion-based mehod, and Secion 6 repors resuls of numerical experimens. 2. PROBLEM FORMULAION hroughou his paper we assume ha,f, P) is a complee probabiliy space on which is defined a d-dimensional Brownian moion W = W ). Le F =F denoe he naural filraion generaed by W, augmened by he P-null ses of F; and le F = F. Furhermore, we use he noaions =, x = x,..., x n ), and 2 = xx = x 2 i x j ) i, n j=,for, x) [, ] IR n. Noe ha if ψ = ψ,...,ψ n ) :IR n IR n hen x ψ = xi ψ j ) i, n j= is a marix. he meaning of xy, yy, ec., should be clear from he conex. Consider he following marke model: here are d risky asses and riskless asse, whose prices a ime are denoed by S = S,...,Sd ) and S, respecively. We assume ha he prices follow he SDE 2.) ds = S r d; ds i = S i [ r d + d j= γ ij ] dw j, i =,...,d. We suppose ha he volailiy marix γ = [γ ij ] is inverible and he discouned sock prices are maringales. Le us recall he sandard opion pricing framework. Suppose ha he seller hereafer called he invesor) of he opion is rying o replicae he opion payoff by invesing in he marke. We denoe by π i he amoun of money he invesor holds in sock i a ime, and we denoe π = π,...,π d ). ypically, he payoff of he opion is given by gs ) for some funcion g, and, by definiion, he opion) price process is equal o he wealh process, which replicaes he opion a he mauriy ime. he discouned price process saisfies Y = Y + R sπ s γ s dw s, where R = exp r s ds. Since γ and R are boh inverible, we can simply se Z = R π γ so ha he discouned wealh process Y is now described by a simple form: 2.2) Y = Y + Z s dw s. In his paper we shall assume ha r and γ are componens of a finiely dimensional diffusion process. o be more precise, we shall consider a sae process X of he form X = S, R, X d+2,...,x n ) and assume X is an n-dimensional diffusion ha saisfies he following SDE 2.3) dx i = b i, X ) d + d j= σ ij, X ) dw j, i =,...,n, where X,...,X d ) = S and X d+ = R. We can hen se he discouned payoff funcion o be gx ) = R gs), and hus he discouned price process Y of he opion a each ime
BS3-9 MAFI.cls Ocober 4, 22 5:28 38 J. CVIANIĆ, J. MA, AND J. ZHANG is given by 2.4) Y = EgX ) F =Y + Z s dw s = gx ) Z s dw s. Noe ha he riple X, Y, Z) isnowanf -adaped soluion o he forward-backward SDE 2.3) and 2.4). We refer he readers o El Karoui, Peng, and Quenez 997) and Ma and Yong 999) for a complee accoun regarding he heory of backward/forwardbackward SDEs and heir applicaions in finance. In order o perform hedging in our model, we have o find an efficien numerical mehod for compuing he porfolio π, or, equivalenly, o compue he process Z ha makes Y = R gs ) = gx ). We are paricularly ineresed in he case where g is a disconinuous funcion. A ypical example is he so-called digial opion or binary opion) ha is, gs) = s K, for some K >. he main difficuly in he numerical compuaion is ha he disconinuiy of g will cause many echnical problems using sandard argumens via he PDE heory. Secondly, since he dimension of X is n, his increases he dimension of he corresponding PDE o n sae variables plus one ime variable), which makes he PDE mehods very slow for n > 2. A mehod ha can circumven hese difficulies has been developed by Fournié e al. 999); however, in ha paper i is assumed ha he number of facors n is equal o he number of Brownian moions and he number of socks) d. his is no he case in many models used in pracice, such as he case of one Brownian moion, bu wih r or γ being random. o conclude his secion we give he sanding assumpions ha we will use hroughou his paper. ASSUMPION 2.. n d. he funcions b C, b [, ] IR n ;IR n d ), σ C, b [, ] IR n ;IR n ), and all he parial derivaives of b and σ wih respec o x) are uniformly bounded by a common consan K >. Furhermore, we assume ha sup b, ) + σ, ) K. ASSUMPION 2.2. he funcion g :IR n IR is a measurable funcion; and here exiss a consan K > such ha gx) K + x ). We also inroduce he following noaion: We represen he n d) marix σ as [ ] σ, x) 2.5) σ, x) =, σ 2, x) where σ is a d d marix. 3. REPRESENAIONS OF HEDGING PORFOLIOS Recall from he previous secion ha we are considering he following sysem of sochasic differenial equaions: X = x + bs, X s) ds + σ s, X s) dw s, 3.) Y = gx ). Z dw. Le he dimension of X be n, and ha of W be d. We use Assumpions 2. and 2.2 and ha he dimension of Y is. Our mehod of compuing hedging porfolios is based heavily
BS3-9 MAFI.cls Ocober 4, 22 5:28 EFFICIEN COMPUAION OF HEDGING PORFOLIOS 39 on he Feynman-Kac ype represenaion of he process Z of a backward SDE, as in Ma and Zhang 2). However, our payoff funcion may no even be coninuous, much less uniformly Lipschiz, as was assumed in Ma and Zhang. For he sake of compleeness, we begin by he following modified represenaion heorem for he process Z, which can be regarded as a special case of he Clark-Ocone formula. Noe ha we do no require ha he marix σ be a square marix. HEOREM 3.. Use Assumpion 2. and furher assume ha he funcion g is coninuous. Denoe A =x IR n : x gx) does no exis. Assume furher ha PX A = and ha x g is uniformly bounded ouside A. hen, we have 3.2) Z = E [ x gx ) X / A][ ] σ, X ). Here, is he soluion o he variaional equaion 3.3) = I n n + x bs, X s ) s ds + x σ s, X s ) s dw s, where I n n denoes he n n ideniy marix, and E is he expecaion condiional on F. Proof. Le g be a sequence of modifiers of g. ha is, g s are smooh funcions such ha x g are uniformly bounded, g g uniformly, and x g x) x gx) for all x / A, as. Since g X ) gx ), by sandard sabiliy resuls for backward SDEs cf., e.g., Ma and Yong 999) one has 3.4) E Z Z 2 d, as. Furhermore, since g is differeniable wih bounded derivaives, we can apply he represenaion heorem of Ma and Zhang 2) or he Clark-Ocone formula) o ge Z = E [ x g X ) ] [ ] σ, X ). On he oher hand, le us denoe he righ-hand side of 3.2) by Z. I is easy o see ha [ Z Z E x g X ) x gx ) X / A] [ ] σ, X ) [ +E x g X ) X A] [ ] σ, X ), where v = [ v,..., v n ] whenever v = [v,...,v n ]. Noing ha PX A =, and ha x g X ) x gx )forx / A, applying he dominaed convergence heorem we have E Z Z 2 d. his, ogeher wih 3.4), implies ha Z = Z, d dp-almos surely. Finally, noe ha being he produc of a maringale and a coninuous process Z has a càdlàg version. hus as a modificaion of Z, we conclude ha Z hasacàdlàg version as well, compleing he proof of he heorem.
BS3-9 MAFI.cls Ocober 4, 22 5:28 4 J. CVIANIĆ, J. MA, AND J. ZHANG We remark ha in heorem 3. he assumpion PX A = plays a crucial role. However, in pracice such an assumpion is no easy o verify, especially in he case when d < n. he following sufficien condiion is herefore useful for our fuure discussion. HEOREM 3.2. Use Assumpion 2. and ha g is uniformly Lipschiz in all variables, and differeniable wih respec o x d+,...,x n ). Assume furher ha deσ, X )). hen, PX A =. In paricular, 3.2) holds. Proof. Le Xˆ = X,...,X d ). We firs show ha he law of Xˆ is absoluely coninuous wih respec o he Lebesgue measure on IR d, denoed by d. o his end, le Â= Proj IR d A) be he projecion of se A on IR d, where A is he se defined in heorem 3.. ha is, ˆ A = ˆ X = x,...,x d ): x d+,...,x n ), such ha x = x,...,x n ) A. Since g is Lipschiz coninuous on x,...,x d ), and differeniable on x d+,...,x n ), we see ha A ˆ d =. Nex, noe ha by sandard argumens one shows ha X whence Xˆ ) is Malliavin differeniable; ha is, X ID,2 and 3.5) D X = [ ] σ, X ). In paricular, we have D X = σ, X ) and D ˆ X = σ, X ). Define γ = D ˆ X D Xˆ ) d. From 3.5) i is readily seen ha D X ) is coninuous in, and ha ded Xˆ ) = deσ, X )), a.s herefore, for x IR d such ha x, xd Xˆ D Xˆ ) x is nonnegaive for all [, ] and is posiive for close o. hus we have x D X )D X ) d x >, which implies ha he symmeric marix γ has posiive deerminan. Now we can apply heorem 2..2. of Nualar 995) o conclude ha he law of Xˆ is absoluely coninuous wih respec o d, which, combined wih he fac ha A ˆ d =, implies ha P Xˆ A) ˆ =. Since g is differeniable wih respec o x d+,...,x n, we see ha PX A) =, and he resul follows from heorem 3.. Moivaed by he digial opion, we now consider he case where g is allowed o be disconinuous. o he bes of our knowledge, he represenaion heorem in such a siuaion is new. We sae he heorem specifically for an opion wih one disconinuiy poin, bu he resul can easily be exended o an opion wih finiely many disconinuiies. he proof is moivaed by Proposiion 2.. in Nualar 995). HEOREM 3.3. Assume d =, σ, x) c, and σ, b C,2 wih bounded firs and second derivaives. Assume ha g is differeniable wih respec o x 2,...,x n, wih bounded derivaives; and ha g is uniformly Lipschiz coninuous wih respec o x, excep for he poin x = x, and boh gx +, x 2,...,x n ) and gx, x 2,...,x n ) exis and are differeniable. hen for [, ), we have Z = E x gx ) u X / A + X >x δ F u ),
BS3-9 MAFI.cls Ocober 4, 22 5:28 EFFICIEN COMPUAION OF HEDGING PORFOLIOS 4 where A is he same as in heorem 3.; Xˆ 2 = X 2,...,X n ) ; = o 3.3); δ ) is he indefinie Skorohod inegral over [, ]; ) ˆ is he soluion 2 3.6) g x F =, )[ ] ˆX 2 u, u = [ ] σ, X ), [, ]; DX 2 [, ] and DX 2 = [, ] Ds X 2 ds, g x, ˆ X 2) = g x +, ˆx 2) g x, ˆx 2). o prove he heorem, we need a echnical lemma, whose proof we omi. LEMMA 3.4. Assume ha all he assumpions of heorem 3.3 are in force. hen for any <, he process F u s s is Skorohod inegrable over [, ]. Proof of heorem 3.3. Firs we denoe x = x, x 2,...,x n ) = x, x 2 ) and recall ha A =x IR : g is no differeniable a x. Define g o be a modificaion of g as follows. gx, x 2 ), x x g x, x 2 ) = >; 3.7) x +) x gx 2, x 2) + x x ) gx 2 +, x 2), oherwise. hen, clearly, lim ɛ g x) gx) = for all x excep for x = x, x 2). Now by heorem 3.2 we have Z = E x g X ) X / A u, where A = A x x >) x +, x. Since Z is he maringale inegrand in he soluion of backward SDE 3.) wih g being replaced by g, he sabiliy resul of backward SDEs ells us ha lim E Z Z 2 3.8) d =. ɛ Now noe ha 3.9) x g X ) X / A = [ g X ) + x 2 g X ) ˆX 2 ] X / A = 2 [ g x +, ˆX 2 ) g x, ˆX 2 )] X x < + g X ) X x x >\A) + x 2 g X ) ˆX 2 X / A = 2 g x, ˆX 2 ) X x < + 2 X x < [ g x +, ˆX 2 ) g x +, ˆX 2 )) + g x, X 2 ) g x, ˆX 2 ))] + g X ) X x x >\A) + x 2 g X ) ˆX 2 X / A = I + I 2 + I 3 + I 4,
BS3-9 MAFI.cls Ocober 4, 22 5:28 42 J. CVIANIĆ, J. MA, AND J. ZHANG where I,...,I 4 are defined in he obvious way. Clearly, lim ɛ I 3 = gx ) X / A; lim ɛ I 4 = x 2 g X ) ˆX 2 X / A. Now similar o heorem 3.2 we can show ha he law of X So by 3.8) i suffices o show ha has a densiy, hus lim ɛ I 2 =. 3.) lim ɛ E I u =E X >x δ F u ). o do his, for a < b, wedefineψx) = [a,b] x) and ϕx) = x s, D s ϕ X ) = ψ X ) Ds X. ψy) dy. hen for Muliplying boh sides above by D s X and hen inegraing over [, ], we have D s ϕ X) Ds X ds = ψ X ) Ds X 2 ds. Since F u is Skorohod inegrable over [, ], hanks o Lemma 3.4, applying inegraion by pars formula we ge ) ) E ψ X g x, Xˆ 2 u = E D s ϕ X ) D s X g x, Xˆ ) 2 u DX 2 ds [, ] = E D s ϕ X) ) F u s ds = E ϕ X δ F u ). On he oher hand, by Fubini s heorem we have ) E ψ X g x, ˆX 2 ) b 3.) = E X >y δ F u ) dy. a Noe again ha he law of X has a densiy, hus he inegrand in he righ-hand side of 3.) is coninuous wih respec o y. herefore, leing [a, b] = [x, x + ], dividing boh sides of 3.) by, and hen sending we obain 3.), whence he heorem. If, in fac, he volailiy marix σ is squared, hen he following heorem gives a simpler represenaion resul. We omi he proof. he resul was given under sronger condiions in Fournié e al. 999) and exended in Ma and Zhang 2). HEOREM 3.5. Assume d = n, Assumpions 2. and 2.2, and ha he marix σ is nondegenerae. Assume furher ha A d =, where A IR d is he se of all disconinuiy poins of g. hen 3.2) Z = E gx )N σ, X ), where N [ = ] σ r, X r ) r ) [ dw r ]. In paricular, 3.3) Z = E gx )N σ, x).
BS3-9 MAFI.cls Ocober 4, 22 5:28 EFFICIEN COMPUAION OF HEDGING PORFOLIOS 43 4. COMPUAION OF HE SKOROHOD INEGRAL In he case of a digial opion, we see from heorem 3.3 ha he represenaion of he hedging porfolio involves a Skorohod inegral, which needs o be deal wih numerically so ha he represenaion is useful in pracice. In his secion we propose a scheme o compue his Skorohod inegral explicily. Due o he Markovian naure of our seing, we shall consider only = : 4.) Z = E x gx ) σ, x) X / A + X >x δfu ), where δ = δ, DX 2 H = DX 2 [, ], and F = F = g x, )[ ˆX 2 σ, x) ]. DX 2 H We remark here ha alhough he Skorohod inegral can be approximaed by Riemann sums see Nualar 995), in our case he Riemann summand will sill conain he Malliavin derivaive DX, which is quie undesirable in pracice. We now ry o derive a scheme ha involves only compuaions of Iô inegrals and Lebesgue inegrals, which can be simulaed simulaneously wih he underlying asses. o begin our analysis, le us firs use inegraion by pars formula for Skorohod inegrals and, noing ha gx, ˆX 2)[ σ, x)] is a scalar, we have 4.2) I = δfu ) = F u dw D F)u d. I is easy o see ha he only unusual par in F, which involves he Malliavin derivaive, is DX 2 H. However, i can be calculaed as follows: DX 2 H = [ ] σ, X ) 2 [ d = u u 4.3) d ]. I remains o calculae he inegral D F)u d. o his end, le us denoe 2 g = x2,..., xn ) g). A direc compuaion shows ha 4.4) D F)u = DX 2 H [ 2 g x, ˆX ) 2 ˆX 2 u ] [ σ, x)][ u ] + g x, ˆX )[[ 2 D ] ][ ] σ, x) u + g x, ˆX )[ 2 σ, x) ][ D u ] g x, ˆX )[ 2 σ, x) ][ u ] D DX 2 DX = I ) + I 2 ) + I 3 ) I 4 ), where I i s are defined in he obvious way. We now analyze I i) d separaely. Firs, I ) d = σ, x) 2 g x, ˆX 2 ) 2 [ u u 4.5) DX d ]. 2 H 4 H H
BS3-9 MAFI.cls Ocober 4, 22 5:28 44 J. CVIANIĆ, J. MA, AND J. ZHANG Nex, since is he soluion o he variaional SDE 3.3), we have 4.6) [ ] = Id D u = x σ, X ) + + u [ s ] x b x σ ) 2) s, X s ) ds u Now applying Iô s formula we ge 4.7) Ɣ u [ s ] x σ s, X s ) dw s ; [D x bs, X s ) s + x bs, X s )D s ] ds [D x σ s, X s ) s + x σ s, X s )D s ] dw s, u. = [ u ] D u ) = [ ] x σ, X ) + + u u [ s ] [D x bs, X s ) x σ s, X s )D x σ s, X s )] s ds [ s ] D x σ s, X s ) s dw s. Noe ha he i, j)h enry of he n n marix is j X i. hus for i =,...,n we deduce from 4.7) ha D u i = u iɣ u. Moreover, since D x bs, X s )) = n k= [ k x b)] s, X s ) [ s ku ] ; 4.8) D x σ s, X s )) = n k= [ k x σ )] s, X s ) [ s ku ], where, for ψ = b,σ, k x ψ)isann n marix whose i, j)h enry is xk xj ψ i,wemay rewrie 4.7) as n u Ɣu = γ [ ][ 4.9) + k s u α k s ds + βs k dw s], where 4.) α k s k= = [ s ] [ k x b) x σ k x σ )]s, X s ) s β k s = [ s ] k x σ )s, X s ) s ; γ = [ ] x σ, X ). Combining 4.9) and D u i = u iɣ u, from 4.4) we obain I 2 ) d = g x, ) ˆX 2 [ ] [ ] u D dσ, x) 4.) DX 2 H = g x, ) ˆX 2 n j X I j 2 σ, x), DX 2 H where I j 2 = u j D d = u j Ɣ d n [ s ] [α = u j γ d + s k u j k u d s ds + βk s dw s], k=
BS3-9 MAFI.cls Ocober 4, 22 5:28 EFFICIEN COMPUAION OF HEDGING PORFOLIOS 45 hanks o Fubini s heorem. Using analogous argumens, we ge I 3 ) d = g x, )[ ˆX 2 σ, x) ] 4.2) DX 2 D u d H + g x, )[ ˆX 2 σ, x) ] DX 2 γ u d H n [ s ] [α + j Xs k u j k u d s ds + βk s dw s]. k, j= I remains o analyze I 4) d. Firs, by 4.3) we have 4.3) D DX 2 H = 2D [ ] + 2 u r ur dr[ ] D u r )u r dr[ ]. From r u r = σ r, X r ), we have D [ r ])u r + r D u r ) = x σ r, X r ) r u, and 4.4) D u r = [ r ] [ x σ r, X r ) r u D r )u r ]. By 4.4) and 4.3), and applying Fubini s heorem again, we have I 4 ) d = 2 g x, ˆX 2 )[ σ, x) ] n 4.5) DX 4 j X H j= u j D d u r ur dr [ r ] [ + u j D u r d ur dr ]. Using 4.4), we can rewrie 4.3) as 4.6) I 4 ) d = 2 g x, ˆX 2 )[ σ, x) ] n DX 4 j X H j= u j D d u r ur dr + [ r ] r = 2 g x, ˆX 2 )[ σ, x) ] DX 4 H γ r r ) [ u j D r d u r ur dr ] n j= ) u j u d ur dr j X j, I 4 + I j,2 4 I j,3 [ ] 4, where I j,l 4, l =, 2, 3, are defined in an obvious way. In paricular,
BS3-9 MAFI.cls Ocober 4, 22 5:28 46 J. CVIANIĆ, J. MA, AND J. ZHANG 4.7) I j,2 4 = γ r r ) u j u d ur dr. Recalling 4.9), 4.7), and D i u = i u Ɣ u,wehave 4.8) n [ I j, 4 = u j γ d + k= r n I j,3 4 = u j γ d + k= k s r [ s k s ] [α u j k u d s ds + βk s dw ] s u r ur ); dr s ] [α u j k u d s ds + βk s dw s] u r ur dr. Finally, since 4.6) can be calculaed via 4.8) and 4.7), we can compue D F)u d by compuing 4.5), 4.), 4.2), and 4.6). Consequenly, combined wih 4.3) we can compue he Skorohod inegral I in 4.2). 4.. Summary of he Algorihm We have obained he following explici scheme for compuing he Skorohod inegral in 4.2). Recall he processes X,, and [ ] and define u = [ ] σ, X ); A = u s dw s ; B = u s u s ds; α k = [ ] [ k x b x σ k x σ ], X ) ; β k = [ ] k x σ, X ) ; γ = [ ] x σ, X ) ; H = γ s u s ds + hen we have n j,k= C j = u j s γ s ds + n k= j Xs k B s j [ α k s ds + βs k dw j s] ; L = [ k s Bs j ][ α k s ds + βs k dw ] s ; [ γs A j s C j s u s] u s ds. 4.9) I = B [ ] g x, ˆX )[ 2 σ, x) ][ A ] [ σ, x)] 2 g x, ˆX ) 2 ˆX 2 B [ ] g x, ˆX 2 ) n j X C j σ, x) j= g x, ˆX 2 )[ σ, x) ] H + 2 g x, ˆX 2 )[ σ, x) ] n [ ] B ) 2 j X j C B + L j ] [. j=
BS3-9 MAFI.cls Ocober 4, 22 5:28 EFFICIEN COMPUAION OF HEDGING PORFOLIOS 47 5. A PERURBAION MEHOD In his secion we propose anoher mehod ha can be applied when d n. Our numerical experimens show ha his mehod may be more efficien han he one described in previous secions. I is also concepually easier o undersand and o program. However, i is sensiive o he choice of he perurbaion size. Le W be an n d)-dimensional sandard Brownian moion defined on,f, P), such ha i is independen of W. Le W = W, W ) ), and denoe F o be he filraion generaed by W. For each >wedefine [ ] σ, x) = σ, x) 5.). σ 2, x) I n d) n d) hen i is clear ha σ, x) > for all, x) [, ] IR n, and σ, x) σ o, x), as, where σ, x) = [σ, x)... ]. Now, consider he following perurbed version of 3.): 5.2) X Y = x + b s, Xs = g ) X ) ds + σ ) s, Xs d W s, Z d W. By he sabiliy resuls for boh forward and backward SDEs we know ha, as, he following limi mus hold e.g., see Ma and Yong 999): 5.3) E sup X X 2 + sup Y Y 2 + E where X, Y, Z ) saisfies he following SDE: Z Z 2 d, 5.4) X Y = x + b ) s, Xs ds + = g ) X Z d W. σ s, Xs ) d W s, Since σ, X ) d W = σ, X ) dw, by uniqueness of he soluion o he SDE we have X X,, almos surely. hus by he uniqueness of he backward SDE or he maringale represenaion heorem) we conclude ha Z mus be of he form Z = Z, ), where Z is he soluion of 3.). Hence 5.4) is indeed 3.), and consequenly we mus have E Z, Z 2 5.5) d, as, where Z = Z,, Z,2 ) is he soluion o 5.2). We have hus proved he firs par of heorem 5.. HEOREM 5.. Assume Assumpion 2. and ha g is bounded and piecewise coninuous. hen he hedging porfolio process Z can be approximaed by Z,, in he sense of 5.5). Furhermore, if g is Lipschiz coninuous no necessarily bounded), hen here exis a consan K >, independen of, such ha 5.6) E Z, Z 2 d K 2 e K.
BS3-9 MAFI.cls Ocober 4, 22 5:28 48 J. CVIANIĆ, J. MA, AND J. ZHANG he proof of he above inequaliy is sandard in backward SDE heory, using Gronwall s inequaliy. he problem can now be reduced o compuing Z,. Hence, we can use he resuls for d = n from Fournié e al. 999) and Ma and Zhang 2). In paricular, assuming ha σ is nondegenerae and ha g is Lipshiz coninuous, we have 5.7) where 5.8) Z, Nr, = [ r r and saisfies he linear SDE 5.9) i X = e i + = E g X ) N, σ, X ), [ σ ] s, Xs )] ) s d W s ) ; x b s, Xs ) s ds + d j= [ x σ j s, Xs )] i Xs dw j, i =,...,n, where e i =,...,,...,) i is he ih coordinae vecor of IR n and σ j ) ishejh column of he marix σ ). When g is no Lipshiz coninuous, we consider he case d =, and assume ha he discouned payoff funcion is of he form gx ) = R gs), where g is a piecewise linear funcion K 5.) gx) = g i x) [ai,a i )x), i= where g i x) = A i x + B i, i =,...,K. We firs give he following approximaion lemma, whose proof we omi. LEMMA 5.2. Under he above assumpion here exiss a sequence of smooh funcions g k x) such ha i) g k x) C + x ), x, for some C >. ii) For each k, sup x g k x) C k, for some C k >. iii) For all x IR \ K i= a i, g k x) g x) and g k x) gx), as k. COROLLARY 5.3. Assume Assumpions 2. and 2.2, ha σ is nondegenerae, and ha he forward BSDE 5.2) has a unique adaped soluion X, Y, Z ). hen he following relaion holds: Z = E g X ) N, σ, X ). Proof. Le g k be he smooh sequence ha approximaes gp), as in Lemma 5.2. Le X,k, Y,k, Z,k ) be he soluion o 5.2) wih gx ) being replaced by g kx ) = R g kp ). Since he forward equaion is no changed, we have X = X,k. hus, applying 5.7) we have Z,k = E gk X ) N, σ, X ). We claim ha E g k X ) gx ) 2, as k. Indeed, since X is a imehomogeneous) diffusion, we le p x, y, ) be is ransiion densiy. hen E gk,x X ) gx ) 2 = gk y) gy) 2 p x, y, ) dy. IR n
BS3-9 MAFI.cls Ocober 4, 22 5:28 EFFICIEN COMPUAION OF HEDGING PORFOLIOS 49 Since g k g holds excep for only finiely many poins, and g k s have uniform linear growh Lemma 5.2i)), one can easily check ha, for any x IR n, i holds ha g k x) + gx) K + x 2 ). Recalling 5.2) and noing ha b and σ are uniformly Lipschiz, by sandard SDE argumens one can easily show ha E,x X 4 <, which implies ha IR n y 4 p s, y, ) dy <. Now we can apply he dominaed convergence heorem o conclude ha g k X ) gx )inl2 ) for each >. hus, by he sabiliy resuls for backward SDEs, we know ha Y,k, Z,k ) Y, Z ) in he sense of 5.3). herefore, P-almos surely, one has Z = lim Z,k ) = lim E gk X N, k k 6. NUMERICAL EXPERIMENS σ, X ). We compare here four mehods for compuing he hedging porfolio of a digial opion:. he Malliavin calculus mehod, called he M mehod, for Malliavin. We use heorem 3.3 and 4.9) o do he compuaions. 2. he Malliavin calculus mehod using he approximaion of σ. Le us call i he M mehod. We use he formulas 5.7) 5.9). 3. he Rerieval of Volailiy Mehod of Cvianic e al. 2), called he RVM mehod. his mehod is based on he fac ha he hedging porfolio process Z can be rerieved from he quadraic variaion process of he process Y. 4. he sandard finie difference mehod, called here. In his mehod one compues by cenral differences approximaion and simulaion) he derivaives of he price process Y a iniial ime wih respec o he iniial condiions X,...,X n) = x,...,x n ), and uses he fac from Io s lemma) ha he value Z is deermined from hese derivaives and he marix σ see Boyle e al., 997, for a survey). We consider a sochasic volailiy exension of he Black-Scholes model: S = S + σ r S r dw r ; σ = σ + k σ σ σ r ) dr + ρ σ σ r dw r. he chosen parameer values are: S =,σ =., k σ =.695,ρ σ =.2, σ =. Noe ha we assume, for simpliciy, ha he ineres rae is zero. We consider he digial opion wih payoff gs ) = S >K wih =.2, K =. he simulaion of he pahs of he underlying processes is done using he firs-order Euler scheme. Denoing he number of ime seps by N, he number of simulaed pahs is se o N 2 his is of he opimal order for he Euler scheme; see Duffie and Glynn 995). he mean porfolio value and he sandard error are obained by repeaing he procedure imes. he processing ime is he oal ime for all he repeiions. he resuls are repored in ables 6. and 6.2. he column labeled Porfolio gives he number of shares o be held by he hedging porfolio. able 6. is compued so ha all he processing imes are similar. he RVM mehod involves compuing an addiional condiional expecaion, condiioned on he firs ime period d, using K RVM simulaed pahs. We se N M = 45, N RVM = 8, K RVM = 2, N =, N M = 8, =.5, S =.S, σ =.σ, d =.5. Numerical experimens show ha he sandard error of he mehod is very much sensiive o he choice
BS3-9 MAFI.cls Ocober 4, 22 5:28 5 J. CVIANIĆ, J. MA, AND J. ZHANG ABLE 6. Similar Processing ime Mehod Error Processing ime Porfolio M.39 344.8896 RVM.257 338.8875 M.647 3447.88843.884 3379.886 ABLE 6.2 Similar Sandard Error Mehod Error Processing ime Porfolio M.623 995.88832 RVM.58 63.88755 M.495 4888.88832.53 659.88372 of S, σ. In paricular, if hese are no chosen carefully, we may achieve very small error, bu wih high bias. Wih he above choice of parameers, he mehod and he M mehod have somewha smaller sandard errors han he oher wo mehods, for a similar processing ime. However, i seems ha he M mehod is no ha sensiive o he choice of. Moreover, in his example we have only wo-dimensional facor process S,σ). he compuaional ime of he mehod increases linearly wih he number of facors, since i has o compue approximae derivaives of he opion price wih respec o all he facors. he RVM mehod has somewha larger errors han he and M mehods, bu no much. I is, however, also very sensiive o he choice of parameers d and K RVM. Surprisingly, he M mehod does wors here, even hough no significanly worse. On he oher hand, we do no have o worry abou he choice of any small parameers for he M mehod. In fac, he M mehod uses he smalles number of ime seps and simulaed pahs, bu apparenly akes a lo of ime compuing addiional quaniies such as he Skorohod inegral. able 6.2 was compued so as o achieve a similar sandard error for all mehods. his was done wih N M = 8, N RVM =, K RVM = 2, N = 25, N M = 9, =.5, S =.S, σ =.σ, d =.5. he resuls are similar o hose in able 6.: he M mehod requires he longes ime, even hough i uses a smaller number of seps and simulaed pahs. he oher hree mehods are comparable hey all need processing ime of he same order. he resuls are somewha surprising considering ha he direc mehod of heorem 3. is ypically more efficien han he mehod. In his example he mehod is no inferior if is parameers are carefully chosen o avoid high bias and high sandard error. REFERENCES BOYLE, P., M. BROADIE, and P. GLASSERMAN 997): Mone Carlo Mehods for Securiy Pricing, J. Econ. Dynam. Conrol 2, 267 32.
BS3-9 MAFI.cls Ocober 4, 22 5:28 EFFICIEN COMPUAION OF HEDGING PORFOLIOS 5 CVIANIĆ, J., L. GOUKASIAN, and F. ZAPAERO 2): Hedging wih Mone Carlo Simulaion; in Compuaional Mehods in Decision-Making, Economics and Finance, eds., E. Konoghiorghes, B. Rusem, and S. Siokos. Dordrech: Kluwer Academic Publishers, forhcoming. Q DUFFIE, D., and P. GLYNN 995): Efficien Mone Carlo Simulaion of Securiy Prices, Ann. Appl. Prob. 5, 897 95. EL KAROUI, N., S. PENG, and M. C. QUENEZ 997): Backward Sochasic Differenial Equaions in Finance, Mah. Finance 7, 72. FOURNIÉ, E., J.-M. LASRY, J. LEBUCHOUX, and P.-L. LIONS 2): Applicaions of Malliavin Calculus o Mone Carlo Mehods in Finance II, Finance Soch., o appear. Q2 FOURNIÉ, E., J.-M. LASRY, J. LEBUCHOUX, P.-L. LIONS, and N. OUZI 999): Applicaion of Malliavin Calculus o Mone Carlo Mehods in Finance, Finance Soch. 39, 39 42. GLASSERMAN, P., and X. ZHAO 999): Fas Greeks by Simulaion in Forward LIBOR Models, J. Compua. Finance 3, 5 39. Q3 MA, J., and J. YONG 999): Forward-Backward Sochasic Differenial Equaions and heir Applicaions, Lecure Noes Mah. 72. MA, J., and J. ZHANG 2): Represenaion heorems for Backward Sochasic Differenial Equaions, Ann. Appl. Prob., o appear. NUALAR, D. 995): he Malliavin Calculus and Relaed opics. Springer. Q4
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