MEAN-VARIANCE HEDGING FOR STOCHASTIC VOLATILITY MODELS

Transcription

1 MAN-VARIANC HDGING FOR SOCHASIC VOLAILIY MODLS FRANCSCA BIAGINI, PAOLO GUASONI, AND MAURIZIO PRALLI Absrac. In his paper we discuss he racabiliy of sochasic volailiy models for pricing and hedging opions wih he mean-variance hedging approach. We characerize he variance-opimal measure as he soluion of an equaion beween Doleans onenials: lici examples include boh models where volailiy solves a diffusion equaion, and models where i follows a jump process. We furher discuss he closedness of he space of sraegies. Inroducion he mean-variance hedging approach o pricing and hedging coningen claims was inroduced in he maringale case by Föllmer/Sondermann 1986 : subsequen exensions o he general semimaringale case were done by Duffie/Richardson 1991, Schweizer 1992, 1996, Schäl 1994, Gouriéroux e al. 1998, Pham e al and Rheinländer/Schweizer he paper of Schweizer 1999 conains a general overview of he subjec, and a complee bibliography. he aim of his paper is o analyse he mean-variance hedging crierion in sochasic volailiy models : we develop a general framework inroduced by Föllmer/Schweizer 1991 where a sochasic volailiy model is seen as a model wih incomplee informaion. his model would be complee wih respec o some larger filraion usually including all informaion on pas and fuure volailiy, bu no under he filraion available o he hedging agen who usually observes only he asse price hisory. his framework is general enough o include boh he diffusion models such as Hull- Whie, Heson, Sein and Sein, only o menion a few, and less common models where volailiy jumps. We begin our analysis wih a characerizaion of he se of equivalen maringale measures in erms of Doléans onenials: his provides a one-one correspondence beween equivalen maringale measures and a class of predicable processes. xploiing resuls of Schweizer 1996 and Delbaen/Schachermayer 1996, we hen idenify he variance-opimal maringale measure as he soluion of an equaion involving onenial maringales. Our resuls are illusraed by several examples: he deailed analysis of all hese examples can be found in Biagini/Guasoni In he case of diffusion sochasic volailiy models we recover some resuls of Lauren/Pham 1999 wih a differen mehod: in fac, while hey use dynamic programming echniques, we essenially focus on sochasic inegraion. Also he 1991 Mahemaics Subjec Classificaion. 6H3, 9A9; JL Classificaion: G1. Key words and phrases. Hedging in incomplee markes, sochasic volailiy models, meanvariance opimal measure, change of numéraire. 1

2 2 FRANCSCA BIAGINI, PAOLO GUASONI, AND MAURIZIO PRALLI recen paper by Heah e al conains a deailed analysis of he mean-variance hedging crierion compared o he locally risk-minimizing crierion in sochasic volailiy models. In order o keep noaions simple, we only consider one-dimensional models; however our resuls can easily be exended o he mulidimensional case. 1. Saemen of he problem For all definiions on sochasic inegraion and maringale represenaion, we refer o Proer 199 and Dellacherie/Meyer 1982 in paricular, all filraions are supposed o saisfy he so-called usual hypohesis. We have wo complee filered probabiliy spaces denoed by Ω, F W, F W, P W and,,, P. We assume ha W is a sandard Brownian Moion on Ω = C, ], R, P W is he sandard Wiener measure, F W = F W, and F W is he P W - augmenaion of he filraion generaed by W. We have wo asses: he risky asse S, and he riskless asse B = r sds, where r is a deerminisic funcion. he risky asse is represened by a process S w, η on he produc space Ω, whose dynamics is given by he following equaion: {ds, ω, η = S, ω, η µ, ω, ηd + σ, ω, ηdw ω S = S We shall make he following assumpions: i on he space we have a possibly d-dimensional maringale M which has he predicable represenaion propery wih respec o he filraion, ]. ii he informaion available a ime is given by he filraion F W. iii he probabiliy P on Ω is he produc probabiliy P W P. Remark 1.1. In many applicaions, he mos naural filraion available o he agen is he one generaed by S: le us see how ii ranslaes in his case. If σ has a righconinuous version, i is F S -adaped: in fac we recall ha see Föllmer/Schweizer 1991 page 41 S = σ 2 ss 2 s ds = lim Si+1 S i 2 sup i i+1 i is F S -adaped. If µ, ω, η is also F S -adaped, i is easy o see ha he filraion generaed by S coincides wih he one generaed by W, µ, σ. herefore he assumpion ii boils down o F W,µ,σ = F W Remark 1.2. Since he echnicaliies involved in he definiion above may hide he idea of incomplee informaion, we provide a simple lanaion. his marke would be complee if he agen had access o he larger filraion F = F, which conains a any ime all he informaion on pas and fuure drif and volailiy. As poined ou by Föllmer/Schweizer 1991, his is a consequence of he fac ha all

3 MAN-VARIANC HDGING FOR SOCHASIC VOLAILIY MODLS 3 F -maringales can be wrien in he form N ω, η = N η + H s ω, ηdw s ω for some F-predicable process H. his resul is an exercise on sochasic inegraion: we provide a proof in he appendix Proposiion 5.1, for he sake of compleeness. he discouned value of he risky asse follows he equaion: {dx = X µ r d + σ dw X = S We assume µ and σ are such ha X L 2 P for all, ], and denoe by µ, ω, η r λ ω, η = he so-called marke price of risk. σ, ω, η xample 1.3. his example was inroduced by Harrison/Pliska 1981, and invesigaed laer by Föllmer/Schweizer 1991, page 142. µ and σ are consan unil a fixed ime, hen hey jump simulaneously, he pair µ, σ having wo possible oucomes. In oher words { µ η = 1 {< }µ + 1 { }µ η σ η = 1 {< }σ + 1 { }σ η where = {, 1}, = {, } for < and = P for. A fundamenal maringale is given by M = 1 { }1 {η=1} p, where p = P η = 1. his example was generalized by Föllmer/Leuker 1999, where he values of µ and σ afer he jump ime have a coninuous disribuion: in his case = R and he maringale M has o be replaced by a random measure see Biagini/Guasoni 1999 for deails. xample 1.4. he previous example can be exended in several ways: we consider in paricular a model proposed in discree ime fashion in RiskMerics Monior see Zangari 1996 as an improvemen of he sandard lognormal model for calculaing Value a Risk. More precisely, we have muliple independe jumps a fixed equispaced ime inervals. We can se = {, 1} n and, denoing η = {a 1,..., a n }, is equal o he pars of {a i } i where i = i n+1. One obains he following dynamics: { µ η = 1 {<1 }µ + i 1 { i < i+1 }µ ai + 1 { n }µ an σ η = 1 {<1 }σ + i 1 { i < i+1 }σ ai + 1 { n}σ an Since in his model η is binomially disribued in fac he numbers a i are independen and P a i = 1 = p, i is eviden he exisence of a maringale wih he represenaion propery. xample 1.5. his example was sudied in deail by Biagini/Guasoni We have µ η = 1 {<} µ { } µ 2 σ η = 1 {<} σ { } σ 2

4 4 FRANCSCA BIAGINI, PAOLO GUASONI, AND MAURIZIO PRALLI where is a sopping ime whose law resriced o, has a densiy f and P = = 1 fsds. In his case, =, ], = B, ], ], and a fundamenal maringale can be found in he form M = 1 { } a, where a. is an increasing funcion which can be liciely deermined in erms of f. xample 1.6. he previous example can be generalized in he following way: afer he jump ime, µ and σ have a general probabiliy disribuion independen from. he space is, in his case,, ] R and he maringale M is replaced by he random measure ν ν p, where ν p is he compensaor of he random measure νη, d, dx = ɛ η,αη d, dx and αη = λ 2 η λ 2 1. xample 1.7. A number of diffusion sochasic volailiy models have been proposed in he lieraure, mos of hem being paricular cases of he following { dx = σ, X, v X λ, X, v d + dw 1 dv = α, X, v d + β, X, v dw 1 + γ, X, v dw 2 where W 1 and W 2 are wo independen Brownian Moions. We se = C, 1], R, and is he augmenaion of he filraion generaed by W 2 : he naural choice for a maringale wih he represenaion propery on is clearly W 2. In he general framework described above, an agen wishes o hedge a cerain uropean opion H iring a a fixed ime : his goal is o minimize he risk, defined as he variance of he racking error a iraion. herefore we look for a soluion o he minimum problem 1 min H c G c R θ 2] θ Θ where G θ = θ s dx s and Θ = { θ LX, G θ S 2 P } Here LX denoes he space of X-inegrable F -predicable processes, and S 2 he space of semimaringales Y decomposable as Y = Y + M + A, where M is a squareinegrable maringale, and A is a process of square-inegrable variaion. Definiion 1.8. We define he following spaces of signed maringale measures M s = {Q P : X is a Q-local maringale} M e = {Q M s, Q P } { M 2 s = Q M s, dq } dp L2 P M 2 e = M e M 2 s If Q is a signed probabiliy wih densiy Z wih respec o P, by definiion X is a Q-maringale if X Z is a P -maringale, where Z = Z F]. he exisence of a minimizer for 1 was shown for any H L 2 P independenly by

5 MAN-VARIANC HDGING FOR SOCHASIC VOLAILIY MODLS 5 Gouriéroux e al 1998 and Rheinländer/Schweizer 1997 under he wo sanding assumpions which need o be checked for each paricular model: i M 2 e ; ii G Θ is closed. While i is equivalen o a no-arbirage condiion see Delbaen/Schachermayer 1996 and holds for very general models, ii ofen fails even for models commonly used in pracice. However we shall reurn o his issue laer. If 1 has a soluion, he opimal value for c can be wrien as c = Ẽ H] where Ẽ denoes he ecaion under a new signed measure P, he so-called variance-opimal maringale measure. By definiion, P is he elemen of minimal norm in M 2 s which evidenly exiss as soon as M 2 s : see for insance Schweizer 1996 for furher deails. Our firs sep owards an lici formula for d P is he characerizaion of he se dp M 2 e of he square-inegrable equivalen maringale measures. We sar by recalling he following: Definiion 1.9. he Doléans onenial Z of a semimaringale Z is defined as Z = Z 1 2 Zc + Z s Z s s 1 where Z c denoes he coninous par of Z, while Z s = Z s Z s. We prove now he following lemma. Lemma 1.1. Le Z be a local maringale wih Z = 1. he following condiions are equivalen: i Z X is a local maringale ii Z = λ sdx s 1 + k sdm s for some predicable process k s such ha he sochasic inegral k sdm s is a local maringale. Proof. We recall ha he pair W, M has he predicable represenaion propery see Proposiion 5.2. herefore By Iô s formula, we have Z = 1 + h s dw s + k s dm s dz X = Z µ r + h σ ] X d + Z σ X + h X ] dw + k X dm he process Z X is a local maringale if and only if h = µ r Z. More σ precisely, if λ = µ r, Z saisfies he following sochasic differenial equaion: σ dz = λ Z dw + k dm

6 6 FRANCSCA BIAGINI, PAOLO GUASONI, AND MAURIZIO PRALLI which has a unique soluion see Proer 199 for deails. I can easily be verified ha Z = λ sdx s 1 + k sdm s is he soluion of he above equaion. If Z is sricly posiive, hen N = 1 + k sdm s can be wrien as he Doléans k s onenial N = dm s. N s An immediae consequence of he above lemma is he characerizaion of M 2 s and M 2 e. Proposiion For every Q M 2 s dq dp = λ ω, ηdw c + k dm where k is a process such ha he above ression is square inegrable. 2. For every Q M 2 e dq dp = λ ω, ηdw k ω, ηdm wih k such ha k M > 1 and λ sdw s + k s dm s is a squareinegrable maringale. Recall ha λ sdw s + k s dm s = λ sdw s k sdm s since W, M] = see Proer 199 pag. 79. Condiion k M > 1 guaranees he posiiviy of k dm. Remark A similar characerizaion holds for he probabiliies Q P such ha X is a Q-maringale wih respec o he enlarged filraion F : more precisely dq dp = Gη λ s ηdw s wih G such ha he above ression is square inegrable and G] = 1. Q is a rue probabiliy if G >. Before we find an equaion o idenify P, we need anoher Definiion We define he wo processes Ŵ and W as follows: Ŵ = W + W = W + 2 λ s ds λ s ds Remark By he heorem of Girsanov, if λ sdw s and 2 λ sdw s are uniformly inegrable maringales, hen Ŵ and W are Brownian Moions respecively under he measures P and P, defined as d P dp = λ dw and dp dp = 2 λ dw

7 MAN-VARIANC HDGING FOR SOCHASIC VOLAILIY MODLS 7 We recall ha P if i exiss is called he minimal maringale measure. Lemma Le h, k be wo predicable sochasic processes whose sochasic inegrals h sdws and k sdm s are defined. he following condiions are equivalen: λ 2 sds = c h sdws 2 k sdm s 3 λ s dw s + k s dm s = c λ s + h s dŵs where c is he same consan in boh equaions. Proof. We will use he properies of he Doléans onenial lised in Proer 199 pag. 79. Saring from he lef-hand side of 3, we have λ s dw s + k s dm s = = λ s dŵs k s dm s λ 2 sds Conversely, saring from he righ-hand side of 3, we have λ s + h s dŵs = = λ s dŵs h s dŵs λ s h s ds = = λ s dŵs h s dws he conclusion is now immediae. From now on, we suppose ha M 2 e. By Schweizer 1996, Lemma 1 page 21, and Delbaen/Schachermayer 1996 lemma 2.2 and heorem 1.3, we obain he following characerizaion of he variance-opimal maringale measure: P is an elemen of M 2 e i.e. P is a rue probabiliy and i is he unique elemen of M 2 s which can be wrien in he form d P dp = c + γ s dx s wih c 1. In he above equaion, γ is a predicable sochasic process which does no necessarily belong o Θ ; however he inegral process γ sdx s is a square inegrable maringale for every probabiliy measure Q M 2 e. In paricular, γ sdx s is an elemen of G Θ. Since d P is sricly posiive, i can be wrien as a Doléans onenial. From he dp previous resul, we obain he following :

8 8 FRANCSCA BIAGINI, PAOLO GUASONI, AND MAURIZIO PRALLI heorem Le h, k be wo predicable processes such ha he onenial maringale λ sdw s + k sdm s is square-inegrable. hen h, k are soluions of he equaion 2 of Lemma 1.15 if and only if d P dp = λ s dw s + k s dm s = β sdx s β ] sdx s where β s = λ s h s σ s X s. he equaliy d P dp = c is useful o characerize he opimal sraegy see Rheinländer/Schweizer 1997 ; we also recall ha β is he so-called hedging numéraire of Gouriéroux e al β s dx s 2. xplici Soluions We have seen ha a soluion o he equaion: λ 2 sds = c h sdws k sdm s provides an lici form for he densiy of he variance-opimal maringale measure. Definiion 2.1. We recall he definiion of he mean-variance radeoff process K see, for insance, Schweizer 1996: K = λ 2 sds From 2 we can immediaely see he following: Proposiion 2.2. K is a consan if and only if P = P, and β = λ σx. his was firs poined ou by Pham e al and, for Iô processes, by Lauren/Pham In more realisic siuaions, a soluion o 2 can easily be found in wo cases: α λ s ω, η = λ s ω: in his case we se k =, and solve he equaion h dw = λ2 d λ2 d which, provided ha exiss, and he above ecaion is finie, admis a soluion by he represenaion propery of W and hus of W on Ω. his case covers he so-called almos complee models, where P = P, while β s = λ s h s σ s. In a ypical example, H is an opion on wo observable asses, bu rading is allowed in only one of hem. As a resul, F S is sricly smaller han F W, unlike in he usual sochasic volailiy models, where hese filraions are equal. For a discussion on almos complee models, see for insance Pham e al or Lauren/Pham ]

9 MAN-VARIANC HDGING FOR SOCHASIC VOLAILIY MODLS 9 We only remark ha in his case M 2 e if and only if P exiss and d P dp is in L2. Since d P dp 2 = 2 λ s dx s λ 2 d, his condiion is saisfied if he probabiliy P exiss and is P -inegrable. λ2 d β λ s ω, η = λ s η : in his case we can simply se h =, and hen solve he equaion λ2 ηd k dm = ] λ2 ηd which always admis a soluion, since M has he represenaion propery on. his case covers all examples considered in Biagini/Guasoni 1999 : β s =, σ s X s and P is generally differen han P, unless K is deerminisic for diffusion processes, his is proved Pham e al. 1998, heorem 11. We remark ha if λ2 ηd is finie almos surely, hen M 2 e. Namely, in his case we obain d P dp 2 = 2 λ 2 ηdw λ2 ηd ] 2 λ2 ηd he process 2 λ2 dw is acually a sochasic inegral depending on he parameer η see Proer 199 for deails: herefore for every fixed η we have ha Ω 2 λ2 ηdw dp ω = 1, and consequenly we ge d P ] dp 2 = 1 ] λ2 ηd When λ s ω, η = λ s η, i may be hard o find k licily ; bu in fac i is ofen sufficien o know ha i exiss, since P can be obained hrough he equaliy d P dp = λ2 d 4 λ s dw s ] λ2 d Below we have some examples: xample 2.3. If we consider he example 1.4, under he probabiliy P he numbers a i are sill independen, bu a i = 1 wih a new probabiliy p, where, if = λ s, n+1 p = p e λ2 1 p e λ p e λ2 2

10 1 FRANCSCA BIAGINI, PAOLO GUASONI, AND MAURIZIO PRALLI xample 2.4. If we consider he example 1.6, under he new probabiliy P he ime jump and he new values of µ and σ afer are no more independen: in Biagini/Guasoni 1999 one can find he lici form of he law of under P and of he laws of µ and σ condiional o { = }. In some models, however, i may be desirable o find k : his is he case, for insance, for sochasic volailiy models defined by diffusion processes. In example 1.7, if β, x, y = and if α, γ, σ don depend on X, we have λ s ω, η = λ s η, and P can be wrien as in 4: however, his does no clarify he dynamics of v under P. On he oher hand, if k is known, hen one can apply Girsanov s heorem, and ge { dx = X σ, v d W 1 dv = α, v k d + γ, v d W 2 where W 1 and W 2 are independen Wiener processes under P. If he model is in some sense Markovian, we obain he following resul which coincides wih Proposiion of Lauren/Pham 1999, bu i is proved in a compleely differen way : Proposiion 2.5. Assume ha ] λ 2 ss, v s ds F = G, v, and ha he funcion G, x is C 1 in and C 2 in x. hen we have k s dws 2 = λ2 s, v s ds λ2 s, v s ds ] iff k = G γ x G Proof. By maringale represenaion, here exiss a process g such ha λ 2 s, v s ds = G + g s dws 2 herefore: G = G F ] = G + g s dws 2 = = λ 2 s, v s ds Applying Iô s formula, we obain: dg = g dw 2 = ] λ 2 s, v s ds F = = G λ 2 s, v s ds x γ, v dw 2,v λ 2 s, v s ds G, v where, in he las equaliy, he sum of he erms of finie variaion vanishes since G is a maringale. herefore, g = λ2 s, v s ds G γ, v x. However, we

11 also have MAN-VARIANC HDGING FOR SOCHASIC VOLAILIY MODLS 11 G = G + g s dws 2 g s = G dws 2 G s herefore k = g G, and he proof is complee. = G k s dws 2 3. Condiions for he closedness of G Θ he closedness of he space G Θ in L 2 P plays a key role in mean-variance hedging, since i guaranees he exisence of an opimal hedging sraegy in he space Θ. A sufficien condiion for G Θ o be closed is he boundedness of K, as shown by Pham e al In some sense, we now show ha in cases α and β, he boundedness of K is almos necessary. We will show ha his condiion is no saisfied for some commonly used models. Firs we recall, and sae as a heorem, a shor version of a necessary and sufficien condiion esablished by Delbaen e al : heorem 3.1. Le X be a coninuous semimaringale: suppose ha M 2 e and d e le Z = P F ]. he following condiions are equivalen: dp i G Θ is closed in L 2 P ; ii Z saisfies he following reverse Hölder inequaliy: Z ] 2 F C for all sopping imes and for some consan C. We shall now see how his condiion ranslaes for α and β. Z Proposiion 3.2. Assume ha M 2 e : i If λ s ω, η = λ s ω, hen G Θ is closed if and only if here exiss some M such ha, for all sopping imes, ] λ 2 ωd F < M ii If λ s ω, η = λ s η, hen G Θ is closed if and only if here exiss some ɛ > such ha, for all sopping imes, ] λ 2 ηd F > ɛ Proof. From 3.1, i follows ha G Θ is closed if and only if condiion ii in heorem 3.1 is saisfied. For ii, we have Z = Z F ] = λ dw ] λ2 ηd F ] = λ2 ηd

12 12 FRANCSCA BIAGINI, PAOLO GUASONI, AND MAURIZIO PRALLI I follows ha Z Z = λ dw λ dw herefore: Z ] 2 F = Z 2 λ2 ηd ] = λ2 ηd F = λ dw λ2 ηd λ2 ηd λ dw ] λ2 ηd F ] 2 = F λ2 ηd = λ2 ηd F ] ] λ2 ηd F ] 2 F However, since λ depends only on η, we find ha he projecion of P on F coincides wih P, and hus ] λ2 ηd F = ] λ2 ηd F. Hence Z ] 2 Z F 1 = ] λ2 ηd F as claimed. For i, calculaions are more sraighforward: Z = λ dw and hus Z Z Finally Z ] 2 F = Z and he proof is complee. = λ dw λ dw 2 λ dw = We shall give some models where G Θ is no closed. λ dw ] λ 2 ωd F = ] = λ 2 ωd F xample 3.3. Consider example 1.3 or beer he generalizaion of Föllmer and Leuker, where: λ = λ 1 {< } + λη1 { }

13 MAN-VARIANC HDGING FOR SOCHASIC VOLAILIY MODLS 13 As menioned before, here = R : G Θ is closed if and only if he disribuion of λη has compac suppor. In fac, if las condiion is saisfied, hen K is bounded ; conversely, for we have ] λ 2 sηds F = λ 2 η By Proposiion 3.2, he conclusion is immediae. xample 3.4. We now examine he Heson model, ha is a sochasic volailiy model described by he equaions { dx = X λ v d + v dw 1 dv = α βv d + v dw 2 Here we have see, for insance, Lauren/Pham 1999 : ] 5 λ 2 ηd F = A λ 2 v B where A = 1 + ζ δ 1 e δ δ = β 1 + 2λ2 1 + ζe δ β 2 ζ = δ β δ + β Since δ, ζ >, i follows ha A >, and herefore 5 is bounded from below if and only if v is bounded from above. However, his is never he case, since in Heson model v is he square of a Bessel process wih an appropriae change of ime. Analogous calculaions can be carried ou in he Sein and Sein model see Heah e al. 1999, example for deails showing ha also in his case G Θ is no closed. We poin ou ha he drawback of he non-closedness of he space G Θ has been overcome by Schweizer 1999 : by loiing he approach inroduced by Gouriéroux e al. 1998, Schweizer has proved he exisence of an opimal meanvariance sraegy no in he space Θ, bu in he space Θ of all predicable processes θ such ha he sochasic inegral θ sdx s is a square inegrable maringale for every Q M 2 e. 4. Conclusions We have seen ha a simple equaion involving sochasic onenials can idenify he variance opimal probabiliy P and he mean-variance hedging sraegy in a general class of sochasic volailiy models. All examples inroduced are analysed in Biagini/Guasoni We furher poin ou ha he change of numéraire approach inroduced by Geman e al can be adaped o give he lici form of he mean-variance hedging sraegy for a call opion see Biagini/Guasoni 1999.

14 14 FRANCSCA BIAGINI, PAOLO GUASONI, AND MAURIZIO PRALLI 5. Appendix Proposiion 5.1. Any square-inegrable maringale wih respec o he filraion F can be wrien as N ω, η = N η + where H is F -predicable and such ha H s ω, ηdw s ω ] H2 s ds <. Proof. Denoe by M he se of maringales which admi a represenaion in he desired form. We begin by showing ha M conains all maringales N such ha N ω, η = F ωgη, wih F, G square-inegrable and measurable funcions. In fac, if F ω = F + ] H sωdw s ω, wih F 2 ] = F 2 + H2 s ds, i is easily seen ha: F ωgη = F Gη + H s ωgηdw s ω he sochasic process H s ω, η = H s ωgη is F -predicable, and F 2 G 2] = F 2 G 2] ] + Hs 2 G 2 ds M is obvoiusly sable under linear combinaions, hence he se {N : N M} is dense in L 2 Ω, F, P. However, if N = N + H sdw s, he map N N 2] = N 2 + ] is an isomeric injecion from M ino L 2 Ω H2 s ds, F, P. his concludes he proof. Proposiion 5.2. Any square-inegrable maringale N wih respec o he filraion = F can be wrien in he form F W N ω, η = N + wih H, K predicable and such ha: Hs 2 ds + H s ω, ηdw s ω + K 2 s dm] s ] < K s ω, ηdm s η Proof. Denoe by M he se of maringales which admi a represenaion in he desired form. We sar showing ha M conains all maringales N such ha N ω, η = F ωgη. We wrie F ω = F + H sωdw s ω, Gη = G + K sηdm s η, and consider he maringales R = F F ] and V = G F ]. By Iô s formula, and recalling ha W, M] =, we have F ωgη = F G + V s H s dw s + R s K s dm s Again, M is sable under linear combinaions, and {N : N M} is dense in L 2. Since he map N N 2 + H2 s ds + ] K2 s dm] s is an isomeric injecion from M in L 2, he proof is complee.

15 MAN-VARIANC HDGING FOR SOCHASIC VOLAILIY MODLS 15 References 1] Biagini F. and Guasoni P Mean-variance hedging wih random volailiy jumps, Preprin. 2] Delbaen F., Mona P., Schachermayer W., Schweizer M. and Sricker C Weighed norm inequaliies and hedging in incomplee markes Finance and Sochasics 1 3, ] Delbaen F. and Schachermayer W he variance-opimal maringale measure for coninuous processes Bernoulli 2, ] Dellacherie C. and Meyer P.A Probabiliies and Poenial B: heory of maringales Norh-Holland, Amserdam. 5] Duffie D. and Richardson H. L Mean-variance hedging in coninuous ime Annals of Applied Probabiliy 1, ] Föllmer H. and Leuker P Quanile hedging Finance and Sochasics 33, ] Föllmer H. and Schweizer M Hedging of coningen claims under incomplee informaion In R.J. llio and M.H.A. Davis, ediors, Applied Sochasic Analysis Gordon and Breach. 8] Föllmer H. and Sondermann D Hedging of non-redundan coningen claims In W. Hildebrand and A. Mas-Colell, ediors, Conribuion o Mahemaical conomics, ] Geman H., l Karoui N. and Roche J.C Changes of numéraire, changes of probabiliy measures and opion pricing J. Appl. Probab. 32, ] Gouriéroux L., Lauren J.P. and Pham H Mean-variance hedging and numéraire Mah. Finance 8, ] Harrison J.M. and Pliska R.S Maringales and Sochasic Inegrals in he heory of coninuous rading Soc. Processes and heir Appl. 11, ] Heah D., Plaen. and Schweizer M A comparison of wo Quadraic Approaches o hedging in Incomplee Markes Preprin 13] Lauren J.P. and Pham H Dynamic programming and mean-variance hedging Finance and Sochasics 31, ] Pham H., Rheinländer. and Schweizer M Mean-variance hedging for coninuous processes: new proofs and examples Finance and Sochasics 22, ] Proer P. 199 Sochasic Inegraion and Differenial quaions: A new approach Springer-Verlag. 16] Rheinländer. and Schweizer M On l 2 -projecions on a space of sochasic inegrals Annals of Probabiliy 25 4, ] Schäl M On quadraic cos crieria for opion hedging Mahemaics of Operaions Research 19, ] Schweizer M Mean-variance hedging for general claims Annals of Applied Probabiliy 2, ] Schweizer M Approximaion pricing and he variance-opimal maringale measure Annals of Probabiliy 64, ] Schweizer M A Guided our hrough Quadraic Hedging Approaches Preprin. 21] Zangari P An improved meodology for measuring VaR Risk-Merics Monior 2, Dip. di Maemaica Piazza Pora S. Donao 41 Bologna Ialy -mail address: biagini@dm.unibo.i Scuola Normale Superiore P.zza dei Cavalieri, Pisa Ialy -mail address: guasoni@sns.i Dip. di Maemaica Via Buonarroi, Pisa Ialy -mail address: praelli@dm.unipi.i