Working Paper 5-81 Business Economics Series 21 January 25 Deparameno de Economía de la Empresa Universidad Carlos III de Madrid Calle Madrid, 126 2893 Geafe (Spain) Fax (34) 91 624 968 Opion-Pricing in Incomplee Markes: The Hedging Porfolio plus a Risk Premium-Based Recursive Approach Alfredo Ibáñez (his draf: January, 3, 25) Absrac Consider a non-spanned securiy C T in an incomplee marke. We sudy he risk/reurn radeoffs generaed if his securiy is sold for an arbirage-free price bc and hen hedged. We consider recursive one-period opimal self-financing hedging sraegies, a simple bu racable crierion. For coninuous rading, diffusion processes, he one-period minimum variance porfolio is opimal. Le C () be is price. Self-financing implies ha he residual risk is equal o he sum of he oneperiod orhogonal hedging errors, P T Y ()e r(t ). To compensae he residual risk, a risk premium y is associaed wih every Y. Now le C (y) be he price of he hedging porfolio, and P T (Y (y)+y ) e r(t ) is he oal residual risk. Alhough no he same, he one-period hedging errors Y () and Y (y) are orhogonal o he rading asses, and are perfecly correlaed. This implies ha he spanned opion payoff does no depend on y. Le C b = C (y). A main resul follows. Any arbirage-free price, bc, is jus he price of a hedging porfolio (such as in a complee marke), C (), plus a premium, C b C (). Tha is, C () is he price of he opion s payoff which can be spanned, and C b C () is he premium associaed wih he opion s payoff which canno be spanned (and yields a coningen risk premium of P y e r(t ) a mauriy). We sudy oher applicaions of opion-pricing heory as well. I am graeful o seminar paricipans a Universidad Carlos III and CEMFI (Madrid), Nova (Lisboa), USC and UCLA (Los Angeles), he 14h Derivaive Securiies Conference (New York), and o Alejandro Balbás, Tony Bernardo, Manuel Domínguez, Ming Huang, Francis Longsaff, Fernando Zapaero, and, especially, o Eduardo Schwarz for commens. Any remaining errors are of course my own. Par of his research was done when I was a 23/4 Visiing Scholar in he Finance Deparmen, he Anderson School a UCLA. Dpo. de Adminisración. Insiuo Tecnológico Auónomo de México, ITAM. Col Tizapán-San Angel. 1 México D.F., México. Phone: + 52-55-56284 (X 3415). Fax: + 52-55-5628449. e-mail: ibanez@iam.mx.
1 Inroducion In an incomplee marke here is no a replicaing porfolio for hose non-spanned securiies, and hus, one canno apply he law of one price, and obain a unique soluion. On he conrary, here are an upper and a lower arbirage bound, which conain he non arbirage prices (see Meron (1973)). One mus make furher assumpions o selec one of hese prices, or o consrain he arbirage bounds. Consider a non-spanned opion C T wih mauriy T. Le C and C+ be he arbirage bounds. We sudy he risk/reurn rade-offs generaed if his securiy is sold for C b C,C+ and hen hedged. We consider recursive one-period opimal self-financing hedging sraegies, a simple bu racable crierion. For coninuous rading, diffusion processes, he one-period hedging errors are Gaussian. This implies ha he one-period minimum variance porfolio, b h +1,isopimal,and,in his case, is he unique one-period hedging crierion. Le X b h +1 be is price a ime. We now connec pricing wih hedging. For every {,,..., T }, we recursively define an opion price process C (y) =X b h +1 +y, where y is a risk premium associaed wih he one-period hedging error Y b h +1 (y) =X b h +1 +1 C +1(y). Noe ha C () and Y b h +1 () are he opion price and he hedging error, respecively, if all risk premiums are zero (i.e., y =). Self-financing implies ha he residual risk is equal o he sum (financed o he riskless rae r) of he one-period orhogonal hedging errors and heir associaed risk premiums, P T 1 = Y b h +1 (y)e r(t ) + P T 1 = y e r(t ), which can be considered separaely. Alhough he errors Y b h +1 (y) and Y b h +1 () are no he same if y 6=, hey are orhogonal o he rading asses, and are perfecly correlaed. This implies ha he spanned opion payoff does no depend on he process y. Now, le bc = C (y). Two main resuls follow. 1. Any arbirage-free price, bc, is jus he price of a hedging porfolio (such as in a complee marke), C (), plus a premium, bc C (). Thais,C () is he price of he opion s payoff which can be spanned, and bc C () is he premium associaed wih he opion s payoff which canno be spanned (and yields a coningen risk premium of P T 1 = er(t ) y a mauriy). 2. As we do no advocae a specific risk premium y, we do no provide a unique price bc.we derive (as b h is one-period opimal) an opimal fronier in he non arbirage opion prices/risk premiums space (i.e., y C (y)), and i is he final user who provides he risk premium y. In brief, our model reduces pricing in incomplee markes o he explici valuaion of a one-period orhogonal diffusion risk. One can consrain C (y) by consraining y; paramerize y(λ), λ R; 1 dc compue he elasiciy (y(λ)) C (y) dλ ;define an upper (lower) bound if y (y ). In addiion, we explicily obain he laer price decomposiion which can be applied o a complee marke or o risk-neural pricing. Moreover, y does no need o depend on a price of risk, differing from a complee marke and risk-neural pricing, a flexibiliy which can be used o fi volailiy smiles. 2
Our model is racable because i is based in recursive one-period opimal porfolios, and allows us o sudy oher opion-pricing applicaions such as risk managemen, American-syle payoffs, porfolio consrains, ec. Our model is relaed o oher approaches as will be demonsraed. There are imporan pricing approaches in incomplee markes, including he equilibrium or uiliy maximizaion-based approach (see Rubinsein (1976)); o consider prices of risk associaed wih nonraded sae variables (see Heson (1993)); o compue an opimal hedging porfolio, whose price is hen he desired incomplee marke price (Meron (1998)); o use a risk/reward crierion, such as he gains-o-losses raio (Bernardo and Olivier (2)) or he Sharpe raio (Cochrane and Saá-Requejo (2)). See also Carr e al. (21), Cerný (23), and references herein for oher models. Nex, we describe he main resuls of our approach. Firs, recursive prices are consisen wih recursive one-period opimal self-financing hedging sraegies. A relaed approach is ha based on fully opimal dynamic sraegies. 1 This lieraure has indeed raised he issue of parially or fully opimal hedging, and focuses more on hedging han pricing. The main limiaion of a fully opimal crierion is racabiliy. In our approach, e.g., mulifacor models are feasible o analyze. Second, he residual risk is equal o he sum of he one-period errors plus he risk premiums, and can be used for risk managemen (o compue momens, ails, ec.). In addiion, i conveys a coningen risk premium. If we specify ha larger one-period risk premiums are associaed wih more volaile or risky hedging errors, hen a sample pah of large (small) one-period hedging errors is associaed, in probabiliy, wih a pah of large (small) risk premiums. Therefore, hese risk premiums are a risk managemen ool. Furher, he residual risk can be used empirically o sudy marke incompleeness, o exrac prices of risk, or o sudy risk/reurn rade-offs in derivaive markes. Third, in coninuous-ime, diffusion processes, he hedging sraegy is unique and hus recursive prices differ only in he one-period risk premiums. This resul allows us o relae our model o opion-pricing heory. In a complee marke, he risk premium of every sae variable depends on a price of risk o avoid arbirage. Here, his consrain does no necessarily apply o he residual risk. This pricing flexibiliy differeniaes a complee marke, and risk-neural pricing, from an incomplee marke, and allows us o beer fi volailiy smiles in opion markes. (Cerainly, we are pricing securiies separaely. If, laer, hese securiies are raded, hese risk premiums will depend on a price of risk.) In Meron (1998) and Cochrane and Saá-Requejo (2), which are recursive mehods also, he risk premium y is equal o zero and is proporional o he residual risk volailiy, respecively. 2 Fourh, recursive prices are easily characerized hrough PDE s equaions or a risk-neural dy- 1 See Duffie and Richardson (1991), Schweizer (1992), Heah e al. (21), Bersimas e al. (21), among ohers; e.g., Duffie and Richardson (1991) sudy lognormal processes. Heah e al. (21) focus on sochasic volailiy models. 2 Meron (1998, 333) argues ha he one-period risk premium should be zero, since he residual risk is orhogonal wih he raded asses and herefore wih he equilibrium marke porfolio. Equilibrium models, however, are no empirically suppored in general. I hank Rober L. McDonald for poining ou his reference. 3
namic. Then, opion-pricing applicaions are like hose of coninuous-ime complee markes models, sudied by he lieraure in deail. For example, boh he hedging porfolio and he volailiy of he residual risk depend linearly on he opion s Delas, which can be used for risk managemen. 3 Fifh, recursive prices are equal o he price of a hedging porfolio, C (), plus a premium, C b C (). We explicily obain his decomposiion. C () is given by a proper risk-neural expecaion of he discouned opion payoff. This pricing measure is he so-called Minimal Maringale measure, Q h. This resul allows us o explicily separae he pricing of he risk which can be hedged from he pricingofheriskwhichcannobehedgedinadynamicconex. We derive furher resuls from his decomposiion. Le r be he risk-free ineres rae. Then, Z bc = E Qh e rt T C T + E Q h e r y d. The firs par is he price of he hedging porfolio. The second is he premium which depends on y. h Le y depend on λ R. The opion price elasiciy is given by dc R i dλ = d T dλ EQh e r y (λ)d. By using his decomposiion, we can define an upper and a lower bound by i Z T C = E Qh he r(t ) C T + ae Qh e r(s ) y s ds, where a = +1 (a = 1) for he upper (lower) bound. Assume ha y s, s [,T]. The upper bound is larger han he lower bound. Moreover, under echnical condiions, under he Q h probabiliy measure, he discouned upper (lower) bound is a super-maringale (sub-maringale), and he discouned price of he hedging porfolio, E Qh e rt C T, is he maringale componen. This resul is relaed o Ross (1978) and Harrison and Kreps (1979), bu in incomplee markes. 4 We are also ineresed in opion-pricing from a porfolio perspecive. In a complee marke, he price of a porfolio of (any kind of) n securiies is equal o he sum of he n individual prices, as linear pricing. In an incomplee marke, he porfolio can be less expensive if here is some diversificaion. We apply he decomposiion o a porfolio of n European securiies, C p T = C (1) T,C(2) T,...,C(n) T, " nx nx h i nx Z # T (i) C (i) = E Qh e rt C (i) T + E Qh e r y (i) d and i=1 i=1 i=1 nx h i Z d T C p = E Qh e rt C (i) T + E Qh e r y p, i=1 where y (i) and y p are he risk premium of every securiy and of porfolio p, respecively. Boh prices differ only in he valuaion of he residual risk. If he underlying securiies are no (insananeously) perfecly correlaed, here is some diversificaion in he (insananeous) residual risk of porfolio p, and consequenly, y p P n i=1 y(i) is a sensible specificaion for all. Thais,C p is cheaper. 3 See also Bersimas e al. (2) and Heah e al. (21) for more on muliperiod residual risks. 4 In a complee marke, a decomposiion resul, in an equivalen European opion plus an early exercise premium, also holds for American-syle securiies which are super-maningales under he pricing measure (see Carr e al. (1992)). 4
American-syle securiies and porfolio consrains are problems which have no been addressed in incomplee markes in general. For insance, one mus rely on numerical mehods o price American opions even in a complee marke. A recursive muliperiod model is a series of one-period models, easily o analyze. We adap he one-period definiion o American opions. The one-period porfolio hedges he maximum of he opion value and he exercise value in he nex period. The same way, he one-period risk premium compensaes he remaining one-period residual risk. 5 Assume shor-selling consrains. We can derive a volailiy smirk for in-he-money pu opions in he sandard Black-Scholes and Cox-Ross-Rubinsein binomial models. In-he-money pu opions canno be perfecly hedged (especially shor-erm opions) if here are shor-selling consrains. I is an example of marke incompleeness due o marke fricions. Thus, i is inuiive o assume a posiive risk premium associaed wih he residual risk. The deeper he opion in-he-money, he larger he residual risks and consequenly he larger he risk premiums and he volailiy smirk. The paper is organized as follows. Secion 2 presens he one-period incomplee marke model. Secion 3 sudies recursive bounds in muliperiod discree-ime incomplee markes. Secion 4 sudies recursive bounds in coninuous-ime incomplee markes for diffusion processes, and Secion 5 solves a few examples under basis risk, sochasic volailiy, and shor-selling consrains. Secion 6 concludes. 2 The One-period Model Assume he sandard one-period model of financial economics. Le be he iniial period and +1 be he final period. Le K be he number of saes, Ω = {ω 1,ω 2,..., ω K } he sae space, and P he rue probabiliy measure, wih P (ω) > for all ω Ω. There exiss a risk-free asse wih price process S =1and S+1 =1+r>, and N risky asses wih iniial prices S = {S 1,S 2,..., S N } and final prices S +1 = {S+1 1,S2 +1,..., SN +1 },whicharedefined on he space sae Ω. Assume ha here are no arbirage opporuniies. The objecive is o deermine he price C of a coningen claim, wih a final-period payoff given by C +1. Le H be he porfolio s or rading sraegy s space. In paricular, here are no consrains on his space (i.e., H = R N+1 ). Le h +1 =(h +1,h1 +1,..., hn +1 ), for h +1 H and h +1 chosen a ime, be a (hedging) porfolio wih value process X h = {X h,x h +1 }; i.e., Xh = h +1 + P N n=1 hn +1 Sn and X+1 h = h +1 (1 + r)+p N n=1 hn +1 Sn +1. Assume ha his marke is incomplee (i.e., K>N+1)and ha he payoff C +1 is no replicable. Equivalenly, here does no exis a porfolio h +1 such ha X h +1 (ω) =C +1(ω) for all ω Ω. LeC and C + be he wo arbirage bounds of his securiy, which solve wo linear programs (see Ingersoll (1987) or Pliska (1997)). Then he price C mus saisfy ha C <C <C + o avoid arbirage opporuniies. 6 5 American opions can be priced by simulaion (e.g., Longsaff and Schwarz (21)) using he risk-neural dynamic. 6 An example and source of marke incompleeness are porfolio consrains and ransacion coss. However, o 5
2.1 A Hedging Porfolio plus a Risk Premium-Based Approach Le Y h +1, he hedging error or residual risk produced by he porfolio h +1, bedefined as Y h +1 = X h +1 C +1. (1) Le b h +1 be an opimum porfolio associaed wih a hedging crierion f(y+1 h ); i.e., and X b h is price. We do no specify he funcion f(). b h+1 =arg min {h +1 } f(y h +1), (2) I is convenien o assume (we do i in he nex subsecion) ha b h +1 saisfies C <X b h <C +, which is equivalen o ha he hedging error Y b h +1 is posiive for some saes and negaive for ohers if he model is arbirage free, as we assume. For example, a hedging porfolio wih a zero expeced hedging error (i.e., E P Y h +1 =)saisfies his consrain as i has boh posiive and negaive errors. Le y be a risk premium associaed wih Y b h +1. Assume ha y does no inroduce arbirage opporuniies, i.e., C X b h <y <C + X b h. Equivalenly, C <X b h + y <C + and hus X b h + y is an arbirage free price. In paricular, y can be equal o zero if his risk is no priced. Le C <X b h + y <C +. This paper defines he incomplee marke price of C as he price of he hedging porfolio, X b h, plus he risk premium, y ; i.e., C = X b h + y. (3) Noe ha he price C in equaion (3) is compued in wo seps. Firs, X b h corresponds formally wih he applicaion of he law of one price nonarbirage condiion if we assume ha he residual risk is zero. Second, we add he risk premium y o compensae he residual risk Y b h +1. Moreover, y should be invesed in riskless bonds o no change he properies of he opimal porfolio b h +1. Again, we define he hedging sraegy as b h + y bonds and b h n risky asses for n =1, 2,..., N. Therefore, he oal risk assumed by he wrier of his securiy is X b h +1 C +1 + y (1 + r) =Y b h +1 + y (1 + r). (4) In an incomplee marke, one is ineresed in defining wo bounds, an upper (lower) bound, C s (C), l obained when hedging he shor (long) posiion; i.e., C +1 (C +1 ). Moreover, hese bounds should saisfy C s C l o make economic sense. Consider wo opimal hedging porfolios, b h +1 (s) and b h+1 (l), and wo risk premiums, y s and y, l associaed wih he hedging errors Y b h(s) +1 = X b h(s) +1 C +1 and Y b h(l) +1 = X b h(l) +1 C +1 for he shor and he long posiion s, respecively. Then hese wo bounds can be defined as in equaion (3); i.e., C s = X b h(s) + y s and C l = X b h(l) y l. (5) formalize hese cases (in H), we have o include new assumpions and definiions, and we lose he simpliciy of he acual formulaion. We prefer o deal wih hese problems on a case-by-case basis as we show in he examples below. 6
In paricular, X b h(s) X b h(l) and y s y l are sufficien condiions for C s C. l For example, b h+1 = b h +1 (s) = b h +1 (l) and y = y s = y l, i.e., he same porfolio and he same nonnegaive risk premium, imply ha C s = X b h + y C l = X b h y. 2.2 A Risk-Neural Pricing Formulaion In sandard fricionless markes, for boh complee and incomplee markes, an arbirage free price can be expressed as an expecaion under a risk-neural probabiliy measure (henceforh, RNP measure). By using RNP measures also, we are going o derive a relaed bu novel resul, for which is imporance is eviden in he muli-period model. Le Q be a RNP measure, and E Q [.] be he condiional expecaion operaor. Q saisfies S n = E Q Sn +1,n=1, 2,..., N. (6) 1+r Recall he implicaions of a RNP measure Q > (see, e.g., Pliska (1997)). Firs, he exisence of Q is equivalen o nonarbirage. Second, he uniqueness of Q is equivalen o marke compleeness. h i Third, le C = {C,C +1 } be he value process of an arbirary securiy. Then, if C = E Q C+1 1+r, C is an arbirage-free price. In paricular, if Q is unique, hen C is he unique arbirage-free price. Therefore, Q is a ool ha allows us o compue he price C as a simple risk-neural expecaion. This las poin is he one ha is imporan in he presen nonarbirage, incomplee marke conex. Recall ha porfolio b h +1 saisfies C <X b h <C +, hen here is a RNP measure Qb h such ha X b h = E Qb h C+1, (7) 1+r where he noaion Q b h highlighs he dependence on porfolio b h.thais,q b h allows us o compue he price of he hedging porfolio X b h by he risk-neural expecaion of he discouned payoff C +1. Consequenly, from equaion (3), he incomplee marke price of C can also be expressed as C = E Qb h C+1 + y. (8) 1+r In sum, from equaions (5) and (7), C = X b h + ay = E Qb h where a =+1( 1) for he shor (long) posiion and upper (lower) bound. C+1 + ay, (9) 1+r On he oher hand, if y 6=, here exiss a differen RNP measure Q b h,y such ha C = X b h + y = E Qb h,y C+1, (1) 1+r which can be used for pricing purposes, or o prove ha C is arbirage free. Noe ha Q b h Q b h,y depend on he payoff C +1. Therefore, differen from a complee marke where he unique RNP measure prices any securiy, in our model Q b h and Q b h,y mayonlypricehesecuriyc +1. 7 and
Remark 1. In addiion o he definiion of he price of an arbirary securiy in incomplee markes in equaion (3), equaion (8) is he main resul of he one-period model. The measure Q b h will allow us o derive an imporan resul on he decomposiion of he price C in a proper hedging porfolio plus a muliperiod risk premium in muliperiod markes. Remark 2. Alhough we have assumed a fricionless marke, he definiion of equaion (3) is independen of marke fricions such as porfolio consrains or ransacion coss. For equaions (7)(and(8))o hold,i is necessary o find a probabiliy measure which allows us o compue he price X b h as he discouned expecaion of C +1. For a fricionless marke, we have shown (since C <X b h <C + )haqb h is a RNP measure. For a fricion marke, we prefer o sudy his problem on a case-by-case basis. Remark 3. The resuls of he one-period model do no depend on a specific porfolio, h +1,and on a specific risk premium, y. We only assume ha C is arbirage free. In an incomplee marke and, in pracice, here can be differen funcions for b h and y, which depend of he problem of ineres. They can depend on he model s saisical properies (such as jumps, skewness or kurosis) or on economic facors (such as defaul, iniial wealh or regulaion issues). Noe ha y can also be sudied in an equilibrium or porfolio conex (e.g., if he hedging error can be diversified; see Meron (1976)). This is relevan in pracice, because C can be jus a derivaive from a larger porfolio of securiies. 7 Examples of opion pricing in incomplee markes are lef for he coninuous-ime model. 3 The Muliperiod Model Consider a discree-ime muliperiod model wih iniial ime, final ime T, andm rading daes such ha =, 1,..., M 1, and = T M. This muliperiod model is defined over a probabiliy space (Ω, F,P,{F }), wihω = {ω 1,ω 2,..., ω K } finie, where he sochasic processes S n are adaped and he hedging sraegies h n +1 are predicable wih regard o he filraion F, respecively, for =, 1,..., M and n =, 1,..., N. The risk-free asse corresponds wih a bank accoun wih value process S = {S,S 1,..., S M } = {1,er,..., e rt }. 8 Assume ha he model is arbirage free and incomplee. The objecive is o price a coningen claim C, whose payoff C M occurs in he las period and is no replicable. Le h = {h 1,h 2,...,h M } be a self-financing dynamic porfolio wih value process X h = {X h,xh 1,..., X h M }, where Xh = P N n= hn 1 Sn and Xh = P N n= hn S n for =1, 2,..., M. The aserisk denoes 7 See Hansen and Jaganahan (1991), Cerný and Hodges (21), Jaschke and Kücheler (21), and Longarela (21) for oher models of incomplee markes. See Arzner e al. (1999) for risk measures. 8 See Pliska (1997, chaper 3) for deails. In paricular, he same resuls hold if he ineres rae r is a predicable sochasic proces, i.e., he one-period shor-erm ineres rae r +1 is F measurable. 8
discouned values. I is well known ha a porfolio h is self-financing if i holds ha where X h M 1 X XM h = X h + X+1, h (11) = = e r X h, X+1 h = P N n=1 hn +1 Sn +1,and Sn +1 = e r(+1) S+1 n er S n is he discouned gain process for every risky asse n =1, 2,..., N. 3.1 A Hedging Porfolio plus a Risk Premium-Based Recursive Approach For a self-financing porfolio h, he hedging error is defined by Y h T = a X h M C M,wherea =+1 (a = 1) is he shor (long) posiion. Le us rewrie his hedging error, which is imporan for deriving he opimal hedging porfolio ha follows. Tha is, le Y h T = e rt Y h T Y h T M 1 X = a XM h CM = a X h C + a X+1 h C+1 = = be wrien as MX = Y h, (12) where C = {C,C 1,..., C M } is a F adaped sochasic process (o be specified below, excep for he mauriy payoff C M ). C = C C 1,C = e r C,and Y+1 h = e r(+1) Y+1 h = a X+1 h C +1 for =, 1,..., M 1 and Y h = a X h C. Consequenly, he oal hedging error, Y h, can be undersood as he sum of one-period replicaion errors Y h, =, 1,..., M. T In pracice, o find an opimum dynamic porfolio b h = { b h 1, b h 2,..., b h M } which minimizes a proper hedging crierion f YT h could be difficul. Moreover, he risk of he hedging error and is associaed risk premium, could also be difficul o quanify. Therefore, because of racabiliy (see foonoe 1) we consider only recursive bounds, which allows us o sudy general problems in incomplee markes. The opimal hedging crierion min f YT h = min f Y h + Y1 h +... + YM h e rt {h 1,h 2,...,h M } {h 1,h 2,...,h M } is changed as follows. Firs, consider a more simple recursive series of one-period hedging problems 9 min f Y+1 h = min f a X +1 h C +1 a X h C r e {h +1 } {h +1 } wherewedefine X h 1 = C 1 =. Second, for every define ha X h (13) for = M 1,M 2,...,, 1, (14) = C from he applicaion of he law of one price (since oherwise (14) produces an addiional hedging error, which can be hedged 9 Noe ha à N! X Y+1 h = a X+1 h C+1 e r(+1) = a h n +1 S+1 n C+1 à X N = a h n +1S+1 n e r n= X N h n +1S n n= n=1! a C +1 e r C e r(+1) = a X +1 h C +1 a X h C e r. 9
wih he bank accoun). In paricular, for = 1, X h = C. Then, he problems o solve are min f Y+1 h = min f a X +1 h C +1 {h +1 } {h +1 } which are similar o he previous one-period problem, wherein b h is compued. for = M 1,M 2,...,, (15) Therefore, his problem is solved recursively from = M 1 unil =. For every period, aking as given he previously solved C +1, wo iems are compued: he porfolio b h +1,which solves equaion (15), and he opion price C = X b h +1 + ay, where y is he risk premium per period associaed wih Y b h +1. Noe ha his recursion is well-defined since C M is known a mauriy, and ha C is F adaped. Noe also ha hese recursive prices (i.e., he value process) C = {C,C 1,...,C M 1 } are arbirage free if and only if each price deermined in each one-period model is arbirage free (equivalenly i does exis a RNP measure, see Pliska (1997)). However, his recursive porfolio b h derived from equaion (15) is no self-financing if he risk premium y +1 6=, or if he hedging error Y b h +1 6=, since b h +1 and b h +2 arechoseninwoindependen seps. The new noaion X b h +1 (insead of X b h = P N n= b h n S n ) is o disinguish beween X b h +1 +1 = P N b n= h n +1 Sn +1 and Xb h +2 +1 = P N b n= h n +2 Sn +1, and applies only o his non-self-financing porfolio. Consequenly, he non-self-financing porfolio b h mus be changed o a self-financing porfolio denoed by e h andwihvalueprocessx eh. Recall he definiions C = X b h +1 + ay and Y b h +1 +1 = a X b h +1 +1 C +1 for =, 1,...,M 1, and noe ha a Y b h +1 +1 = X b h +1 +1 C +1 since a 2 =1. Recall ha X h = e r X h is he discouned value. Tha is, a he iniial ime =, X e h = C = X b h 1 + ay, and e h n 1 = b h n 1 for n =1, 2,..., N and e h 1 = b h 1 + ay. A ime =1, A ime =2, X e h 2 = X b h 2 X e h 1 = X b h 1 1 + ay = X b h 1 1 + ay C1 + = X b h 2 1 + a Y b h 1 + y + y1,and e h n 2 = b h n 2 for n =1, 2,..., N and e h 2 = b h 2 + a 2 + a = X b h 3 2 + a X b h 2 1 + ay1 Y b h 1 + y + y 1 Y b h 1 + y + y1 = X b h 2 2 + a Y b h 1 + y + y1 Y b h 1 + y + Y b h 2 + y1 + y2,and e h n 3 = b h n 3 for n =1, 2,..., N and e h 3 = b h 3 + a. Y b h 1 + y + Y b h 2 + y 1 + y 2 C2 + X b h 3 2 + ay2. 1
In general, for any ime =, 1,..., M 1, and X e h X e h = X b h +1 X 1 + i= a Y b h i+1 + yi + ay, and (16) e h n +1 = b h n +1 for n =1, 2,..., N and e h +1 = b X 1 h +1 + M = X b h M M M 2 + X = i= a Y b M 1 X h +1 + y +aym 1 C M +CM = CM + a Y b h i+1 + yi + ay (17) = a Y b h +1 + y. (18) The following proposiion summarizes hese resuls and gives he muliperiod hedging error. Proposiion 1 Recursive prices based on every recursive price is defined as he price of a one-period hedging porfolio plus a risk premium associaed wih he one-period hedging error are consisen wih recursive one-period opimal self-financing hedging sraegies, which depend on he previously compued recursive prices, and where he one-period hedging errors, and risk premiums, are financed or invesed a he riskless rae. Thus, he muliperiod hedging error Y e h T, is he sum of he one-period hedging errors, plus he associaed risk premiums, financed or invesed a he riskless rae r, Y e h T = a X e M 1 X h M C M = a = a Y b h +1 + y e rt = M 1 X = Y b h +1 + y e rt. (19) Tha is, ac M = ax e h M Y e h T,whereaX e h M is he risk ha can be hedged and Y e h T ha canno be hedged. is he (residual) risk Recursive prices are defined as a generalizaion of equaions (3) and (8) in he one-period model, where b h is he previous non-self-financing porfolio solving (15). Tha is, C = X b h + ay, and (2) C = E Qb h + ay, (21) C+1 e r =, 1,..., M 1, and a =+1(a = 1) defines he upper (lower) price bound. As he one-period model, we assume ha C <X b h <C + and ha C <X b h + ay <C +, =, 1,...,M 1. The nex resuls show imporan properies of hese muliperiod recursive prices. A mauriy, he price is equal o C M. One period before mauriy M 1, he price is equal o C M 1 = E Qb h CM M 1 e r + ay M 1, or equivalenly, = X b h M 1 + ay M 1. 11
Two periods before mauriy M 2, C M 2 = E Qb h M 2 = E Qb h M 2 CM 1 e r CM e r2 + ay M 2 + a µ h y M 2 + E Qb h ym 1 M 2 by using he law of he ieraed expecaion E Qb h M 2 [C M]=E Qb h M 2 C M 2 = X b h M 2 + ay M 2 = X b h M 2 ae Qb h M 2 i, er, or equivalenly, E Qb h M 1 [C M] h ym 1 i µ h e r + a y M 2 + E Qb h ym 1 i M 2. er And, recursively, a he iniial period, we have he following resul. Theorem 2 If muliperiod prices are derived recursively and if one-period prices are equal o he price of a hedging porfolio plus a risk premium, as in equaions (2) and (21), hen C is as follows. " M 1 # C = E Qb h CM e rt + ae Qb h X y, or equivalenly, (22) er = " M 1 # " X M 1 # X = X b h ae Qb h =1 y er + ae Qb h = y er. (23) Similar o he one-period model, C can be divided in wo pars. Firs, " M 1 # E Qb h CM e rt = X b h ae Qb h X y, (24) er and second, which depends on he risk premium y. To undersand he erm E Qb h ae Qb h " M 1 X = =1 # y er, (25) h i h i CM we need an addiional assumpion. We advance ha E Qb h CM e rt e rt is relaed o he price of a hedging porfolio. We require ha every Q b h is independen of all he previous risk premiums y +1,y +2,..., y M 1. Noe ha C +1, and herefore b h +1, depend on all hese risk premiums because recursive pricing. Then, denoe by b h +1 (y =)he opimal porfolio when all risk premiums are zero, i.e., y +1 = y +2 =... = y M 1 =,andbyq b h(y=) measure. The Q b h(y=) Noe ha E Qb h(y=) implies ha E Qb h h CM e rt i is he appropriae RNP measure and we have he desired resul. h i CM = X b h(y=) e rt = E Qb h(y=) he associaed RNP from equaion (24). Therefore, he assumpion Q b h = Q b h(y=) h CM e rt i, and from equaion (24), E Qb h h CM e rt i = X b h(y=) is unique and well-defined. Tha is, he hedging porfolio corresponds wih he iniial recursive one-period hedging porfolio compued when all risk premiums are zero, b h(y =). If here are muliple RNP 12
n o n o n o measures Q bh and Q bh(y=),werequireha Q b h = Q b h(y=), where Q b h means he se of RNP measures which verify equaion (2) and (21). n o Consequenly, by assuming ha Q b h = C = X b h(y=) + ae Qb h(y=) " M 1 X = n Q b h(y=) # y = E Qb h(y=) er o,hen " M 1 CM e rt + ae Qb h(y=) X = # y er, (26) which is he muliperiod exension of equaion (9) in he one-period model. The condiion ha n o n o Q b h = Q b h(y=) depends on he one-period hedging crierion and on he risk premiums specificaion. We show ha his condiion holds in he coninuous-ime model for diffusion processes. 1 Remark 4. Theorem 2, and equaion (26), give a novel decomposiion of muliperiod recursive prices in incomplee markes. If Q hª = Q h(y=)ª, he recursive price of a non American-syle securiy, C, is equal o a risk-neural expecaion of he discouned payoff a mauriy (i.e., he price of a hedging porfolio) plus a risk-neural expecaion of he discouned one-period risk premiums (i.e., he muliperiod risk premium). In he nex secion, we show ha he spanned opion payoff does no depend of y, and hus (26) is no only a mahemaical decomposiion bu economic meaningful. We can derive an alernaive expression for C. Specify he risk premium as proporional o he price C,i.e.,y = α C. Now, insead of discouning he risk premiums, we reinves hem in he h i (1 aα ) 1 C +1,=, 1,..,M 1, e r opion C. Then, by again using equaion (21), i.e., C = E Qb h C = E Qb h (1 aα ) 1 C 1 e r (1 aα1 ) 1 C 2 = E Qb h =... = E Qb h EQbh 1 1 (1 aα ) à M 1 Y (1 aα ) = e r! 1 CM e rt e r = E Qb h (1 aα ) 1 (1 aα 1 ) 1 C 2 e r2. (27) The one-period risk premium α C is similar o a sochasic dividend flow paid by C,and from (27), C is equal also o he risk-neural expecaion of he discouned payoff a mauriy QM 1 adjused by hese reinvesed risk premiums. Noe ha = (1 aα ) 1 can be approximaed as exp{a P M 1 = α }. For example, if α is consan (α = α), C can approximaed as C e aαt E Qb h CM e rt. (28) 1 Neverheless, assume ha he densiy raio is finie, i.e., E Qb h CM e rt = E Qb h(y=) " dq b h C M dq b h(y=) e rt = X b h(y=) + E Qb h(y=) "à # = E Qb h(y=) dq b h dq b h(y=) 1 " h dq b < for all ω Ω, dqh(y=) b # dq b h dq b h(y=)! µ CM e rt E Qb h(y=) X b h(y=) #. CM e rt à + cov Qb h(y=) dq b h dq, CM b h(y=) e rt! 13
As equaion (1) in he one-period model, here exiss a differen RNP measure Q b h,y such ha C = X b h + ay = E Qb h,y h C+1 e r i he law of he ieraed expecaion, for =, 1,..., M 1 (since C C = E Qb h,y Consequenly, he muliperiod risk premium also saisfies ha ae Qb h " M 1 X = # y = E Qb h,y er <X b h + ay <C + ). Therefore, from CM e rt. (29) CM e rt E Qb h from equaions (22) and (29). Noe ha Q h = Q h,y if y =; i.e., Q h = Q h,. CM e rt, (3) Finally, if all he risk premiums are zero because, for example, he marke is complee (and b h is he replicaing porfolio), hen in all he equaions above we obain he very well-known resul, if y = y 1 =... = y M 1 =,henc = E Qb h CM e rt = X b h. (31) In sum, Proposiion 1 and Theorem 2 are wo imporan resuls, which disinguish our paper from he exan lieraure. For example, Cochrane and Saá-Requejo (2) bounds verify equaion (2) and are recursive (see heir Proposiions 1 and 2, respecively), and herefore, verify hese properies. 3.1.1 Recursive Prices in Coninuous Time Since e r =1+r + O 2, equaion (21) can be rewrien as 1 EQ bh [C +1 C ]+O ( ) =rc ay,=, 1,...,M 1. (32) Assume ha he N risky asses follow diffusion processes. Therefore, wih he help of Iô s Lemma and oher sochasic calculus ools, when lim, from (32), recursive bounds can be characerized hrough PDE s, once he sochasic processes for he sae variables, he hedging sraegy, and he risk premiums are specified. This PDE is derived in wo simple seps: firs, he risk-neural drif, h 1 EQb [C +1 C ],isequalorc, and second, he erm ay is subraced from his riskless reurn. This is formally proved in he nex secion. Noe ha equaion (32) can also be exended o sudy problems where he N risky asses follow sochasic processes more complex han diffusions. 11 3.1.2 American-syle Securiies in Incomplee Markes The valuaion of American-syle securiies in incomplee markes is a problem ha has no been addressed in he lieraure in general. The join deerminaion of an opimal dynamic self-financing porfolio e h and an opimal sopping-ime (or exercise policy) is a complex problem. However, he 11 The sudy of more general process is an imporan subjec of fuure research given he evidence of non-normaliy in reurns. Beyond diffusion models, see affine jump-diffusionmodelsinduffie e al. (2), or Carr and Wu (22). 14
recursive approach allows us o price American securiies easily, in a manner similar o oher recursive numerical mehods in a complee marke. Le I (S,E) be he inrinsic payoff wih C M = I (S M,E) and E he srike price. I is enough o subsiue he erm C +1 in he recursive hedging equaion (15) and hen in he recursive pricing equaions (2) and (21) wih max{i (S +1,E),C +1 }. Tha is, f a X+1 h max{i (S +1,E),C +1 },henx b h = E Qb h e r max{i (S +1,E),C +1 } and C = E Qb h e r max{i (S +1,E),C +1 } + ay, for =, 1,..., M 1, respecively. The oal residual risk is given now by P M 1 = Y b h +1 + y 1 {+1 bτ},wherebτ {1, 2,..., M} is he opimal sopping-ime defined by he firs τ such ha I (S τ,e) C τ. 3.1.3 Empirical Applicaions of he Hedging Errors We can devise empirical applicaions of he muliperiod hedging error Y e h T in equaion (19) as well. The muliperiod hedging error produces wo main empirical esable implicaions. Noe ha Y e h T = P M 1 = Y b h +1 + y has wo erms: he one-period hedging errors and risk premiums. Assume h i ha he condiional expecaion E P Y b h +1 =for =, 1,..., M 1. Firs, if he model being sudied is complee hen (here exiss a porfolio b h such ha) Y b h +1 =and he risk premium y =o avoid arbirage opporuniies. Consequenly, a period-by-period hedged porfolio has a zero (expeced) hedging error since Y e h T =. Second, if he model is incomplee and y 6=,hen h i a period-by-period hedged porfolio has an expeced hedging error E P Y b h T = P M 1 = EP [y ], which can be differen from zero. We have he following resul. h i Proposiion 3 Assume, firs, E P Y b h +1 =for =, 1,..., M 1, second, y =if and only if Y b h +1 =,andhirdy is a posiive or a negaive risk premium, bu is sign does no change from ime o M 1. Then, given N +1 hedging asses S n, n =, 1,..., N, he marke is incomplee if and h i only if E P Y b h T = P M 1 = EP [y ] 6=, i.e., a period-by-period hedged porfolio has an expeced hedging error differen from zero. This resul is dependen upon he number of hedging asses S n, n =, 1,..., N. For insance, assume a sochasic volailiy model. If he hedging asses are only he bond and he sock, hen he residual risk is he volailiy risk (i.e., Y b h +1 6=)andicanholdhay 6=, which is a volailiy risk premium. On he oher hand, if he hedging asses are he bond, he sock and a second opion, hen he residual risk is zero (i.e., Y b h +1 =)andy =in order o avoid arbirage opporuniies. h i 12 Finally, anoher ineresing applicaion is as follows. Assume ha E P Y b h +1 =for =, 1,..., M 1. Then every one-period hedging error Y b h +1 + y canberegressedonaseriesofvariables o analyze if he risk premium y is relaed o such variables. See he basis risk example below. 12 For insance, Fan, Gupa, and Richken (23) sudy how many facors are necessary o price swapions from he perspecive of hedging effeciveness insead of he sandard approach of pricing performance. 15
4 Recursive Prices in Coninuous Time Assume a vecor of K sae variables, S() =(S 1 (),S 2 (),..., S K ()), which follows he following diffusion process ds() =µ (, S()) d + Σ (, S()) dz, (33) where µ is a (K 1) vecor,σ is a (K K) marix, and z is a (K 1) vecor of independen Wiener processes. We assume ha µ (, S()) and Σ (, S()) saisfy growh and regulariy condiions such ha he process ds is well defined and has a unique soluion (see Duffie (21)). Le r() be he insananeous shor ineres rae, and r() =r be consan o save noaion. We assume ha only he firs N sae variables S 1 (),S 2 (),..., S N (), wih N K, are radable and consequenly he marke is incomplee if N<K.Forinsance,heS N+1 (),S N+2 (),..., S K () correspond wih illiquid asses, sochasic volailiy, ec., (see he nex secion s examples). We consider he pariion of he volailiy marix Σ =[A B ], where A and B conain he firs N and he las K N rows of Σ, respecively. We assume ha he rank of he marix A is equal o N (almos sure), i.e., here are no redundan radable asses. In paricular, his implies ha he model is arbirage free, and equivalenly, here exis muliple risk-neural probabiliy measures for he N radable asses (under echnical condiions, see Duffie (21)). We assume ha here are no porfolio consrains, which are a real source of marke incompleeness. Because he recursive approach is based on one-period opimizaion, porfolio consrains can be easily incorporaed (e.g., we price a pu opion under shor-selling consrains). Finally, we assume ha he rank of Σ is equal o K (almos sure), every sae variable represens a differen risk. Two special cases are N =where he only hedging insrumen is he risk-free asse (we define Σ = B), and N = K where he marke is complee (Σ = A and A is inverible). Alhough recursive prices can be derived in he limi from equaion (32) when, wefind easier o derive hem along he lines of Black and Scholes (1973) and Meron (1973). 4.1 The Hedging Sraegy Le C(, S()) and X h (S()) be he price of a derivaive securiy and he price of a hedging porfolio h, respecively, where X h (S()) = P N n=1 h n()s n (), h () =,andc(t,s(t )) is he European opion payoff a mauriy (wih he noaion slighly changed). By Io s lemma, dc and dx h saisfy dc = µ c d + C SΣdz and (34) dx h = µ h d + h Adz, (35) wih µ c = C + 1 2 KX i=1 j=1 Ã KX K! X C SS(i,j) Σ i,k Σ j,k + µ C S and µ h = µ (1:N) h, (36) k=1 16
and h() =(h 1 (),h 2 (),..., h N ()). Noe ha C S is he (K 1) vecor of firs derivaives and C SS is he (K K) marix of second derivaives, and we have suppressed he dependence of all variables on and S(). We assume ha C(T,S(T )) depends on all sae variables S. If C does no depend on some sae variables, he corresponding parial derivaives C S and C SS are equal o zero. Define he hedging error dy h = a h A CSΣ dz = a h A CS A B dz = a (h B C S(1:N) ) A CS(N+1:K) dz. (37) Because dc and dx follow diffusion processes and because of he fac ha coninuous rading is allowed, he infiniesimal one-period hedging errors are (condiionally) normally disribued and consequenly he appropriae, and unique, hedging crierion is o minimize he variance. Therefore, he hedging crierion f(dy ) is given by minimizing f(dy h )= 1 h i 2 d EP dy h (h = CS(1:N) ) A CS(N+1:K) B 2, (38) where k.k 2 is he Euclidean norm. Le denoe g = h C S(1:N). Then, he N orhogonaliy condiions h i (i.e., E P ds n dy b h =,n=1, 2,..., N) for his problem imply ha bg = AA 1 AB C S(N+1:K), and b h = C S(1:N) + bg (39) is he opimal minimum variance porfolio, and he marix AA is inverible since he rank of A is equal o N. Then, dx bh = µ bh d + b h Adz is he dynamics of he opimal hedging porfolio, and dy b h is he remaining residual risk. This residual risk dy b h = a bg B A CS(N+1:K) dz = acs(n+1:k) A B AA 1 A I dz (4) has hree componens: C S(N+1:K) measures he non-raded opion s risk, B are he volailiies of he non-raded variables, and (A (AA ) 1 A I) is relaed o he marke incompleeness. Noe ha B(A (AA ) 1 A I)dz is he risk which is non-spanned by S (1:N),and noe ha he opion s Delas, C S(N+1:K), can be used for risk managemen. 4.2 The PDE Equaion: he Law of One Price and he Risk Premium Firs, if we forge he residual risk dy b h, he law of one price implies ha, similar o he Black- Scholes-Meron model, he reurn of a riskless porfolio mus be equal o he riskless rae; i.e., Second, if dy b h a µ bh µ c = a Xbh C r, (41) 6=, we add a risk premium y o compensae he residual risk; i.e., a µ bh µ c = a Xbh C r + y, (42) 17
which is equaion (2) bu in coninuous ime. applicaion of non arbirage argumens wih complee markes. In oher words, he risk-reurn rade-off of y d + dy b h, i.e., N aracive for he wrier (or buyer) of he opion, where σ Y k If dy b h =, y =and we have he sandard µ y d, = C S(N+1:K) B q PK k=1 σ Y k 2 d,is A (AA ) 1 A I (k), k =1, 2,..,K, is he vecor of volailiies of he residual risk. The invesor in C obains an exra premium y d for carrying exra risk on dy b h. (Noe ha, under echnical condiions, his premium is arbirage-free due o he coninuous suppor of N (, d) over all he real line R). h i Noe furher ha he orhogonaliy condiions, E P ds n dy b h =, imply ha he risk premium y can be specified independenly of he radable asses ds n, highlighing he imporance of an opimal hedging porfolio. From he PDE equaion (42), a µ bh rx bh + y is he opion risk premium and y is he risk premium when he opion is hedged by porfolio b h (and recall ha a =+1(a = 1) for a shor (long) posiion). Moreover, µ bh rx bh is an endogenous risk premium relaed o he raded asses, whereas y is he exogenous risk premium associaed wih he residual risk. Equivalenly, since a 2 =1, he laer PDE equaion can be rewrien as µ c µ bh rx bh = rc ay, (43) which is similar o he risk-neural equaions (21) and (32) bu in coninuous ime. Moreover, we are ineresed in wo prices C s and C l,wihc s C l. Then, from he PDE equaion (43), a sufficien condiionisifha < y s y l < + (or equivalenly, + >y s y l >, and in paricular, + >y s = y l ), where y s (y) l is he risk premium associaed wih C s (C l ). Inuiively, a lower (higher) erm ay in he PDE equaion (43) implies a higher (lower) opion price. Noe ha i is he same condiion on equaion (5) in he one-period model. Finally, subsiuing µ c and b h in equaion (43), and noing ha µ C S µ bh rx h b = µ C S µ b (1:N) h rx h b = µ C S (µ rs) (1:N) bg + CS(1:N) = rs(1:n) C S(1:N) + µ (N+1:K) C S(N+1:K) (µ rs) (1:N) bg = rs(1:n) C S(1:N) + µ (N+1:K) (µ rs) (1:N) AA 1 AB C S(N+1:K), (44) hen equaion (43) is given explicily by KX C + 1 2 i=1 j=1 rs(1:n) C S(1:N) + µ (N+1:K) (µ rs) (1:N) Ã KX K! X C SS(i,j) Σ i,k Σ j,k + k=1 AA 1 AB C S(N+1:K) = rc ay. (45) 18
As noed before, he recursive approach can be generalized o include porfolio consrains or more general process han diffusions for he sae variables; i.e., an equivalen equaion o (43) can be derived along he same lines. For example, for jump-diffusion models, one needs a generalized version of Io s Lemma(see Duffie (21)) o derive equaion (34). Then, he one-period hedging problem is no simply he sandard minimum variance problem in equaion (38). By choosing differen risk premiums y, we can connec several opion-pricing models. Meron (1998) where y =. Recall ha σ Y (1:K) is he vecor of volailiies of he residual risk. In he risk-neural approach based on prices of risk, where we define y = P K k=1 σy k λ k. We can disinguish several cases. Firs, N =and here are no radable asses. Then, we define bg =, dy b h = acs(n+1:k) Bdz,andσ Y k C = S(N+1:K) B. In his case, he λ k (, S()) are naurally he (k) marke prices of risk associaed wih each orhogonal facor dz k.second, <N<K.Inhiscase, i is possible ha some of he λ k correspond wih radable asses (and herefore, σ Y k =), whereas oher are exogenous (e.g., he volailiy price of risk in a sochasic volailiy model). Third, complee markes, where N = K and A inverible. Then, dy b h =, σ Y k =for all k, and consequenly, y =. Cochrane and Saá-Requejo (2) bounds, where y = ea q PK k=1 σ Y k 2. Cochrane and Saá- Requejo show ha ea is a parameer relaed o a bound on he pricing kernel volailiy, and equivalenly, on he maximum Sharpe raio. Indeed, from our equaion (43), ea is he Sharpe raio of he hedged opion, i.e., he marke price of risk of he residual risk. In he local risk minimizaion approach (see Heah, Plaen and Schweizer (21)), where y =. 4.3 The RNP Measures Q h and Q h,y From equaions (33) and (41) we exrac he RNP measure Q h. This is one of he innovaions of our paper. The lieraure exracs he RNP measure Q h,y from equaion (33) and (43). Q h allows o separae he recursive price in he price of a hedging porfolio plus a premium, which has a naural inerpreaion in incomplee markes. Q h,y is bes used for pricing purposes and o prove no arbirage. Noe ha Q h = Q h,y if y =; i.e., Q h = Q h,. Le Q be a RNP measure. Q can be characerized hrough he Radon-Nikodyn derivaive, i.e., where ξ T is he sae price densiy, dq dp = ξ T, ξ =1and dξ = λ ξ dz for [,T], and λ is a vecor of prices of risk (and he Novikov s condiion holds, E hexp 12 R i T λ λ d < ). For compleeness, we recall a sandard resul of pricing by arbirage in fricionless markes (see Duffie (21, 111-114) for echnical deails). Under echnical condiions, he following hree properies 19
are equivalen, (a) a well-defined marke prices of risk process λ, (b) he exisence of a risk-neural probabiliy measure Q, and (c) non arbirage. I holds for boh complee and incomplee markes. Clearly, for he N radable asses, he risk-neural drif mus be equal o he riskless rae r o avoid arbirage opporuniies, and herefore, µ (1:N) Aλ = rs (1:N). (46) For he res of nonradable asses, he risk-neural drif is implici in he PDE pricing equaion (45). To exrac i, we mus look a he loading of he vecor C S. Tha is, hey are obained from he erm rs(1:n) C S(1:N) + µ (N+1:K) (µ rs) (1:N) AA 1 AB C S(N+1:K) + ay. (47) Noe ha he risk-neural drif of he N radable asses is equal o r; i.e., he PDE equaion derived from he opimal minimum variance porfolio, b h, is consisen wih equaion (46). Define he vecor D k N = ay C α S(k) k 1 {CS(k) 6=}, k = N +1,..., K, andp K k=n+1 α k1 {CS(k) 6=} =1. Then, ay = D C S(N+1:K) (in paricular, D =if y =) and equaion (47) can be rewrien as rs(1:n) C S(1:N) + µ (N+1:K) (µ rs) (1:N) AA 1 AB D C S(N+1:K). (48) Therefore, he risk-neural drif of he nonradable asses is given by he loadings of C S(n+1:K) ;i.e., µ (N+1:K) Bλ 1 {CS(N+1:K) 6=} = µ (N+1:K) BA AA 1 (µ rs)(1:n) D 1 {CS(N+1:K) 6=}, (49) wih 1 {CS(N+1:K) 6=} a vecor of indicaor funcions. Equivalenly, A I λ = N N B1 {CS(N+1:K) 6=} BA 1 (µ rs) (AA (1:N) + (1:N). ) 1 {CS(N+1:K) 6=} D1 {CS(N+1:K) 6=} (5) Since he rank of he marix A and Σ is equal o N and K, respecively, hen AA and Σ =(A B ) are inverible and he sysem has well defined soluions. Noe ha he marke prices of risk, λ, do no depend on he drif of he nonradable sae variables, µ (N+1:K). Le us show a few examples. If here are no hedging asses, N =,andb1 {CS(N+1:K) 6=} λ = D1 {C S(N+1:K) 6=}. If he marke is complee, K = N, andλ = A 1 (µ rs). y C S(N+1) If K = N +1, λ and Q donodependonheopionbeingsudiedif does no (e.g., if y =). If K = N +1, λ and Q are unique. For example, if K = N +1and C S(N+1) 6=,he sysem (5) simplifies o Aλ =(µ rs) (1:N) and Bλ = BA AA 1 y (µ rs) (1:N) a. (51) C S(N+1) 2
However, if K>N+1, λ and Q depend on he opion being sudied from he erm 1 {CS(N+1:K) 6=}. Moreover, if K>N+1,fory =, λ and Q are unique if he rank of C S(N+1:K) is equal o K N. Fory 6=, λ and Q are no unique even if he rank of C S(N+1:K) is equal o K N since he weighs α in D are arbirary. For insance, assume ha he N radable asses only depend on he firs N sae variables, i.e., A =[A 1 A 2 ] where A 1 conains he firs N columns of A and A 2 is a marix of zeros (i.e., A 2 = N K N ). Pariion he marix B =[B 1 B 2 ] where B 1 conains he firs N columns of B. Then, 13 he sysem (5) simplifies o λ (1:N) = A 1 1 (µ rs) (1:N) and B 2 λ (N+1:K) 1 {CS(N+1:K) 6=} = D1 {C S(N+1:K) 6=}. (52) Moreover, if 1 {CS(N+1:K) 6=} 6=, wih λ (N+1:K) = (1:K N) if y =. λ (1:N) = A 1 1 (µ rs) (1:N) and λ (N+1:K) = B2 1 D, (53) Finally, if y =hen D =and equaion (5) is given by A I λ = N N B1 {CS(N+1:K) 6=} BA 1 (µ rs) (AA (1:N), (54) ) 1 {CS(N+1:K) 6=} which is independen of he risk premiums y s, < s T, since his equaion does no depend on C(, S()). Consequenly, he recursive price can be divided in he price of he iniial one-period hedging porfolio wih all zero risk premiums, plus a muliperiod risk premium, which is shown nex. 4.4 Risk Managemen Firs, he oal hedging error is simply he sum of he one-period hedging errors plus he one-period risk premiums, financed or invesed a he riskless rae r. The dynamic of a self-financing porfolio b h saisfies X bh (T )=Xb () + R T h dx b(), and C (T,S(T )) = C (,S()) + R T h dc (, S()). Wedefine Xb () = C (,S()). Therefore, h 13 Bλ1 {CS(N+1:K) 6=} = BA AA 1 (µ rs) (1:N) 1 {CS(N+1:K) 6=} + D1 {C S(N+1:K) 6=} B1λ (1:N) + B 2λ (N+1:K) 1{CS(N+1:K) 6=} = B1A 1 1 A 1λ (1:N) 1 {CS(N+1:K) 6=} + D1 {C S(N+1:K) 6=} B 2λ (N+1:K) 1 {CS(N+1:K) 6=} = D1 {C S(N+1:K) 6=}. 21