Jump-Diffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach

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1 ump-diffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono, On., M5X 9, Canada, phone: (03) , ohn Molson School of Business, Concordia Universiy, 455 de Maisonneuve Blvd. W, Monreal, QC H3G M8, Canada, Phone: (+-54) ex. 963, Fax: (+-54) , Perrakis acknowledges he suppor of SSHRC, he Social Sciences and Humaniies Research Council of Canada, and boh acknowledge he suppor of IFM, he Insiu de Finance Mahémaique de Monréal.

2 ump-diffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach Absrac We presen a new mehod of pricing plain vanilla call and pu opions when he underlying asse reurns follow a jump-diffusion process. The mehod is based on sochasic doance insofar as i does no need any assumpion on he uiliy funcion of a represenaive invesor apar from risk aversion. I develops discree ime muliperiod reservaion wrie and reservaion purchase bounds on opion prices. The bounds are valid for any asse dynamics and are such ha any risk averse invesor improves her expeced uiliy by inroducing a shor (long) opion in her porfolio if he upper (lower) bound is violaed by he observed marke price. The bounds are evaluaed recursively and he limiing forms of he bounds are found as ime becomes coninuous. I is found ha he wo bounds end o he common limi equal o he Black-Scholes-Meron (BSM) price when here is no jump componen, bu o wo differen limis when he jump componen is presen. Keywords: opion pricing; incomplee markes; sochasic doance; jump-diffusion

3 I. Inroducion The presence of exreme evens or jumps in he probabiliy disribuion of sock reurns is one of he earlies examples in he financial lieraure of a dynamically incomplee marke. As Meron (976) was he firs o poin ou, in such markes arbirage mehods are unable by hemselves o price coningen claims wihou addiional assumpions abou financial marke equilibrium. In modeling such equilibrium he exreme evens, which are by definiion rare, are priced by inroducing a represenaive invesor whose porfolio decisions define he marke s pricing mechanism. The resuling opion prices are almos always funcions of he invesor s aiudes owards risk, as well as of he parameers of he reurn disribuions of he underlying asse. In his paper we inroduce a differen approach o coningen claims pricing in he presence of rare even risk. We derive opion prices ha are funcions of he parameers of he underlying asse reurn disribuion and do no depend on he aiude owards risk of a represenaive invesor. Since he financial markes are incomplee he prices ha we derive are no unique bu lie in an inerval whose widh is a funcion of he risk premium of he underlying asse; 3 excep for his parameer, he prices ha we derive use he same informaion se as he exising approaches. We firs derive resuls for index opions and hen exend hem o individual sock opions. Our numerical resuls indicae ha for reasonable values of he parameers of he reurn process he widh of he inerval does no exceed 8% for a-he-money S&P 500 index opions. The main moivaion for our approach is he fac ha invesor aiudes owards risk are no direcly observable and heir esimaes are nooriously unsable and unreliable. Esimaes exraced from empirical research show a complee disconnec beween opionbased models and sudies ha model direcly he consumpion of represenaive invesors, wih he laer esimaes being as much as en imes as large as he former! 4 We discuss in he nex secion he effecs of hese large esimaes on he radiional approach o opion pricing by he represenaive invesor. On he oher hand we noe ha he realized reurn of he underlying asse is cerainly observable and ha here are saisical mehods o exrac efficien esimaes of he parameers of he acual disribuion of he reurns of ha asse, which in many empirical sudies is assumed o be a mixure of a diffusion process (possibly wih sochasic volailiy), and rare evens following Poisson arrivals wih log-normally disribued jump ampliudes. 5 3 While he risk premium is iself deered in equilibrium by he aiudes owards risk of he invesors in he marke, we adop very weak assumpions for his equilibrium, such as no requiring he exisence of a represenaive invesor or paricular ypes of uiliy funcions. 4 Baes (99) uses a value of for his opion pricing resuls, while economeric sudies in opion markes by Engle and Rosenberg (00) and Bliss and Panigirzoglou (004) repor findings of o and.97 o 7.9 respecively. On he oher hand Kandell and Sambaugh (99) give esimaes as high as 30, while he equiy premium puzzle lieraure repors empirical findings of 4 in Mehra and Presco (985) and more han 35 in Campbell and Conchrane (999). See also he survey aricle by Kocherlakoa (996). 5 There is a large economeric lieraure on he esimaion of he parameers of such mixed processes from hisorical daa. See, in paricular, Ai-Sahalia (004), Baes (000), Eraker e al (003), and Tauchen and Zhou (007). Noe ha our resuls do no depend on he assumpion of lognormal jump ampliude disribuions. 3

4 Our resuls are developed in a muliperiod discree ime formulaion of marke equilibrium in an incomplee markes seup and for any ype of asse dynamics. We assume ha here exiss a cerain class of risk averse raders in he financial markes who hold only he underlying asse, he riskless asse and (possibly) he opion. 6 For such raders we derive sochasic doance reservaion wrie and reservaion purchase opion prices, implying ha any risk averse rader would improve her uiliy funcion by inroducing he corresponding shor or long opion in her porfolio if he marke price exceeds he corresponding bound. Since marke equilibrium is incompaible wih he exisence of such doan sraegies, i follows ha in equilibrium he observed opion prices should lie beween he derived bounds. For our assumed class of raders he marke equilibrium condiions imply ha he pricing kernel in he discree ime seup is monoone decreasing wih respec o he underlying asse reurn wihin a single rading period, bu no necessarily when here are more han one periods o opion expiraion. From his propery we exrac wo boundary risk neural disribuions under which he expeced opion payoffs evaluaed recursively for any ype of periods o expiraion define he region of admissible opion prices. We also inroduce a suiable discreizaion of he reurn process ha ends o a desired coninuous ime process a he limi of coninuous rading. The inerval of opion values in coninuous ime under jump-diffusion asse dynamics is hen found by considering he coninuous ime limis of hese expeced payoffs under he wo boundary disribuions. We also show ha he inerval shrinks o a single value, he Black-Scholes-Meron (BSM, 973) opion pricing model, when he jump componen of he reurn is se equal o zero. We noe ha he assumpion abou he monooniciy of he pricing kernel in a single rading period, he only propery ha is necessary for he derivaion of our bounds, is also saisfied in all he models of coningen claims pricing under jump-diffusion processes ha have appeared in he lieraure so far. In he equilibrium models ha use a represenaive invesor wih a CPRA uiliy his monooniciy is implied by he join dynamics of he sock reurn and he wealh of he represenaive invesor, as well as by he shape of he indirec uiliy funcion. 7 These feaures also preserve he monooniciy in equilibrium models wih more elaborae assumpions ha include behavioral consideraions such as uncerainy aversion and recursive uiliy. 8 Unlike sochasic doance, hese approaches assume specific expressions for he pricing kernel. Our resuls are closely relaed o earlier sudies of opion pricing in incomplee markes in a discree ime framework. These were inroduced by Perrakis and Ryan (984), wih imporan exensions by Richken (985), Levy (985), Perrakis (986, 988) and Richken and Kuo (988). Consaninides and Perrakis (00, 007) inroduced proporional ransacion coss ino hese models and showed ha hey were capable of producing useful resuls under such condiions, unlike arbirage- or equilibrium-based models. Empirical applicaions of his discree ime approach o he pricing of S&P This assumpion is evenually relaxed o allow he derivaion of sock opions. 7 See, for insance, Baes (99), A and Ng (993) and A (993). 8 See Liu, Pan and Wang (005). 4

5 index opions were presened in Consaninides, ackwerh and Perrakis (009). This paper provides a linkage beween hese discree ime models and he coninuous ime framework in which mos opion pricing work was done. The paper is organized as follows, wih all proofs consigned o he appendix. In he nex secion we presen he discree ime represenaion of he sock reurns and inroduce he Lindeberg condiion, he weak convergence crierion o coninuous ime processes. We show ha under his crierion our represenaion converges o a generalized version of he mixed jump-diffusion process in coninuous ime, wih possibly ime- or sae-dependen parameers and independen Poisson arrivals of he jump componens wih any ype of disribuion for heir jump ampliudes. In secion 3 we presen he discree ime sochasic doance opion pricing resuls and apply hem o our represenaion of he sock reurns. We inroduce he discree ime financial marke equilibrium based on he monooniciy of he pricing kernel under general condiions abou he uiliy funcion of a risk averse invesor and for an unspecified reurn disribuion. We hen derive he wo maringale probabiliy disribuions, ransforms of he physical reurn disribuion, ha define he upper and lower bounds in a muliperiod conex for a disribuion wih imedependen reurns wihin which he admissible opion prices lie. These ransforms are applied in secion 4 o he discree ime represenaion of our reurn process inroduced in secion in order o derive he wo maringale probabiliies defining he upper and lower bounds of admissible opion prices in discree ime. We hen apply he convergence crierion presened in secion o hese wo boundary risk neural disribuions, firs for he simple diffusion case and hen for jump-diffusion. We show ha in he diffusion case he wo bounds end o he same limi, he BSM model, hus esablishing he discree ime opion bounds as generalizaions of he Cox-Ross-Rubinsein (979) binomial model. For jump-diffusion we show ha he opion bounds arise in closed form as soluions o wo disinc parial differenial equaions. Secion 5 exends hese resuls o sock opions by allowing he raders o hold marginal posiions in individual socks and heir opions, while he las secion discusses he exension of he resuls o reurns combining sochasic volailiy wih jumps and concludes. In he remainder of his secion we complee he lieraure review. ump-diffusion processes were firs inroduced ino opion pricing models by Meron (976), who derived a unique opion price by assug ha he rare even risk was fully diversifiable and hus no priced. This assumpion was clearly unenable following he 987 sock marke crash, and he nex generaion of opion pricing models refleced aemps o relax i. Baes, (988, 99) inroduced a general equilibrium model wih a represenaive invesor wih CPRA uiliy, a modified version of Cox-Ingersoll-Ross (CIR, 985) ha incorporaed correlaed jump-diffusion processes in boh he underlying asse and he invesor wealh. Essenially he same model was also used by A and Ng (993) and A (993). A sandard feaure of hese valuaion models was he dependence of he derived opion prices on several parameers over and above he ones defining he jump-diffusion reurn disribuion: he mean and variance of he log-ampliudes of he jump componens in he invesor wealh disribuion; he covariance of hese same ampliudes wih he 5

6 corresponding measures in he reurn disribuion; he risk aversion parameer of he represenaive invesor. 9 These parameers modify he jump inensiy and expeced logampliude parameers of he physical reurn process. In he more general model by Liu, Pan and Wang (005) of opion pricing under jump-diffusion ha includes uncerainy aversion he maringale probabiliy used in valuing coningen claims is also modified by he uncerainy aversion parameers of he uiliy funcion of he represenaive invesor. 0 In empirical ess he jump-diffusion model is ofen included in a nesed model ha also includes sochasic volailiy. Many of hese ess were moivaed by he need o explain he well-known volailiy smile originally documened by Rubinsein (994); only a few will be menioned here. Baes (996) applied he nesed models o Deusche mark currency opions, Baes (000) o S&P 500 fuures opions, Pan (00) and Rosenberg and Engle (00) o S&P 500 index opions, and Bliss and Panigirzoglou (004) o FTSE 00 and S&P 500 index fuures opions. In hose ess he parameers of he implied risk neural disribuion are exraced from cross secions of observed opion prices and aemps are made o reconcile hese opion-based disribuions wih daa from he marke of he underlying asse. All sudies sress he imporance of jump risk premia in hese reconciliaion aemps. The resuls of his paper, by inroducing an addiional se of requiremens in he valuaion of opions under jump-diffusion have obvious implicaions for fuure empirical work. II. The Sock Reurn Model We consider a marke wih an underlying asse (he sock) wih curren price S and a riskless asse wih reurn per period equal o R. There is also a European call opion wih srike price K expiring a some fuure ime T. Time is iniially assumed discree r = 0,,..., T, wih inervals of lengh, implying ha R = e = + r + o( ), where r denoes he ineres rae in coninuous ime. In each inerval he underlying asse has S+ S reurns S z +, whose disribuion may depend on S. 9 See, for insance, equaion (0) in Baes (99), equaion (7) of A and Ng (993), or equaion (33) of A (993). If he models are applied o index opions wih he index considered a proxy for invesor wealh hen he need for separae esimaes of wealh disribuion parameers disappears, bu he risk aversion parameer remains in he expressions. 0 See heir equaions (9)-(). As Eraker e al (003, p. 94) poin ou, he join esimaion of parameers from boh sock and opion daa does no necessarily reduce he uncerainy in he esimaes, unless he jump risk premia are arbirarily resriced. Our resuls provide objecive mehods of resricing such premia and, hus, improve he esimaes. More complex dependence of he reurns on oher underlying sae variables like, for insance, sochasic volailiy will be discussed in subsequen secions. 6

7 We model he reurns as a sum of wo componens, one of which will end o diffusion and he oher o a jump process. Wih probabiliy λ he reurn has he following form 3 z = [ µ ( S, ) λ µ ] + + σ ( S, ) ε. (.) In his expressionε has a bounded disribuion of mean zero and variance one, ε D(0,) and 0 < ε ε ε max, bu oherwise unresriced. Wih probabiliy λ here is a jump in he reurn. The jump is a random variable wih disribuion D wih mean µ and varianceσ. In mos of he lieraure i is assumed ha = and D is lognormal, ha is ln( + ) has a normal disribuion. Alhough our resuls may be exended o allow for dependence of boh jump inensiy and jump ampliude disribuion on S, we shall adop he assumpion ha he jump process is saeand ime-independen, wih λ = λ, =. On he oher hand he disribuion D is resriced only by he requiremen ha he reurn canno be less han -, alhough we assume, wihou loss of generaliy, ha he variable akes boh posiive and negaive values wih < 0 < max. Wih his specificaion he reurn becomes, if we se µ ( S, ) µ, σ ( S, ) σ z = ( µ λµ ) + + σ ε + N. (.) where N is a Poisson couning process wih inensiy λ. An alernaive represenaion of he discreizaion (.), which will also be used in he proofs, is he descripion of is oucomes ( µ λµ ) + σ ε wih probabiliy λ z + =, (.a) ( µ λµ ) + σ ε + wih probabiliy λ The radiional approach o he pricing of index opions when he index reurns are given by (.) or (.a) is o value he opions as he discouned expecaions of heir payoffs under a risk-adjused disribuion of he form * z+ = ( r λ * µ ) + σ ε + * N, where boh λ * and * µ have been disored by he risk aversion parameerγ of a CPRA uiliy funcion; see Baes (99, p. 034). In paricular λ* = λ exp( γµ + γ ( + γ ) σ ), which is approximaely equal o λ if γ andσ are small. For he esimaed parameer values of he jump-diffusion esimaes for he S&P 500 of he recen sudy of Tauchen and Zhou (007) of 3 For simpliciy dividends are ignored hroughou his paper. All resuls can be easily exended o he case where he sock has a known and consan dividend yield, as in index opions. In he laer case he insananeous mean in (.) and (.) is ne of he dividend yield. 7

8 λ = 0.3, σ = 0.54, µ = 0.05 he parameer λ * becomes very large for all bu he smalles risk aversion esimaes menioned in foonoe 4 of he previous secion and corresponds o unreasonably large opion values. In his secion we presen he condiions ha esablish he convergence of he processes described by (.) and (.) respecively o diffusion and o a mixed jump-diffusion process. In he nex secion we discuss he marke equilibrium and derive he discree ime bounds on admissible opion values suppored by such equilibrium. The convergence crieria presened here are hen applied o he processes under which hese bounds are derived in order o find he opion values under coninuous ime diffusion and mixed processes. To prove he convergence of opion prices, we rely on he weak convergence of he underlying price process, firs o a diffusion and hen o a jump diffusion. For any number m of ime periods o expiraion we define a sequence of sock prices { S, m} and an associaed probabiliy measure P m. The weak convergence propery for such processes 4 sipulaes ha for any coninuous bounded funcion f we mus m have E P f ( S m ) E P [ f ( S )] T T, where he measure P corresponds o diffusion limi of m he process, o be defined shorly. Pm is hen said o converge weakly o P and S T is said o converge in disribuion o S. A necessary and sufficien condiion for he T convergence o a diffusion is he Lindeberg condiion, which was used by Meron (99) o develop crieria for he convergence of mulinomial processes. In a general form, if φ denoes a discree sochasic process in d-dimensional space he Lindeberg condiion saes ha a necessary and sufficien condiion ha φ converges weakly o a diffusion, is ha for any fixed δ > 0 we mus have lim Q ( φ, dϕ) = 0 (.3) 0 φ ϕ > δ where Q ( φ, dϕ) is he ransiion probabiliy from φ = φ o φ + = ϕ during he ime inerval. Inuiively, i requires ha φ does no change very much when he ime inerval goes o zero. When he Lindeberg condiion is saisfied, he following limis exis lim ( ) (, ) ( ) (.4) 0 i i Q d i ϕ φ φ ϕ µ φ ϕ φ < δ = 4 For more on weak convergence for Markov processes see Ehier and Kurz (986), or Srook and Varadhan (979). 8

9 lim ( )( ) (, ) ( ) (.5) 0 i i j j Q d ij ϕ φ ϕ φ φ ϕ σ φ ϕ φ < δ = The condiions (.3), (.4) and (.5) are equivalen o he weak convergence of he discree process o a diffusion process wih he generaor 5 d d A = σ + µ. (.6) ij i i=, j= φi φ j i= xi By he definiion of he generaor, for each bounded, real valued funcion f we have Ef ( φ ) f ( φ ) = A f (.7) lim + 0 In our case he sae variable vecor is one-dimensional andφ = we have he following resul, proven in he appendix. S. Wih hese definiions Lemma. For λ = 0 he discree process described by equaion (.) converges weakly o he following diffusion (.8), where W is a Wiener process wih E( dw ) = 0 and Var( dw ) = d ds S = µ d + σ dw. (.8) I can be easily seen ha he process given by (.) does no saisfy he Lindeberg condiion, since lim Q ( ) dd ( ) lim dd( ) δ = λ + ε 0 z > δ 0 z > δ As shown in he proof of Lemma for he diffusion case, he second inegrand is zero for sufficienly low. However, he firs inegrand is sricly posiive for any, implying ha he process does no converge o diffusion in coninuous ime. The limi resul for (.) is given by he following lemma, proven in he appendix. Lemma. The discree process described by (.) converges weakly o he jumpdiffusion process (.9): ds S = ( µ λµ ) d + σ dw + dn. (.9) Having esablished ha (.) and (.) are valid discree ime represenaions of a mixed 5 See for insance Meron (99) for a discussion on he generaors of diffusions and jump processes. 9

10 process and is diffusion componen, we now urn o coningen claims pricing for such processes. III. The Opion Pricing Model in Discree Time Excep for he rivial case where he random variableε akes only wo values he marke for he sock wih reurns given by (.) is incomplee in a discree ime conex. The same is of course rue a foriori for he sock wih reurns given by (.). The valuaion of an opion in such a marke canno yield a unique price. Our marke equilibrium is derived under he following se of assumpions ha are sufficien for our resuls: There exiss a leas one uiliy-maximizing risk averse invesor (he rader) in he economy who holds only he sock and he riskless asse 6 This paricular invesor is marginal in he opion marke The riskless rae is non-random 7 Each rader holds a porfolio of x in he riskless asse and y in he sock by maximizing recursively he expeced uiliy of final wealh 8 over he periods = 0,,..., T ' of lengh. The curren value funcion is Ω ( x + y S ) = Max E[ Ω(( x v ) R + ( y + v )( + z ) S ], v + where v denoes he opimal porfolio revision or sock purchase from he riskless accoun. If he rader also has a marginal open posiion in a given call opion wih value C ( S ) and wih eral condiion CT = ( ST K ) + a opion expiraion ime T < T ' 9 hen he following relaions characerize marke equilibrium in any single rading period (, + ), assug no ransacion coss and no axes: E Y z S R E z Y z S [ ( + ) ] =, [( + + ) ( + ) ] =, C ( S ) = E[ C ( S ( + z )) Y ( z ) S ] + + +, (3.) In (3.) Y ( z + ) represens he pricing kernel, he sae-coningen discoun facor or normalized marginal rae of subsiuion of he rader evaluaed a her opimal porfolio choice. Because of he assumed risk aversion and porfolio composiion of our raders i can be easily seen ha he pricing kernel would possess he following propery: 6 This assumpion is relaxed in Secion V. 7 Alhough his assumpion may no be jusified in pracice, is effec on opion values is generally recognized as or in shor- and medium-lived opions. I has been adoped wihou any excepion in all equilibrium based jump-diffusion opion valuaion models ha have appeared in he lieraure. See he commens in Baes (99, p. 039, noe 30) and A and Ng (993, p. 89). In order o evaluae he various feaures of opion pricing models, Bakshi, Cao and Chen (997) applied wihou deriving i a riskneural model feauring sochasic ineres rae, sochasic volailiy and jumps. They found ha sochasic ineres raes offer no goodness of fi improvemen. 8 The resuls are unchanged if he raders are assumed o maximize he expeced uiliy of he consumpion sream. 9 All resuls in his paper are derived for call opions. They are applicable wihou reformulaion o European pu opions, eiher direcly or hrough pu-call pariy. 0

11 Propery: The pricing kernel Y ( z + ) is monoone (eiher non-increasing or nondecreasing) in he sock reurn z + for every = 0,,..., T. This propery is sufficien for he derivaion of igh opion bounds for all sock reurn disribuions and no only hose given by (.). I can be easily seen from he second relaion in (3.) ha under such an assumpion Y ( z + ) mus be non-increasing if he opioned sock is a posiive bea one, wih expeced reurn exceeding he riskless rae, since his implies ha he rader will always hold a posiive amoun of he sock. Since his is he case for he overwhelg majoriy of socks, his is he assumpion ha will be adoped here. The same se of assumpions also underlies he sochasic doance opion bounds of Consaninides and Perrakis (00, 007) ha were derived under general disribuional assumpions and included proporional ransacion coss. These assumpions may be resricive for opions on individual socks, bu heir validiy in he case of index opions canno be doubed, given ha fac ha numerous surveys have shown ha a large number of US invesors follow indexing sraegies in heir invesmens. 0 These marke equilibrium assumpions are quie general, insofar as hey allow he exisence of oher invesors wih differen porfolio holdings han he rader. They do no assume he exisence of a represenaive invesor, le alone one wih a specific ype of uiliy funcion. The resuls presened in his secion are derived for unspecified discree ime asse dynamics, and are applied o he specific case of jumpdiffusion in he nex secion. A more resricive se of marke equilibrium assumpions underlies he well-known jumpdiffusion opion valuaion models of Baes (988, 99) and A and Ng (993), and heir more recen exension by Liu, Pan and Wang (005) ha include behavioral consideraions. In he case of index opions hese sudies assume ha here is a represenaive consumer wih a ime-addiive CPRA uiliy funcion of consumpion over a finie or an infinie horizon. In all hose models he indirec uiliy is a concave funcion of invesor wealh, implying ha he marginal uiliy is decreasing. If, as is commonly he case, he correlaions beween boh diffusion and jump componens of he sock and he wealh processes are posiive hen he condiional expecaion of he marginal uiliy given z + is decreasing, implying clearly ha Y ( z + ) is non-increasing. Thus, he bounds derived by our marke equilibrium assumpions are also applicable o hese earlier models as well. 0 Bogle (005) repors ha in 004 index funds accouned for abou one hird of equiy fund cash inflows since 000 and represened abou one sevenh of equiy fund asses. In he case of sock opions he marke equilibrium assumpions mus also include he following: boh sock and aggregae wealh (or consumpion) are jump-diffusion processes wih sae- and imeindependen jump componen; (one plus) he percenage jump ampliudes in boh wealh and sock follow a bivariae lognormal disribuion.

12 The derivaion of opion pricing bounds under a non-increasing pricing kernel Y ( z + ) can be done wih a leas wo differen approaches, he expeced uiliy comparisons under a zero-ne-cos opion sraegy inroduced by Perrakis and Ryan (984) and he linear programg (LP) mehod pioneered by Richken (985). In he former approach an opion upper bound is found by having he rader open a shor posiion in an opion wih price C, wih he amounsα C and ( α)c added respecively o he riskless asse and he sock accoun. For he opion lower bound a long posiion is financed by shoring an amoun β S, β < of sock, wih he remainder invesed in he riskless asse. Boh bounds are found as limis on he call price C such ha he value funcion of he invesor wih he open opion posiion would exceed ha of he rader who does no rade in he opion if he wrie (purchase) price of he call lies above (below) he upper (lower) limi on C. This approach yields resuls ha are idenical o he ones of he LP approach, which is he one presened here. The disribuion of he reurn z + is assumed discree, wih he coninuous case arising as he obvious limi as he number of saes becomes progressively denser. We denoe his disribuion by P( z+ S ), which may depend on S in he mos general case; for noaional simpliciy his dependence is suppressed in he expressions ha follow. For he discree disribuion case a any ime he sock s reurn is z j in sae j, where j is an index, such ha z... z zn.the probabiliies of he n saes are p, p,..., p n. The pricing kernel, he sae-coningen discoun facors, are denoed byy, K, Y, and i is assumed ha Y Y... Yn. Le also C ( S ) and C ( S ) denoe respecively he upper and lower bounds on admissible call opion prices suppored by he marke equilibrium (3.), he asse dynamics and he monooniciy of he pricing kernel assumpion. n If he opion price funcion C( S ) = C( S ( + z )) is known hen bounds on C( S ) are + + found by solving he following LP, for given disribuions ( z j+, p j+ ). This LP evaluaes he reservaion wrie and reservaion purchase prices of he opion under marke equilibrium ha excludes he presence of sochasically doan sraegies, namely sraegies ha augmen he expeced uiliy of all raders. Violaions of he bounds given by he LP imply ha any such rader can improve her uiliy by inroducing a corresponding shor or long opion in her porfolio. I is presened in an appendix, available from he auhors on reques.

13 j= n max C ( S ( + z )) p Y Yj+ + j+ j+ j+ j= subjec o : ( C ( S ( + z )) p Y ) n j= j= n Yj+ + j+ j+ j+ j= = ( + z ) p Y n R = p j+ Yj+ j= Y Y... Y j+ j+ j+ + + n+ (3.) Define also he following condiional expecaions: z p j i= i+ i+ zˆ j+ = = E z z z j, S j i= p i+, j=,..., n (3.3) Wih hese definiions i is clear ha zˆ = E[ z S ] zˆ and by n+ + zˆ + z + z + assumpion+ zˆ n+ R. 3 Similarly we have = =,, he lowes possible reurn, which will be iniially assumed sricly greaer han -. The consrains in he LP differ from he general marke equilibrium relaions (3.) by he las se of inequaliies in (3.) ha correspond o he monooniciy of he pricing kernel assumpion. In he absence of his inequaliy se i can be shown ha he resuls of he LP yield he well-known no-arbirage bounds derived by Meron (973), 4 he only bounds on admissible opion prices ha rely only on absence of arbirage and on no oher assumpion abou he marke equilibrium process. The following imporan resul, proven in he appendix, characerizes he soluion of he LP in (3.): Lemma 3: If he opion price C ( S ) is convex for any hen i lies wihin he following bounds 3 Similar expressions as he ones presened in Lemma 3 and Proposiion also hold when we have a negaive bea sock, wih Y ( z + ) is non-decreasing and + zˆ < R. This case is presened in an n+ appendix, available from he auhors on reques. The limiing resuls of he nex secion also hold for his case as well, wih or modificaions. 4 See Richken (985, secion III). Acually, he upper bound in ha LP is equal o he sock price us he srike price discouned by he highes possible reurn; his las erm goes o 0 in he muliperiod case. 3

14 L U E [ C+ ( S ( + z + ))] C ( S ) E C + ( S ( + z+ )), (3.4) R R where U L E and E denoe respecively expecaions aken wih respec o he disribuions U R zˆ zˆ + R = p zˆ zˆ + zˆ zˆ +, (3.5a) R zˆ U = p, j =, K, n + j j+ zˆ zˆ + zˆ + R p R zˆ p L = +, j =,..., h h +, + j+ h + j+ j h h+ zˆ h+, + zˆ h+ k = pk+ zˆ h+, + zˆ h+ k= pk+ R zˆ p L =, L = 0, j > h + h+ h+, + h+, h+ j zˆ h+, + zˆ h+ k = pk + In he expressions (3.5b) h is a sae index such ha zˆ ˆ + R < z +, +. h h For a coninuous disribuion P( z+ S ) of he sock reurn he expecaions are aken wih respec o he following disribuions P( z ) wih probabiliy U ( z ) = R z + S E( z ) z, + +, + + E( z+ ) + R z wih probabiliy E( z ) z, + +, + * * ( + ) ( +, + ), ( +, + ) L z = P z S z z E + z S z z = R (3.5b) Q,. (3.6) Wih his LP i can be shown ha he bounds C ( S ) and C ( S ) may be derived recursively by a procedure described in Proposiion. This procedure yields a closed form soluion, which relies heavily on he assumed convexiy of he opion price C ( S ), iself a consequence of he convexiy of he payoff. The convexiy propery clearly holds for he diffusion and jump-diffusion cases exaed in his paper. 5 Proposiion : Under he monooniciy of he pricing kernel assumpion and for a discree disribuion of he sock reurn z all admissible opion prices lie beween he upper and lower bounds C ( S ) and C ( S ), evaluaed by he following recursive 5 The convexiy of he opion wih respec o he underlying sock price holds in all cases in which he reurn disribuion had iid ime incremens, in all univariae sae-dependen diffusion processes, and in bivariae (sochasic volailiy) diffusions under mos assumed condiions; see Meron (973) and Bergman, Grundy and Wiener (996). 4

15 expressions CT ( S ) = C ( S ) = ( S K ) T T T T U C ( S ) = E [ C + ( S ( + z + )) S ] R L C ( S ) = E [ C + ( S ( + z+ )) S ] R +, (3.7) U L where E and E denoe expecaions aken wih respec o he disribuions given in (3.5ab) or (3.6). Proof: We use inducion o prove ha (3.7) yields expressions ha form upper and lower bounds on admissible opion values. I is clear ha (3.7) holds a T and ha CT ( S ) and CT ( S T ) are boh convex in S T. Assume now ha C + ( S ( + z+ )) and C ( S ( + z )) are respecively upper and lower bounds on he convex + + funcion C + ( S ( + z )), implying ha C + ( S ( + z+ )) C+ ( S ( + z + )) C ( S ( z )) R R (3.8) By Lemma 3 we also have T L U E [ C+ ( S ( + z + ))] C ( S ) E C + ( S ( + z+ )) R R (3.9) (3.8) and (3.9), however, imply ha L U C( S ) = E [ C + ( S ( + z+ ))] C ( S ) E C + ( S ( + z+ )) = C( S ), (3.0) R R QED. An imporan special case arises when z = z, =, implying ha he sock can + + become worhless wihin a single elemenary ime period (, + ). In such a case he lower bound given by expecaions aken wih (3.5b) or (3.6) remains unchanged, bu he P upper bound akes he following form, wih E denoing he expecaion under he acual 5

16 reurn disribuion P( z S ) + : + CT ( S ) = ( S K), C ( S ) = T T P E [ C + ( S ( + z+ )) S ]. (3.) E[ + z S ] + When he reurns are iid hen (3.) corresponds o he expeced payoff given S discouned by he risky asse s reurn. The upper bound of (3.) has been exended o allow for proporional ransacion coss. The same is rue for he lower bound given by (3.6). 6 I can be easily seen from boh (3.5ab) and (3.6) ha he disribuions are risk neural, U L wih E ( + z+ ) = E ( + z+ ) = R. These disribuions were derived from he LP in (3.), and are independen of opion characerisics such as he srike price or ime o expiraion. Noe also ha he pricing kernel Y ( z + ) corresponding o he upper bound has a spike a z, + and is consan hereafer, while he kernel of he lower bound is consan and posiive ill a value * z such ha E[( + z ) z z ] = R, and becomes * + + * zero for z+ > z. These pricing kernels are boundary marginal uiliies ha do no correspond o a CPRA uiliy funcion or, indeed, o any class of uiliy funcions wih coninuously decreasing marginal uiliies. The disribuions U and L are he incomplee marke counerpars of he risk neural probabiliies of he binomial model, he only discree ime complee marke model. If, in addiion o payoff convexiy, he underlying asse reurns are iid henu and L are imeindependen and independen of he sock price S. In all cases, however, he disribuions U and L depend on he enire acual disribuion of he underlying asse, and no only on is volailiy parameer, as in he binomial and he BSM models. In paricular, hey depend on he mean ẑ of he disribuion. If + zˆ = R hen (3.5ab) and (3.6) imply ha he wo disribuions U and L coincide. As ẑ increases above R he bounds widen, reflecing he incompleeness of he marke. The dependence of U and L on convexiy and on he enire reurn disribuion may appear resricive, bu in fac he approach is quie general. The sochasic doance assumpions may sill be used o find he ighes bounds ha can be suppored by he marke equilibrium monooniciy condiion by solving he LP (3.) recursively when convexiy does no hold, wih he bounds now depending in general on opion characerisics. Recall ha arbirage and equilibrium models are able o provide expressions for opion prices only under specific assumpions abou asse dynamics. By conras he sochasic doance approach can accommodae any ype of asse dynamics, including ime- and sae-varying disribuions, provided a suiable discree ime represenaion similar o (.) can be found. As shown in he nex secion, he dependence on many parameers of he disribuion, including ẑ in he 6 See Proposiions and 5 of Consaninides and Perrakis (00). 6

17 diffusion case, disappears a he coninuous ime limi. We provide a numerical example of he widh of he bounds for a case in which he underlying asse reurn is given by (.), wih λ = 0 and wih mean and volailiy parameers similar o hose normally prevailing in he marke for he S&P 500 index. Le S = 00, T = 0.5, r = 4%, µ = 8%, σ = 0% in (.), wih he parameer ε uniform in [ 3, 3], and consider an a-he-money call opion wih one period o 0. expiraion. The opion is in he money forε. Applying now equaions (3.6) we 0.5 see ha he upper bound is equal o he expeced payoff discouned by he riskless rae.0 and muliplied by he probabiliy Q, yielding C = For he lower bound we firs idenify he value ε * =.449 such ha E( + µ T + σε T ε ε *) = R =.0 and hen we find he lower bound as he condiional payoff expecaion given ε ε * discouned by.0, yielding C = These correspond o a widh of 7.6% of he midpoin for hese single period bounds, which is expeced o decrease in he presence of inermediae rading. 7 In he nex secion we explore he limis of he expressions in (3.5ab) and (3.6) when z + is given by he coninuous ime processes (.8) and (.9). IV. Opion Pricing for Diffusion and ump-diffusion Processes The recursive procedure described in (3.4) and (3.6) can be applied direcly o he sock reurns z + given by (.) or by (.) in order o generae he upper and lower bounds a ime zero. Of paricular ineres, however, is he exisence of a limi o hese bounds as 0 and (.) ends o (.9). These limis are expressed by he following proposiion whose proof is in he appendix. Alhough his proposiion does no conain any new opion pricing resuls, i does provide a link beween he opion bounds approach and he coninuous ime resuls ha underlie mos of opion pricing. I is also necessary for he proof of he jump-diffusion resuls in Proposiions 3 and 4, which are novel. Proposiion : When he underlying asse follows a coninuous ime process described by he diffusion (.8) hen boh upper and lower bounds (3.4)-(3.6) of a European call opion evaluaed on he basis of he discreizaion of he reurns given by (.) converge o he same value, equal o he expecaion of he eral payoff of an opion on an asse whose dynamics are described by he process ds S = rd + σ ( S, ) dw, (4.) 7 Numerical resuls for he bounds are also in Richken (985) for single period lognormal reurns and in Richken and Kuo (988) for muliperiod rinomial reurns. An empirical applicaion of he bound (3.) under ransacion coss o S&P 500 index opions is in Consaninides, ackwerh and Perrakis (009). 7

18 discouned by he riskless rae. This resul esablishes he formal equivalence of he bounds approach o he prevailing arbirage mehodology for plain vanilla opion prices whenever he underlying asse dynamics are generaed by a diffusion or Io process, no maer how complex. Noe ha he univariae Io process is he only ype of asse dynamics, corresponding o dynamically complee markes, for which opions can be priced by arbirage consideraions alone. The wo bounds (3.4)-(3.6), herefore, by defining he admissible se of opion prices for any discree ime disribuion corresponding o such a dynamic compleeness, generalize he binomial model o any ype of discree ime disribuion. From he proof of Proposiion i is clear ha he resul holds because a he limi he pricing kernels of boh upper and lower bound coninue o play heir risk-neuralizing role, while heir effec on he insananeous variance of he process disappears. 8 This propery does no exend o he jump-diffusion case, as shown furher on in his secion. (Figure 4. abou here) Figure 4. illusraes he convergence of he wo bounds o he BSM value for an a-hemoney call opion wih K = 00 and T = 0.5 years for he following insananeous annual parameers: r = 3%, µ = 5% o 9%, σ = 0%. The diffusion process was approximaed by a 300-period rinomial ree consruced according o he algorihm of Kamrad and Richken (99). The wo opion bounds were evaluaed as discouned expecaions of he payoffs under he risk neural probabiliies obained by applying he expressions (3.4)-(3.6) o subrees of he 300-period rinomial ree. Fas Fourrier Transforms were applied for he derivaion of he eral disribuions of he underlying asse and he bounds, given ha he reurns are iid. As he figure shows, he wo bounds converge o heir common limi uniformly from below and above respecively. The speed of convergence varies inversely wih he size of he risk premium, bu convergence is essenially complee afer 300 periods even for he larges premium of 6%. This speed of convergence may also be helpful for he cases where no closed-form expression for he opion price exiss, as in complex cases of saedependen univariae diffusions, like he Consan Elasiciy of Variance (CEV) model. 9 In such cases valuaion of he opion by Mone Carlo simulaion of he bounds is 8 Mahur and Richken (999) show ha he BSM opion price arises as he lower bound of he LP program (3.) in a single period model, in which he pricing kernel has been resriced o saisfy he Decreasing Relaive Risk Aversion (DRRA) propery; his bound does no ighen when he ime inerval is subdivided. In our case he discree ime lower bound of he LP is always lower han he one corresponding o DRRA since he class of admissible kernels has been resriced, bu i ighens wih denser subdivisions and becomes equal o he DRRA and he BSM model a he limi. 9 See Cox and Rubinsein (985, pp ). 8

19 cerainly an alernaive o an opion value compued as a discouned payoff of pahs generaed by he Mone Carlo simulaion of (4.). While here may no be any compuaional advanages in going hrough he bounds roue o opion valuaion, he fac ha boh upper and lower bound end o he same limi from above and from below may provide a benchmark for he accuracy of he valuaion, in conras o he direc simulaion of (4.). Nex we exae he limiing behavior of he sochasic doance bounds ha can be derived from he discree ime process (.) ha was shown by Lemma o end o a jump-diffusion. For such a process a unique opion price can be derived by arbirage mehods alone only if σ = 0 and akes exacly one value when a jump occurs. In such a case he process (.) is binomial and i can be readily verified ha he disribuionsu and L coincide, and he sochasic doance approach yields he same unique opion price as he binomial jump process in Cox, Ross and Rubinsein (979). Oherwise, we mus exae he wo bounds separaely. For he opion upper bound we apply he ransformaion (3.6) o he discreizaion (.), aking ino accoun ha he variable akes boh posiive and negaive values, or ha < 0 < max. For such a process we noe ha as decreases, here exiss h, such ha for any h, he imum oucome of he jump componen is less han he imum oucome of he diffusion componen, < µ + σ ε. Consequenly, for any h, he imum oucome of he reurns disribuion is, which is he value ha we subsiue z in (3.6). Wih such a subsiuion we have now he following resul, proven in for,+ he appendix. Proposiion 3: When he underlying asse follows a jump-diffusion process described by (.9) he upper opion bound is he expeced payoff discouned by he riskless rae of an opion on an asse whose dynamics are described by he jump-diffusion process where, ds S U U = r ( λ + λ U ) µ d + σ dw + dn λ µ r (4.) U = (4.3) and U is a mixure of jumps wih inensiy λ λu + and disribuion and mean wih probabiliy U = wih probabiliy U λ λu µ = µ + λ + λ λ + λ U U λ λ+ λ U λu λ+ λ U (4.4) 9

20 By definiion of he convergence of he discree ime process, Proposiion 3 saes ha he call upper bound is he discouned expecaion of he call payoff under he jump-diffusion process given by (4.), which implies ha he ransformed jump componen is fully diversifiable. We may, herefore, use he resuls derived by Meron (976) for opions on asses following jump-diffusion processes wih he jump risk fully diversifiable. 30 Applying Meron's approach o (4.) we find ha he upper bound on call opion prices for he jump-diffusion process (.9) mus saisfy he following parial differenial equaion (pde), wih eral condiion C( S, T ) = max{ S K,0} : T T U U C C C U U r ( λ + λ U ) µ S + σ S + ( λ + λ ) [ ( ( )) ( )] 0 U E C S + C S rc =. S T S (4.5) An imporan special case is when he lower limi of he jump ampliude is equal o 0, in which case = and he reurn disribuion has an absorbing sae in which he sock becomes worhless and z = z, = ; his is he case described in (3.), in which + + as we saw he opion price is he expeced payoff wih he acual disribuion, discouned by he expeced reurn on he sock. Hence, his is idenical o he Meron (976, equaion (4)) case wih r replaced by µ, yielding C C C [ ] µ λµ S + σ S + λ E[ C( S( + )) C( S)] µ C = 0 S T S (4.6) If (4.6) holds and we assume, in addiion, ha he diffusion parameers are consan and he ampliude of he jumps has a lognormal disribuion wih ln( + ) ~ N ( µ, σ ), he disribuion of he asse price given ha k jumps occurred is condiionally normal, wih mean and variance k µ k = µ kλµ + ln( + µ ) T k σ k = σ + σ T (4.7) Hence, if k jumps occurred, he opion price would be a Black-Scholes expression wih µ k replacing he riskless rae r, or BS( S, K, T, µ k, σ k ). Inegraing (4.6) would hen yield he following upper bound, which can be obained direcly from Meron (976) by replacing r by µ. k [ λ ( + µ ) T ] C = exp[ λ ( + µ ) T ] BS( S, K, T, µ k, σ k ) (4.8) k! k= 0 When he jump disribuion is no normal, he condiional asse disribuion given k jumps 30 Noe ha we do no assume here ha he jump risk is diversifiable. 0

21 is he convoluion of a normal and k jump disribuions. The upper bound canno be obained in closed form, bu i is possible o obain he characerisic funcion of he bound disribuion. Similar approaches can be applied o he inegraion of equaion (4.5), which holds whenever 0 > >. Closed form soluions can also be found whenever he ampliude of he jumps is fixed as, for insance, when here is only an up and a down jump of a fixed size. 3 A pde similar o (4.5) also holds if he process has only up jumps, in which case = 0 and he lowes reurn z in (3.6) comes from he diffusion componen. In such a case he key probabiliy Q of (3.6) is he same as in he proof of Proposiion and (4.5) sill holds wih λ U = 0, implying ha he opion upper bound is he Meron (976) bound, wih he jump risk fully diversifiable. The opion lower bound for he jump-diffusion process given by (.9) and is discreizaion (.) is found by a similar procedure. We apply L( z + ) from (3.6) o he process (.) and we prove in he appendix he following resul. Proposiion 4: When he underlying asse follows a jump-diffusion process described by (.9), he lower opion bound is he expeced payoff discouned by he riskless rae of an opion on an asse whose dynamics is described by he jump-diffusion process where ds S L L = r λµ d + σ dw + dn L is a jump wih he runcaed disribuion The mean µ of he jump and he value of can be obained by solving he equaions L µ λµ + λµ = r L µ = E( ) L (4.9) (4.0) Observe ha (4.0) always has a soluion since µ > r by assumpion. The limiing disribuion includes he whole diffusion componen and a runcaed jump componen. Unlike simple diffusion, he runcaion does no disappear as 0. As wih he upper bound, we can apply he Meron (976) approach o derive he pde saisfied by he opion lower bound, which is given by L C C C L L r λµ S + σ S + λe [ C( S( + )) ( )] 0 C S rc = S T S (4.) wih eral condiion CT = C( ST, T ) = max{ ST K, 0}. The soluion of (4.) can be obained in closed form only when he jump ampliudes are fixed, since even when he 3 See, for insance, he example in Masson and Perrakis (000).

22 jumps are normally disribued, he lower bound jump disribuion is runcaed. Observe ha he jump componens in boh C ( S ) and C ( S ) are now sae-dependen if µ, he diffusion componen of he insananeous expeced reurn on he sock, is saedependen., even hough he acual jump process is independen of he diffusion. In many empirical applicaions of jump-diffusion processes, which were on he S&P 500 index opions, he uncondiional esimaes are considered unreliable. On he oher hand here is consensus ha he uncondiional mean is in he 4-6% range; 3 his is refleced in he numerical resuls below. Observe also ha for normally disribued jumps he only parameers ha ener ino he compuaion of he bounds are he mean of he process, he volailiy of he diffusion and he parameers of he jump componen. Hence, he informaion requiremens are he same as in he more radiional approaches, wih he imporan difference ha he mean of he process replaces he risk aversion parameer. We presen in Table 4. and Figure 4. esimaes of he bounds under a jump-diffusion process for an a-he-money opion wih K = 00 and mauriy T = 0.5 years for varying subdivisions of he ime o expiraion, and wih he following annual parameers: r = 3%, µ = 5% o 9%, σ = 0%, λ = 0.3, µ = 0.05, σ = 7%. Table 4. and Figure 4.3 presen he bounds for µ = 7% and wih he oher parameers unchanged for various degrees of moneyness of he opion. The jump-diffusion process was approximaed by a 300-ime sep ree buil according o he mehod inroduced by A (993). The jump ampliude disribuion was lognormal, which was runcaed for numerical purposes in building he ree. The bounds were compued by aking he discouned expecaions of he payoff under he ime-varying risk neural probabiliies of (3.4) applied o subrees. The risk neural price is he Meron (976) price for his process. (Table 4. abou here) (Figure 4. abou here) (Table 4. abou here) (Figure 4.3 abou here) The resuls shown in Table 4. show a maximum spread beween bounds of abou 0%, a spread ha is an increasing funcion of µ. In Table 4. he spread is much lower for inhe-money opions and reaches abou 8% for he mos ou-of-he money opions. Noe 3 See Fama and French (00), Consaninides (00) and Dimson, Marsh and Saunon (006).

23 ha he range of values of µ implies an ex-dividend risk premium range from % o 7%, a range ha covers wha mos people would consider he appropriae value of such a premium in many imporan cases; he corresponding widh of he bounds ranges from 4.5% o 0%. For he mos commonly chosen risk premium of 4%, corresponding o µ = 7%, he spread a-he-money is abou 7.6%. This range of allowable opion prices in he sochasic doance approach is he exac counerpar of he inabiliy of he radiional arbirage-based approaches o produce a single opion price for jump diffusion processes wihou an arbirarily chosen risk aversion parameer, even when he models have been augmened in his case by general equilibrium consideraions. V. Exensions o Sock Opions In his secion we exend he se of assumpions abou marke equilibrium, by defining he risky asse held by he rader as a marke index porfolio and allowing he rader o adop addiional marginal posiions in a single sock, as well as in opions on ha sock. Le now I denoe he curren value of he index and S he value of he sock, wih he I+ I S+ S reurns z+ and v +. The marke equilibrium condiions (3.) are I S now as follows E Y z I S R [ ( + ), ] =, E[( + z ) Y ( z ) I, S ] = E[( + v ) Y ( z ) I, S ] =, C ( S, I ) = E[ C ( S ( + v ), I ( + z )) Y ( z ) I, S ] (5.) Assume now a join discree disribuion of he wo reurns, and se E[ v z = z, S, I ] = v j ( z ) v j+. The equilibrium relaions (5.) imply + + j+ j+ cerain resricions on he parameers of he join disribuion. These are expressed by he following Lemma. I covers he case of diffusion and can be exended o cover jumpdiffusion for ha join reurn disribuion. Lemma 4: If he funcion j( ) is linear, ν( z ) = θ+ ζz, hen he following relaion mus hold: ν z j + j j+ j+ ( R )( ζ) = θ. (5.) Furher, if in addiion E[ C ( S ( + v ), I ( + z )) z ] can be wrien as a funcion Cˆ ( S ( + ν ( z )), I ( + z )) hen C ( S, I ) akes he form C ( S ), independen of he index level I. Proof: We wrie he las relaion in (5.) 3

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