Virtual Arbitrage Pricing Theory

Transcription

1 1 arxv:cond-mat/ v1 [cond-mat.stat-mech] 3 Feb 1999 Vrtual Arbtrage Prcng Theory Krll Ilnsk School of Physcs and Space Research, Unversty of Brmngham, Edgbaston B15 2TT, Brmngham, Unted Kngdom Abstract We generalze the Arbtrage Prcng Theory (APT) to nclude the contrbuton of vrtual arbtrage opportuntes. We model the arbtrage return by a stochastc process. The latter s ncorporated n the APT framework to calculate the correcton to the APT due to the vrtual arbtrage opportuntes. The resultng relatons reduce to the APT for an nfntely fast market reacton or n the case where the vrtual arbtrage s absent. Correctons to the Captal Asset Prcng Model (CAPM) are also derved. 1 Introducton Along wth the Captal Asset Prcng Model (CAPM) [1, 2, 3] the Arbtrage Prcng Theory (APT) [4] s a standard model for determnng the expected rate of return on ndvdual stocks and on a portfolo of stocks [5]. In contrast to the CAPM, the APT does not rely on any assumptons about utlty functons or the assumpton that agents consder only the mean and varance of possble portfolos. Takng nto account crtcal remarks about the expected utlty approach to the theory of the decson makng under uncertanty, and the long runnng dscusson about the defnton of rsk, ths seems to be a consderable achevement. What the APT does mply are homogeneous expectatons, assumptons of lnear factor weghtngs, a large enough number of securtes to elmnate the specfc rsk and no-arbtrage condton. In smplfed terms, the latter means that the return on a rskless portfolo should be equal to the rskless rate of return whch can be taken for a moment equal to the rate of the return on a bank depost (here, for sake of smplcty, we assume perfect captal market condtons,.e. sngle rate of lendng and borrowng wthout restrctons, absence of transacton costs and bd-ask spread; we touch on possble generalzaton n concluson). Beng a bass for the APT and dervatve prcng [6], the no-arbtrage assumpton appears to be very reasonable and robust. Indeed, f any arbtrage possblty would exst then agents (arbtrageurs) would use the opportunty to make an abnormal rskless proft whch tself wll brng the system to the equlbrum and elmnate the arbtrage opportunty. Thus E-mal: kn@th.ph.bham.ac.uk

2 2 the arbtrage cannot exst for long and ts lfetme depends on a lqudty of the market. Ths, however, does not mean that the arbtrage opportuntes do not exst at all and cannot nfluence the asset prcng volatng the APT assumpton. Moreover, there exst emprcal studes whch pont out the exstence of vrtual arbtrage opportuntes even for very lqud markets (order of 5 mnutes for futures on S&P [7] and can be much longer for less lqud markets such as bonds market and dervatve market [8]). That s why n ths paper we try to overcome the no-arbtrage assumpton and suggest a model to account for the exstence of vrtual arbtrage opportuntes and ther nfluence on asset prcng n the framework of the APT. To ths end we follow the lne ntroduced n Ref. [9] to account the vrtual arbtrage n dervatve prcng. One of the possble approaches to the problem of handlng of vrtual arbtrage has been suggested n Ref [10] and based on a feld-theoretcal descrpton of arbtrage opportuntes and the correspondng money flows. Beng a consstent theory, the approach s however extremely complcated and does not allow one yet to get smple analytcal results whch would be easy understandable and handleable. That s why n ths paper we develop smplfed tractable analytcal verson to account for vrtual arbtrage opportuntes whch does not use complcated technques and results n smple enough fnal formulas. The paper s organzed as follows. In next secton we derve an effectve equaton for a prce of a rskless portfolo n presence of vrtual arbtrage opportuntes. There t s shown how the local arbtrage opportuntes can be ntroduced n the model and how they change the rate of return. In secton 3 we derve equatons of the APT under vrtual arbtrage. These equatons generalzes the APT relatons and converge to them n the lmt of absence of the arbtrage opportuntes or nfntely fast market reacton. Secton 4 s devoted to the correctons to the CAPM. In concluson we dscuss the drawbacks of the model and possble ways to mprove t. 2 Effectve equaton for rskless portfolo n presence of vrtual arbtrage In ths secton we ntroduce a model for vrtual arbtrage fluctuatons and descrbe ther nfluence on return on rskless portfolo. Under no-arbtrage assumpton the rate of return on a rskless portfolo shall be equal to the sngle rskless rate of return, whch s equal to, for example, the return on a bank depost. However, f we allow exstence of vrtual arbtrage opportuntes, the rate of return on a portfolo does not have to be equal the rate of return on bank depost, t s random (snce unpredctable) and mght depend on the structure of a portfolo. Indeed, f, for example, the portfolo conssts of only rskless assets (bank depost or bond) there should be no arbtrage opportuntes wth ths portfolo and the rate of return on ths portfolo, by defnton, shall be equal to the rate of return on a bank depost r 0. In the general case of a rskless portfolo whch conssts of dversfed rsky assets the exstence of the arbtrage fluctuatons would depend on the portfolo composton. To gve an ntutve example, we can magne that f some market sector s more attractve for nvestors, the trades and resultng arbtrage fluctuatons wll occur more often. Ths wll result n more arbtrage

3 3 opportuntes for portfolos contanng the assets of the sector and the correspondng composton dependence. Let us consder rskless portfolo Π whch s created by N +1 assets wth fractons {x } N =0. Inthecase ofno-arbtragetheportfoloπwould satsfythefollowngequaton: dπ dt r 0Π = 0, Π(0) = 1 (1) where r 0 s rskless nterest rate on bank depost. However, n case of the vrtual arbtrage RHS of Eq(1) shall be changed to R(t, Π)Π where R(t, Π) represents the vrtual arbtrage return. To fnd an expresson for R(t, Π) let us magne that at some moment of tme τ < t a fluctuaton of the return (an arbtrage opportunty) appeared n the market. We then denote ths nstantaneous arbtrage return as ν(τ,π). Arbtragers would react to ths crcumstance and act n such a way that the arbtrage gradually dsappears and the market returns to ts equlbrum state,.e. the absence of the arbtrage. For small enough fluctuatons t s natural to assume that the arbtrage return R (n absence of other fluctuatons) evolves accordng to the followng equaton: dr dt = λr, R(τ) = ν(τ,π) (2) wth some parameter λ whch s characterstc for the market. Ths parameter can be ether estmated from a mcroscopc theory lke [10] or can be found from the market usng an analogue of the fluctuaton-dsspaton theorem [11]. In the last case the parameter λ can be estmated from the market data as λ = 1 (t t ) log[ (r Π r 0 )(t)(r Π r 0 )(t ) market / (r Π r 0 ) 2 (t) ],t > t market (3) and may well be a functon of tme, portfolo composton and even prces of assets. In what follows we however consder λ as a constant to get smple analytcal formulas for APT correctons. The generalzaton to the case of tme-dependent parameters s straghtforward. The solutonofeqn(2) gves usr(t,π) = ν(τ,π)e λ(t τ) whch, aftersummng over all possble fluctuatons, leads us to the followng expresson for the arbtrage return R(t,Π) t e λ(t τ) ν(τ,π)dτ. (4) To specfy the stochastc process ν(t, Π) we assume that the fluctuatons at dfferent tmes are ndependent and form the whte nose wth a varance Σ 2 (Π) whch depends on the structure of the portfolo Π: ν(t,π) ν = 0, ν(t,π)ν(t,π) ν = Σ 2 (Π) δ(t t ), (5)

4 4 where, agan, the parameter Σ 2 (Π) can be taken from the market: Σ 2 (Π)/2λ = (r Π r 0 ) 2 market. (6) To smplfy the consderaton we assume here that the quantty does not depend on tme but ths lmtaton may be straghtforwardly overcome (see dscusson n the concluson). Snce we ntroduced the stochastc arbtrage return R(t, Π), Eqn(1) has to be substtuted wth the followng equaton: or, n the ntegral form, dπ dt r 0Π = R(t,Π)Π (7) Π = t G(t,t )R(t,Π)Π(t )dt (8) where G(t,t ) = Θ(t t )e r 0(t t ) s Green functon of the problem: ( d dt r 0)G(t,t ) = δ(t t ), G(t,t ) t<t = 0. We can terate Eq(7) and substtute Eq(8) n RHS of Eq(7) whch gves dπ t dt r 0Π = R(t,Π) G(t,t )R(t,Π)Π(t )dt. (9) Our next step s to average Eq (9) over vrtual arbtrage fluctuatons and to obtan the effectve prcng equaton for the average prce Π. At the frst order n 1/λ t can be wrtten as d Π t dt r 0 Π = G(t,t )K(t,t, Π) Π(t )dt. (10) where the kernel K(t,t, Π) = R(t,Π)R(t,Π) ν s gven by the expresson [9]: K(t,t ) = Σ2 ( Π) 2λ θ(t t )e λ(t t ) + Σ2 ( Π) 2λ θ(t t)e λ(t t) whch can be easly obtaned from Eq(5). Collectng everythng together, Eqs(10,11) are used n place of Eq(1) n stuatons where vrtual arbtrage opportuntes exst. To solve Eq(10) we frst notce that ts RHS can be wrtten as t G(t,t )K(t,t, Π) Π(t )dt = t e λ(t t ) e r 0(t t ) Σ2 ( Π) 2λ Π(t )dt. We now make an approxmaton Π(t ) = e r 0t because the correcton to ths expresson s order of Σ2 ( Π) and s rrelevant for our consderaton of RHS of Eq(10) snce t was 2λ (11)

5 5 derved wthn ths order of accuracy. Such approxmaton results n the followng relaton: t G(t,t )K(t,t, Π) Π(t )dt = Σ2 ( Π) e r0t. 2λ 2 If we substtute t nto Eq (10) ths gves us the followng approxmate dfferental equaton for the average portfolo prce: d Π dt r 0 Π = Σ2 ( Π) 2λ 2 e r 0t whch has the soluton: Π(t) = (1+ Σ2 ( Π) t)e r 0t 2λ 2 or, to the same level of accuracy, (12) Π(t) = e (r 0+ Σ2 ( Π) 2λ 2 )t. (13) The latter expresson we use n the next secton n place of the no-arbtrage expresson Π(t) = e r 0t n the Arbtrage Prcng Theory. Let us emphasze that we, as well as an nvestor, are nterested n the average value of the portfolo and the correspondng rate of return on the average portfolo rather than n average rate of return on the rskless portfolo whch s exactly equal to r 0 and, n fact, s not nfluenced by the arbtrage. Indeed, the only valuable and materal thng for an nvestor s the average value of her nvestment. She s not nterested n a mathematcal quantty whch s not actually connected wth her wealth. The dfference between the two returns s the frst effect whch s solely due to the presence of the vrtual arbtrage. 3 Correctons to APT Let us frst remnd ourselves of the standard dervaton of Arbtrage Prcng Theory relatons. The frst assumpton below APT s the exstence of M fluctuatng leadng parameters, or factors, {ξ j } M j=1 whch defne the prces of all N +1 assets such that ξ j ξ = 0, ξ ξ j ξ = δ j and r = r + j b j ξ j +ǫ (14) wth b j = r r,ξ j ξ as a measure of nfluence of j-th parameter on -th asset and ǫ s a resdual rsk whch s ndependent on the factors and can be elmnated n large portfolos. To form a rskless portfolo we pck up such fractons {x } that x b j = 0

6 6 for any j = 1,...,M and they obey normalzaton condton x = 1. We assume that 0-th asset s rskless, e.g. bank depost or treasury bonds. The return on the portfolo s r x = r x and has to be equal to the rsk-free nterest rate r 0 n the case of no-arbtrage. Ths leads to the equaton or, after a change of varables x = y +δ 0, r x = r 0 (15) r y = 0 subject to the constrants y b j = 0 j, y = 0. The soluton of the equaton for r then can be wrtten as r = α+ j b j γ j. (16) where α and{γ j } areset of ndependent parameters. Snce the 0-thasset was chosen as rsk-free, the coeffcents b 0j = 0 for all j and α = r 0. Eq(16) consttutes the Arbtrage Prcng Theory relatons. To generalze the APT relatons to the case of vrtual arbtrage we have to deal wth average portfolos consdered n the prevous secton. We defne the rates of the return { r k }N k=0 on k-th component of an average rskless portfolo as Π(t) = e r x t. These are the quanttes whch have to substtute the rates of return n the APT relatons. To fnd an expresson for { r k} N k=0 we have to substtute r 0 n Eq(15) on the return to the rskless portfolo derved under vrtual arbtrage consderaton n the prevous secton,.e. on (r 0 + Σ2 ( Π) 2λ 2 ) as follows from Eq(13). Ths gves us the equaton for r : subject to constrants y b j = 0 j, r y = Σ2 ( Π) 2λ 2 (17) y = 0. (18) To smplfy the followng consderaton we ntroduce a convenent bass n the portfolo space such that the above constrants wll take the form η = 0, 0 M

7 7 n ths new bass whle {η j } N j=m+1 wll be coordnates n the subspace of rsk-free portfolos. To ths end we frst ntroduce vectors {e }M =0 : e 0, = 1, e,j = b j,1 j M. These M +1 vectors are lnear ndependent (otherwse t would be lnear dependence for the factors and M +1 constrants (18) would not be ndependent). The rest of N M lnear ndependent vectors {e }N =M+1 can be chosen arbtrary but lnear ndependent wth {e } M =0. For example, f then one possble choce could be b j 0, j and 0 < < M +1 e,j = δ j j,m +1 N +1. The next step s the Gram-Schmdt orthogonalsaton and normalsaton of the vector set {e }N =0 whch gves us the bass set {e } N =M+1. We shall start the orthogonalsaton wth the vector e 0: e 0 e 0 = (e 0,e 0), proceed wth the vector e 1 as e 1 = e 1 (e 1,e 0)e 0 (e 1 (e 1,e 0 )e 0,e 1 (e 1,e 0 )e 0 ) and carry on followng the standard procedure: j 1 e j = e j j 1 (e j,e k)e k / (e j,e j ) (e j,e k) 2 k=0 k=0 1/2. It s easy to check now that n ths new bass {e } N =0 the constrants (18) can be represented as a zero projecton along the frst M +1 bass vectors,.e. as η = 0, 0 M where η are coordnates n the new bass. Indeed, for any vector y the condtons (y,e j ) = 0, 0 j M are exactly equvalent to the constrants (18). Ths remans true also after the orthogonalsaton (y,e j ) = 0, 0 j M due to step-by-step nature of the procedure. Therefore any vector y = N+1 j=m+1 η j e j

8 8 represents a rsk-free portfolo and the set {e j } N j=m+1 s a bass n the subspace of rskless portfolos. To express the rskless portfolos bass n terms of the orgnal assets we ntroduce the rotaton matrx U = U k : U = e 0 e 1... e N. (19) Usng ths matrx the portfolo represented by the bass vector e can be constructed by collectng j-th assets (j = 0,..,N +1) wth the fractons e,j +δ 0j. Now we are ready to dscuss the possble form of Σ 2 ( Π) as a functon of portfolo structure,.e. as a functon of {y } N =0 or {η } N =M+1. Frst of all we wll not allow the exstence of arbtrage fluctuatons for a portfolo whch conssts of pure bonds,.e. when all η equal to zero. Ths provdes that the rate of return on the bank depost s actually equal to r 0, does not have any arbtrage correctons and our use of r 0 s selfconsstent. It means that seres expanson of Σ 2 (η) cannot contan a constant term. Second, snce the vrtual arbtrage return s brought n by portfolo η, the arbtrage return on the portfolo η has to be equal to mnus the latter and Σ 2 (η) = Σ 2 ( η). Ths results n the absence of all odd terms. Therefore the seres expanson of Σ 2 ( Π) has the form: N Σ 2 ( Π) = Σ 2 j η η j +..., or Σ 2 ( Π) = N,j=M+1,j=M+1 Σ 2 j η η j +..., Σ 2...k = 0 for 0,...,k M. Inwhatfollowswekeeponlyfrstnontrvalterm, (η,σ 2 η),σ 2 Σ 2 k ntheexpanson though the fnal result wll be vald for the general case too. Returnng back to prcng equaton (17) we substtute y = Uη, Σ 2 ( Π) = (η,σ 2 η), r y = ( r,y) we come to the prcng equaton for r: subject to constrants ( r,uη) = (η,σ 2 η) η = 0, 0 j M. The soluton of the problem can be easly found as r = α+ j b j γ j + N,k,l=0 U k Σ 2 kl 2λ 2η l. (20) At ths pont we face an unpleasant problem: a return on an asset n the portfolo depends on the structure of the portfolo. Ths s the second effect whch stems from the vrtual arbtrage presence. However unpleasant t s, t s hardly surprsng snce t was derved under the assumpton of vrtual arbtrage whch s portfolo-dependent.

9 9 Intutvely t s also clear: f an asset consttutes a part of a hot arbtrage portfolo ts rate of return wll be dfferent from assets n quet portfolos ceters parbus. r(π) To fnd an average growth factor for the -th asset we have to take a sum of e weghted wth probabltes of appearance of the portfolo Π,.e. to calculate the average value e r (Π) Π e r whch defnes therateofgrowth r oftheaverage growthfactor. Ths r sacounterpart of the average rate of return under the vrtual arbtrage presence snce t characterzes how fast -th asset grows n average. It s easy to see that n the frst nontrval order e r = e α+ j b jγ j 1+ N k,m,l,l =0 U k U m Σ 2 kl Σ2 ml 8λ 4 η l η l Π and, hence, wth the same measure of accuracy the rate of growth s gven by r = α+ j b j γ j + N k,m,l,l =0 U k U m Σ 2 klσ 2 ml 8λ 4 η l η l Π (21) Snce the 0-th asset was chosen as rsk-free, the coeffcents β 0j = 0 for all j and r = r 0 + N Σ 2 kl b j γ j + U k U Σ2 N ml Σ 2 kl m η j 8λ k,m,l,l =0 4 l η l Π U 0k U Σ2 ml 0m η 8λ k,m,l,l =0 4 l η l Π. (22) The last equaton looks qute complcated. It can be presented n a more compact form usng matrx notatons and the followng matrx : k = 1 4λ 2 Σ 2 (η) η The prcng relatons then can be rewrtten as r = r 0 + j Σ 2 (η) η k Π b j γ j + 1 2λ 2(U U 1 ) 1 2λ 2(U U 1 ) 00. (23) Eqns (22,23) gves a generalzaton of the APT relatons n the case where vrtual arbtrage opportuntes exst. It s easy to see that the correctons are proportonal to the square of the product of the varance of the arbtrage fluctuatons and the square of a characterstc lfetme of the arbtrage. Ths results n the dsappearance of the correlatons n the case of nfntely fast market reacton (λ ) or an absence of the arbtrage (Σ 0) and the reducton of Eqns(22,23) to the APT expresson (16). 4 Smplest applcaton: correcton to CAPM In ths short secton we concentrate on a partcular case of APT wth the only one leadng parameter whch wll be chosen to be a random part of a return on a market portfolo. In ths case the standard APT relatons are reduced to the CAPM equatons..

10 10 In the case of only one leadng parameter chosen to be a random part of a return on the market portfolo ξ (r m r m )/σ m, Eqn (23) can be rewrtten as r = r 0 +b γ 1 2λ 2(U U 1 ) 1 2λ 2(U U 1 ) 00 (24) where b = r r,r m r m /σ m. The next step s to ntroduce the market portfolo fractonsθ and fnd anexpresson for the average rate ofreturn on the market portfolo θ r usng Eqn (24): r m = r 0 + θ b γ + 1 2λ 2um 1 2λ 2(U U 1 ) 00 where we ntroduce a notaton u m = (U U 1 ) θ. Ths allow us to defne a value of the varable γ: γ = 1 ( r m r σ m 2λ 2(U U 1 ) 00 1 ) 2λ 2um and to fnd a fnal expresson for the CAPM generalzed to the vrtual arbtrage assumpton:, r = r 0 +β ( r m r 0 )+ β 2λ 2 ( (U U 1 ) 00 u m) + (U U 1 ) 2λ 2 (U U 1 ) 00 2λ 2 (25) wth the standard defnton β = r r,r m r m /σ 2 m. The frst two terms represent the CAPM under the no-arbtrage assumpton and the remanng terms are the vrtual arbtrage correctons. Everythng sad n the last paragraph of the prevous secton can be repeated wth respect to Eqn(25) for the vrtual arbtrage generalzaton of the CAPM. 5 Dscusson In concluson, we ntroduced a vrtual arbtrage opportuntes n the framework of the Arbtrage Prcng and derved correctons to the APT prcng relatons. These correctons dsappear as Σ 0 or λ 0.e. when there s no arbtrage fluctuatons or the speed of market reacton to the msprcng s nfnte (whch corresponds to extremaly lqud market). In the course of the analyss we faced two major effects whch are specfc for the presence of the arbtrage. The frst one s a necessty to analyze growth factors rather than the the correspondng rates of return. The second effect s a dependence of a growth factor of an asset on the structure of a rskless portfolo whch ncludes the asset. Ths forced us to ntroduce an average over all rskless portfolos and reduced the problem to a calculaton of the matrx k = 1 Σ 2 (η) 4λ 2 η Σ 2 (η) η k Π (26)

11 whch s essentally a new element. We already mentoned that Σ 2 (Π) can be obtaned fromthe market usng Eqn(6). It means that, n prncple, can also be obtaned from the market followng (26). However the most economcal procedure of the evaluaton stll has to be worked out. It mght, for example, be reasonable to consder some most frequent arbtrages and reduce the space of the relevant rskless portfolos. Another opton s to study thhe man factors generatng arbtrage opportuntes and carry out a factor analyss smlar to the orgnal APT. At the moment we leave ths as an open queston whch requres further nvestgaton. Now let us dscuss some obvous drawbacks of the model and ways to mprove t. Frst of all, we have to admt mmedately that t would be more dffcult to carry out emprcal tests of the Vrtual Arbtrage Prcng Theory than the APT because the addtonal terms n Eqn(22) do not make lfe easer n any respect whle even emprcs for the CAPM and the APT [12] have not produced yet any decsve result. Furthermore, all crtcal comments of APT analyss can be forwarded to the presented model, except for the no-arbtrage constrant. Homogeneous expectatons, absence of clear recpe of how to measure the factors ξ and possble tme dependence of the parameters of the model are three obvous ponts to crtcze. These are not new problems and many efforts to overcome these dffcultes have been undertaken. It s possble to demonstrate that the vrtual arbtrage model can be mproved n the same manner by these methods as they succeed for no-arbtrage APT analyss. Ths, n prncple, allows one to nclude the transacton costs, taxes and market mperfecton n the present model by redervng the bare APT relatons and then to add the vrtual arbtrage n the consderaton wth mnor modfcatons. The second pont concerns the market reacton to the arbtrage opportunty, or, qualtatvely the form of R(t,Π) n Eqn(4). It may be argued that the market reacton s not exponental as assumed n Eqn(2), but has another functonal dependence. Ths dependence can be found from statstcal analyss of the stock prces and then ncluded n the equaton for R(t, Π) (n partcular, the functonal dependence can change wth tme, for example λ n (2) can be a functon of τ). It certanly complcates the model but leaves the general framework ntact. Another pont to consder s the absence of correlatons between vrtual arbtrage opportuntes whch we assumed n the text,.e. the whte nose character of the process ν(t,π). It s clear that some correlatons can be easly ncluded n the model by a proper substtuton of the relatons n Eqn(5). Such generalzatons, though makng the analytcal study almost mpossble, allow one to proceed wth numercal analyss for the model. Fnally, themodelcontansnewparameterssuchasσandλdefnedbyeqns(6)and (3) from the market data. It mght appear that tme-ndependent parameters are not good enough approxmaton and both Σ and λ are functons of tme as a consequence of market ntermttent busts of actvty. Furthermore, the parameter λ mght actually depends on a structure of the portfolo exactly n the same manner as Σ does because of partcular sector preferences of arbtrageurs. Both stuatons can be processed n a smlar manner to the smplest case we consdered above but t wll make the fnal results much more complcated. 11

12 12 References [1] W.F. Sharp: Captal Asset Prces: A theory of Market Equlbrum Under Condtons of Rsk, J.of Fnance, 19, N3 (1964), ; [2] J. Lntner: The valuaton of Rsk Assets and the Selecton of Rsky Investments n Stock Portfolo and Captal Budgets, Revew of Economcs and Statstcs, 41, N1 (1965) 13-37; [3] J.Mossn: Equlbrum n Captal Asset Market, Econometrca, 34, N4 (1966) ; [4] S.A. Ross: The Arbtrage Prcng Theory of Captal Asset Prcng, J.of Economc Theory, 13 (1976) ; [5] E.J. Elton, M.J. Gruber, Modern portfolo theory and nvestment analyss, Jonh Wley & Sons, 1995; [6] F. Black, M. Scholes, Journal of Poltcal Economy, 81 (1973) 637; [7] G. Sofanos: Index Arbtrage Proftablty, NYSE workng paper 90-04; J.of Dervatves, 1, N1 (1993); [8] F. Black and M. Scholes: The Valuaton of Opton Contracts and a Test of Market Effcency, Journal of Fnance, 27 (1972) ; D. Gala: Tests of Market Effcency and the Chcago Board Opton Echange, Journal of Busness, 50 (1977) ; [9] K. Ilnsk and A. Stepanenko: Dervatve Prcng wth Vrtual Arbtrage, submted to J.of Dervatves; [10] K. Ilnsk: Physcs of Fnance, n: J. Kertesz & I. Kondor (Eds.): Econophyscs: an emergng scence, Dordrecht: Kluwer, 1998; avalable also at [11] H.B. Callen, Thermodynamcs, John Wley& Sons, 1960; [12] F. Black, M.C. Jensen and M. Scholes: The Captal Asset Prcng Model: Some Emprcal Tests, n M.C.Jensen (ed.) Studes n the Theory of Captal Markets, NY, Praeger, 1972; E.F. Fama and MacBeth, J.of Fnancal Economcs, 1, N1, (1974) 43-66; R. Roll and S.A. Ross, J.of Fnance, 35, N5, (1980) ; R. Roll and S.A. Ross, J.of Fnance, 39, N5, (1984) ; J.Shanken, Revew of Fnancal Studes, 5, (1992) 1-33; A.D. Clare and S.H.Thomas, J.of Busness Fnance and Accountng, 21, (1994) ;