THIRTY YEARS AGO marked the publication of what has come to be

Transcription

1 ONE NO ARBITRAGE: THE FUNDAMENTAL THEOREM OF FINANCE THIRTY YEARS AGO marked the publcaton of what has come to be known as the Fundamental Theorem of Fnance and the dscovery of rsk-neutral prcng. The earler opton prcng results of Black and Scholes (973) and Merton (973) were the catalyst for much of ths work, and certanly one of the central themes of ths research has been an effort to understand opton prcng. Hstory alone, then, makes the topc of these monographs the neoclasscal theory of fnance partcularly approprate. But, hstory asde, the basc theorem and ts attendant results have unfed our understandng of asset prcng and the theory of dervatves, and have generated an enormous lterature that has had a sgnfcant mpact on the world of fnancal practce. Fnance s about the valuaton of cash flows that extend over tme and are usually uncertan. The basc ntuton that underles valuaton s the absence of arbtrage. An arbtrage opportunty s an nvestment strategy that guarantees a postve payoff n some contngency wth no possblty of a negatve payoff and wth no ntal net nvestment. In smple terms, an arbtrage opportunty s a money pump, and the canoncal example s the opportunty to borrow at one rate and lend at a hgher rate. Clearly ndvduals would want to take advantage of such an opportunty and would do so at unlmted scale. Equally clearly, such a dsparty between the two rates cannot persst: by borrowng at the low rate and lendng at the hgh rate, arbtragers wll drve the rates together. The Fundamental Theorem derves the mplcatons of the absence of such arbtrage opportuntes. It s been sad that you can teach a parrot to be an economst f t can learn to say supply and demand. Supply, demand, and equlbrum are the catchwords of economcs, but fnance, or, f one s beng fancy, fnancal economcs, has ts own dstnct vocabulary. Unlke labor economcs, for example, whch The central tenet of the Fundamental Theorem, that the absence of arbtrage s equvalent to the exstence of a postve lnear prcng operator (postve state space prces), frst appeared n Ross (973), where t was derved n a fnte state space settng, and the frst statement of rsk neutral prcng appeared n Cox and Ross (976a, 976b). The Fundamental Theorem was extended to arbtrary spaces n Ross (978) and n Harrson and Kreps (979), who descrbed rsk-neutral prcng as a martngale expectaton. Dybvg and Ross (987) coned the terms Fundamental Theorem to descrbe these basc results and Representaton Theorem to descrbe the prncpal equvalent forms for the prcng operator.

2 2 CHAPTER specalzes the methodology and econometrcs of supply, demand, and economc theory to problems n labor markets, neoclasscal fnance s qualtatvely dfferent and methodologcally dstnct. Wth ts emphass on the absence of arbtrage, neoclasscal fnance takes a step back from the requrement that the markets be n full equlbrum. Whle, as a formal matter, the methodology of neoclasscal fnance can be ftted nto the framework of supply and demand, dependng on the ssue, dong so can be awkward and may not be especally useful. In ths chapter we wll eschew supply and demand and develop the methodology of fnance as the mplcaton of the absence of arbtrage. No Arbtrage Theory: The Fundamental Theorem The assumpton of no arbtrage (NA) s compellng because t appeals to the most basc belefs about human behavor, namely that there s someone who prefers havng more wealth to havng less. Snce, save for some anthropologcally nterestng socetes, a preference for wealth appears to be a ubqutous human characterstc, t s certanly a mnmalst requrement. NA s also a necessary condton for an equlbrum n the fnancal markets. If there s an arbtrage opportunty, then demand and supply for the assets nvolved would be nfnte, whch s nconsstent wth equlbrum. The study of the mplcatons of NA s the meat and potatoes of modern fnance. The early observatons of the mplcatons of NA were more specfc than the general theory we wll descrbe. The law of one prce (LOP) s the most mportant of the specal cases of NA, and t s the bass of the party theory of forward exchange. The LOP holds that two assets wth dentcal payoffs must sell for the same prce. We can llustrate the LOP wth a tradtonal example drawn from the theory of nternatonal fnance. If s denotes the current spot prce of the Euro n terms of dollars, and f denotes the currently quoted forward prce of Euros one year n the future, then the LOP mples that there s a lockstep relaton between these rates and the domestc nterest rates n Europe and n the Unted States. Consder ndvduals who enter nto the followng seres of transactons. Frst, they loan $ out for one year at the domestc nterest rate of r, resultng n a payment to them one year from now of ( + r). Smultaneously, they can enter nto a forward contract guaranteeng that they wll delver Euros n one year. Wth f as the current one-year forward prce of Euros, they can guarantee the delvery of ( + r)f Euros n one year s tme. Snce ths s the amount they wll have n Euros n one year, they can borrow aganst ths amount n Europe: lettng the Euro nterest rate be r e, the amount they wll be able to borrow s

3 NO ARBITRAGE 3 ( + rf ) ( + ). Lastly, snce the current spot prce of Euros s s Euros per dollar, they can convert ths amount nto ( + rf ) ( + r) s dollars to be pad to them today. Ths crcle of lendng domestcally and borrowng abroad and usng the forward and spot markets to exchange the currences wll be an arbtrage f the above amount dffers from the $ wth whch the nvestor began. Hence, NA generally and the LOP n partcular requre that ( + r)f = ( + r e )s, whch s to say that havng Euros a year from now by lendng domestcally and exchangng at the forward rate s equvalent to buyng Euros n the current spot market and lendng n the foregn bond market. Not surprsngly, as a practcal matter, the above party equaton holds nearly wthout excepton n all of the foregn currency markets. In other words, at least for the outsde observer, none of ths knd of arbtrage s avalable. Ths lack of arbtrage s a consequence of the great lqudty and depth of these markets, whch permt any perceved arbtrage opportunty to be exploted at arbtrary scale. It s, however, not unusual to come across apparent arbtrage opportuntes of msprced securtes, typcally when the securtes themselves are only avalable n lmted supply. 2 Whle the LOP s a nce llustraton of the power of assumng NA, t s somewhat msleadng n that t does not fully capture the mplcatons of removng arbtrage opportuntes. Not all arbtrage possbltes nvolve two dfferent postons wth dentcal cash flows. Arbtrage also arses f t s possble to establsh two equally costly postons, one of whch has a greater set of cash flows n all crcumstances than the other. To accommodate ths possblty, we adopt the followng framework and defntons. Whle the results we obtan are qute general and apply n an ntertemporal settng, for ease of exposton we wll focus on a one-perod model n whch decsons are made today, at date 0, and payoffs are r e e 2 I once was nvolved wth a group that specalzed n mortgage arbtrage, buyng and sellng the obscure and arcane peces of mortgage paper strpped and created from government pass-through mortgages (pools of ndvdual home mortgages). I recall one such pece a specal type of IO whch, after extensve analyss, we found would offer a three-year certan return of 37 percent per year. That was the good news. The bad news was that such nvestments are not scalable, and, n ths case, we could buy only $600,000 worth of t, whch, gven the hgh-prced talent we had employed, barely covered the cost of the analyss tself. The market found an equlbrum for these very good deals, where the cost of analyzng and accessng them, ncludng the rents earned on the human captal employed, was just about offset by ther apparent arbtrage returns.

4 4 CHAPTER receved n the future at date. By assumpton, nothng happens n between these two dates, and all decsons are undertaken at the ntal date, 0. To capture uncertanty, we wll assume that there s a state space, Ω, and to keep the mathematcs at a mnmum, we wll assume that there are only a fnte number of possble states of nature: Ω={θ,...,θ m }. The state space, Ω, lsts the mutually exclusve states of the world that can occur, m. In other words, at tme the uncertanty s resolved and the world s n one and only one of the m states of nature n Ω. We wll also assume that there are a fnte number, n, of traded assets wth a current prce vector: Lastly, we wll let p = (p,...,p n ). (η,...,η n ) denote an arbtrage portfolo formed by takng combnatons of the n assets. Each element of η, say η, denotes the nvestment n asset. Any such portfolo wll have a cost, and we wll refer to such a combnaton, η, as an arbtrage portfolo f t has no postve cost, pη 0. We represent the Arrow-Debreu tableau of possble securty payoffs by G = [g j ] = [payoff of securty j f state θ j occurs]. The rows of G are states of nature and the columns are securtes. Each row of the matrx, G, lsts the payoffs of the n securtes n that partcular state of nature, and each column lsts the payoffs of that partcular securty n the dfferent states of nature. Wth the prevous notaton, we can defne an arbtrage opportunty. Defnton: An arbtrage opportunty s an arbtrage portfolo wth no negatve payoffs and wth a postve payoff n some state of nature. Formally, an arbtrage opportunty s a portfolo, η, such that and pη = p η, pη 0 Gη > 0,

5 NO ARBITRAGE 5 where at least one nequalty for one component or the budget constrant s strct. 3 We can smplfy ths notaton and our defnton of NA by defnng the stacked matrx: p A = G. Defnton: An arbtrage s a portfolo, η, such that Aη > 0. Formally, then, the defnton of no arbtrage s the followng. Defnton: The prncple of no arbtrage (NA): NA {η Aη > 0} =, that s, there are no arbtrage portfolos. The precedng mathematcs captures our most basc ntutons about the absence of arbtrage possbltes n fnancal markets. Put smply, t says that there s no portfolo, that s, no way of buyng and sellng the traded assets n the market so as to make money for sure wthout spendng some today. Any portfolo of the assets that has a postve return no matter what the state of nature for the world n the future, must cost somethng today. Wth ths defnton we can state and prove the Fundamental Theorem of Fnance. The Fundamental Theorem of Fnance: are equvalent: The followng three statements. No Arbtrage (NA). 2. The exstence of a postve lnear prcng rule that prces all assets. 3. The exstence of a (fnte) optmal demand for some agent who prefers more to less. Proof: The reader s referred to Dybvg and Ross (987) for a complete proof and for related references. For our purposes t s suffcent to outlne the argument. A lnear prcng rule, q, s a lnear operator that prces an asset when appled to that asset s payoffs. In ths fnte dmensonal setup, a lnear prcng rule s smply a member of the dual space, R m and the requrement that 3 We wll use to denote that each component s greater than or equal, > to denote that holds and at least one component s greater, and >> to denote that each component s greater.

6 6 CHAPTER t s postve s just the requrement that q >> 0. The statement that q prces the assets means smply that q satsfes the system of equatons: p = qg. It s easy to show that a postve lnear prcng operator precludes arbtrage. To see ths let η be an arbtrage opportunty. Clearly, 0 pη = (qg)η = q(gη). But, snce q >> 0, ths can only be true f Gη < 0, whch s nconsstent wth arbtrage. The converse result, that NA mples the exstence of a postve lnear prcng operator, q, s more sophstcated. Outlnng the argument, we begn by observng that NA s equvalent to the statement that the set of net trades, S = {x η, x = Aη}, does not ntersect the postve orthant snce any such common pont would be an arbtrage. Snce S s a convex set, ths allows us to apply a separatng hyperplane theorem to fnd a y that separates S from the postve orthant, R +. Snce yr + > 0, we have y >> 0. Smlarly, for all η, we have ys 0 yaη 0 yaη = 0 (snce f the nequalty s strct, η wll volate t,.e., S s a subspace.) Defnng mples that q = ( q,..., qm) = ( y2,..., ym+ ) y p = qg, hence q s the desred postve lnear prcng operator that prces the marketed assets. Relatng NA to the ndvdual maxmzaton problem s a bt more straghtforward and constructve. Snce any agent solvng an optmzaton problem would want to take advantage of an arbtrage opportunty and would want to do so at arbtrary scale, the exstence of an arbtrage s ncompatble wth a fnte demand. Conversely, gven NA, we can take the postve lnear prcng rule, q, and use t to defne the margnal utlty for a von Neumann-Morgenstern expected utlty maxmzer and, thereby, construct a concave monotone utlty functon that acheves a fnte maxmum.

7 NO ARBITRAGE 7 Much of modern fnance takes place over tme, a fact that s usually modeled wth stochastc dffuson processes or wth dscrete processes such as the bnomal. Whle our smplfed statement of the Fundamental Theorem s nadequate for that settng, and whle the theorem can be extended to nfnte dmensonal spaces, some problems do arse. However crtcal these dffcultes are from a mathematcal perspectve, as a matter of economcs ther mportance s not yet well understood. The nut of the dffculty appears to be that the operator that prces assets may not have a representaton as a vector of prces for wealth n dfferent states of nature, and that makes the usual economc analyss of trade-offs problematc. 4 A complete market s one n whch for every state θ there s a combnaton of the traded assets that s equvalent to a pure contngent state clam, n other words, a securty wth a payoff of the unt vector: one unt f a partcular state occurs, and nothng otherwse. In a complete market G s of full-row rank, and the equaton has a unque soluton, p = qg q = pg. Ths determnancy s one reason why market completeness s an mportant property for a fnancal market, and we wll later dscuss t n more detal. By contrast, n an ncomplete market, the postve prcng operator wll be ndetermnate and, n our settng wth m states and n securtes, f m > n, then the operator wll be an element of a subspace of dmensonalty m n. Ths s llustrated n fgure. for the m = 3 state, n = 2 securty example. In fgure., each of the two securtes has been normalzed so that R and R 2 represent ther respectve gross payoffs n each of the three states per dollar of nvestment. The Representaton Theorem The postve lnear operator that values assets n the Fundamental Theorem has several mportant alternatve representatons that permt useful restatements of the theorem tself. 4 Mathematcally speakng, separaton theorems requre that the set of net trades be fat n an approprate sense, and n, say, the L 2 norm, the postve orthant lacks an nteror. Ths prevents the applcaton of basc separaton theorems and requres some modfcatons to the defnton of arbtrage and no arbtrage (see Ross [978a], who extends the postve lnear operator by fnessng ths problem, and Harrson and Kreps [979], who fnd a way to resolve the problem).

8 8 CHAPTER R q R 2 Fgure. Defnton: Martngale or rsk-neutral probabltes are a set of probabltes, π *, such that the value of any asset wth payoffs of z = (z,...,z m ) s gven by the dscounted expectaton of ts payoffs usng the assocated rskless rate to dscount the expectaton, Defnton: Vz () r E * z = [] = * z. + + r π A state prce densty, also known as a prcng kernel, s a vector, φ = (φ,...,φ m ), such that the value of any asset s gven by Vz ()= E[ φz] = πφz. These defntons provde alternatve forms for the Fundamental Theorem.

9 NO ARBITRAGE 9 Representaton Theorem: The followng statements are equvalent:. There exsts a postve lnear prcng rule. 2. The martngale property: the exstence of postve rsk-neutral probabltes and an assocated rskless rate. 3. There exsts a postve state prce densty or prcng kernel. Proof: () (2) From the Fundamental Theorem we know that there exsts a postve prcng vector, q >> 0, such that Consder, frst, the sum of the q V(z) = qz. V() e = qe = q. We have wrtten ths as V(e), where e s the vector of snce t s the value of recevng for sure, that s, of gettng n each state. We can defne a rate of return, r, as the return from nvestng n ths securty, thus, or r = Ve () Ve () = = qe = q r. + Notce that r s unquely defned f a rskless asset s traded explctly or mplctly, that s, f e s n the span of the marketed assets. Next we defne the rsk-neutral probabltes as Notce that, just as probabltes should, the π * sum to. Hence, we have π * = >. q 0 Vz () qz q q = = ( ) q z q = * z r r E * z = [], + π +

10 0 CHAPTER where the symbol E* denotes the expectaton taken wth respect to the rskneutral probabltes (or measure ). Conversely, f we have a set of postve rsk-neutral probabltes and an assocated rskless rate, r, t s clearly a postve lnear operator on R m, and we can smply defne the state space prce vector as q = π * > 0. + r () (3) Defnng the postve state densty as we have φ q = > 0, π Vz () = qz = q π z π = πφz = E[ φz], and the converse s mmedate. Much of the research n modern fnance has turned on extendng these results nto mathematcally deeper terran. The martngale, or rsk-neutral, approach n partcular has been the focus of much attenton because of the ablty to draw on the extensve probablty lterature. Interestngly, the rsk-neutral approach was dscovered ndependently of the Fundamental Theorem (see Cox and Ross 976a, 976b) n the context of opton prcng, and only later was the relaton between the two fully understood. Gven the tradton of Prnceton n the physcal scences and gven the role of fnance as, perhaps, the most scentfc of the socal scences, t s only approprate to pont out the close relaton between these results and some fundamental results n physcs. Wthout much magnaton, the Representaton Theorem can be formally stated n a general ntertemporal settng n whch a securty wth a random payoff, z T, across tme, T, has a current value, p, gven by T rds, T p = E* [ 0 e z ], where r s denotes the nstantaneous rsk-free nterest rate and the astersk denotes the expectaton under the martngale measure. Ths s the usual modern statement of rsk-neutral prcng. In ths computaton, the short nterest rate

11 NO ARBITRAGE can be stochastc, n whch case all lkely paths must be consdered. More generally, too, the payoff tself could be path dependent as well. In practce, ths expectaton can be computed numercally by a summaton over Monte Carlo smulatons generated from the martngale measure. Such an ntegral, generated over paths, s equvalent to the Feynman-Kac ntegral of quantum mechancs. Opton Prcng Black-Scholes-Merton In the prevously shown form, rsk-neutral prcng (Cox and Ross 976a, 976b) provdes the soluton to the most famous and mportant examples of the mplcatons of NA: the classc Black-Scholes-Merton (973) optonprcng formula and the bnomal formula of Cox, Ross, and Rubnsten (979). 5 Wthout gong nto great detal, we wll outlne the argument. The tme T payoff for a call opton on a stock wth a termnal prce of S T at the opton s maturty date, T, s: C(S, T) = max{s T K, 0}, where K s referred to as the strke or exercse prce. 6 To compute the current value of the opton we take the expectaton of ths payoff under the rskneutral measure, that s, the martngale expectaton. Ths s exactly the rskneutral valuaton that s obtaned by assumng that the expected return on the stock s the rsk-free nterest rate, r, and takng the ordnary expectaton. The Black-Scholes model assumes that the stock process follows a lognormal dstrbuton. In the formalsm of Ito s stochastc calculus, ths s wrtten as ds = µ dt + σ dz, where µ s the local drft or expected return, σ s the local speed or standard devaton, and the symbol dz s nterpreted as a local Brownan process over the nterval [t, t + dt] wth mean zero and varance dt. Assumng that the call opton has a current value of C(S, t), where we have wrtten ts arguments to emphasze the dependence of value on the current stock prce, S, and the current tme, t, and applyng the ntegral valuaton equaton, we obtan C(S, 0)= E*[e r(t t) max{s T K, 0}]. The approach of Black and Scholes was less drect. Usng the local equaton, they observed that the return on the call, that s, the captal gan on the call, would be perfectly correlated locally wth the stock snce ts value depends on 5 See Black and Scholes (973) and Merton (973) for the orgnal analyss wth a dffuson process, and Cox, Ross, and Rubnsten (979) for the bnomal prcng model. 6 We are assumng that the call s on a stock that pays no dvdends and that t s a European call, whch s to say that t can be exercsed only at maturty and not before then.

12 2 CHAPTER the stock prce. They used ths result together wth a local verson of the Captal Asset Prcng Model (dscussed later) to derve a dfferental equaton that the call value would follow, the Black-Scholes dfferental equaton: 2 2 σ S C rsc rc C 2 SS S t + + =. Notce the remarkable fact that the assumed expected return on the stock, µ so dffcult to measure and surely a pont of contenton among experts plays no role n determnng the value of the call. Merton ponted out that snce the returns on the call were perfectly correlated wth those on the underlyng stock, a portfolo of the rsk-free asset and the stock could be constructed at each pont n tme that would have dentcal returns to those on the call. To prevent arbtrage, then, a one-dollar nvestment n ths replcatng portfolo would be equvalent to a one-dollar nvestment n the call. The resultng algebra produces the same dfferental equaton, and, as a consequence, there s no need to employ an asset-prcng model. Black and Scholes and Merton appealed to the lterature on dfferental equatons to fnd the soluton to the boundary value problem wth the termnal value equal to the payoff on the call at tme T. Cox and Ross observed that once t was known that arbtrage alone would determne the value of the call opton, analysts were handng off the problem to the mathematcans prematurely. Snce arbtrage determnes the value, they argued that the value would be determned by what they called rskneutral valuaton, that s, by the applcaton of the rsk-neutral formula n the Representaton Theorem. In fact, the Black-Scholes dfferental equaton for the call value s the backward equaton for the log-normal stochastc process appled to the rsk-neutral expected dscounted value ntegral, and, conversely, the dscounted ntegral s the soluton to the dfferental equaton. The assumed expected return on the stock, µ, s rrelevant for valuaton snce n a rsk-neutral world all assets have the rskless rate as ther expected return. It follows that the value s smply the dscounted expected value of the termnal payoff under the assumpton that the drft on the stock s the rsk-free rate. Applyng ths analyss yelds the famous Black-Scholes formula for the value of a call opton: C(S, 0)= SN(d ) e rt KN(d 2 ), where N( ) denotes the standard cumulatve normal dstrbuton functon and where and d 2 ln( SK / ) + ( r+ 2 σ ) T = σ T d 2 = d σ T.

13 NO ARBITRAGE 3 The Bnomal Model The crucal features of the Black-Scholes-Merton analyss were not mmedately apparent untl the development of the bnomal model by Cox, Ross, and Rubnsten. Was t the lognormal dstrbuton, the contnuty that allowed contnuous tradng or some other feature of the model, and would the analyss fall apart wthout these features? The bnomal model resolved these questons, and because of ts flexblty t has now become the preferred structure for valung complex dervatves n the practcal world of fnance. The bnomal analyss s llustrated n fgure.2. The value of the opton at tme t s gven by C(S, t) where we explctly recognze ts dependence on S, the stock prce at tme t. There are two states of nature at each tme, t, state a and state b, representng the two possble futures for the stock prce, as or bs where a > + r > b. The fgure dsplays the gross returns on the three assets, + r for the bond, a or b for the stock, and the formula C(aS, t + )/C(S, t) for the opton f state a occurs and C(bS, t + )/C(S, t) f b occurs. Wth two states and three assets, one of the assets may be selected to be redundant, and ths s represented by the lne through the gross returns of the assets. The lne s the combnaton of returns across the states, avalable by combnng a unt nvestment n the three assets. To prevent arbtrage these three ponts must State a /q a <C(bS, t+)/c(s,t),c(as,t+)/c(s,t)> <b,a> <+r, +r> /q b State b Fgure.2

14 4 CHAPTER all le on a lne or else a portfolo of two of them would domnate the thrd. The equaton of ths lne provdes the fundamental dfference equaton, whch can be solved subject to the boundary condton of the payoff at maturty, T, to gve the current value of the opton: ( + rcs ) (, τ ) = + r b (, ) + + (, ), a b CaSτ a r a b CbSτ where τ s the number of perods left to maturty, T t. The coeffcents of the next perod values n the equaton are called the rsk neutral probabltes for the bnomal process, that s, they are the probabltes for the process under the assumpton that the expected return s the rsk-free rate. Notce that parallel wth the absence of a role for the drft n the dffuson model, because the market s locally complete wth two states and three assets, one of the assets s redundant and, gven the values of the other two, the probabltes for the bnomal jumps are not needed to value the thrd securty. If we let the tme dfference between nodes grow small and ncrease n number so as to mantan a constant volatlty for the stock return, ths dfference equaton converges to the Black-Scholes dfferental equaton. Alternatvely, we can derve the dfference equaton drectly from the Fundamental Theorem by nvokng NA drectly to prce a dervatve securty such as a call opton. Snce the stock and the opton are perfectly correlated assets locally, we can form a portfolo of the two of them that s rskless, for example, we can short the stock and go long the opton n just the correct proportons so as to elmnate all rsk n the portfolo. Equvalently, we could construct a replcatng portfolo composed of the rsk-free rate and the stock whose return s locally dentcal to the return on the call opton. NA n the form of the LOP requres that the portfolo, as a rskless asset, have the unque rsk-free rate of return, r. The resultng equaton s the dscrete equaton of the bnomal model and the famous partal dfferental equaton of Black and Scholes. In ether case, the soluton s the prcng equaton derved from rskneutral prcng. 7 What, then, are the crucal features that allow for arbtrage to determne the prce of the dervatve securty or opton as a functon of traded securtes? Clearly, local spannng s suffcent (and except n unusual cases t s necessary) n the sense that the return of the dervatve securty must be a (locally) lnear combnaton of the returns on the stock and any other marketed securtes that are requred to span ts return. For example, f the volatlty of the 7 Whle the above s the most straghtforward dervaton, t s not the most powerful. In fact, there s no need to assume that the opton nherts the stochastc propertes of the underlyng asset snce t s possble to replcate the payoff of the opton wth a portfolo of the stock and the bond. Through NA ths would force the value of the opton to be the orgnal cost of such a portfolo. See Ingersoll (987) for ths argument.

15 NO ARBITRAGE 5 stock s tself random, then another nstrument such as, say, the movement on a traded volatlty ndex or on a market ndex that s perfectly correlated wth changes n the volatlty, would have to be used; and the value of the opton would depend not only on the stock on whch t was drectly wrtten, but also on the value of these other securtes. As another example, we could have an opton whose payoff depended not smply on the return of a sngle stock but rather on two or more assets, S and P wth a payoff at tme T of S 2 + P. In response to such complcatons, n dscrete tme the bnomal mght gve way to a trnomal or an even more complex process (see Cox and Ross 976a, 976b). Some Further Applcatons of the Representaton Theorem Asset Prcng Hstorcally and currently the prcng of assets has been a focus of neoclasscal fnance and, not surprsngly, a number of asset-prcng theores have been developed. Whle these dffer n ther form, they share some common ntutons. The theory of asset prcng began hundreds of years ago wth a smple formulaton drawn from the ntutons of gamblng. The common sense vew was that an asset would have a far prce f ts expected return was smply the rskless rate, E E[x ] = r, where x denotes the expected rate of return on asset. 8 If ths were so, then, on average, the asset would nether excessvely reward nor penalze the holder. Wth the work of Hcks and Samuelson and others, t was recognzed that ths was not an adequate representaton. After all, ndvduals are generally assumed to be averse to rsk and, therefore, assets whose returns are rsky should command a lower prce than rskless assets. Ths was captured by postng a return premum, π, for rsky assets, E = r + π, where π > 0 to capture the need for addtonal return to compensate for rsk. Both theoretcal and emprcal works have focused on the determnants of the rsk premum. Presumably π should be determned by two effects. On the one hand, the rsker the asset the greater the premum should be for bearng that rsk and, on the other hand, the more averse nvestors are to rsk, the greater the premum they would demand for bearng that rsk. If we let rsk 8 Ths s usually credted to Bernoull.

16 6 CHAPTER averson be captured by a parameter, R, whch could denote some average of the von Neumann-Morgenstern rsk-averson parameter for ndvdual agents, and f we represent uncertanty by the smple varance of the asset return, σ 2, then we would have π Rσ wth, perhaps, other factors such as asset supply and the lke enterng n. It took decades from ths ntal set of concepts to the development of mean varance portfolo theory and the ntutve leap of the Captal Asset Prcng Model (CAPM) to recognze that ths way of thnkng about the rsk premum was not qute correct. The contrbuton of modern neoclasscal theory to ths queston comes wth the recognton that snce an asset fts nto a portfolo, what matters for determnng ts rsk premum s how t relates to the other assets n that portfolo, and not smply ts overall volatlty. We can llustrate ths wth a look at the three man asset-prcng theores. 2 Arbtrage Prcng Theory (APT) The Arbtrage Prcng Theory (APT) s the most drect descendant of the Representaton Theorem (see Ross 973, 976a). It begns wth the smple observaton that f one wants to say more about asset returns than can be obtaned from arbtrage alone, one must make some assumptons about the dstrbuton of returns. Suppose, then, we post that returns follow an exact lnear model, x = E + β f, where f s a common factor (or vector of factors) that nfluences the returns on all assets and β s the beta, or loadng, for asset on that factor. The vector f captures the nnovatons n the state varables that nfluence the returns of all the assets and, mathematcally, n a statc settng ths s equvalent to a rank restrcton on the matrx of returns. Snce all the assets are perfectly correlated, n ths exact one-factor world we would expect the force of NA to dctate a prcng result. In fact, the logc s dentcal to that whch we employed to derve the bnomal prcng model for optons. Snce a unt nvestment n any asset gves a payoff equal to the gross return on that asset, the value of any gross return must be, = + + r E * [ x ] = ( + E* [ E + f + β ]) r = + E + β E* [ f ]. + r

17 NO ARBITRAGE 7 Rearrangng the equaton, we obtan the famlar statement of the rsk premum, the Securty Market Lne (SML): where E r = E*[ f ]β = π f β, π f E*[ f ]. Observng from the SML that π f s the rsk premum on an asset wth a beta of, we can rewrte the SML n the tradtonal form as E r = (E f r)β, where E f denotes the expected return on any portfolo wth a beta of on the factor. In other words, the rsk premum depends on the beta of the asset that captures how the asset relates or correlates to the other assets through ts relaton to the common factor(s) and the premum, π f, on those factor(s). The exact specfcaton of the foregong statement s too stark for emprcal practce because outsde of opton prcng and the world of dervatve securtes, assets such as stocks are not perfectly correlated and are subject to a host of dosyncratc nfluences on ther returns. We can modfy the return generatng mechansm to reflect ths as x = E + β f + ε, where ε s assumed to capture the dosyncratc, for example, company- or ndustry-specfc forces on the return of asset. If the dosyncratc terms are suffcently ndependent of each other, though, by the law of large numbers asymptotcally as the number of assets n a well-dversfed portfolo s ncreased, the portfolo return wll approach the orgnal exact specfcaton. Such arguments lead to the SML holdng n an approxmate sense. 9 Whle the ntuton of the APT s clear, namely that factors exogenous to the market move returns and that prcng depends on them, and, further, that we can capture these by usng endogenous varables such as those created by formng portfolos wth unt loadngs on the factors, exactly what these factors mght be s unspecfed by the theory. Next we turn to the tradtonal Captal Asset Prcng Model (CAPM) and ts cousn, the Consumpton Beta Model (CBM), to get more explct statements about the SML. 9 These condtons have led to a wde lterature and some debate as to exactly how ths return s accomplshed and what s to be meant by an approxmate prcng rule. See, for example, Shanken (982) and Dybvg and Ross (985).

18 8 CHAPTER The Captal Asset Prcng Model (CAPM) and the Consumpton Beta Model (CBM) Hstorcally, the CAPM (Sharpe 964; Lntner 965) preceded the APT, but t s a bt less obvous what ts relaton s to NA. By assumng that returns are normally dstrbuted (or locally normal n a contnuous tme-dffuson model), or by assumng that ndvduals all have quadratc utlty functons, t s possble to develop a beautful theory of asset prcng. In ths theory two addtonal concepts emerge. Frst, snce the only focus s on the mean and the varance of asset returns, ndvduals wll choose portfolos that are mean-varance effcent, that s, portfolos that le on the fronter n meanvarance space. Ths s equvalent to what s called two-fund separaton (see Ross [978b], who observes that all effcent portfolos le on a lne n the n-dmensonal portfolo space). Second, a consequence of mean-varance effcency s that the market portfolo, m, that s, the portfolo n whch all assets are held n proporton to ther values, wll tself be mean-varance effcent. Indeed, the mean-varance effcency of the market portfolo s equvalent to the CAPM. 0 Not surprsngly, n any such model where valuaton s by a quadratc, the prcng kernel wll be n terms of margnal utltes or dervatves of a quadratc and wll be lnear n the market portfolo. It can be shown that the prcng kernel for the CAPM has the form Hence any asset must satsfy Settng = m, we can solve for λ, ϕ = [ λ( m E + r m )]. = E[ ϕ( + x)] = E [ λ( m E + m)]( x) + r = + E λ cov( x m + r +, ). r E m r = λσ 2, m 0 See Stephen Ross (977). Rchard Roll (977) ponted out the dffcultes that the potental nablty to observe the entre market portfolo rases for testng the CAPM. The astute reader wll recognze that ths operator s not postve, and, therefore, the CAPM admts of arbtrage outsde of the doman of assets to whch the model s delberately restrcted. See Dybvg and Ingersoll (982) for a dscusson of these ssues.

19 NO ARBITRAGE 9 whch allows us to rearrange the prcng equaton to the SML: E r = λcov(x,m) = (E m r)β, where β s the regresson coeffcent of asset s returns on the market, m. The SML verfes a powerful concept. Indvduals who hold a portfolo wll value assets for the margnal contrbutons they wll make to that portfolo. If an asset has a postve beta wth respect to that portfolo, then addng t nto the portfolo wll ncrease the volatlty of the portfolo s returns. To compensate a rsk-averse nvestor for ths ncrease n volatlty, n equlbrum such an asset must have a postve excess expected return, and the SML verfes that ths excess return wll, n fact, be proportonal to the beta. Ths s all vald n a one-perod world where termnal wealth s all consumed and end-of-perod consumpton s the same as termnal wealth. Once we look at a multperod world, the possblty of ntermedate consumpton separates the stock of wealth from the flow of consumpton. Not surprsngly, though, wth a focus only on consumpton and wth a restrcton that preferences depend only on the sum of the utltes of the consumpton flow, the SML agan holds, wth β as the regresson coeffcent of asset on ndvdual or, when aggregaton s possble, on aggregate consumpton, and wth the E m nterpreted as the expected return on a portfolo of assets that s perfectly correlated wth consumpton. Ths result s called the Consumpton Beta Model (CBM) (see Merton 97; Lucas 978; and Breeden 979). More generally, n the next chapter we wll explot the fact that the Representaton Theorem allows prcng to take the form of the securty market lne where the market portfolo s replaced by the prcng kernel, that s, excess expected return on assets above the rsk-free return s proportonal to ther covarance wth the prcng kernel. In a complete market where securtes are traded contngent on all possble states and where agents have addtve separable von Neumann-Morgenstern utlty functons, ndvduals order ther consumpton across states nversely to the kernel, whch mples that aggregate consumpton s smlarly ordered. Ths fact allows us to use aggregate consumpton as a state varable for prcng and replaces covarance wth the prcng kernel n the SML wth covarance wth aggregate consumpton. Corporate Fnance Corporate fnance s the study of the structure and the valuaton of the ownershp clams on assets through varous forms of busness enterprse. The famous Modglan-Mller (MM) Theorem (Modglan and Mller 958) on the rrelevance of the corporate fnancal structure for the value of the frm

20 20 CHAPTER emerges as a corollary of the NA results. The MM Theorem s to modern corporate fnance what the famous Arrow Impossblty Theorem s to the Theory of Socal Choce. Lke a great boulder n our path, t s too bg to move and all of current research can be nterpreted as an effort to crcumvent ts harsh mplcatons. The MM Theorem teaches us that n the prstne envronment of perfect markets wth no frctons and no asymmetres n nformaton, corporate fnance s rrelevant. As wth the Impossblty Theorem, appealng axoms have seemngly appallng consequences. Consder a frm wth an array of clams aganst ts value. Typcally we model these clams as debt securtes to whch the frm owes a determned schedule of payoffs wth bankruptcy occurrng f the frm cannot pay these clams n full. The resdual clamant s equty, whch receves the dfference between the value of the frm at tme and the payoffs to the debt holders provded that the dfference s postve, and whch receves nothng f the frm s n bankruptcy. Notce that the equty clam s smply a call on the value of the frm wth an exercse prce equal to the sum of the face value of the debt clams. The statement of the MM Theorem, though, s much more general than ths partcular case. The Modglan-Mller (MM) Theorem: of ts fnancal structure. The value of a frm s ndependent Proof: Suppose a frm has a collecton of clams aganst ts value, wth payoffs of z,..., z k at tme. Snce we have ncluded all of the clamants, we must have z z k = V. As a consequence, at tme 0 we have k value of frm = L( z ) + + L( z ) k = Lz ( + + z) = LV ( ), whch s ndependent of the partculars of the ndvdual clams. Concluson There s, of course, much more to neoclasscal fnance than the materal or even the topcs presented n ths chapter. We have omtted a detaled examnaton of ndvdual portfolo theory and have hardly mentoned the fascnatng

21 NO ARBITRAGE 2 ntertemporal analyss of both ndvdual optmal consumpton and portfolochoce problems as well as the attendant equlbra. We wll rectfy a small porton of ths n the next chapter, but we wll stll not have tme or space adequately to treat the ntertemporal analyss. Havng read that, though, a cync mght observe that almost all of contemporary ntertemporal analyss s really just an extenson of the statc analyss of ths chapter. Ths concdence occurs for two rather opposte reasons. On the one hand, most ntertemporal models are smply sequences of one-perod statc models. Wth some rare exceptons, there are really no true ntertemporal models n whch consumpton or producton nteracts over tme n a fashon that s not addtvely separable nto a sequence of statc models. Secondly, a marvelous nsght by Cox and Huang (989) demonstrates that n a complete market the optmzaton and equlbrum problems may be treated as though the world were statc the ntegral vew. If the market s complete, then any future pattern of consumpton can be acheved by choosng a portfolo constructed from the array of state space-contngent securtes. Such a portfolo wll have a representaton as a dynamc polcy and t wll produce the desred ntertemporal consumpton profle. In a sense, f the number of states of a statc problem s augmented suffcently, t can replcate the results of a complete ntertemporal world. We wll expand a bt on these ssues n the next chapter.