Contngent Clams and the Arbtrage Theorem Paul J. Atzberger Paul J. Atzberger Please send any comments to: atzberg@math.ucsb.edu
Introducton No arbtrage prncples play a central role n models of fnance and economcs. The assumpton of no arbtrage essentally states that there s no free lunch n a market. In other words, there are no zero cost nvestment strateges that allow for a market partcpant to make a proft wthout takng some rsk of a loss. We shall manly be nterested n prcng dervatve contracts, whch derve ther value from an underlyng asset or ndex such as a stock or nterest rate. In ths context no arbtrage can be used to determne a prce for a contract by constructng portfolos whch replcate the payoff n complete markets or gve a bound on permssble prces for a contract by constructng portfolos whch bound the payoff n ncomplete markets. Usng results from lnear programmng, the prcng theory obtaned by constructng such replcatng portfolos can be shown to be have as ts dual problem the constructon of a rsk-neutral probablty for the market. The rsk-neutral termnology can be motvated by consderng an ndvdual who values a contract by only consderng ts expected payoff and s unnfluenced by the rskness of obtanng an uncertan payoff. Such an ndvdual we mght call a rsk-neutral nvestor. The rsk-neutral probablty of a market s defned as a probablty measure whch when takng the expected payoff of any dervatve contract and dscountng nto todays dollars gves the prce of the contngent clam consstent wth no arbtrage. The connecton between the exstence of rsk-neutral probablty measures and no arbtrage n a market s establshed n the The Arbtrage Theorem. For complete markets the Arbtrage Theorem states that there s no arbtrage f and only f there exsts a rsk-neutral probablty measure. In other words there s no arbtrage n a market only f the dervatves have a prce whch s gven by the expectaton of ts payoff, under the rsk-neutral probablty measure, dscounted to express the value n today s dollars. Ths theory plays a central role n the prcng of optons n mathematcal fnance by connectng arbtrage consderatons wth probablty theory. Ths theory wll be the man subject of these notes. The materals presented here are taken from the followng sources: Dervatves Securtes: Lecture Notes by R. Kohn, Optons, Futures, and Other Dervatves by Hull, and Dervatves Securtes by Jarrow and Turnbll. One Perod Markets We shall consder prce movements over a sngle tme perod [0,T] for a market consstng of two assets, a stock and a bond. Many models could n prncple be used to model the prce movements, we shall restrct ourselves n these notes to two very smple models, n partcular, a bnomal and trnomal tree to llustrate the basc deas. Throughout the dscusson we shall assume that the nterest rate for the bond s fxed wth rate r. Bnomal Market Model Let the prce of the stock at tme 0 be denoted by s 0. In the bnomal market model we shall allow for only two possbltes for the movement of the prce over the perod [0,T]. Let these be denoted by s 1,s 2 wth s 1 < s 2, see fgure 1. s 2 s 0 s 1 Fgure 1: Bnomal Model 2
In our model no arbtrage requres that there be no zero cost nvestment strateges that would allow for an nvestor to make a proft wthout some rsk of a loss. Ths places constrants on the possble values of s 1,s 2 n our model. In partcular, f we had s 1 > e rt s 0 and s 2 > e rt s 0 then there would be an arbtrage opportunty n the market. We could ensure ourselves a proft by takng out a bank loan for the amount s 0 at tme 0 and buyng the stock. At tme T we would then owe the bank s 0 e rt, whch s less than we would make from sellng the stock n ether outcome of the market, s 1 or s 2. A smlar type of nvestment strategy can be constructed f s 1 < e rt s 0 and s 2 < e rt s 0, except n ths case we sell the stock and buy a bond. From the assumpton of no arbtrage the prces n our model must satsfy: s 1 < s 0 e rt < s 2. We shall now consder the prcng of a dervatve securty whch has a payoff whch depends only on the prce of the stock at tme T. We shall denote the prces of the contract by f 0, f 1, f 2 wth the same ndexng conventon we used for the stock. For example, a call opton wth strke prce K would have the value at tme T gven by f 1 = (s 1 K) + and f 2 = (s 2 K) +. A put opton would have the value f 1 = (K s 1 ) + and f 2 = (K s 2 ) +. Determnng the value of the contract at tme 0 requres a bt more work, whch we shall now dscuss. The basc strategy to determne the value of the contngent clam at tme 0 s to buld a portfolo of the bond and stock whch gves the same payoff as the contract. No arbtrage then requres that the contract be worth at tme 0 the same as the replcatng portfolo. We shall denote the weghts for the assets n our portfolo by w 1 for the stock and w 2 for the bond, so that the portfolo at tme zero has the value: w 1 S 0 + w 2 e rt. Recall a bond whch pays $1 at tme T s worth e rt today. Replcatng the payoff of the contngent clam at tme T requres that we solve the followng lnear equatons for w 1,w 2 : Ths has soluton: w 1 s 1 + w 2 = f 1 w 1 s 2 + w 2 = f 2. w 1 = f 2 f 1 s 2 s 1 w 2 = s 2f 1 s 1 f 2 s 2 s 1. Snce the portfolo wth the weghts w 1 and w 2 has the same payoff as the contngent clam n each outcome of the stock market, by no arbtrage the value of the contngent clam at tme 0 must be: V (f) = w 1s 0 + w 2e rt. Markets n whch for any contngent clam a replcatng portfolo can be constructed are referred to as complete markets. By usng the defnton of w 1 and w 2 and factorng the coeffcents nto common terms of f 1 and f 2 we have: V (f) = e rt [(1 q)f 1 + qf 2 ] where q = s0ert s 1 s2 s1. It can be readly checked that no arbtrage holds for the market, s 1 < s 0 e rt < s 2, f and only f 0 < q < 1. Thus we see that the Arbtrage Theorem holds for the bnomal market model. 3
Trnomal Market Model Let us now consder a market whch at tme T has three dfferent outcomes for the stock prces s 1 < s 2 < s 3, see fgure 2. The assumpton of no arbtrage n ths market requres that at least one outcome does better than the rsk-free return and that at least one outcome does worse. Ths requres: s 1 < s 0 e rt, and, s 0 e rt < s 3. s 3 s 0 s 2 s 1 Fgure 2: Trnomal Model Consder agan a contngent clam wth value f 0 at tme 0 and value f 1,f 2,f 3 at tme T. For a gven outcome of the stock market, we agan know the values f 1,f 2,f 3 of the contract, but do not know the value of f 0 at tme 0. We agan seek to construct a replcatng portfolo. In ths case we must fnd the weghts w 1,w 2 by solvng the followng lnear system for the payoff at tme T: w 1 s 1 + w 2 = f 1 w 1 s 2 + w 2 = f 2 w 1 s 3 + w 2 = f 3. We see that ths lnear system only permts a soluton for a select subset of contngent clams wth payoffs [f 1,f 2,f 3 ] wthn a 2-dmensonal subspace. Thus the payoff of many of the contngent clams can not be replcated by formulatng a portfolo usng only the stock and bond. Such markets are referred to as ncomplete markets. Whle we can not determne an exact prce by ths method, useful nformaton can stll be obtaned. Any portfolo havng a larger value at tme T than the contngent clam s payoff, n each outcome of the stock market, must have a greater value at tme 0 by no arbtrage arguments. A smlar statement follows for portfolos havng lesser value at tme T than the contngent clam s payoff. Ths gves the bounds: V (f) w + 1 + w+ 2 e rt V (f) w 1 + w 2 e rt where the superscrpt + ndcates that the portfolo domnates the payoff of the contngent clam, whle ndcates the portfolo has a smaller value than the payoff of the contngent clam n each outcome of the stock market. More precsely, we can express ths as: mn w 1 s +w 2 f =1,2,3 w 1 s 0 + w 2 e rt V (f) General One Perod Market Models mn w 1 s +w 2 f =1,2,3 w 1 s 0 + w 2 e rt. We shall use the followng notaton to express the structure of a market n general. Let the payoff of asset for outcome of the market be denoted by payoff matrx D,. Let the prce of the th asset at tme 0 be denoted by p. 4
For example, the payoff matrx for the assets n the bnomal market s: [ ] 1 1 D = s 1 s 2 wth asset prces at tme 0: p = [ s 0 e ] rt T. For the trnomal market we have: and asset prces: [ 1 1 1 D = s 1 s 2 s 3 ] p = [ s 0 e rt ] T. In a general market wth N assets wth M possble outcomes we have: 1 1 D 2,1 D 2,1 D =.. D N,1 D N,M p R N. The Dualty of Replcatng Portfolos and Rsk-Neutral Probabltes The problem of constructng the best replcatng portfolo to determne the prce of a contngent clam turns out to be dual to the problem of fndng the best valuaton of the contngent clam under all rsk-neutral probabltes permtted by the market. In partcular, f we consder the upper bound on the prce of the contngent clam n a market wth N general assets we can express the problem of fndng the weghts {w } of the best replcatng portfolo as: V (f) mn = mn w wd, f = max π 0 mn w = max π 0 mn w = max πd, =p π 0 The frst equalty follows by notng that: max π 0 π (f w p max w p + π (f w D, ) π 0 w p + π (f w D, ) w (p π D, ) + π f π f. w D, ) = { 0, f D, f for all, +otherwse 5
The second nequalty n whch we used that mn max = max mn follows from the dualty theorem. For the lower bound a smlar argument can be made to obtan: V (f) max wd, f = mn πd, =p π 0 w p π f. Ths gves the followng bound on the prce of a contngent clam: mn π f V (f) max π f. πd, =p π 0 πd, =p π 0 Now f we assume that one of the assets n the market s a rsk-free bond wth nterest rate r, say the asset wth ndex = 1, then we have e rt = D 1,π. Snce the frst row of the payoff matrx s of the form D 1, = 1 and we have e rt = π, f we defne ˆπ = e rt π then the weghts {ˆπ } are a rsk-neutral probablty measure. Ths allows for us to express the nequaltes as: mn e rt rsk-neutral probablty ˆπ ˆπ f V (f) max e rt rsk-neutral probablty ˆπ ˆπ f. For a complete market ths becomes an equalty and we see that the rsk-neutral probablty terms are precsely the dual values of the weghts of the replcatng portfolo. The General Prncple of No Arbtrage For the general class of markets consdered here, no arbtrage corresponds to the followng: Prncple of No Arbtrage: (a) N =1 w D, 0 for all, mples N =1 w p 0. (b) f we have N =1 w D, 0 for all and N =1 w p = 0 then we must have N =1 w D, = 0. The condton (a) states that f the payoff s non-negatve then the value of the portfolo must be nonnegatve. The statement (b) says that f the payoff of a portfolo s always non-negatve but costs nothng then the non-negatve payoff must be the trval payoff whch s zero. Part (a) s sometmes referred to as the weak no arbtrage prncple whereas (a) and (b) together s referred to as the strong no arbtrage prncple. The Arbtrage Theorem Theorem 1 (Arbtrage Theorem): The market satsfes (a) f and only f there exsts π 0 such that D, π = p, = 1,...,N. (1) The market satsfes both (a) and (b) f n addton the π can be chosen to be all strctly postve. 6
Before provng the theorem we remark that snce the weghts π are non-negatve they can be renormalzed to form probablty weghts (provded they are not all zero). Thus n essence, the theorem states that no arbtrage n a market s equvalent to the exstence of rsk-neutral probablty weghts for the assets. sketch of the proof: One drecton of the theorem follows rather easly. If there are non-negatve weghts π 0 so that equaton 1 holds then f w D, 0 for all t follows mmedately that w D, π = w p 0, whch shows (a). If the weghts are postve π > 0 then (b) follows by a smlar argument. To prove the other drecton of the theorem we shall make use of Farkas s Lemma whch states that: A T y 0 b T y 0, y holds f and only f there s a soluton x 0 satsfyng Ax = b. In other words, ths lemma states that f a collecton of nequaltes mples another nequalty, ths occurs n a rather trval fashon. In partcular, the new nequalty s a postve lnear combnaton of the nequaltes n the collecton. We shall now show that the no arbtrage condton (a) mples the exstence of the non-negatve weghts. Condton (a) can be expressed n vector notaton as statng: D T w 0 for all w mples w T p 0. By Farkas s Lemma we have that there s a soluton π 0 satsfyng Dπ = p. Ths shows that condton (a) mples the exstence of the non-negatve weghts satsfyng equaton 1. We shall now show that f both condton (a) and (b) hold then weghts π can be found whch are strctly postve. To do ths we shall label the weghts so that π 1,...,π M > 0 and π M +1,...,π M = 0. If all the weghts are already postve we are fnshed, so we shall only consder the case when M < M. In the case that there are postve weghts a > 0 for some set of coeffcents b such that: we have: p = = ǫ M =M +1 D, π M =1 M =M +1 a D, = b D, (2) M =1 a D, + D, (π ǫb ). The equalty on the second lne follows by addng and subtractng the term on the left hand sde of equaton 2 and then substtutng the rght hand sde n the second summand. We can obtan postve coeffcents π n ths case by drectly usng that a > 0. In partcular, we have that π = ǫa > 0, for = M + 1,...,M. To obtan postve coeffcents for the remanng ndces we use that π > 0 and make the factor ǫ > 0 suffcently small so that π = π ǫb > 0 for = 1,...,M. Now f there are not postve weghts a for any coeffcents b then the followng statement holds: M =M +1 M =1 a D, = b D,, a 0 a = 0, = M + 1,...,M. M =1 Ths states that the space spanned by the vectors {D,1,...,D,M } does not nclude any non-trval nonnegatve lnear combnaton of the vectors {D,M +1,...,D,M }. Geometrcally, ths corresponds to the set {D,M +1,...,D,M } formng a cone whch ntersects the lnear space spanned by {D,1,...,D,M } only at the vertex pont 0. From ths t can be shown that there s a vector w whch s orthogonal to the lnear space {D,1,...,D,M } and whch for the half space {x R N x,w > 0} contans the cone. In other words, there s a w such that: D,,w = 0, = 1,...,M D,,w > 0, = M + 1,...,M. 7
Wrtng ths out n terms of summands we have: w D, = 0, = 1,...,M w D, > 0, = M + 1,...,M. But then w represents a portfolo that has some postve payoffs wthout any rsk of a loss, and requres zero cost of nvestment snce w p = M =1 w D, π = 0. Ths contradcts the assumpton of no arbtrage. Therefore ths second case, n whch there are no postve weghts a, s excluded by the no arbtrage condtons (a) and (b). 8
References [1] Convex Analyss and Nonlnear Optmzaton by Borwen and Lews, Canadan Mathematcal Socety [2] Numercal Optmzaton by Nocedal and Wrght [3] Dervatves Securtes: Lecture Notes by R. Kohn. [4] Optons, Futures, and Other Dervatves by Hull. [5] Dervatves Securtes by Jarrow and Turnbll. Paul J. Atzberger Please send any comments to: atzberg@math.ucsb.edu