A True Buyer s Risk and Classification of Options

Transcription

1 Inform. Technol. Econom. Managemen No. 1, 21, (1-2) Research Repor No. 386, 1997, Dep. Theore. Sais. Aarhus A True Buyer s Risk and Classificaion of Opions GORAN PESKIR Acceping he classic Black-Scholes model for a financial marke consising of a riskless bank accoun (B ) T and a risky sock (S ) T, and considering he problem of pricing an opion of American ype associaed wih he reward process f = (f ) T, we address and discuss he quesion of he opion risk. Moivaed by he basic facs of he opion pricing heory in complee markes reviewed below, and aking he formal fair-game sandpoin of a rue buyer, we are naurally led o idenify he opion risk (of a firs kind) wih he disribuion law of he raional paymen under he equivalen maringale measure: R(f) = Law f3 B 3 P e where 3 is he opimal sopping ime for he buyer o exercise he opion, and P e is he equivalen maringale measure. Two opions are hen said o be equivalen if hey have he same risk. This is an equivalence relaion, and he se of all opions splis ino equivalence classes, wo opions being in he same class if and only if hey are equivalen. Since from he formal fair-game sandpoin of a rue buyer wo opions belonging o he same equivalence class may be hough of as he same, his relaion offers an exac mahemaical ool for comparing differen opions and classifying hem. A more realisic descripion of he opion risk mus also accoun for a possible random displacemen of he appreciaion rae around he ineres rae r according o a disribuion funcion F which is subjec o saisical observaions of each specific sock. Given ha akes is value independenly from a Wiener process driving he sock price, he opion risk (of a second kind) is obained by replacing P e in he definiion above by he acual probabiliy measure P F. When F frg he wo definiions coincide. All hese facs exend o opions of European ype wih a reward variable f T, if in he preceding definiions one replaces 3 wih T. A naural problem is hen formulaed as follows: Given any probabiliy measure % on R +, consruc an opion wih he opion risk equal o %. This problem has a simple soluion if he opion is of European ype. If he opion is of American ype his problem is referred o as he opimal Skorokhod-embedding problem (see [1]). 1. Fair price There is a large number of financial opions, bu here is no exac concep developed on how o compare hem and classify. In his noe we presen a simple idea of how his could be done. Our consideraions are devoed o complee markes. Considering he classic Black-Scholes model for a financial marke, and saring from basic facs upon which he fair price of an opion is defined, we inroduce and describe a naural concep MR 1991 Mahemaics Subjec Classificaion. Primary 9A9, 9A12, 9A46, 9D5. Secondary 6G4, 6J6, 6G44, 94A17. Key words and phrases: The Black-Scholes model, a complee financial marke, a financial opion (of American ype; of European ype), a rue buyer, he opion risk (of a firs kind; of a second kind; per uni of he fair price), he mean-square risk of an opion, he opion enropy, he riple of he fundamenal characerisics of an opion, he opion design, he opimal Skorokhod-embedding problem. (Second ediion) goran@imf.au.dk 1

2 of risk which may be associaed wih he opion. All opions can hen be compared and classified according o he risk hey incur, and from a formal poin of view wo opions incurring he same risk are he same. A naural quesion is hen o ask: Given a risk, can one consruc an opion wih his risk? While in he conex of European opions his quesion has a simple answer (Secion 3), in he conex of American opions such a problem is more sophisicaed (see [1]). We elaborae our presenaion by considering opions of American ype (Secions 1-2). These consideraions are hen easily exended o opions of European ype (Secion 3). 1. Consider he Black-Scholes model for a financial marke consising of a riskless bank accoun wih value B = (B ) and a risky sock wih value S = (S ). The equaions which govern B and S are respecively given by: (1.1) db = rb d (1.2) ds = S d + S dw where r > (he ineres rae), 2 IR (he appreciaion rae), > (he volailiy coefficien) and W = (W ) is a sandard Brownian moion defined on a probabiliy space (; F; P ). The bank-accoun value is deerminisic, and is given by: (1.3) B = B e r where B >. The sock value is random, and is given as a geomeric Brownian moion: (1.4) S = S exp W + 2 =21 where S >. By convenion we assume ha B = 1 in (1.3). In order o reach he cenral poin of our exposiion as simply as possible, we will drop some regulariy assumpions (on measurabiliy, inegrabiliy, ec.) in he sequel. Such deails, if no self-eviden, may be found in sandard references on he subjec quoed below. 2. American opions. Given a reward process f = (f ) T, consider an opion of American ype as a conrac beween he seller and a buyer which eniles he buyer o exercise he opion a any (sopping) ime 2 [; T ] and receive he paymen f from he seller (if evaluaed a ime ). Afer selling he opion a a price x, he seller has a disposal self-financing sraegies = ( ; ) wih (non-negaive) consumpion C = (C ) which, afer saring wih X ;c = x, a ime brings him he (non-negaive) value: (1.5) X ;c if evaluaed a ime Z = x + = B + S r db r + Z, or equivalenly, he discouned (real) value: r ds r C (self-financing wih consumpion) (1.6) Y ;c if evaluaed a ime. = X ;c B = + S B 2

3 (A) The cenral quesions abou such an opion conrac are: Wha is he fair price x? ( Fair refers o boh he seller and he buyer.) (B) Wha is he opimal sraegy 3 = ( 3 ; 3) wih consumpion C3 = (C 3 ), and wha is he opimal exercise ime 3? The general opion pricing heory ([1]-[6], [8]-[9], [12]-[13]) gives he following answers o hese quesions. 3. Fair price. A self-financing sraegy = ( ; ) wih consumpion C = (C ) is called a hedge ( wih respec o x and f given), if X ;c = x and we have: (1.7) X ;c f P -a.s. for all 2 [; T ]. The minimal x, denoed by V 3 (f ), for which here exiss a hedge (wih respec o f ) is called he fair price of he opion. 4. In order o deermine he fair price V 3 (f ), and answer he quesions (A) and (B) above, he following facs and observaions are shown essenial. Firs, he requiremen (1.7) is invarian under a measure change from P o ep, as long as P e and P are locally equivalen, which means ha P e and P have he same null-ses in F := (W s j s ) for T. By he Girsanov heorem here exiss such a measure ep on F := ([ F ) saisfying ye anoher good propery described below. The measure P e is deermined by is values on F hrough he ideniy: (1.8) d ep = exp r W 1 2 r 2 dp whenever. Under ep he process fw = W + ((r)=) is a sandard Brownian moion, and he process (1.6) admis he following supermaringale represenaion: (1.9) Y ;c = Y ;c + Z r S r B r dfw r where he firs inegral defines a local maringale and he second inegral defines an increasing process. For his reason he measure ep is called an equivalen maringale measure. (In fac, such a measure is unique. Is exisence guaranees ha he marke is arbirage-free, is uniqueness is expressed by saying ha he marke is complee.) Second, since we wan a minimal x for which here is a hedge, i is clear ha we shall look for a hedge wih consumpion C for which he values X ;c saisfying (1.7) (or he values Y ;c saisfying Y ;c f =B ep -a.s. for all 2 [; T ] ) are minimal in some sense. Third, i is well-known ha he smalles supermaringale ( under ep ) which dominaes f =B ep -a.s. on [; T ] is given as he Snell envelope: (1.1) Y 3 = ess sup ee f F B T Z dc r B r for T, where he (essenial) supremum is aken over all sopping imes aking values 3

4 in [; T ]. By he Doob-Meyer decomposiion his furher splis ino: (1.11) Y 3 = Y 3 + M3 A 3 where M 3 = (M 3) is a local maringale ( under ep ), and A 3 = (A 3 ) is an increasing process, boh sared a zero. Moreover, by he Iô-Clark heorem we have: (1.12) M 3 = Z r dfw r for some process = ( ). Finally, he Snell envelope process (Y 3) is generally known o be a maringale unil i his (f =B ), and hus during his period A 3 ep -a.s. 5. Raional performance. From he argumens jus presened i is eviden ha he opimal 3 = ( 3 ; 3 ), C3 = (C 3 ) and 3 are obained by idenifying: (1.13) (1.14) Z Z r S r B r dfw r = M 3 dc r B r = A 3. This gives he following explici answers o he quesions (A) and (B) saed above: (1.15) V 3 (f ) = sup ee f B T (1.16) 3 = Y 3 (1.17) 3 B = S (1.18) C 3 = (1.19) 3 = inf Moreover, we have: Z B r da 3 r > Y 3 = f B (1.2) The sopping ime 3 is opimal for he problem (1.15). I is poinwise he smalles possible wih his propery. From (1.19) and he maringale propery of Snell s envelope noed above, we see ha if he buyer acs raionally and exercises he opion a 3, here will be no consumpion for he seller, ha is: (1.21) C 3 for 2 [; 3 ] e P -a.s. Thus he fair price V 3 (f ) is indeed fair from his poin of view as well. Finally, since Law (S() j P e ) = Law (S(r) j P ), i follows from (1.15) ha he fair price. 4

5 V 3 (f) does no depend on, neiher does he opimal sraegy 3 = ( 3; ) 3 wih consumpion C 3 = (C 3 ), nor does he opimal sopping ime 3. This propery is no surprising, since he requiremen (1.7) is invarian under an equivalen measure change. 6. I should be realized ha he process Y 3 = (Y 3 ) (and herefore M 3 and A 3 oo) is compuable (a leas in principle) and known a priori before he opion conrac has been signed (in much he same way as he fair price V 3 (f) iself). This is imporan since he opimal sraegy 3 = ( 3; ) 3 wih consumpion C 3 = (C 3 ) is expressed in erms of Y 3, and is exisence in an explici form should show he seller how o ac in order o keep up wih he demand of he reward process f, as well as o provide boh he seller and he buyer wih a guaranee (and needed comfor) ha he opion conrac can be realized a he fair price V 3 (f). The problem of an explici compuaion of he Snell envelope Y 3 = (Y 3 ) is closely linked o he problem of compuing he fair price V 3 (f). Solving he opimal sopping problem (1.15) in a Markovian seing (which is o be found in each concree case of he reward process f ), we ge he fair price V 3 (f) as a funcion of he iniial posiion of he underlying Markov process. Composing hen his funcion wih he Markov process iself, we obain (Y 3 ) as he smalles supermaringale which dominaes he gain process (f =B ). Thus he fac (1.2) is in accordance wih he general opimal sopping heory for Markov processes (see [14]). 7. These consideraions show ha he problem of design of an opion is closely relaed o he fac ha he reward process f should be chosen in such a way ha he opimal sopping problem (1.15) admis an explici soluion. Some experience of work wih opimal sopping problems shows ha his is very difficul o achieve, he main wo obsacles in (1.15) being he finie horizon T and he presence of discouning (B ). Moreover, crieria given in he lieraure for he choice of a reward process f are very ofen based raher on a subjecive view han on an exac mahemaical concep. Alhough such an approach may be well suied o he spiri of opion rading, we believe ha he general heory should offer exac crieria. Our main aim in he nex secion is o poin ou and describe a simple concep of risk which may be associaed wih he opion and upon which he choice of a reward process can be based. This concep offers an exac crierion for he choice of a reward process and leads o a classificaion of all opions according o he risk hey incur. I should be noed, however, ha his crierion does no deermine he reward process uniquely, bu raher gives an admissible class of such processes. Oher crieria may hen be used (solvabiliy of he opimal sopping problem (1.15), pah properies of f providing comfor, ec.) o deermine a reward process from his class. From a formal poin of view, however, any oher reward process from he admissible class generaes an opion wih he same risk, which hen may be hough of as he same opion. 2. The concep of he opion risk Having compued (a leas in principle) he fair price V 3 (f), he opimal sraegy 3 = ( 3 ; 3 ) wih consumpion C 3 = (C 3 ), and he opimal sopping ime 3, he opion is ready for use. A new naural quesion which is missing in he analysis above and which we wan o address now may be saed as follows: (C) Wha is he risk? ( Risk refers o a rue buyer before concluding he opion conrac, 5

6 and aferwards, unil i is exercised. By rue buyer we mean a buyer who has no abiliy or desire o sell he opion. Thus every rue buyer will exercise he opion according o he raional performance.) We noe ha despie is appealing simpliciy and necessiy, his quesion has no been considered in such a form before. Perhaps one of he main reasons for his is ha in complee markes he seller and he buyer are usually idenified, i.e. a buyer can always proec himself by selling he same opion o anoher buyer. We find his argumen somewha dubious, as any buyer could proec himself even beer by simply no buying he opion in he firs place. Before making any aemps o answer his quesion, we firs wan o make i clear ha he bank-accoun value B and he sock value S (wih some which we do no have o know a priori) are indeed given by he equaions (1.1) and (1.2) respecively, and we will no be ineresed in answering he quesion as wha a risk would be if some of hese assumpions fail. Such a risk cerainly exiss, bu is descripion will no be he subjec of our discussion. Thus, we wan o answer he quesion on wha he risk is if we know ha he given B and S are auhenic. In oher words, he risk we wan o describe is a risk wihin he given model, and no he risk of having a model which is no genuine, or possibly a combinaion of hese wo. 1. In order o see where he risk is hidden, assume ha he seller and a buyer sign he opion conrac a ime =. Accordingly, he buyer pays he fair price V 3 (f) o he seller, and he opion performance sars. The seller has a disposal self-financing sraegies = ( ; ) wih consumpion C = (C ), and he raionally aemps o hedge he reward process f = (f ) T given by he opion conrac. In fac, in our ideal world, he seller has no oher choice bu o apply he opimal sraegy 3 = ( 3 ; 3 ) wih consumpion C 3 = (C 3 ) given by (1.16)-(1.18), while he buyer has no oher choice eiher bu o exercise he opion conrac a he opimal sopping ime 3 given by (1.19). From he resuls of general heory exposed above, i is clear ha if eiher of hem does no apply he opimal ools jus described, he oher can do so and gain more in he expecaion. Such cases are no of ineres for general heory, however, since we expec a raional behaviour from boh paries as a basic hypohesis. Therefore suppose ha boh he seller and he buyer ac raionally and follow he opimal ools hey have a disposal. Thus, he seller applies 3 wih C 3 and he buyer exercises he opion conrac a 3. A he ime of exercise 3, he seller pays he value Y 3 ; c 3 3 o he buyer which is obained by applying 3 wih C 3 and which by (1.19) equals he discouned reward value f 3 =B 3. The consumpion o he seller up o 3 is idenically zero by (1.21). Moreover, he seller is no exposed o any risk, as he fair price V 3 (f) by is definiion enables him o pay f 3 =B 3 o he buyer as sipulaed by he opion conrac. Since by (1.1) we have Y 3 ; c 3 = V 3 (f), we see by (1.2) ha in he expecaion under e P he buyer receives as much as he paid for he opion conrac hrough he fair price V 3 (f). The opion performance is fair under P e, since he buyer neiher gains nor loses anyhing exra (see Fig.1 above). However, i is necessary o observe ha his saemen is rue only in he expecaion (under he equivalen maringale measure). 2. Here we come up o he cenral poin of our discussion. Imagine wo players A and B playing a random game where he oucome is a random variable R wih zero mean. Posiive values of R correspond o he wealh aken from he player B and given o he player A, while 6

7 Y 3 f R V 3 (f ) f f =B 3 T Figure 1: A schemaic drawing of he raional performance a he opion of American ype as described in Paragraph 1 below. The raional paymen R = Y 3 3 = f 3 =B 3 is assumed o have a densiy funcion fr for convenience. The expecaion of he raional paymen R equals he fair price V 3 (f), and he opion performance is fair. Our undersanding is ha he size and he shape of he displacemen of he raional paymen R around is mean V 3 (f) deermine he opion risk. negaive values of R correspond o he wealh aken from he player A and given o he player B. Firs we may ask: Is he game fair? Second: Wha is he risk? These are exacly he quesions we are rying o answer above. Consider a special case where R akes wo values 1 and 1 wih probabiliy 1=2. Ask he same quesion: Is he game fair? There is no doub ha mos of people would say ha his game is fair. Why? Possibly because neiher of he players can gain somehing by swapping he roles in he game if such a righ is given before he game has been sared. Now ask he second quesion: Wha is he risk which eiher or boh players ake by playing such a game? If he number 1 above represens a uni of money, hen mos of people would say ha such a game is no "risky" (we assume ha he game is played only once). However, if you replace number 1 above by he number 1, hen mos of people would say ha his game is very "risky" and would no play i. I shows ha in our percepion of he word risk we do no hink only of chance (which in his case corresponds o he probabiliies 1=2 ) bu also of consequences (which corresponds o he values 1 or 1 ) and poenial (which corresponds o he oal wealh of he players). As a consequence we obain ha he concep of risk canno be solely expressed in erms of he enropy of R. (The concep of enropy is recalled in Paragraph 6 below.) The special cases jus considered are symmerical, and neiher of he players can gain somehing by swapping he roles in he game. Consider now a ypical asymmerical case where R akes wo values 999 and 1 wih probabiliies 1=1 and 999=1 respecively. Ask he same quesions: Is he game fair, and wha is he risk? While mos of people would say ha his game is fair (alhough no symmeric and he players may like o swap he roles), he quesion of risk 7

8 seems a bi more ricky. Imagine he game is played once. Wha could he players expec from he game? Player A wins he game 999 imes ou of 1 plays on he average, ha is, almos surely if he game is played only once. However, even so, in he win he gains only 1 uni of money. Player B wins he game only once ou of 1 plays. However, when winning he game he gains he forune of 999 unis of money. Is i clear who is in a more risky siuaion? We believe ha mos of people would agree ha his is he player A. Thus, here we have a case where he risk is no he same for boh players. So, wha is he risk, and how o define i? I is now easy o coninue he lis of examples by considering more and more general (asymmerical) disribuions (wih densiies, ec.), and soon one can realize ha hese maers ge raher complicaed. Insead of aemping o give a general answer o hese quesions, we shall confine ourselves o he problem of opion pricing considered above, and noe ha whaever he risk is, i is conained in he disribuion law of R. In oher words, knowing he disribuion law of R boh players can read from i everyhing abou he risk hey wan o know. For his reason we shall idenify he risk wih he disribuion law. I would be perhaps more precise o call his risk by he risk disribuion, or he risk law, and we shall do i now and hen as well. 3. Risk. Moivaed by hese consideraions we reurn back o he problem of opion pricing and define he risk variable as he value of he paymen (obained by a raional behaviour): (2.1) R := f 3 B 3 which also equals he opimal value Y 3 ; c 3 3 wih 3 being he opimal sopping ime. Alhough he expecaion of R equals he fair price V 3 (f ), and he opion game is fair : (2.2) e E R 1 = V 3 (f ) i is clear ha R is a random variable which generally may in is values deviae from is expecaion. For insance, i is no he same if R akes values and 2 wih probabiliy 1=2, or if R akes values 999 and 11 wih probabiliy 1=2. In boh cases he fair price is 1, while clearly he risk in he firs case is much higher for he buyer. By simple observaions presened above, we are naurally led o formulae he following definiion. Definiion 2.1 (A rue buyer s risk of he American opion) The risk of he American opion is he disribuion law of he raional paymen R under he equivalen maringale measure P e. In oher words, he opion risk R(f ) is defined o be: (2.3) R(f ) = Law (R j e P ) where R is given by (2.1) wih 3 from (1.19). Remark 2.2 Observe ha he preceding definiion is made from he sandpoin ha he raional performance defines a fair game for he buyer. The acual probabiliy measure P may be, and usually is, differen from he equivalen maringale measure e P, and in each such a case he raional performance will ypically be eiher more or less favourable o he buyer. For example, consider f = (S K) + for T where K >. Then he case > r is more favourable o 8

9 he buyer as he receives E(f 3=B 3 ) which is sricly larger han he fair price E(f e 3=B 3 ) ha he pays for he opion conrac, and similarly he case < r is less favourable o he buyer for exacly he same reason. In neiher of hese cases he raional opion performance leads o a fair game for he buyer. However, as one would expec ha here should be equal chance for o be above or below r, he hypohesis made in he definiion ha he raional paymen R should be considered under ep is a reasonable approximaion of all risk exposures associaed wih he raional opion performance ha he buyer faces. This idealised approach is very much in he spiri of he Black-Scholes model (where may be unknown) as he opion risk so defined does no depend on in any way. For hese reasons he opion risk of Definiion 2.1 could also be called he American opion risk of a firs kind. A less idealised and somewha more realisic hypohesis could be made by exending he Black-Scholes model and assuming ha akes an arbirary real value a ime = (or laer) independenly of W = (W ) in accordance wih some disribuion funcion F (which is subjec o observaion and saisical esimaes of each specific sock). Assuming ha he random variable is defined on ( ; F ; P ) and ha W is defined on ( ; F ; P ), we could realise he sock price process S = (S ) on (; F; P F ) := ( 2 ; F 2 F ; P 2 P ). A naural hypohesis would be ha is cenered around r, i.e. ha E () = r (alhough real daa seem o indicae ha E () > r in mos cases), and possibly ha E F (f 3=B 3 ) = E(f e 3=B 3 ). (The laer assumpion is very resricive. In he case f = (S K) + for T i is possible o verify using Jensen s inequaliy ha under E () r we mus have E F (f 3=B 3 ) E(f e 3=B 3 ) and ha his inequaliy is always sric unless F frg. This fac discerns a somewha unexpeced bu raher desirable propery of he opion.) In his case i is reasonable o exend (2.3) by seing R F (f ) = Law(R j P F ) and call i he American opion risk of a second kind. Observe ha his definiion of he risk depends on he disribuion F of given a priori. Clearly, when F = r hen he wo definiions coincide. More involved descripions of he appreciaion raes, being also funcions of and s, will lead o more sophisicaed descripions of he opion risk. The knowledge of could also be coninuously updaed as he ime passes by, which in urn will influence he risk and make i a funcion of ime as well. We shall no go here any deeper ino hese more complicaed consideraions, bu will mainly concenrae on he opion risk of a firs kind. Thus, o conclude, in order o obain a more precise descripion of he opion risk, one should addiionally o (2.3) also accoun for a displacemen of around r. This can be illusraed hrough Figure 1 above by viewing he densiy funcion f R as a random oucome of P corresponding o some, and hen averaging over all such s according o a probabiliy disribuion which accouns for all possible displacemens of around r ha are ypical for he given sock. The opion is said o be risk-free if he risk variable R is degeneraed a a poin, ha is, if R = c ep -a.s. for some posiive c. Noe, however, ha any such c mus hen be equal o he fair price V 3 (f ). Thus, he opion is risk-free if he opion risk is degeneraed a he fair price. 4. The mean-square risk of he opion. While i was clear how o define a risk-free opion, i is less obvious how o define a risky opion. This can be achieved, however, by looking a various funcionals of he opion risk. For insance, he mean-square risk of he opion may be defined as he variance of he opion risk: 9

10 (2.4) Var e R(f) 1 = E R e E(R) e 2 = e E f 3 B 3 V 3 (f)! 2. The opion may now be called "risky" (in he mean-square sense), if he mean-square risk (2.4) is "large". Noe ha he mean-square risk quanifies he risk of a big loss and a big gain a he same ime. I reas boh players equally, and no informaion from i can be obained on he size of he individual risks indicaed in he example above. We believe ha i is clear how o exend his concep furher by replacing he square funcion (r; e) 7! (re) 2 in (2.4) wih oher funcions of ineres (which can measure he size of he displacemen of R from is mean). In his way we can obain oher (more sophisicaed) risk funcionals of he opion risk according o which he opion iself may hen be called "risky". We shall omi furher deails in his direcion. Remark 2.3 A parial answer o he quesion above as how o define a risky opion can also be given using he well-developed concep of uiliy funcions (see e.g. [5] pp.1-37). We recall ha an individual s preferences admi an expeced uiliy represenaion if here exiss a uiliy funcion U = U(c) such ha he random consumpion C 1 is preferred o a random consumpion C 2 if and only if E(U(C 1 )) E(U(C 2 )). Individuals who prefer more wealh o less have increasing uiliy funcions. Risk averse individuals are characerized by concave uiliy funcions. The Arrow- Pra coefficiens of absolue and relaive risk aversion are defined respecively as U (c)=u (c) and c U (c)=u (c). These coefficiens measure he individual s aiude oward risk. In his conex he concep of sochasic dominance (see e.g. [5] pp.39-57) arises as a useful ool for comparing he riskiness of risky asses. As an illusraion we shall noe ha he following resul of Rohschild and Siegliz (see e.g. [5] p.49) direcly applies o he problem of risky opion addressed above. Le R 1 and R 2 be wo raional paymens corresponding o wo American opions wih he same fair price. Then he following saemens are equivalen: (i) All risk averse individuals (buyers) having uiliy funcions whose firs derivaives are coninuous excep on a counable se prefer R 1 Law(R 2 ) = Law(R 1 +Z) where E(Z e j R1 ) =. This concep, however, does no allow us o o R 2 ; (ii) e E(R 1 ) = e E(R 2 ) and R z (F R 1 (x)fr 2 (x)) dx for all z ; (iii) compare any wo raional paymens i.e. opions. (For wo similar resuls when e E(R 1 ) e E(R 2 ) see [5] p.45 and p.5). Observe ha (iii) above implies ha be expeced. The converse, however, is no rue. e V ar(r1 ) e V ar(r 2 ), which is o 5. The opion risk per uni of he fair price. Generally, i can also be of ineres o know he opion risk per uni of he fair price. To inroduce such a risk concep one can modify he risk variable (2.1) in he following way: (2.5) b R = R V 3 (f ) and define he opion risk per uni of he fair price as: (2.6) b R(f ) = Law ( b R j e P ). Noe ha E(R) b = 1 and from R b one can no read he fair price V 3 (f ). Noe also ha he opion is risk-free (in he sense described above) if and only if R b is degeneraed a 1, or in oher words, if and only if R(f b ) = 1. In analogy wih (2.4) we can now define he mean-square 1

11 risk of he opion per uni of he fair price as: (2.7) e Var b R(f ) 1 = e E b R 112 = e E b R 21 1 = e E f 3 V 3 (f ) B 3!2 1. One could now inerpre and exend his concep furher in a manner similar o our reamen of he risk variable R following (2.4) above, and in his way obain oher (more sophisicaed) risk funcionals of he opion risk per uni of he fair price. 6. Enropy. Anoher imporan concep associaed wih he opion risk is he opion enropy: (2.8) e H(R) = Z R f (x) log f (x) dx if R is absoluely coninuous ( under e P ) wih a densiy funcion f, or: (2.9) e H(R) = X k p k log p k if R is a discree variable aking some values x k (being irrelevan) wih probabiliies p k ( under ep ). The opion enropy is a measure of uncerainy of he oucome of he raional paymen R. I is a value-free measure, since i is expressed only hrough he law of chance of R. (For more informaion on he concep of enropy we refer o [7].) In order o illusrae why his concep is of ineres for design of opions, we shall quoe wihou proof he following well-known resul. In he firs hree cases below R is assumed absoluely coninuous, in he fourh final case R is assumed discree. (2.1) 1. If Var(R) e = 2 < 1, hen he inequaliy holds: p eh(r) log 2e 2 wih equaliy iff R N (; 2 ) wih 2 R. 2. If R and e E(R) = < 1, hen he inequaliy holds: eh(r) log(e) wih equaliy iff R Exp(1=). 3. If a R b for some a and b, hen he inequaliy holds: eh(r) log(ba) wih equaliy iff R U (a; b). 4. If R akes n values wih probabiliies p k >, hen he inequaliy holds: eh(r) log(n) wih equaliy iff R is uniformly disribued wih p k = 1=n for all k. 11

12 These facs have a beauiful inerpreaion in he problem of opion design. To illusrae his, noe ha he raional paymen R is non-negaive in our model, so if he fair price V 3 is given, hen by (2.1.2) we see ha he mos uncerain opion (in erms of he oucome of he raional paymen) is he opion wih a reward process f = (f ) T for which ( under e P ): (2.11) R = f 3 B 3 1 Exp V 3 V 3 (f) = V 3. I does no mean ha his opion is wih he highes (mean-square) risk. I wih raher means ha if he (mean-square) risk is given and fixed, hen his opion is he mos uncerain wih respec o he oucome of he raional paymen wihin he class of all opions having he given mean-square risk. Noe ha he mean-square risk mus hen be equal o V 3 (f) 2, since evar(r) = ( e ER) 2 if R Exp(1=V 3 ). Similar inerpreaions may now be given, subjec o oher resricions on he raional paymen R, if one uses remaining facs from (2.1). 7. A riple of he fundamenal characerisics of he opion wih a reward process f = (f ) T is defined as he hree-dimensional vecor ( e E(R); e Var(R); e H(R)) consising of he fair price, he mean-square risk, and he opion enropy. From our consideraions above i follows ha hese hree numbers offer a good deal of informaion on he opion characer ha is of ineres o a buyer. 8. Classificaion of opions. In view of he previous consideraions, he following definiion is naural. Two opions O 1 = O 1 (f) and O 2 = O 2 (g) wih reward processes f = (f ) T and g = (g ) T are said o be equivalen (or risk-equivalen) if hey have he same risk: (2.12) Law f3 B 3 ep = Law g3 B 3 ep where 3 and 3 are he opimal sopping imes associaed wih O 1 and O 2 respecively. In oher words, we have: (2.13) O 1 O 2 iff R(f) = R(g). Clearly, his is an equivalence relaion on he se of all opions wihin he given model, and in his way he se of all opions splis ino equivalence classes, wo opions being in he same class if and only if hey are equivalen. By means of his equivalence relaion we obain a ool for comparing differen opions and classifying hem. We noe ha from a formal fair-game sandpoin wo equivalen opions may be hough of as he same. 9. The opimal Skorokhod-embedding problem. As an opion may be idenified wih is reward process, we also see ha he equivalence relaion (2.13) offers an exac mahemaical crierion on how o choose he reward process when designing an opion. In his conex he following quesion appears fundamenal: (2.14) Given a disribuion law % on R +, find a reward process f = (f ) T such ha: Law f3 B 3 P e = % where 3 is he opimal sopping ime (1.19) for he problem (1.15). 12

13 In oher words (and less formally) we ask: Given any risk, is here an opion wih his risk? We noe ha if he answer is posiive, and we are able o indicae a reward process, hen a leas from a formal fair-game sandpoin all opions would be designed. For obvious reasons we shall refer o he problem (2.14) as he opimal Skorokhod-embedding problem. (I is clear ha R + in (2.14) may be replaced by R generally.) We noe ha his problem involves more difficuly han he classic Skorokhod-embedding problem (see [11] p.258), since for a given % we are no only supposed o find a sopping ime 3 a which o sop he underlying process in order o ge %, bu also an opimal sopping problem (a funcional of he underlying process) for which he sopping ime 3 is opimal. I is indeed a highly sophisicaed machinery for design of opions (or games), and alhough his problem is complex and difficul generally, i is shown in [1] ha such a problem in principle can be solved. We noe ha by solving he opimal Skorokhod-embedding problem we also solve he classic Skorokhod-embedding problem. 1. I is an ineresing quesion o deermine a class of reward processes f = (f ) T among which one should ry o find he opimal one in (2.14). (We noe ha a finie horizon T can also be given in advance ogeher wih he law %, and hen one has o find solely he funcional rule f for wihin [; T ]. Similarly, oher consrains may be imposed on he admissible class of reward processes.) Since i is unlikely (if impossible) ha funcionals of he form f = f (S ()) can have he righ power, we resric ourselves o pah-dependen funcionals of he form: (2.15) f = f Su() 1 u where f is some map on he space of pahs. Since Law(S() j P e ) = Law(S(r) j P ), we see ha: (2.16) Law f3 () B 3 () f3 (r) P e = Law P B 3 (r) wih a clear inerpreaion of he noaion. In oher words, since he risk variable (2.1) saisfies: (2.17) R Law f3 (r) B 3(r) P i is enough o consider he pah-dependen funcional: (2.18) f = f Su(r) 1 u under P. This enables one o reformulae he problem (2.14) and consider i under he iniial measure P where from (1.2) has o be replaced by r from (1.1). R The inegral g(s u(r)) du, alhough a pah dependen funcional, does no seem o have he righ power eiher (due o is Markovian naure in he expecaion which follows by Iˆo formula and opional sampling). Therefore i seems reasonable o work wih he maximum funcional: (2.19) f = max S u (r) u as being he nex on he lis of naural funcionals, and which is known o produce enough 13

14 randomness for he soluion of he classic Skorokhod-embedding problem (see [11] p.258). This funcional has also anoher well-known feaure of ineres in opion rading: i provides comfor of exercising a a maximum hus reducing regre for no exercising earlier. 11. In [1] we formulae and solve he problem (2.14) in a relaxed form. We assume ha he horizon T equals +1, and we drop discouning B. Such a simplified seing admis explici compuaions and closed formulas hroughou. A naural sock process is sandard Brownian moion (W ), and a naural reward process is he maximum process discouned linearly: (2.2) f = max W r r Z c(w r ) dr where x 7! c(x) is a posiive funcion. The opion pricing problem in his seing can be formalized wihin he original Bachelier model. Even hough his model is no longer acceped, i gives a good insigh ino (merely) echnically more complicaed maers of he Black-Scholes model. 3. Design of opions of European ype All ha was said above for opions of American ype exends o opions of European ype in an obvious manner. For he sake of compleeness we shall indicae his exension below in deail. We furher noe ha he opimal embedding problem which corresponds o (2.14) is easily solved in his case (Proposiion 3.2), and alhough i may be of some ineres for applicaions, his resul does no require deep mahemaics. (I could be ha some consrains of ineres, being imposed on he choice of he funcional f, would make his problem more ineresing mahemaically as well.) Noneheless, from he formal fair-game sandpoin of a rue buyer his simple resul offers a consrucion of all opions of European ype wihin he given model. A European opion wih he mos uncerain oucome of he raional paymen (in he sense of enropy), wihin he class of opions having he fixed fair price, is hen easily obained as a consequence (Corollary 3.3). 1. The pricing heory for opions of European ype is less sophisicaed han for opions of American ype. We presen he essenial facs of his heory in complee analogy wih he essenial facs on American opions saed in Secion 1 above. Consider he Black-Scholes model for a financial marke consising of a riskless bank accoun wih value B = (B ) and a risky sock wih value S = (S ) which are given by (1.1) and (1.2) respecively. 2. European opions. Given a reward variable f = f T, where T > (he expiraion dae) is given and fixed, consider an opion of European ype as a conrac beween he seller and a buyer which eniles he buyer o exercise he opion a ime T and receive he paymen f T from he seller ( if evaluaed a ime ). Afer selling he opion a a price x, he seller has a disposal self-financing sraegies = ( ; ) which, afer saring wih X = x, a ime brings him he (non-negaive) value: (3.1) X = B + S = x + Z r db r + Z r ds r (self-financing) 14

15 if evaluaed a ime, or equivalenly, he discouned (real) value: (3.2) Y = X B = + S B if evaluaed a ime. The cenral quesions abou such an opion conrac are: (A) Wha is he fair price x? ( Fair refers o boh he seller and he buyer.) (B) Wha is he opimal sraegy 3 = ( 3 ; 3 )? The general opion pricing heory ([1]-[6], [8]-[9], [12]-[13]) gives he following answers o hese quesions. 3. Fair price. A self-financing sraegy = ( ; ) is called a hedge ( wih respec o x and f given), if X = x and we have: (3.3) X T f T P -a.s. The minimal x, denoed by V (f ), for which here exiss a hedge (wih respec o f ) is called he fair price of he opion. 4. In order o deermine he fair price V (f ), and answer he quesions (A) and (B) above, we noe ha he observaions and facs saed in Paragraph 1.4 above may be repeaed here as well wih no essenial difference. Thus, by he Girsanov heorem here exiss an equivalen maringale measure e P given by (1.8), under which he process f W = W + (( r)=) is a sandard Brownian moion, and he process (3.2) admis he following represenaion: (3.4) Y Z = Y + r S r B r where he inegral defines a local maringale. Therefore we consider he smalles maringale ( under ep ) which dominaes f T =B T ep -a.s. and which is given as he Snell envelope: (3.5) Y 3 ft = ee F = Y 3 B + M 3 T dfw r for T. By he Iô-Clark heorem we have: (3.6) M 3 = Z r dfw r for some process = ( ). 5. Raional performance. From he argumens indicaed above i is eviden ha he opimal 3 = ( 3; 3 ) is obained by idenifying: (3.7) Z r S r B r dfw r = M 3. 15

16 This gives he following explici answers o he quesions (A) and (B) saed above: (3.8) V (f ) = e E (3.9) 3 = Y 3 ft B T (3.1) 3 = B S. Since Law(S() j P e ) = Law(S(r) j P ), i follows from (3.8) ha neiher he fair price V (f ), nor he opimal sraegy 3 = ( 3 ; 3 ), does depend on. Finally, i should be noed ha he process Y 3 = (Y 3 ) is compuable (a leas in principle) and known a priori before he opion conrac has been signed (as well as he fair price V (f ) iself). This provides boh he seller and he buyer wih a guaranee (and needed comfor) ha he opion conrac can be realized a he fair price V (f ). 6. Risk. In complee analogy wih American opions in he beginning of Secion 2, a new naural quesion which is missing above is saed as follows: (C) Wha is he risk? ( Risk refers o a rue buyer before concluding he opion conrac, and aferwards, unil i is exercised. By rue buyer we mean a buyer who has no abiliy or desire o sell he opion. Thus every rue buyer will exercise he opion according o he raional performance.) For exacly he same reasons as in he conex of American opions in Secion 2, we are naurally led o define he risk variable as he value of he paymen: (3.11) R := f T a he expiraion dae T. We noe again ha alhough he expecaion of R V (f ), and he opion game is fair : 1 (3.12) E e R B T = V (f ) equals he fair price i is clear ha R is a random variable which generally may deviae in is values from is expecaion. Therefore we are again naurally led o formulae he following definiion. Definiion 3.1 (A rue buyer s risk of he European opion) The risk of he European opion is he disribuion law of he raional paymen R under he equivalen maringale measure P e. In oher words, he opion risk R(f ) is defined o be: (3.13) R(f ) = Law (R j e P ) where R is given by (3.8). The consideraions presened in Remark 2.2 above apply fully in he presen case as well, and for exacly he same reasons he opion risk of Definiion 3.1 could also be called he European 16

17 opion risk of a firs kind. This also leads o a definiion of he European opion risk of a second kind as R F (f) = Law(R j P F ), where F is a disribuion funcion which accouns for possible displacemens of around r. We omi all remaining deails and refer he reader direcly o Remark 2.2 above for a complee accoun. The opion is said o be risk-free if he risk variable R is degeneraed a he fair price V (f). The mean-square risk of he opion, he opion risk per uni of he fair price, he mean-square risk of he opion per uni of he fair price, he opion enropy, and he riple of he fundamenal characerisics of he opion, can now all be defined exacly as in Paragraphs above. 7. Classificaion of opions. In exacly he same way as American opions, all European opions are clearly classified hrough he equivalence relaion: (3.14) O 1 (f) O 2 (g) iff R(f) = R(g) where R(f) = Law(f T =B T j e P) and R(g) = Law(g T =B T j e P). In his way he se of all European opions splis ino equivalence classes, wo opions being in he same class if and only if hey are equivalen. By means of his equivalence relaion we obain a ool for comparing differen European opions and classifying hem. We again noe ha from a formal fair-game sandpoin wo equivalen opions may be hough of as he same. 8. Opion design of European ype. As European opions may be idenified wih is reward variables, we see ha he equivalence relaion (3.11) offers an exac mahemaical crierion on how o choose he reward variable when designing a European opion. In his conex he following reformulaion of he opimal embedding problem (2.14) appears fundamenal: (3.15) Given a disribuion law % on R +, find a reward variable f = f T such ha: Law ft B T e P = % where T is he expiraion ime of he opion. In oher words (and less formally) we ask: Given any risk, is here an opion wih his risk? In he nex proposiion we noe ha he answer o his quesion is posiive. This simple resul shows ha a leas from a formal fair-game sandpoin all opions of European ype are easily designed. Below we use he sandard noaion: Z (3.16) 8(x) = 1 x 2 p =2 2 1 e d o denoe he disribuion funcion of a sandard normal random variable X N(; 1). Proposiion 3.2 Consider he Black-Scholes model for a financial marke consising of a riskless bank accoun wih value B = (B ) and a risky sock wih value S = (S ) which are given by (1.1) and (1.2) respecively. Le % be a probabiliy measure on R + associaed wih a sricly increasing and coninuous disribuion funcion F. 17

18 1. Then here exiss an opion of European ype associaed wih he reward variable: (3.17) f T = f S T 1 such ha he opion risk R(f ) from (3.1) is equal o %. 2. The funcion f in (3.17) is explicily given by he following formula: e (3.18) f (s) = e F rt 1 G rt s 1 where he map G is defined as follows: (3.19) G(s) = 8 for all s >. p 1 T + 2 p T log(s) Proof. Consider he random variable: (3.2) Z = exp W T 2 T 2. p Then by using W T T W1 i is easily verified ha he funcion G given in (3.19) above is he disribuion funcion of Z. Noe ha: (3.21) S T (r) = exp W T + r 2 T = e rt Z. Therefore G(e rt S T (r)) U (; 1), and hence F 1 (G(e rt S T (r))) F. Thus: (3.22) f T (r) B T f ST (r)1 = = F 1 (G(e rt S T (r))) F. B T I remains o recall ha Law(S() j e P ) = Law(S(r) j P ) so ha: (3.23) R(f ) = Law and he proof is complee. ft () B T 2 ft (r) P e = Law P B T Remarks: 1. In he heorem above i is assumed ha he ineres rae r > and he expiraion ime T > are given and fixed a priori (before he opion has been consruced). Therefore he reward funcion (3.18) depends on r and T. If r > and T > are no given a priori, hen for given % F one may consider he reward funcion: (3.24) f (s) = c F 1 1 G s=c where c > is a consan given and fixed. The opion which correspond o he reward variable (3.17) has hen he propery ha whenever he ineres rae r > and he erminaion ime T > are aken o saisfy rt = log(c), he opion risk R(f ) will be equal o %. 18

19 2. These consideraions can be furher exended and formalized. Suppose we are given a family of measures % r wih he corresponding disribuion funcions F r for r >. (Such a family is ypically obained by any reward variable f T = f T (r) which generally depends on he ineres rae r > under P, and herefore he risk R(f ) depends also on r so ha R(f ) = % r for r >.) Then he map (3.18) could also be seen as a map from ]; 1[ ino he space of (coninuous) funcions on ]; 1[2]; 1[ upon idenifying: (3.25) 1 f (s) r; T = e rt F 1 1 r G e rt s for s >. Wih his generalized inerpreaion of he reward variable (3.13) we see ha whenever r > and T > are given before he opion conrac has been sared, we may hink of he reward funcion f o be given by (3.24) above wih c = e rt (which is jus a consan). Thus, from a formal fair-game sandpoin we may look a (3.18) as a generalized reward funcion which generaes all opions of European ype in regard o he risk equivalence (3.14). 3. We noe ha % in he heorem above is assumed o have a sricly increasing and coninuous disribuion funcion for simpliciy (he inverse is hen easily wrien down). I is clear how his resul can be exended o he case of more general % by means of sandard echniques. 9. We close his secion by considering he quesion of consrucing an opion wih he mos uncerain oucome of he raional paymen R (in he sense of mahemaical enropy as explained in Paragraph 2.6 above). If he fair price V is given a priori and fixed, hen by (2.1.2) we see ha he mos uncerain opion (wih respec o he oucome of he raional paymen R ) is he opion wih a reward variable f T for which ( under P e ): (3.26) R = f T B T Exp 1 V We noe ha he mean-square risk of such an opion mus hen be equal o V 2. The answer o his quesion is now easily obained from he resul of Proposiion 3.2. Corollary 3.3 Consider he Black-Scholes model for a financial marke consising of a riskless bank accoun wih value B = (B ) and a risky sock wih value S = (S ) which are given by (1.1) and (1.2) respecively. A European opion wih he mos uncerain oucome of he raional paymen R (in he sense of mahemaical enropy) wihin he class of all opions having he fair price V (which is given and fixed) is he opion wih he reward process: 1 (3.27) f T = f S T. where f is given explicily as follows: (3.28) f (s) = V e rt log! G e rt s for s > wih G from (3.19). 19

20 Proof. By (3.22) above we see ha F in Proposiion 3.2 should saisfy F Exp(1=V ). Thus F (x) = 1e x=v and hence F 1 (y) = V log(1=(1y)). Hence we see ha (3.27)+(3.28) follows from (3.17)+(3.18). This complees he proof. Acknowledgmens. I am indebed o Alber Shiryaev for suggesing o consider he raional paymen per uni of he fair price which inspired he conens of Paragraph 2.5, and o Donna Salopek for helping me o coin he erm a rue buyer. REFERENCES [1] BENSOUSSAN, A: (1984): On he heory of opion pricing: Aca Appl: Mah: 2 ( ). [2] BLACK, F: and SCHOLES, M: (1973): The pricing of opions and corporae liabiliies: J: Poliical Economy 81 ( ). [3] HARRISON, J: M: and KREPS, D: (1979): Maringales and arbirage in muliperiod securiy markes: J: Econom: Theory 2 (381-48). [4] HARRISON, J: M: and PLISKA, S: (1981): Maringales and sochasic inegrals in he heory of coninuous rading: Sochasic Process: Appl: 11 (215-26). [5] HUANG, C: and LITZENBERGER, R: H: (1988): Foundaions for Financial Economics: Elsevier Science Publ: Co: [6] KARATZAS, I: (1988): On he pricing of American opions: Appl: Mah: Opim: 17 (37-6). [7] MARTIN, N: F: G: and ENGLAND, J: W: (1981): Mahemaical Theory of Enropy: Addison- Wesley. [8] MERTON, R: C: (1973): Theory of raional opion pricing: Bell J: Econom: Manage: Sci: 4 ( ). [9] MYNENI, R: (1992): The pricing of he American opion: Ann: Appl: Probab: 2 (1-23). [1] PESKIR, G. (1997). Designing opions given he risk: The opimal Skorokhod-embedding problem. Research Repor No. 389, Dep. Theore. Sais. Aarhus (18 pp). Sochasic Process. Appl. 81, 1999 (25-38). [11] REVUZ, D: and YOR, M: (1994): Coninuous Maringales and Brownian Moion: Springer- Verlag (Second Ediion). [12] SAMUELSON, P: (1965): Raional heory of warran pricing: Indus: Manage: Rev: 6 (13-31). [13] SHIRYAEV, A: N:, KABANOV, YU: M:, KRAMKOV, D: O:, and MEL NIKOV, A: V: (1994): Toward he heory of pricing of opions of boh European and American ypes: I: Discree ime & II: Coninuous ime: Theory Probab: Appl: 39 (14-6) & (61-12). [14] SHIRYAEV, A: N: (1978): Opimal sopping rules: Springer-Verlag, Berlin-New York. Goran Peskir Deparmen of Mahemaical Sciences Universiy of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus home.imf.au.dk/goran goran@imf.au.dk 2

21 Summary We examine he quesion of he opion risk in complee markes from he sandpoin of a rue buyer. By rue buyer we mean a buyer who has no abiliy or desire o sell he opion; hus every rue buyer will exercise he opion according o he raional performance. We show ha his approach offers exac mahemaical means for comparing differen opions and classifying hem. As an applicaion of his mehodology we presen a simple consrucion of opions wih he mos uncerain oucomes. While hese consideraions have some pracical implicaions which are ye o be fully esed, he sudy iself has gone some way owards undersanding wo new avenues for research: classificaion of opions and opion design. 21