This paper is a substantially revised version of an earlier work previously circulated as Theory

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1 General Properies of Opion Prices Yaacov Z Bergman 1, Bruce D Grundy 2 and Zvi Wiener 3 Forhcoming: he Journal of Finance Firs Draf: February 1995 Curren Draf: January he School of Business and Cener for Raionaliy, Hebrew Universiy, Jerusalem msyberg@pluomscchujiacil 2 he Wharon School, Universiy ofpennsylvania grundy@wharonupennedu 3 he School of Business, Hebrew Universiy, Jerusalem mswiener@pluomscchujiacil his paper is a subsanially revised version of an earlier work previously circulaed as heory of Raional Opion Pricing: II he auhors are graeful for helpful discussions wih Kaushik Amin, Giovanni Barone-Adesi, Avi Bick, Peer Carr, Domenico Cuoco, Sanjiv Das, Darrell Due, Phil Dybvig, Mark Garman, Seve Grenadier, Sanford Grossman, Jon Ingersoll, Alexander Nabuovsky, Seve Ross and Mark Rubinsein, and for he commens of workshop paricipans a he Wharon School, Berkeley, Carnegie-Mellon, Chicago, Harvard, Hebrew, Houson and Maryland Universiies, Washingon Universiy in S Louis, he NBER Financial Risk Assessmen and Managemen Conference, he Sixh Annual Conference in Financial Economics & Accouning, he 1995 European Finance Associaion Meeings, he Sixh Summer Insiue on Game heory a SUNY Sony Brook and he 1996 American Finance Associaion Meeings he auhors graefully acknowledge nancial suppor from he H Krueger Cener for Finance a he Hebrew Universiy (Bergman and Wiener), a Baerymarch Fellowship and he Geewax-erker Program in Financial Insrumens (Grundy) and a Rohschild Fellowship (Wiener)

2 General Properies of Opion Prices Absrac When he underlying price process is a one-dimensional diusion, as well as in cerain resriced sochasic volailiy seings, a coningen claim's dela is always bounded by he inmum and supremum of is dela a mauriy Furher, if he claim's payo is convex (concave), hen he claim's price is a convex (concave) funcion of he underlying asse's value However when volailiy is less specialized, or when he underlying price follows a disconinuous or non-markovian process, hen call prices can have properies very dieren from hose of he Black-Scholes model: a call's price can be a decreasing, concave funcion of he underlying price over some range; increasing wih he passage of ime; and decreasing in he level of ineres raes Much of he nancial opions lieraure derives precise opion prices, when he underlying asse price process is compleely specied Since i is empirically dicul o ascerain wha he rue underlying process is, anoher par of ha lieraure is concerned wih deriving general properies of opion prices, when he underlying price process is no fully specied, bu insead is assumed o belong o some general class of sochasic processes (Meron (1973), Cox and Ross (1976), Jagannahan (1984)) In paricular, in he absence of arbirage opporuniies and assuming ha he risk-neuralized sock price follows a proporional sochasic process (a risk-neural reurn disribuion independen of price), Meron (1973) and Jagannahan (1984) show ha a call opion's price is an increasing, convex funcion of he sock price Cox and Ross (1976) generalize his resul and show ha, under he same proporionaliy assumpion of he sock price process, he price funcion of any European coningen claim, no jus a call opion, inheris qualiaive properies of he claim's conracual payo funcion Meron (1973) noes ha alhough convexiy is usually assumed o be a propery ha holds for calls, proporionaliy of he sock price process is no a necessary condiion for ha He hus implicily conjecures ha quie general bu dieren condiions exis, ha generae he same resuls 1

3 A class of sochasic processes, ha radiionally have played a prominen role in modeling he dynamics of underlying prices, are he diusions I is hen imporan o verify wheher he increasing and convex call price resul of Meron (1973) and Jagannahan (1984), and he Cox and Ross (1976) generalizaion hereof, is valid for ha class as well Indeed, when he underlying process belongs o a quie general class of diusions, we obain he above resuls and more Specically, we esablish ha whenever he underlying asse follows a diusion whose volailiy depends only on ime and he concurren sock price, hen a call price is always increasing and convex in he sock price, decreasing wih he passage of ime, and increasing in he level of ineres raes However, when volailiy is sochasic, or he sock price process is no a diusion, bu is insead eiher disconinuous or non-markovian, we show ha a call price can be a decreasing, concave funcion of he sock price over some range, ha a call can be a `bloaing' no a `wasing' asse, and ha an increase in ineres raes can lead o a decline in a call price Our analysis is no limied o call opions We esablish properies of any European-syle coningen claims given deerminisic ineres raes and various specicaions of he underlying asse price process Whenever he underlying asse follows a one-dimensional diusion, properies of he sock and bond posiions in a coningen claim's replicaing porfolio are shown o be inheried from hose a mauriy Specically, he posiions are bounded by he inmum and supremum of hose posiions a mauriy If he posiion in sock a mauriy is increasing (decreasing) in he underlying mauriy-dae price, hen ha posiion he claim's `dela' is increasing (decreasing) in he concurren underlying price Equivalenly, if he claim's conracual payo funcion is convex (concave) in he underlying price a expiraion, hen he claim's price is convex (concave) in he concurren underlying price Analogous resuls are esablished for cerain resriced mulidimensional diusion (sochasic volailiy) seings Armed wih hese resuls abou general properies of coningen claim prices, we are able o exend comparaive saic resuls familiar in a Black-Scholes seing o ha of a general onedimensional diusion We consider he eecs of changes in ineres raes, in dividend raes, and in volailiy on he prices of call opions In doing so we derive hree addiional resuls Alhough 2

4 an upward shif in he enire erm srucure will always increase call prices, a wis in he ermsrucure ha decreases he presen value of a call's exercise price can decrease is value We esablish a new bound on he relaive values of opions on oherwise equivalen dividend-paying and non-dividend-paying asses in erms of he fracion of he dividend-paying asse's price ha is due o dividends expeced beyond he call's mauriy Our analysis of diering volailiy funcions esablishes ha when he underlying asse's volailiy is bounded above (below), hen, whaever he funcional form of he relaion beween volailiy, ime, and he conemporaneous sock price, he call price is bounded above (below) by is Black-Scholes value calculaed a he bounding volailiy level One can hen place bounds on he sock posiion necessary o hedge a given opion posiion using only knowledge of he bounds on he underlying asse's volailiy he plan of he paper is as follows Secion I esablishes properies of general coningen claims ha are saised in all one-dimensional diusion seings, and in cerain resriced forms ofamuli-dimensional diusion seing Secion II applies hose properies o he pricing of call opions Secion III conains he comparaive saics analyses Secion IV esablishes ha call opions need no possess any of heir familiar properies, if eiher volailiy is sochasic and no resriced in he manner considered in Secion I, or he underlying process is no a diusion, bu is insead eiher disconinuous or non-markovian Secion V summarizes our resuls I A Diusion Process for he Underlying Asse Consider a European coningen claim mauring a ime he ime price of he raded underlying asse is denoed by s We assume ha he sochasic process describing changes in s admis no arbirage opporuniies and is eiher a one- or muli-dimensional diusion, as dened nex Deniion 1 he underlying asse will be said o follow a one-dimensional diusion when ds = ()d + (s ;)s db : (1) he insananeous volailiy, (), is a funcion of s and only, while he drif parameer, (), is no necessarily so resriced B denoes a sandard Brownian moion 3

5 We follow he nance lieraure and refer o (), raher han ()s, as he volailiy Following Karlin and aylor (1981, p 159) we refer o he produc ()s as he diusion parameer he funcions () and () are assumed o saisfy whaever regulariy condiions are necessary for (1) o be a well-dened sochasic dierenial equaion 1 We refer o he special case when volailiy is deerminisic, (), as a Black-Scholes seing Deniion 2 he underlying asse will be said o follow a wo-dimensional diusion when ds = ()d + (s ;y ;)s db 1 ; dy = (s ;y ;)d + (s ;y ;)db 2 ; (2a) (2b) and db 1 db 2 = (s; y; )d [Superscrips on db are indices { no powers] he underlying asse price dynamics in (2) are usually referred o in he nance lieraure as a seing wih sochasic volailiy, alhough, clearly, he volailiy in he one-dimensional case need no be deerminisic Our resuls are valid for a vecor y as well, bu for ease of exposiion, only a one-dimensional diusion y is considered, so ha changes in s are driven by awo-dimensional diusion We use he erms one-dimensional case and wo-dimensional case o mean, respecively, deniions 1 and 2 Unless oherwise noed, i is assumed ha underlying asses pay no dividends over he life of he coningen claim Ineres raes are assumed o be deerminisic; funcions of ime, a mos 2 Le v(s; ) denoe he ime value of a coningen claim in he one-dimensional case (When, in paricular, a call opion is considered, c insead of v will be used o denoe is price) Numerical subindices denoe parial derivaives hus, for example, v 11 (s; ) is he second parial wih respec o he rs argumen; he sock price We consider only limied liabiliy underlying asses (hus zero mus be an absorbing barrier for he underlying sock) Hence for a call opion, c(0;) = 0 he conracual payo funcion is g(), meaning ha a expiraion ime, when he underlying 1 See Chaper 6 of Arnold (1992) for a discussion of Lipschiz and growh condiions 2 American-ype coningen claims, sochasic ineres raes, and oher exensions are considered in our ongoing work 4

6 price is s, he coningen claim conracs o pay g(s) dollars herefore, o preven arbirage, v(s; )=g(s) We assume ha he value of he claim can be expressed, using he Feynman-Kac heorem, as he discouned expecaion of is payo under a risk-neural probabiliy measure 3 In he one-dimensional case v(s; ) =E ne, R r( )d g, s; o ; (3) where s; ; which will be ermed he risk-neuralized process, denoes he diusion ha a ime sars a he level s; and hen obeys he sochasic dierenial equaion (SDE) d = r() d + ( ;) db : (4) In he wo-dimensional case, v(s; y; ) = g(s), and by assumpion, he price of volailiy risk akes he form (s; y; ) he ime value of a coningen claim will hen have he form v(s; y; ) A he Inuiive Link Beween a Diusion Process and Properies of Opion Prices he following lemma provides he inuiive basis for he properies of coningen claim prices esablished in his secion Lemma (No-Crossing) In he one-dimensional case, s 0 s 00 implies ha, wih probabiliy 1, s0 ; s00 ; for all : Proof: According o (4), a xed sample pah (realizaion) of he Brownian moion B ; deermines, for each pair (s; ); a unique sample pah for he process s; (saring from he level s a ime ) Consider wo such sample pahs, s0 ; and s00 ; ; where s 0 s 00 : (For simpliciy, we use he same noaion for a sochasic process and is sample pahs; no confusion should arise) he claim in he lemma is hen ha he s0 ; sample pah ha sars ou a he higher level s 0 never crosses sricly below he s00 ; sample pah ha sars ou a he lower level s 00 Suppose oherwise 3 Raher han assuming some paricular se of resricions on he diusion parameers in (1) and (2) known o be sucien for he applicabiliy of he Feynman-Kac heorem, we prefer o implicily consider he full se of diusion parameers consisen wih he heorem Noe also ha our assumpion ha he value of he claim can be expressed as in (3) should be read as: \can be correcly expressed" We are no, for example, assuming ha we can price (by (3), or oherwise) a claim wih g(s) =1=s when he underlying asse has a posiive probabiliy ofs =0 5

7 hen he wo sample pahs, which are coninuous (wih probabiliy 1) mus have inerseced (for he rs ime afer ) a some ime, 0 Bu given he Markovian naure of in (4), he wo sample pahs will have become idenical from ime 0 onward herefore, if one sample pah sars higher han anoher, i remains higher 4 his is rue for almos every sample pah of B We call his he no-crossing propery of one-dimensional diusions he no-crossing propery is illusraed in Figure 1 An immediae consequence of he lemma is ha, in he one-dimensional diusion seing, a claim's price inheris monooniciy from he conracual payo funcion For if s 0 s 00 ; hen by he no-crossing lemma, s0 ; if g() is non-decreasing, hen g s0 ; and hence, by (3), v(s 0 ;) v(s 00 ;): g s00 ; : his implies E n g s0 ; o s00 ; n E g : herefore, he no-crossing propery in he one-dimensional case, upon which he preceding demonsraion of inheried monooniciy relies, requires ha, he risk-neuralized process for s,be boh coninuous and Markovian, namely, a diusion (see Karlin and aylor (1981, p 157)) A sochasic process ha is no a diusion need no feaure he no-crossing propery, and coningen claims hereon need no exhibi inheried monooniciy In fac, Secion IV provides a number of such examples Also, in conras o he one-dimensional case, he no-crossing propery need no hold in he wo-dimensional case and Secion IV shows ha inheried monooniciy is no always saised in ha seing s00 ; o ; B Sucien Condiions for Inheried Monooniciy heorem 1 bounds he slope of a coningen claim's price in oher words, he claim's dela by he bounds on he slope of he claim's conracual payo funcion 5 4 Or else i inersecs wih he lower sample pah and boh merge onward, which can happen, for insance, a an absorbing boundary 5 For an ineresing reamen ofpoins of non-diereniabiliy ing(s) see Bick (1982) 6

8 heorem 1 Le he payo funcion g be diereniable on is domain 6 (i) Suppose s follows a one-dimensional diusion hen, for all s and, inf q g 1(q) v 1 (s; ) sup g 1 (q): q (ii) If s followsawo-dimensional diusion wih he propery ha he drif and diusion parameers of he risk-neuralized process for y do no depend on s, hen v 1 (s; y; ) is similarly bounded Proof: Given he condiion in (i), he claim price can be expressed as in (3) and (4) For s 0 s 00 ; he random variable X := s0 ;, s00 ; Furhermore, EfX g = e R X g 1 ( ) g s00 ; v(s 0 ;)=E ne, R r( )d g =v(s 00 ;) + inf q r( )d (s 0, s 00 ) For every sample pah, g s00 ;; s00 ; + X : herefore, + X inf g 1(q), where 2 q o s00 ; + X g 1(q)(s 0, s 00 ); ie, Similarly, one can demonsrae ha E ne, R r( )d g v(s 0 ;), v(s 00 ;) s 0, s 00 is non-negaive by he no-crossing propery s00 ; + X = g s00 ; + v(s 0 ;), v(s 00 ;) s 0, s 00 s00 ; sup g 1 (q): q o inf q + E g 1(q): e, R r( )d inf g 1(q)X q he proof of par (ii) consiss of showing ha he no-crossing propery is saised under he assumed condiions, and hen repeaing he seps in par (i) he deails are in he appendix C Sucien Condiions for Inheried Convexiy he work of Meron (1973), Cox and Ross (1976), and Jagannahan (1984), has esablished ha, when he underlying asse's risk-neuralized price process is proporional, convexiy (concaviy) of he conracual payo funcion implies convexiy (concaviy) of he coningen claim price 7 6 Noe ha g mus sill saisfy all he regulariy condiions ha jusify he mainained assumpion ha he price of a coningen claim can be represened by he Feynman-Kac Inegral In he Appendix, heorem 1 is generalized o he case where he payo funcion g has a lef and a righ derivaive everywhere on is domain, where he wo need no be equal, and where one of hem may be plus inniy (as is he lef derivaive a a jump disconinuiy upward, when he funcion is coninuous on he righ here) or minus inniy (as wih a jump disconinuiy downward) In paricular, he generalizaion implies ha he poins, where he lef and righ derivaives of g are no equal, do no maer I also implies ha a jump-disconinuiy upwards (downwards) yields inniy (minus inniy) as he upper (lower) bound on v 1 (s; ) 7 A proporional one-dimensional diusion for he risk-neuralized process implies a deerminisic volailiy funcion; ie, a Black-Scholes seing 7

9 he nex heorem saes ha he condiion ha he underlying price be a diusion is also sucien for he coningen claim price o inheri convexiy (concaviy) from he conracual payo funcion heorem 2 Suppose ha s is eiher (i) a one-dimensional diusion, or (ii) awo-dimensional diusion feauring he win properies: (a) he drif and diusion parameers of he risk-neuralized process for y do no depend on he level of s, and (b) he covariance beween insananeous percen changes in s, and changes in y does no depend on he level of s hen, if a claim's conracual payo funcion is convex (concave), he claim price is convex (concave) as a funcion of he concurren underlying asse price Proof: See he Appendix he mehod of proof is analogous o ha of heorem 1 and proceeds by combining he Feynman-Kac heorem and a `no-crossing' propery of he relevan SDE's An alernae geomeric proof, based on he sochasic maximum principle, is available upon reques Condiion (ii) of heorem 2 is more resricive han condiion (ii) of heorem 1 o guaranee inheried convexiy we require he addiional resricion, condiion (ii)(b), ha he insananeous covariance beween percen changes in s and changes in y does no depend on s he insananeous covariance beween percen changes in s and changes in y is given by (s; y; )(s; y; )(s; y; ) Condiion (ii)(a) already requires ha he funcion () no depend on s Hence he addiional resricion on he insananeous covariance imposed by (ii)(b) is ha he produc (s; y; )(s; y; ) does no depend on s his could occur in hree ways Firs, and pahologically, boh () and () may depend on s, bu in such inverse ways ha heir produc does no Second, () may be zero for all s, y and Finally, boh () and () may no be dependen ons In his hird case, he risk-neuralized process for s is a proporional sochasic process 8 heorem 2 coninues o apply when for all 2 [; ], he underlying asse pays a coninuous proporional dividend a he rae U () and he coningen claim pays a coninuous proporional 8 In an appendix o Meron (1973), B Goldman provides an example where he payo funcion is convex, bu he call-price funcion is no I hen follows from heorem 2, ha he erminal sock price disribuion assumed in ha example canno be generaed by eiher a one-dimensional diusion process or a wo-dimensional diusion saisfying condiion (ii) of he heorem 8

10 dividend a he rae O () 9 In ha case, heorem 1 [case (ii)] akes he form: e R ( O ( ), U ( ))d inf q R g 1(q) v 1 (s; y; ) e Case (i) akes he same form, bu wihou he y ( O ( ), U ( ))d sup g 1 (q): q D Sucien Condiions for Inheried Bounds on he Elemens of a Replicaing Porfolio When a coningen claim can be replicaed hrough a dynamic sraegy of rading he underlying sock and oher asses, he posiion in sock is given by v 1 (s; y; ) I follows hen ha he value of he non-sock posiion in a replicaing porfolio is v(s; y; ), v 1 (s; y; )s Noing his, heorem 1 can be reinerpreed as saing ha he sock posiion prior o mauriy is bounded by he bounds on he sock posiion a mauriy Likewise, he nex heorem esablishes sucien condiions for he value of he non-sock posiion o be similarly bounded heorem 3 Le he payo funcion g be diereniable on is domain 10 (i) If s follows a one-dimensional diusion, hen for all s and, e, R r( )d inf [g(q), g 1(q)q] v(s; ), v 1 (s; )s e, R q r( )d sup[g(q), g 1 (q)q]: q (ii) If s is a wo-dimensional diusion saisfying he win properies of case (ii) of heorem 2, hen v(s; y; ) is similarly bounded Proof: See he Appendix II Monooniciy, Convexiy and he Pricing and Replicaion of Call Opions In Secion I, condiions are esablished, under which cerain properies of a quie general coningen claim's price and is replicaing porfolio are inheried from he corresponding properies of he conracual payo funcion In his secion we examine he specic implicaions of hese resuls for he pricing and replicaion of call opions hroughou, we assume ha he limied 9 heorem 2 need no be applicable when dividends are non-proporional For example, consider a zero exercise price European call wrien on a sock paying a coninuous version of he non-proporional dividend discussed in foonoe 16 of Chaper 4 of Cox and Rubinsein (1985) 10 Like heorem 1, heorem 3 can be generalized o he case where he payo funcion g has a lef and a righ derivaive everywhere on is domain, where he wo need no be equal, and where one of hem may be plus inniy or minus inniy 9

11 liabiliy underlying asse, which pays no dividends, follows a one-dimensional diusion, ha ineres raes are deerminisic, and ha a call price, c(s; ), is given by he soluion of he pde, r()c 1 (s; )s, r()c(s; )+c 2 (s; )+ 1 2 [(s; )s]2 c 11 (s; ) =0; (5) subjec o he erminal condiion c(s; ) = max[0;s, K] A Relaions Beween a Call's Replicaing Porfolio and he Underlying Asse Price A call opion, like any coningen claim in he curren seing, is replicaed by a dynamic sraegy which, a ime, when he sock price is s; mainains c 1 (s; ) shares of he sock and c(s; ), c 1 (s; )s dollars in bonds Proposiions 1 and 2 esablish ha he replicaing porfolio consiss of a levered posiion in he sock Proposiions 1 and 2 also provide bounds on he sock and bond posiions in he replicaing porfolio and, in addiion, show how hose posiions change wih he value of he underlying sock Proposiion 1 If he underlying asse price follows a one-dimensional diusion, hen (i) he sock posiion in a call's replicaing porfolio is always long, bu never by more han one share, and (ii) he sock posiion is non-decreasing in s For all s and such ha max[0;s,ke, R he sock posiion is sricly increasing in s r( )d ] <c(s; ) <s, Proof: he bounds in heorem 1 apply o any coningen claim For a call opion hey become 0 c 1 (s; ) 1, which proves (i) heorem 2 similarly implies c 11 (s; ) 0, and he non-decreasing claim in par (ii) is esablished he appendix conains a proof of he \sricly increasing" claim in par (ii) Proposiion 2 If he underlying asse follows a one-dimensional diusion, hen (i) a call's replicaing porfolio consiss of a levered posiion in sock, wih he amoun of borrowing being no greaer han e, R r( )d K; (ii) he amoun of borrowing is non-decreasing in s; and (iii) he call's elasiciy, (s; ), saises 1 (s; ) 1+e, R r( )d K/c(s; ) Proof: From heorem 3,,e, R r( )d K c(s; ), c 1 (s; )s 0; 10

12 which proves par (i) From heorem ), c 1 (s; =,sc 11 (s; ) 0; and, since s 0, par (ii) is esablished Rearranging he bounds on he bond posiion gives he par (iii) bounds on he call's elasiciy: 1 (s; ) = c 1(s; )s c(s; ) 1+e, R r( )d K/c(s; ): One immediae implicaion of Proposiions 1 and 2 is ha, since in any one-dimensional diusion seing a call opion is equivalen o a levered posiion in he sock, he absolue value of he risk premium on a call always exceeds ha on he underlying sock in ha seing 11 B he Relaion Beween a Call's Price and he Passage of ime A call's convexiy, coupled wih is implici leverage in any one-dimensional diusion seing, causes a call opion o always be a wasing asse in he one-dimensional case his is shown nex Proposiion 3 If he underlying asse follows a one-dimensional diusion, hen for all s and, c 2 (s; ) 0 For all s and such ha eiher (i) max[0;s, e, R r( )d K] <c(s; ) <sand (s; ) > 0, or(ii) 0 <c(s; ) <sand r() > 0, a call is a sricly wasing asse, ie, c 2 (s; ) < 0 Proof: Rewriing he pde in (5) gives c 2 (s; ) =, 1 2 [(s; )s]2 c 11 (s; ), r()c(s; )((s; ), 1) : (6) heorem 2 implies ha he rs erm on he RHS of (6) is non-posiive for all s and Proposiion 2 implies ha (s; ) 1 Since r() 0 for all, he second erm on he RHS of (6) is non-posiive for all s and Since he RHS of (6) is non-posiive, so is he LHS of (6) 11 Grundy (1991) shows ha opion prices conain informaion no only abou he riskneuralized disribuion of he underlying asse, bu also abou is rue disribuion, provided he underlying asse follows a one-dimensional diusion and he risk premium on he opion can be bounded Proposiion 1 has esablished ha he paricular bound examined in ha paper is always saised for a one-dimensional diusion 11

13 urning o he srong inequaliy claim, we have from he sric convexiy resul of Proposiion 1 ha for all s and such ha max[0;s, Ke, R r( )d ] <c(s; ) <s, he rs erm on he RHS of (6) is sricly negaive provided (s; ) > 0 For 0 <c(s; ) <s;proposiion 1 implies ha c(s; ) =c(0; )+ herefore (s; ) = c 1(s;)s c(s;) Z s RHS of (6) is also sricly negaive 0 c 1 (x; )dx = Z s 0 c 1 (x; )dx < Z s 0 c 1 (s; )dx = c 1 (s; )s: > 1 hus, when 0 <c(s; ) <sand r() > 0, he second erm on he Proposiions 1, 2 and 3 exend resuls familiar from a Black-Scholes seing (deerminisic volailiy) o any one-dimensional diusion seing 12 While hese resuls are inuiive, i is imporan o recognize ha hey need no be rue when he underlying price does no follow a one-dimensional diusion I is shown in Secion IV ha if he underlying price process is eiher amuli-dimensional diusion, or non-markovian, or disconinuous, hen i can be ha in some s range, eiher c 1 (s; ) < 0 holds or c 11 (s; ) < 0 holds, or boh hold When c 1 (s; ) < 0, a call's replicaing porfolio will be shor sock and long bonds When c 11 (s; ) < 0, replicaion requires reducing he posiion in sock as he underlying price rises Secion IV also shows ha when c 1 (s; ) < 0orc 11 (s; ) < 0, a call can be a `bloaing' asse C he Relaion Beween a Call's Dela and is Exercise Price Proposiion 4 If he underlying asse follows a one-dimensional diusion, hen c 1 (s; ), orhe call's dela, does no increase wih he exercise price exercise price K, 0 If c 1 (s; ) is also diereniable wr he Proof: Consider a bullish money spread; long a call wih exercise price K 1, and shor a call wih a larger exercise price K 2 he minimum derivaive of his spread's nal payo funcion is zero Denoe he price of he spread by M(s; ) :=c(s; ; K 1 ), c(s; ; K 2 ): hen, by heorem 1, a any ime before expiraion, M 1 (s; ) 0 In oher words, c 1 (s; ; K 1 ), c 1 (s; ; K 2 ) 0 12 In he deerminisic volailiy seing of Black-Scholes, i is also well known ha for all s and, 1 (s; ) < 0 and 2 (s; ) > 0 hese wo properies do no necessarily generalize o a onedimensional diusion wih non-deerminisic volailiy Sill, he wo inequaliies will be saised for some sucienly large s and sucienly large (, ) 12

14 Noe ha a (one-dimensional) diusion underlying price process is only a special sucien condiion for he resul in Proposiion 4 A more general sucien condiion is ha monooniciy be inheried from he conracual payo funcion hus, by he Cox and Ross (1976) resul, he exercise price proposiion is rue also for proporional underlying processes, even when hose are no diusions We have considered he implicaions of heorems 1, 2, and 3 for call opions, bu clearly, similar implicaions apply o pu opions, as well For example, i is an immediae corollary of heorem 1 ha a pu's dela is bounded beween 0 and,1 Similarly, by heorem 2, a pu price is always a convex funcion of he underlying asse he following call opion resuls also apply o pu opions wih he appropriae modicaions III he Comparaive Saics of Ineres Raes, Dividends, and Volailiy We coninue o assume ha he underlying, limied-liabiliy asse follows a one-dimensional diusion, and ha ineres raes are deerminisic A he Comparaive Saics of Ineres Rae Changes We wish o compare he prices of call opions across wo economies In economy A he ineres rae is r A () he ineres rae in an oherwise equivalen economy B is r B () he underlying asse pays no dividends prior o he opion's expiraion A1 An upward shif in he erm srucure increases call prices heorem 4 Consider wo economies, A and B, diering in heir insananeous ineres raes, R R r A () and r B () Suppose ha for all 2 [; ], r B () r A () and r B ()d > r A ()d hen, for all s and, c B (s; ), he price of a call in economy B expiring a, is a leas as large as c A (s; ), he price of an oherwise equivalen call in economy A For all s and, if0 <c B (s; ) <s, hen c B (s; ) >c A (s; ) Proof: See he Appendix 13

15 A2 A decrease in e, R r( )d K can decrease call prices I is imporan o recognize wha we have no esablished in heorem 4 In order o guaranee ha c B (s ;) >c A (s ;)iisno enough ha he erm srucures dier across he wo economies in such away ha he ime value of a riskless bond mauring a ime is smaller in economy B heorem 4 requires no only ha R r B () r A () for all 2 [; ] 13 r B ()d > R r A ()d, bu, in addiion, ha Consider he following wo oherwise equivalen economies In economy A he ineres rae is given by he sep-down funcion ( R; for 2 [;, 1 r A (, )]; 2 () = 0; for 2 (, 1 (, );] 2 In economy B he ineres rae is given by he sep-up funcion ( 0; for 2 [;, 1 r B (, )]; 2 () = R; for 2 (, 1 (, );] 2 Noe ha Z Z r A ()d =, R = r B ()d: 2 Suppose ha he underlying asse follows a diusion of he form ds = r()s d + (s ;)s db ; where he volailiy depends on he underlying asse price and ime in he following way: For some H>0 and some sricly non-zero $(s; ), ( $(s; ); if s>h and 2 [;,, 2 (s; ) = ]; 0; oherwise Now suppose ha s A = s B = s and ha s, H, and K are such ha 13 If R he inegral, R r B ()d > R s <H<e R, 2 s <K: r A ()d and volailiy is deerminisic, hen c B (s ;) >c A (s ;) Only r()d, eners he Black-Scholes formula 14

16 In economy A, s A will grow deerminisically a he ineres rae R unil, a ime +ln(h=s )=R, he underlying asse price reaches he level H Afer ha ime, s A will follow he diusion ds A = ( Rs A d + $(s A ;)s A db ; for 2 ( +ln(h=s )=R;,, 2 ]; 0; for 2 (,, 2 ;] A possible sample pah for s A is depiced in Figure 2 Wih posiive probabiliy s A hence c A (s ;) > 0 >K, and In economy B, s B will remain equal o s unil ime,,, Afer ime,, 2 2 sb will grow deerminisically a he rae R unil i reaches he level s e R, 2 a he opion's mauriy he sample pah for s B is also depiced in Figure 2b Wih cerainy s B <K, and hence cb (s ;)=0 hus, in his seing, which is no a Black-Scholes seing, shifs in he erm srucure ha leave he curren price of a bond mauring a he opion's expiraion unaeced can aec he opion's price In his example, simply `reordering' he ineres rae prole aecs opion prices B he Comparaive Saics of Dividend Rae Changes he nex heorem compares he price of a call on an asse ha pays dividends o he price of a similar call on an asse ha does no heorem 5 Consider wo one-dimensional-diusion priced underlying asses (raded in he same economy wih he same deerminisic ineres rae dynamics) such ha, for all 2 [; ], asse A R pays a dividend a he rae () 0, wih ()d > 0, while asse B pays no dividends prior o ime Suppose ha A (s; ) = B (s; ) for all s and hen c A (s; ) e, R A ( )d c B (s; ), for all s and his inequaliy becomes sric when 0 <c B (s; ) <s Proof: See he Appendix No surprisingly, given s A = s B, a call on he asse ha pays dividends is no more valuable han a call on he asse ha does no Wha is less obvious, is ha one can place an upper bound on he relaive prices of he wo calls purely in erms of he fracion of he dividend-paying sock's price ha is due o dividends o be paid beyond he call's expiraion When dividends are paid a he coninuous rae (), he presen value a ime of he sock price a ime is e, R ( )d s Equivalenly, he ime value of he dividend disribuions beyond ime is e, R ( )d s 15

17 he bound in heorem 5 is familiar from a Black-Scholes seing Given deerminisic reurn volailiy, he value of a call on asse A when s A = s 0 is equal o he value of an oherwise equivalen call wrien on he fracion e, R ( )d of asse B when s B = s 0 In urn, such a call has he same value as an oherwise equivalen call wrien on one whole uni of asse B when s B = e, R ( )d s 0 c A (s; ) =c B (e, R ( )d s; ) (7) From he sric convexiy of call prices in a Black-Scholes seing, we hen have c A (s; ) <e, R ( )d c B (s; ): (8) In a seing where he reurn volailiy depends on he (conemporaneous) sock price, he equaliy in (7) need no longer be saised Sill, as is shown in heorem 5, he inequaliy in (8) is always saised in any one-dimensional diusion seing C he Comparaive Saics of Volailiy Changes Jagannahan (1984) claries he relaion beween he value of a call and he riskiness of he underlying sock Consider wo socks A and B, bu his ime A (s; ) 6= B (s; ) while, for all, A () = B () = 0 Given s A = s B, a sucien condiion for calls on sock B mauring a ime o be a leas as valuable as oherwise equivalen calls on sock A, is ha he risk-neural probabiliy disribuion of s B is riskier han he risk-neural probabiliy disribuion of sa in he Rohschild-Sigliz (1970) sense 14 We generalize and exend Jagannahan's resuls in he nex wo heorems heorem 6 Le B (s; ) A (s; ) for all s and wih sric inequaliy in some region Le v B and v A denoe he respecive prices of wo coningen claims (no necessarily calls) wih he same expiraion dae and idenical, convex conracual payo funcions hen for all s and, v B (s; ) v A (s; ) 14 e Y is riskier han e X in he Rohschild-Sigliz sense if ey d = e X + e"; and Efe"jXg = 0 for all X: 16

18 Proof: See he Appendix As an example, if he volailiy were always bounded by ha of a paricular CEV process, he call price would be bounded by he corresponding CEV opion pricing model heorem 7 Suppose ha for all s and, B (s; ) A (s; ) For s A = s B, he risk-neural disribuion of s B s A is riskier, in he Rohschild-Sigliz sense, han he risk-neural disribuion of Proof: Le A s; and B s; be he ime risk-neuralized prices of asses A and B saisfying d A = r()a d + A ( A ;) A db and wih a common iniial condiion, s; a ime v i (s; ) =e, R d B = r()b d + B ( B ;) B db ; Consider wo coningen claims wrien, respecively, on he wo asses, feauring he same convex payo funcion g() n r( )d E g v B (s; ) v A (s; ) I hen follows ha E By he Feynman-Kac represenaion, he prices of he claims are o i s; ; (i = A; B): For his seing, heorem 6 has esablished ha n g o n B s; E g o A s; for arbirary convex g()'s his condiion is equivalen o A s; being riskier han B s; in he Rohschild-Sigliz sense D Condiions under which Black-Scholes provides Bounds on Opion Prices Ineresing special cases of heorem 6 occur when, for all s and, eiher B (s; ) =() A (s; ), or B (s; ) A (s; ) =() Le c bs() (s; ) denoe he Black-Scholes value of a call on a sock wih deerminisic volailiy (s; ) = () for all s and heorem 8 If for all s and, B (s; ) =() A (s; ), hen c A (s; ) c bs() (s; ) If for all s and, B (s; ) A (s; ) =(), hen c B (s; ) c bs() (s; ) Proof: heorem 8 is a special case of heorem El Karoui, Jeanblanc-Picque and Visvanahan provide an alernae proof of he special case resul in heorem 8 We hank Darrell Due for bringing his paper o our aenion Independen derivaions of varians of heorem 6 can be found in El Karoui, Jeanblanc-Picque and Shreve (1995) and Avellaneda, Levy and Paras 17

19 Of major pracical relevance o anyone charged wih hedging an opion posiion is ha, despie a lack of knowledge of he funcional form of he relaion (s; ), knowledge of bounds on ha relaion over he opion's life, and, provides bounds on he opion's dela for any s and hese bounds are an immediae implicaion of heorem 8 and he convexiy propery of opion prices Proposiion 5 If for all s and, () (s; ) (), hen c bs() 1 (s 00 ;) c 1 (s; ) c bs() 1 (s 0 ;), where s 00 solves c bs() (s; ) =c bs() (s 00 ;)+c bs() 1 (s 00 ;)(s,s 00 ) and s 0 solves c bs() (s; ) =c bs() (s 0 ;), c bs() 1 (s 0 ;)(s 0, s) Proof: As depiced in Figure 3, if he lower bound on dela were violaed, hen, even if he opion ook on is minimal possible value, convexiy would imply ha for values of he underlying asse less han s 00, he opion's value would violae is upper bound Similarly, if he upper bound on dela were violaed hen, even if he opion ook on is minimal possible value, convexiy would imply ha for values of he underlying asse greaer han s 0, he opion's value would violae is upper bound When he opion's value is known, he bounds on is dela can be srenghened as follows Proposiion 6 If for all s and, (s; ) (), hen for any s and such ha one knows c(s; ), c bs() 1 (s 00 ;) c 1 (s; ) c bs() 1 (s 0 ;), where s 00 solves c(s; ) =c bs() (s 00 ;)+c bs() 1 (s 00 ;)(s, s 00 ) and s 0 solves c(s; ) =c bs() (s 0 ;), c bs() 1 (s 0 ;)(s 0, s) Proof: he logic is he same as ha of Proposiion 5 IV Pricing when he Underlying Asse does no follow a One-Dimensional Diusion Secions II and III examin he pricing of call opions in a one-dimensional diusion seing, and esablish ha much of he inuiion familiar from he Black-Scholes model carries over o he case where he underlying asse's volailiy depends on boh he conemporaneous price and ime his secion considers he many ineresing seings in which call prices need no possess anyof heir familiar properies In paricular, we demonsrae ha if he process describing changes in he value of he underlying asse is eiher disconinuous or non-markovian, or he underlying asse 18

20 follows a muli-dimensional diusion, hen replicaing a call's payo can involve shoring sock and lending, and selling addiional sock as he sock price rises hroughou hese examples he `no-crossing' propery is violaed A Call Prices in a Disconinuous Markovian Seing Suppose he underlying asse follows a non-proporional process, such ha below a cerain level H i behaves like a mixed diusion-jump process, 16 and above H i grows a he ineres rae Formally, ds = r()s d for s H r(), (J, 1) s d + s dq for 0 <s <H; where q isapoisson process governing jumps in he sock price, is he mean number of jumps per uni ime, and (J, 1) is he percenage price increase in he sock, if he Poisson even occurs Possible sock price pahs are depiced in Figure 4a Consider wo possible ime sock prices, s 0 and s 00, wih 0 <s 00 <H<s 0 For a given realizaion of he random componen of he process, he sock price pah saring a s 00 can `jump hrough' he pah saring a s 0 which is larger han s 00 Coninuiy precludes his in he one-dimensional case; recall he `no-crossing' propery of Secion IA Now consider a call opion on his asse wih an exercise price K and s 0 e R r( )d <K<s 00 J When s = s 0, he opion will, wih cerainy, nish ou-of-he-money When s = s 00, he opion has a posiive probabiliy of nishing in-he-money hus we have 0=c(0;) <c(s 00 ;) >c(s 0 ;)=0; (9) ie, he call price is no everywhere increasing, and herefore canno be everywhere convex, in he underlying asse price Anoher example of a Markovian underlying process for which call prices are non-increasing in he underlying price is depiced in Figure 4b 17 he non-recombining binomial ree depiced illusraes a seing where a rm's managemen faces he following incenive problem Suppose 16 See Meron (1976) for he developmen of an opion pricing model applicable when he underlying follows a proporional mixed diusion-jump process wih a diversiable jump componen 17 For ye anoher example, see foonoe 14 of Chaper 4 of Cox and Rubinsein (1985) 19

21 ha managemen will be evaluaed on he basis of he sock price a dae relaive o some goal, G Failure o mee he benchmark level, G, will resul in erminaion Exceeding he goal will bring forh a bonus If a dae, 1 he rm has done poorly and he sock price is low, say s,1 = s 00, he rm mus swich o high variance projecs in order for here o be any chance of meeing he benchmark necessary for managemen o reain heir poss Alernaely, if he sock price is high a dae, 1, say s,1 = s 0, managemen can, and will, eecively lock in heir fuure bonuses by swiching o a low risk invesmen sraegy he `no-crossing' propery is violaed Now consider he dae, 1value of a call opion wih a dae mauriy wrien on he sock of his company When =, 1, and K is as depiced in Figure 4b, he call price again saises he se of relaions in (9) Ineresingly, if back a ime, 2 he sock price is equal o s 000, hen he replicaing sraegy a ha ime involves shoring he underlying sock and lending Furher, an increase in he ineres rae from, 2o, 1 will decrease he call's value We now urn o a coninuous bu non-markovian seing, which also fails o saisfy he `no-crossing' propery 18 B Opion Prices in a Coninuous Non-Markovian Seing When he underlying asse's insananeous volailiy depends no only on he conemporaneous price and ime, bu also on pas prices, he process is said o be rearded 19 While rearded processes are relaively unexplored in he derivaives lieraure, hey arise quie naurally when he volailiy of a sock reecs he underlying rm's invesmen and nancing decisions ha a sock's volailiy is relaed o he conemporaneous sock price as a reecion of he rm's invesmen and nancing policy is familiar from he Displaced Diusion Opion Pricing Model of Rubinsein (1983) and he Compound Opion Pricing Model of Geske (1979) In boh hese models 18 Figure 4b can be viewed as depicing a coninuous, bu non-markovian process Suppose ha beween rading daes one could observe (bu no rade along) he rajecory of prices Now suppose ha a ime, one observed a rajecory level of s One could no hen characerize he disribuion of s given only he knowledge ha s =s One would also need o know apas sock price as well (eg, wheher s,1 was equal o s 0 or s 00 ) 19 For a discussion of such processes, see Mohammed (1978) 20

22 he rm's nancing and invesmen decisions predae he opion's issue dae Bu when nancing and invesmen decisions occur during he opion's life, and adjusmen coss are such ha he opimal conrols are no coninuous funcions of he underlying rm value a each insan in ime bu insead exhibi hyseresis, he sock price process can be non-markovian 20;21 he ime line below depics he seing wehave in mind he invesmen-nancing decision made a ime 0 will depend on he value of he rm a 0 as proxied by s 0 For simpliciy, we assume ha prior o ime 0, he sock's volailiy is a consan 0 Opion Valuaion Dae,,,,,,,,,,,,,,,,,,,,,,,,!,,,,,,,, (s ;s 0;),,,,,,,,! Capial Srucure/ Invesmen Choice Opion Mauriy I is rue ha afer s 0 has been realized a ime 0, i will be possible o represen he volailiy of he sock a all imes 2 [ 0 ;] as some funcion (s ;) Bu he funcional form of () canno be deermined ex ane Ex ane, he volailiy a all imes 2 [ 0 ;] akes he form (s ;s 0;) Ex ane, he process is non-markovian As an analyically racable example, consider an unlevered rm ha will pay no dividends prior o ime and will replace is asses a ime 0 Managemen will choose he replacemen asses in a manner ha reecs an incenive problem similar o ha underlying Figure 4b he lower he value of s 0, he higher he volailiy of he replacemen asses Assume ha for 2 [ 0 ;], ( p, ln s (s ;s 0;)=(s 0)= 0=G, 0 ; for s 0 G (10) 0; oherwise 20 For a formal model of opimal invesmen policy given he irreversibiliy ofinvesmen, see Dixi and Pindyck (1994) Fischer, Heinkel and Zechner (1989) model a rm's opimal dynamic capial srucure choice given recapializaion coss 21 ha he rm faces adjusmen coss is no inconsisen wih our implici assumpion ha he securiies issued by he rm, and coningen claims hereon, are raded in fricionless capial markes 21

23 For s 0 G, he rm chooses replacemen asses ha subsequenly have consan volailiy he level of ha consan volailiy is a decreasing funcion of s 0, reaching zero for s 0 = G For all s 0 > G, he rm chooses riskless replacemen asses For a given realizaion of he Brownian moion driving s, he risk-neuralized sock price pah wih iniial condiion s 00 a ime 0 can poenially cross he pah wih iniial condiion s 0 >s 00 Now consider he ime 0 value of a call opion on his sock wih a ime expiraion and an exercise price K such ha e, R 0 r(w)dw K = G Since afer 0 he sock will have a consan volailiy, he level of ha volailiy being deermined by s 0 as in (10), we have c(s; 0 )=c bs((s 0 )) (s; 0 ): he funcion c(s; 0 ) is depiced in Figure 5a he plo consiss of he locus of poins where a verical line drawn from he x-axis a, say, he poin s 0,inersecs he dashed convex curve which plos c bs((s0 )) (s; 0 ) Noice how he verical line drawn from s 00 <s 0 simulaneously inersecs he locus and a second dashed convex curve ha lies everywhere above c bs((s0 )) (s 0; 0 ) his higher convex curve plos c bs((s00 )) (s; 0 ) From he volailiy specicaion in (10) we have (s 00 ) >(s 0 ) since s 00 <s 0 For all s 0 >e, R 0 r(w)dw K he locus of poins is very easy o consruc Wih zero volailiy in he fuure, he call is cerain o nish-in-he-money, and is worh s 0, e, R 0 r(w)dw K a ime 0 he humped shape of he plo is deermined by wo opposing forces As s 0 increases, he underlying sock becomes more valuable (which ends o increase he call price); bu he underlying sock also becomes less volaile over he opion's remaining life (which ends o decrease he call price) In a (Markovian) one-dimensional diusion seing, heorem 1 saes ha he rs force is always he sronger In conras, in his non-markovian seing he second force can overwhelm he rs, hereby creaing he hump When, as here, a call price can be decreasing and concave ins, a call can also have oher properies quie dieren from hose of a Black-Scholes seing A call price can be increasing wih he passage of ime and a call price can decrease when ineres raes increase Firs consider he eec of he passage of ime by comparing he call price a ime 0 o is price a some prior o 0 22

24 We assume ha for all 2 [; 0 ), he sock's volailiy is a consan During he inerval from o 0 he call price saises he pde c 2 (s; ) =, 1 2 [s]2 c 11 (s; ), r()c(s; ) (s; ), 1 : (11) When a call is increasing and convex in s for all s and, he wo erms on he righ-hand-side of (11) are non-posiive Proposiions 1 and 2 esablish ha c 11 (s; ) 0 and (s; ) > 1ina one-dimensional diusion seing Bu when, as here, he call is sricly decreasing and concave in s over some region, boh erms are sricly posiive in ha region hus he call can be a `bloaing' asse Figure 5b illusraes he non-wasing naure of he call over some range of s, by depicing boh c(s; ) and c(s; 0 ) when 0, =1year,, 0 =1year, = 30% per annum, K = G = $3, and r() = 10% per annum for all 2 [; ] Consider he eec on he call price of an increase in ineres raes Given he above parameer values, c(s; ) = 10:69/c when s = $1 Suppose ha he ineres rae during he period from o 0 increases from 10% o 11% his upward shif in he erm srucure will cause a decline in he call price o 10:38/c C Call Prices and Sochasic Volailiy Subsecions A and B of IV provide illusraions of how he `no-crossing' propery can be violaed when he underlying process is eiher disconinuous or non-markovian he proof of heorem 1 esablishes ha he `no-crossing' propery is never violaed in a one-dimensional diusion seing heorem 1 also provides sucien resricions on he drif and diusion parameers of awo-dimensional diusion o guaranee ha he `no-crossing' propery remains saised when volailiy is sochasic Bu when hese resricions are no me, he `no-crossing' propery can be violaed in a diusion seing, and a call price can be decreasing in he value of he underlying Recall from Secion I he deniion of a wo-dimensional diusion Suppose ha ineres raes are zero and ha he drif of he risk-neuralized process for y is zero he risk-neuralized processes for s and y are hen given by: ds = (s ;y ;)s db 1 ; and dy = (s ;y ;)db 2 : 23

25 One could hink of s and y as he share prices of wo rms in a duopoly whose compeiive sraegies and share prices reec heir marke shares As depiced in Figure 6, if s is high and y is low, hen (s; y; )=(s; y; ) = 0 If s is low and y is low, hen (s; y; ) = 0 and (s; y; ) > 0 If y is high, hen (s; y; ) > 0 and (s; y; ) > 0 Noe ha condiion (ii) of heorem 1 is no me in his example In paricular, wheher (s ;y ;) is posiive or zero depends on he value of s Wih iniial condiion fs 0 ;y 0 g, s = s 0 : Wih iniial condiion fs 00 ;y 0 g, here is a posiive probabiliy ha s >s 0 even hough s 00 <s 0, and hence he `no-crossing' propery is violaed in his example Now consider a call opion wih K > s 0 For his opion c(s 0 ;y 0 ;)=0and c(s 00 ;y 0 ;) > 0 hus, over a range of s values, he opion price is decreasing in s, and canno hen be everywhere convex in s VI Conclusions his paper examines he general properies of prices of European coningen claims We show ha when he underlying sock follows a one-dimensional diusion and ineres raes are deerminisic, he sock posiion in he dynamic porfolio ha replicaes a coningen claim (equivalenly, he claim's price slope or \dela") is bounded by he inmum and he supremum of ha posiion a mauriy Similar bounds also hold for he bond posiion in he claim's replicaing porfolio If he claim's payo a mauriy isconvex (concave) in he price of he underlying asse a expiraion, hen prior o expiraion he sock posiion in a replicaing porfolio is increasing (decreasing) in he underlying asse's price he bounds on a claim's dela also apply in a muli-dimensional diusion seing, provided ha he drif and diusion parameers of he risk-neuralized version of he process driving he sochasic changes in volailiy are independen of he underlying asse price Wih appropriae furher resricions, he inheried convexiy resul can also be exended o amuli-dimensional diusion seing In sum, under sipulaed, quie general diusion condiions, properies of he conracual payo funcion, like monooniciy and convexiy, are inheried by he coningen claim price (as a funcion of he underlying price) prior o expiraion 24

26 he bounds and inheried convexiy resuls allow us o underake comparaive saic analyses of he eecs of changes in ineres raes, in dividend raes, and in volailiy on he prices of call opions in a one-dimensional diusion seing Firs, we show ha i is only in a deerminisic volailiy (Black-Scholes) seing ha a decrease in he presen value of he exercise price necessarily implies an increase in a call price In general, coningen claim prices are deermined by he enire erm srucure of insananeous ineres raes hrough he claim's mauriy dae, and no merely by heir inegral A wis in he erm-srucure ha leaves he presen value of a call's exercise price unchanged can change he call's value Second, we develop a new bound on he relaive values of call opions on wo underlying asses; one ha pays dividends, anoher ha does no, bu which are oherwise equivalen hird, we show ha when he underlying asse's volailiy is bounded above (below), hen, whaever he funcional form of he relaion beween volailiy, ime, and he conemporaneous sock price, he opion's price is bounded above (below) by is Black-Scholes value calculaed a he bounding volailiy level We also show how o incorporae bounds on he underlying asse's volailiy ino he deerminaion of bounds on an opion's dela Finally, we underake a comparaive saics analysis of he relaion beween he exercise price of a call and is dela in a one-dimensional diusion seing We esablish ha he call's dela is always non-decreasing in is exercise price he bounds on a coningen claim's dela esablished in a one-dimensional diusion seing, and in cerain resriced sochasic volailiy seings, are shown o be a reecion of he fac ha, for a given realizaion of he Brownian moion driving he risk-neuralized sock price process, he realized value of he process a he claim's mauriy dae is increasing in is saring value We dub his he `no-crossing' propery We demonsrae ha if we relax eiher he coninuiy or Markovian properies inheren in a diusion, or we consider an unresriced sochasic volailiy seing, hen he `no-crossing' propery can be violaed I is hen shown ha he price of a call opion can be decreasing or concave over some underlying price range, increasing in he passage of ime, and decreasing in he level of ineres raes 25

27 Appendix Proof of par (ii) of heorem 1 (hroughou he proof, superscrips `1' and `2' denoe indices { no powers) Using he Feynman-Kac heorem he value of he coningen claim can be expressed as v(s; y; ) =E ne, R o r( )d g 1 s;y; ; where 1 s;y; and 2s;y; solve he sysem of SDE's d 1 = r()1 d + (1 ;2 ;)1 db1 ; d 2 =, ( 1 ;2 ;), (1 ;2 ;)(1 ;2 ;) d + ( 1 ;2 ;)db2 ; (A1a) (A1b) wih db 1dB2 = (1 ;2 ;)d and iniial condiions s and y a ime he SDE's in (A1) describe he `risk-neuralized' processes for s and y he condiions in (ii) imply ha here exis funcions G 1 and G 2, such ha: G 1 ( 2 ;)=( 1 ; 2 ;), ( 1 ; 2 ;)( 1 ; 2 ;); and G 2 ( 2 ;)=[( 1 ; 2 ;)] 2 : In his case, expression (A1) simplies o d 1 = r()1 d + (1 ;2 ;)1 db1 ; (A1a0 ) d 2 = G1 ( 2 ;)d + p G 2 ( 2 ;)db2 : (A1b0 ) hus, for any given realizaion of he B 2 process, condiion (ii) guaranees ha he pah followed by 2 is independen of he iniial condiion for 1 Now consider wo pahs for 1 diering only in heir iniial condiions Given he realizaion of he B 2 process, we consruc each pah for 1 from innovaions db 1, wih he innovaions hemselves consruced as db 1 = (; 1 2 ;)db 2 + p 1, [( 1 ;;)] 2 2 db 3, where B 3 is a sandard Brownian moion independen ofb 2 Suppose hese wo pahs for 1 me a some ime 0 Could hey cross over? Consider he SDE in (A1a 0 ) By consrucion, boh pahs for 1 always share a common realizaion of he 2 process and hence, 26

28 mus merge from ime 0 onward Suppose s 00 <s 0 Saring from f 1 = s 0 ; 2 = yg, he subsequen level of 1 aained a ime is always a leas as grea as he level aained when he processes sar from fs 00 ;yg he res of he proof involves dening X := 1;s0 ;y; as in he proof of par (i), 1;s00 ;y;, and proceeding Generalizaion of heorem 1 Suppose ha he payo funcion g has a lef and a righ derivaive everywhere on is domain, where he wo need no be equal, and where one of hem may be plus inniy (as is he lef derivaive a a jump disconinuiy upward, when he funcion is coninuous on he righ here) or minus inniy (as wih a jump disconuniy downward) his, of course, covers calls, pus, and digial opions) hen heorem 1 generalizes by replacing he double inequaliy by he following inf q [min (g 1(q,);g 1 (q+))] v 1 (s; ) sup [max (g 1 (q,);g 1 (q+))] ; q where, for example, g 1 (q+) sands for he righ derivaive ofg a q Proof: Suppose s follows a one-dimensional diusion he call price can hen be expressed as in (3) and (4) For s 0 s 00 ; he random variable X := s0 ;, s00 ; he no-crossing propery g s00 ; + X g where 2 s00 ; s00 ; ; s00 ; + X is non-negaive by Furhermore, EfX g = e R r( )d (s 0, s 00 ) For every sample pah, + X min (g 1 (,);g 1 ( +)) g s00 ; + X inf q [min (g 1(q,);g 1 (q+))], : Here, we used he generalizaion of he Inermediae Value heorem of dierenial calculus o he case where only he lef and he righ derivaives are guaraneed o exis he res of he proof follows he same seps as in he main ex he Cauchy Problem and he Proof of heorem 2 he heorem 2 condiion ha he claim's payo funcion is convex (concave) implies ha he payo funcion is coninuous Firs consider he Cauchy Problem: For given >0, we wish o nd f 2 C 2;1, IR N [0;) solving Df(x; ), R(x; )f(x; )+h(x; ) =0; (x; ) 2 IR N [0;); (A2) wih he erminal condiion 27

29 f(x; )=g(x); almos everywhere on IR N ; (A3) and Df(x; ) =f N +1 (x;)+ NX f i (x; ) i (x; )+ 1 2 NX NX i (x; ) j (x; ) ij (x; )f ij (x; ); (A4) i=1 i=1 j=1 and where f N +1 (x; ) denoes he parial of f wih respec o, and, for i =1;:::;N, f i (x; ) denoes he parial of f wih respec o he i'h elemen of he vecor x, R :IR N [0;]! [0; 1), h :IR N [0;]! IR, g :IR N! IR, he funcions R, h and g are coninuous, and (x; ) isan N 1vecor whose i'h elemen i (x; ) is such ha i :IR N [0;]! IR, and he superscrips on i (x; ) j (x; ) ij (x; ) do no denoe powers bu are insead indices (x; ) isann 1vecor whose i'h elemen i (x; ): IR N [0;]! IR, for i =1;:::;N Each funcion ij (x; ) = ji (x; ): IR N [0;]! IR, for all i; j =1;:::;N he Feynman-Kac soluion o (A2){(A4), when i exiss, is given by where ' ; = e, R f(x; ) =E Z ' ; h, x; ; d + ' ; g, x; ; (A5) R(x; w ;w)dw he elemens, ix;, of he N 1 vecor x; solve he sysem of SDE's d i = i ( ;)d + i ( ;)db i ; wih iniial condiion x a ime In addiion, db i db j = ij ( ;)d As an example suppose he underlying asse follows he wo dimensional diusion given in (2), ineres raes are deerminisic, and he price of volailiy risk is given by (s; y; ) he value of a coningen claim is given by he soluion of he pde, v 3 (s; y; )+v 1 (s; y; )r()s + v 2 (s; y; )((s; y; ), (s; y; )(s; y; ))+ 1 2 v 11(s; y; )[(s; y; )s] v 22(s; y; )[(s; y; )] 2 + v 12 (s; y; )(s; y; )s(s; y; )(s; y; ), r()v(s; y; )=0; (A6) subjec o he erminal condiion v(s; y; )=g(s) Le he superscrips on he funcions Y 1 ;:::;Y 4 denoe indices no powers, and dene he funcions as: Y 1 (s; y; ) =(s; y; ), (s; y; )(s; y; ): Y 2 (s; y; ) =[(s; y; )s] 2 : Y 3 (s; y; ) =[(s; y; )] 2 Y 4 (s; y; ) =(s; y; )s(s; y; )(s; y; ): 28

30 We can hen rewrie (A6) as v 3 (s; )+v 1 (s; y; )r()s + v 2 (s; y; )Y 1 (s; y; )+ 1 2 v 11(s; y; )Y 2 (s; y; ) v 22(s; y; )Y 3 (s; y; )+v 12 (s; y; )Y 4 (s; y; ), r()v(s; y; )=0: (A7) aking he parial of (A7) wih respec o s gives v 13 (s; y; )+v 11 (s; y; )r()s + v 1 (s; y; )r()+v 12 (s; y; )Y 1 (s; y; )+v 2 (s; y; )Y 1 1 (s; y; )+ 1 2 v 111(s; y; )Y 2 (s; y; )+ 1 2 v 11(s; y; )Y 2(s; y; ) v 122(s; y; )Y 3 (s; y; )+ 1 2 v 22(s; y; )Y 3(s; y; )+v 1 112(s; y; )Y 4 (s; y; )+v 12 (s; y; )Y 4(s; y; ), r()v 1 1(s; y; ) =0: (A8) Le f be he value of he rs parial of a coningen claim's value wih respec o he value of he underlying asse he pde in (A8) can hen be rewrien as f 3 (s; y; )+f 1 (s; y; )[r()s Y 2 1 (s; y; )] + f 2 (s; y; )[Y 1 (s; y; )+Y 4 1 (s; y; )] f 11(s; y; )Y 2 (s; y; )+ 1 2 f 22(s; y; )Y 3 (s; y; )+f 12 (s; y; )Y 4 (s; y; ) +v 2 (s; y; )Y 1 1 (s; y; )+ 1 2 v 22(s; y; )Y 3 1 (s; y; ) =0: (A9) Expression (A9) is in he same form as (A2) wih N =2: s x = y R(s; ) =0: : h(x; ) =v 2 (s; y; )Y 1 1 (s; y; )+ 1 2 v 22(s; y; )Y 3 1 (s; y; ): r()s (x; ) = Y 2 1 (s; y; ) Y 1 (s; y; )+Y 4(s; y; )) : 1 p Y 2 (s; y; ) (x; ) = p : Y 3 (s; y; ) (s; y; ); if i 6= j; ij (x; ) = 1; oherwise hus he Feynman-Kac soluion for v 1 (s; y; ), when i exiss, is given by v 1 (s; y; )=E Z v 2 (^ 1s;y; ; ^ 2s;y; ;)Y v 22(^ 1s;y; 2s;y; ; ^ ;)Y (^ 1s;y; 1s;y; (^ ; ; 2s;y; ^ ;) 2s;y; ^ ;) d + g 1 ^1s;y; ; 29

31 where ^ 1s;y; and ^ 2s;y; solve he sysem of SDE's d^ 1 = r()^ Y 2 1 (^ 1 ; ^ 2 ;) d + d^ 2 = Y 1 (^ 1 ; ^ 2 ;)+Y 4 1 (^ 1 ; ^ 2 ;) q Y 2 (^ 1 ; ^ 2 ;)db1 ; d + q Y 3 (^ 1 ; ^ 2 ;)db 2 ; wih iniial condiion fs; yg a ime In addiion, db 1dB2 = (^ 1 ; ^ 2 ;)d Noe ha hese are no he SDE's describing he risk-neuralized processes for s and y given in (A1) Condiion (i) of heorem 2 implies ha here exiss a funcion Z such ha Z(s; ) =(s; y; )s: When he underlying asse follows a one-dimensional diusion and condiion (i) is saised, he Feynman-Kac soluion for he rs parial of he value of he coningen claim wih respec o he value of he underlying asse hen simplies o v 1 (s; ) =E n g 1 ^s; o ; where ^ s; solves he SDE d^ = r()^ + Z 1 (^ ;) d + Z(^ ;)db ; wih iniial condiion s a ime Suppose s 0 >s 00 hen by he `no-crossing' lemma ^ s0 ; ^ s00 ; wih probabiliy 1 If for all s, g 11 0 hen g 1 is non-decreasing in s, and hence v 1 (s 0 ;)=E n g 1 ^s 0 ; o Similarly, if for all s, g 11 0, v 1 (s 0 ;) v 1 (s 00 ;) E n g 1 ^s 00 ; o = v 1 (s 00 ;): Now consider he wo-dimensional case Condiion (ii) of heorem 2 implies ha here exis funcions G 1, G 2 and G 3, where superscrips denoe indices no powers, such ha G 1 (y; ) = Y 1 (s; y; ), G 2 (y; ) =Y 3 (s; y; ) and G 3 (y; )s = Y 4 (s; y; ) he Feynman-Kac soluion for he value of he rs parial of he coningen claim wih respec o he underlying asse hen simplies o n v 1 (s; y; )=E g 1 ^1s;y; o ; 30

32 where ^ 1s;y; and ^ 2s;y; solve he sysem of SDE's d^ 1 = r()^ Y 2 1 (^ 1 ; ^ ;) 2 d + d^ 2 = G 1 (^ 2 ;)+G3 (^ 2 ;) d + q Y 2 (^ 1 ; ^ 2 ;)db 1 ; q G 2 (^ 2 ;)db2 ; wih iniial condiion fs; yg a ime In addiion, db 1 db 2 = (^ 1 ; ^ 2 ;) Suppose s 0 >s 00 Again ^ 1s0 ;y; non-decreasing in s Hence g 1 ^1s 0 ;y; v 1 (s 0 ;y;)=e n ^ 1s00 ;y; g 1 ^1s 0 ;y; g ^1s 00 ;y; o wih probabiliy 1 If for all s, g 11 0, hen g 1 is E n wih probabiliy one and g 1 ^1s 00 ;y; o = v 1 (s 00 ;y;): Similarly, if for all s, g 11 0, hen g 1 is non-increasing in s Hence v 1 (s 0 ;y;) v 1 (s 00 ;y;) Proof of heorem 3 Combining (A6) and (A7) he quaniy [v(s; y; ), v 1 (s; y; )s] can be shown o saisfy a pde of he form in (A2) subjec o he erminal condiion v(s; y; ), v 1 (s; y; )s = g(s), g 1 (s)s: If y is he price of a raded asse, he quaniy [v(s; y; ), v 1 (s; y; )s] can be inerpreed as he value of he non-sock posiion in a replicaing porfolio Under he condiions of heorem 3 he erms R(x; ) and h(x; ) of(a2) simplify o R(x; ) =r() and h(x; ) =0 hus v(s; y; ), v 1 (s; y; )s = e, R n r( )d E g 1 o s;y;, g 1 1 s;y; 1 s;y; ; where 1 and 2 solve a sysem of correlaed SDE's wih iniial condiions fs; yg a ime Proof of heorem 4 he proof will be made more ransparen by measuring he prices of ime mauriy coningen claims and of he underlying asse relaive o he price of a pure discoun bond mauring a ime ; ie, using he bond as numeraire Using upper (lower) case noaion o denoe relaive (absolue) price levels, he normalized prices ake he form: R r( S = e )d s ; R r( C(S ;)=e )d R r( c(s ;)=e )d c R,r( e )d S ; : 31

33 he normalized call price has he following parial derivaives: C 1 (S; ) =c 1 (s; ): C 11 (S; ) =c 11 (s; ) e, R r( )d : Noe ha convexiy ofc(s; ) ins for all S and implies he convexiy ofc(s; ) ins for all s and, and vice-versa Using he normalized pricing sysem, he opion's value saises he following pde, C 2 (S; )+ 1 2 [(S; )S]2 C 11 (S; ) =0; where (S; ) := e, R r( )d S; = (s; ): (A10) he ransformaion from () o() is non-rivial Oher han in a deerminisic volailiy (Black- Scholes) world, he ransformaion requires knowledge of r() for all 2 [; ] I is emping o view (A10) as implying ha one can deermine he forward price of a coningen claim provided one knows he forward price of he underlying asse and wihou having o know anyhing abou ineres raes Bu when he volailiy of reurns on he underlying asse depends on he spo price of he asse, hen he (S; ) funcion masks, bu does no remove, he relaion beween ineres raes and he claim's forward price A1 Consider he following ransformed price sysems: S = s er r A ( )d ; C A (S ;)=c A S e, R r A ( )d ; C B (S ;)=c B S e, R r A ( )d ; R r e A ( )d ; er ra ( )d : One can hink of he ransformaion as seing he ineres rae o zero in economy A and o () =r B (), r A () in economy B C A (S; ) solves he pde C A 2 (S; )+ 1 2 [(S; )S]2 C A 11(S; ) =0 (A11) A1 If he volailiy of percen changes in he forward price for delivery of he underlying asse a ime depends on he level of he forward price for delivery a ime (and no on he spo price), hen he forward price of a call opion wih a ime mauriy will no depend on he erm srucure of ineres raes However, he forward prices of he se of call opions mauring a daes oher han will depend on he erm srucure of ineres raes 32

34 subjec o C A (S; ) = max[0;s, K], where (S; ) = e, R r A ( )d S; = (s; ) C B (S; ) solves he pde C B 2 (S; )+ 1 2 [(S; )S]2 C B 11(S; ) =,(), C B 1 (S; )S, C B (S; ) (A12) subjec o C B (S; ) = max[0;s, K] Le X(S ;):=C B (S ;), C A (S ;) denoe he dierence beween he ransformed values of he calls across he wo economies Noe ha X(S ;) > 0 implies c B (s ;) >c A (s ;) From (A11) and (A12) we have X 2 (S; )+ 1 2 [(S; )S]2 X 11 (S; )+(), C B 1 (S; )S, C B (S; ) =0: (A13) For a given value of S a mauriy, he ime prices of he wo calls coincide: X(S ;)=C B (S ;), C A (S ;) = max[0;s, K], max[0;s, K] =0: hus X(S; ) is given by he soluion o he pde in (A13) subjec o he erminal condiion X(S; ) = 0 From he Feynman-Kac heorem we have nz X(S; ) =E, () C B 1 ( S; ;) S; o, C B ( S; ;) d ; where S; solves he SDE: d = v( ;) db, wih iniial condiion S a ime For c B (s ;)=s, he weak inequaliy is saised immediaely since c A (s ;) s For c B (s ;) = 0, i follows ha for all 2 [; ], C B ( ;) = 0, and he inegrand, (), C B 1 ( ;)S, C B ( ;), is zero for all 2 [; ] he weak inequaliy is hen saised Finally for 0 < c B (s ;) <s, i follows from heorem 2 ha for all 2 [; ] and all > 0,0 C B ( ;) < For C B ( ;) = 0 he inegrand is zero For 0 < C B ( ;) <, Proposiion 1 implies ha C B 1 ( ;),C B ( ;) > 0, and hence (), C B 1 ( ;), C B ( ;) is non-negaive, and sricly posiive for () > 0 hus for 0 <c B (s ;) <s, since R ()d > 0, nz X(S; ) =E, () C B 1 ( S; ;) S; 33, C B ( S; ;) d o > 0:

35 Raher han apply he Feynman-Kac heorem, he ask of demonsraing ha X(S; ) > 0 can be ransformed ino a familiar, and inuiively posiive, valuaion problem Suppose rs ha in he normalized (zero ineres rae) economy A, we wish o value a coningen claim, V (S ;), wih he following conracual erms: he pary long he conrac will a all imes 2 [; ] receive a coninuous income sream from he shor equal o (), C B 1 (S ;)S, C B (S ;), and nohing hereafer Given he assumpions of heorem 4 such an income sream is always nonnegaive Furher, when 0 <c B (s; ) <s, he sric convexiy resul of Proposiion 1 implies ha he income sream will, wih posiive probabiliy, be sricly posiive over some ime inerval hus a ime his income sream conrac has a sricly posiive value o he long; ie, V (S ;) > 0 A is mauriy, he income sream conrac is valueless, and V (S ;) = 0 he pde and erminal condiion for his income sream coningen claim are idenical o he pde in (A13) and he erminal condiion whose soluion deermines X(S; ) I follows immediaely ha X(S; ) = V (S; ) 0 Furher, provided 0 <c B (s; ) <s, X(S; ) > 0 Proof of heorem 5 Suppose ha changes in he value of a hird underlying asse, U, are also described by a one-dimensional diusion, wih U (s; ) = A (s; ) = B (s; ) =(s; ) Suppose furher ha asse U pays a coninuous proporional dividend a he rae U () for all 2 [; ] he superscrip `U' is a mnemonic for underlying Consider a call opion wrien on asse U, c U (s; ), wih he usual payo a mauriy of max[0;s U, K], and he addiional conracual feaure ha a all imes 2 [; ], he shor pays he long a coninuous dividend, proporional o he value of he call, a he rae O () he superscrip `O' is a mnemonic for opion We inroduce he following noaion o describe his call: c(s; ; U ; O ) Using his noaion we have c U (s; ) =c(s; ; U ; O ), c A (s; ) =c(s; ; A ; 0), and c B (s; ) =c(s; ; 0; 0) o preclude arbirage i mus be ha c U R (s; ) =e O ( )d c(s; ; U ; 0): (A14) he value c U (s; ) is given by he soluion of he following pde c U 2 (s; )+1 2 [(s; )s]2 c U 11 (s; )+r()cu (s; ), U (s; ), 1 + O ()c U (s; ),c U 1 (s; )U ()s =0; (A15) 34

36 subjec o he erminal condiion c U (s; ) = max[0;s, K] Now suppose furher ha for all, U () = O () = A () Subsiuing ino he pde in (A15) gives c U 2 (s; )+ 1 2 [(s; )s]2 c U 11(s; )+r()c U (s; ), U (s; ), 1 + A ()c U (s; ), c U 1 (s; ) A ()s = c U 2 (s; )+1 2 [(s; )s]2 c U 11 (s; )+, r(), A () c U (s; ), U (s; ), 1 =0: (A16) hus, he pde in (A16), whose soluion deermines he value of c U (s; ), is idenical o he pde ha would deermine he value of c B (s; ) in an oherwise equivalen economy in which, a all imes 2 [; ], he ineres rae was lower han r() by he amoun A () I hen follows from heorem 4 ha, for all s and, c B (s; ) c U (s; ) (A17a) and, for all s and such ha 0 <c B (s; ) <s, c B (s; ) >c U (s; ): (A17b) Furher, subsiuing U () = O () = A () ino expression (A14) gives c U R (s; ) =e A ( )d c(s; ; A ; 0) = er A ( )d c A (s; ): (A18) Combining (A17) and (A18) gives, for all s and, c A (s; ) e, R and, for all s and such ha 0 <c B (s; ) <s, c A (s; ) <e, R A ( )d c B (s; ); A ( )d c B (s; ): Proof of heorem 6 Le S A and S B denoe he ime normalized prices of he wo asses ds A = A ()d + A (S A ;)S A db : ds B = B ()d + B (S B ;)S B db : 35

37 For all S and, B (S; ) A (S; ), and, for S and in some region, B (S; ) > A (S; ) Le X(S; ) :=V B (S; ), V A (S; ) denoe he dierence in values of he normalized coningen claim prices V i (S; ) solves V i 2 (S; )+ 1 2 [i (S; )S] 2 V i 11(S; ) =0; subjec o V i (S; )=g(s), i = A; B X(S; ) solves X 2 (S; )+ 1 2 [A (S; )S] 2 X 11 (S; )+ 1 2 [ B (S; )] 2, [ A (S; )] 2 S 2 V B 11(S; ) =0; (A19) subjec o X(S; )=0 From heorem 2 he erm 1 [ B (S; )] 2, [ A (S; )] 2 S 2 V B 2 11(S; ) in he pde in (A19) is non-negaive he remainder of he proof parallels ha of heorem 4 Proof of he Sric Convexiy Claim of Proposiion 1 Le S and C be he normalized prices of he underlying asse and he call as dened in he inroducory paragraph of he Proof of heorem 4 Suppose ha a ime 0 he sric convexiy claim is violaed for prices in some region Le a( 0 ) and b( 0 ) a( 0 ) denoe he normalized prices marking he end poins of ha region; ie, for all S 2 (a( 0 );b( 0 )], max[0;s, K] <C(S; 0 ) <S ye C 11 (S; 0 ) = 0 Suppose a( 0 ) > 0 Since C (a( 0 ); 0 ) > max[0;a( 0 ), K], ye C (a( 0 );)= max[0;a( 0 ), K], here mus exis a se of imes 2 ( 0 ;] a which C 2 (a( 0 );) < 0 Le us hen consider he paricular value of 0 such ha no only is max[0;a( 0 ), K] <C(a( 0 ); 0 ) <a( 0 ) and C 11 (a( 0 ); 0 ) = 0, bu for some 00 > 0 wehave ha for all 2 ( 0 ; 00 ),C 2 (a( 0 );) < 0: Given he pde in (A10) we see immediaely ha for all 2 ( 0 ; 00 ), C 11 (a( 0 );) > 0 and (a( 0 );) > 0 Assume ha (a( 0 ); 0 ) > 0 A2 Since C 2 (a( 0 ); 0 ) = 0 and (a( 0 ); 0 ) > 0, here hen exiss A2 he condiion (a( 0 ); 0 ) > 0 can be relaxed he echniques used o analyze dynamic behavior near a local minimum are discussed in Wiener (1993) Observe ha [(S; )S] 2 is a nonnegaive funcion which can herefore have zero values only a local minimums In any paricular case, one should diereniae he pde in (A10) a sucien number of imes unil he singulariy is resolved 36

38 an arbirarily small posiive " such ha for all " 2 (0; "), no only is C 11 (a( 0 ), "; 0 ) > 0 and (a( 0 ), "; 0 ) > 0, bu C 12 (a( 0 ), "; 0 ) > 0 Furher, since C 2 (a( 0 ); 0 ) = 0 and for all S, C 2 (S; 0 ) 0, i follows ha C 12 (a( 0 ); 0 ) = 0 Sric convexiy for all 2 ( 0 ; 00 ) requires ha for ime 0+, lim "!0 C 1(a( 0 ); 0+ ),C 1(a( 0 ),"; 0+ ) " > 0 Bu since C 12 (a( 0 ); 0 ) = 0, lim "!0 C 1(a( 0 ); 0+ ),C 1(a( 0 ),"; 0+ ) " = lim "!0 C 1 (a( 0 ); 0 ),C 1(a( 0 ),"; 0+ ) ", which, since C 12 (a( 0 ), "; 0 ) > 0, is no greaer han lim "!0 C 1 (a( 0 ); 0 ),C 1 (a( 0 ),"; 0 ) " = C 11 (a( 0 ); 0 )=0,andwehave a conradicion Analogous argumens rule ou a nie value for b( 0 ) Finally, he possibiliy ha a( 0 ) = 0 and b( 0 ) is innie is equivalen o he inernally conradicory claim ha for all S>0, max[0;s, K] <C(S; 0 ) <S,ye C 11 (S; 0 )=0 References Arnold, Ludwig, 1992, Sochasic Dierenal Equaions: heory and Evidence (Krieger Publishing Company, Malabar, Fla) Avellaneda, Marco, Arnon Levy, and Anonio Paras, Pricing and hedging derivaive securiies in markes wih uncerain volailiies, Undaed Working Paper, Couran Insiue of Mahemaical Sciences Bick, Avi, 1982, Commens on he valuaion of derivaive asses, Journal of Financial Economics 10, 331{345 Black, Fischer, and Myron Scholes, 1973, he pricing of opions and corporae liabiliies, Journal of Poliical Economy 81, 637{659 Cox, John C, and Sephen A Ross, 1976, A survey of some new resuls in nancial opion pricing heory, Journal of Finance 31, 383{402 Cox, John C, and Mark Rubinsein, 1985, Opions Markes (Prenice Hall, Inc, Englewood Clis, NJ) Dixi, Avinash K, and Rober S Pindyck, 1994, Invesmen under Uncerainy (Princeon Universiy Press, Princeon, NJ) El Karoui, Nicole, Monique Jeanblanc-Picque, and Ravi Visvanahan, On he robusness of Black- Scholes equaion, Undaed Working Paper, Laboraoire de Probabilies, Universie Pierre e Marie Curie El Karoui, Nicole, Monique Jeanblanc-Picque, and Seven E Shreve, 1995, Robusness of he Black and Scholes Formula, Working Paper, Laboraoire de Probabilies, Universie Pierre e Marie Fischer, Edwin O, Rober Heinkel, and Josef Zechner, 1989, Dynamic capial srucure choice: heory and ess, Journal of Finance 44, 19{40 Geske, Rober, 1979, he valuaion of compound opions, Journal of Financial Economics 7, 63{82 37

39 Grundy, Bruce D, 1991, Opion prices and he underlying asse's reurn disribuion, Journal of Finance 46, Jagannahan, Ravi K, 1984, Call opions and he risk of underlying securiies, Journal of Financial Economics 13, 425{434 Karlin, Samuel, and Howard M aylor, 1981, A Second Course in Sochasic Processes (Academic Press, San Diego, Ca) Meron, Rober C, 1973, heory of raional opion pricing, Bell Journal of Economics and Managemen Science 4, 141{183 Meron, Rober C, 1976, Opion pricing when underlying sock reurns are disconinuous, Journal of Financial Economics 3, 125{144 Mohammed, SEA, 1978, Rearded Funcional Dierenial Equaions: A Global Poin of View (Piman Publishing, London, UK) Rohschild, Michael, and Joseph E Sigliz, 1970, Increasing risk I: A deniion, Journal of Economic heory 2, 225{243 Rubinsein, Mark, 1983, Displaced diusion opion pricing, Journal of Finance 38, 213{217 Wiener, Zvi, 1993, Insabiliy wih wo zero frequencies, Journal of Dierenial Equaions 103,

40 s 0 s Figure 1 Illusraion of he `no-crossing' propery he lowes wo pahs are wo sample pahs of he driving Brownian moion he solid risk-neuralized pah drives he wo solid sample pahs of he risk-neuralized price process he solid risk-neuralized pah saring a s 0 says above he pah saring a s 00 unil, upon meeing, hey fuse and merge onward Similarly he dashed Brownian moion drives he wo dashed sample pahs of he risk-neuralized price process hey also exhibi he `no-crossing' propery 39

41 R 0 Ineres rae in economy A;r A () Ineres rae in economy B;r B () +, 2 Figure 2a Ineres rae ime proles in wo oherwise R equivalen economies R A and B One prole is a ime permuaion of he oher wih r A ()d =,R = 2 r B ()d herefore riskless bonds ha promise he same payo a ime have he same price in boh economies H s Region of posiive volailiy possible sample pah for s A Region of zero volailiy deerminisic sample pah for s B + ln(h=s) R +, 2 K s e R, 2 Figure 2b Sample pahs of an underlying asse's price process in economies A and B Boh pahs commence a he same level s a ime, and boh are driven by he same Brownian moion In economy A he price ends above K a ime Due o he dieren ineres rae ime-prole in economy B and he funcional form of he volailiy, he price always nishes below K in economy B 40

42 s 00 s s 0 c bs() (s 00 ;) c bs() (s; ) c(s; ) c bs() (s 0 ;) O A D B C G F Figure 3 Bounds on a call's price and dela he curve OAC (respecively, OBF) is he Black-Scholes call price compued as a funcion of he underlying price using he upper bound, () (lower bound, ()), on he rue volailiy he wo curves bound he rue call price curve ODG he rue hedge raio given a sock price of s is bounded from above by he slope of BC and from below by he slope of AB; boh angens from B o OAC c(s; ) is he ime value of a call opion wrien on a asse worh s c bs() (s; ) is he ime Black-Scholes value of a call opion wrien on an asse worh s when he asse's volailiy is he deerminisic funcion () c bs() (s; ) is similarly dened 41

43 s 0 s 00 Region of zero jump probabiliy Jump possible sample pah 8 >< >: deerminisic sample pah Region of posiive jump probabiliy K s 0 e R r( ) d H Figure 4a wo sample pahs of a non-proporional mixed diusion-jump process for which he `no-crossing' propery is violaed he pah ha sars ou lower has a posiive probabiliy of nishing above K a ime he pah ha sars ou higher always nishes below K A call opion wih exercise price K and ime mauriy will be such ha c(s 0 ;) = 0, ye c(s 00 ;) > 0 K s 000 s 0 s 00,2,1 Figure 4b A non-recombining binomial ree for which he `no-crossing' propery is violaed If he process reaches s 0, i will always nish below K and a call wih an exercise price of K and a ime mauriy will be valueless Bu if he process reaches s 00, he process can nish above K and he call has a posiive price s G 42

44 c bs((s00 )) (s; 0 ) c bs((s0 )) (s; 0 ) c(s; 0 ) s 00 s 0 Ke, R 0 r(!)d! slope=1 s Figure 5a he relaion beween c(s; 0 ) and s in a non-markovian seing he call price funcion is non-monoonic and non-convex c(s; 0 ) is he ime 0 value of a call opion wrien on a asse worh s c bs((s0 )) (s; 0 ) is he ime 0 Black-Scholes value of a call opion wrien on an asse worh s when he asse's volailiy is consan and equal o (s 0 ) c bs((s00 )) (s; 0 )is similarly dened c(s; ) c(s; 0 ) Ke, R r(!)d! Ke, R 0 r(!)d! slope=1 s Figure 5b he relaion beween c(s; ) and in a non-markovian seing In some region of he underlying asse price, he call opion price is greaer a a laer ime 0 han a an earlier ime Unlike in he Black- Scholes world, he call is no uniformly a \wasing" asse 43

45 s y K s σ = 0 θ > 0 s σ = 0 θ = 0 y σ > 0 θ > 0 Figure 6 wo sample pahs of a wo-dimensional diusion for which he `no-crossing' propery is violaed Shown are wo sample pahs of a zero-drif wo-dimensional diusion: a sock price process s, and he process y which drives he sochasic volailiy of s ds = (s ;y ;)s db 1 : dy = (s ;y ;)db 2 : A ime 0, one pah sars a (s 0 ;y 0 ) in he (ligh grey) region where everyhing is deerminisic I mus hen say consan a is saring values, and he sock price componen s can no reach a value greaer han K he oher sample pah sars from (s 00 ;y 0 ) in he (dark grey) region where he y componen is sochasic From hese saring values here is a posiive probabiliy ha he vecor process will reach he (whie) region where boh s and y have posiive volaiies From ha region i is hen possible for s o reach avalue above K herefore, for a call wih an exercise price of K, c(s 0 ;y 0 ; 0) = 0, ye c(s 00 ;y 0 ; 0) > 0 44