Local time and the pricing of time-dependent barrier options

Transcription

1 Finance Soch 21) 14: DOI 1.17/s Local ime and he pricing of ime-dependen barrier opions Aleksandar Mijaović Received: 3 November 27 / Acceped: 29 Augus 28 / Published online: 21 Ocober 28 Springer-Verlag 28 Absrac A ime-dependen double-barrier opion is a derivaive securiy ha delivers he erminal value φs T ) a expiry T if neiher of he coninuous ime-dependen barriers b ± :[,T] R + have been hi during he ime inerval [,T]. Using a probabilisic approach, we obain a decomposiion of he barrier opion price ino he corresponding European opion price minus he barrier premium for a wide class of payoff funcions φ, barrier funcions b ± and linear diffusions S ) [,T ].Weshow ha he barrier premium can be expressed as a sum of inegrals along he barriers b ± of he opion s delas Δ ± :[,T] R a he barriers and ha he pair of funcions Δ +,Δ ) solves a sysem of Volerra inegral equaions of he firs kind. We find a semi-analyic soluion for his sysem in he case of consan double barriers and briefly discus a numerical algorihm for he ime-dependen case. Keywords Time-dependen single- and double-barrier opions Local ime on curves Volerra inegral equaion of he firs kind Dela a he barrier Mahemaics Subjec Classificaion 2) 6H3 45D5 JEL Classificaion G13 C6 1 Inroducion Barrier opions play an imporan role in modern financial markes. They are a less expensive alernaive o European opions and rade in large volumes paricularly in I should like o hank Peros Spanoudakis for poining ou he problem and for simulaing discussion. Thanks for many useful commens are due o Dirk Becherer, Nick Bingham, Johan Tysk and Michalis Zervos. A. Mijaović ) Deparmen of Mahemaics, Imperial College London, Huxley Building, 18 Queen s Gae, London SW7 2AZ, UK a.mijaovic@imperial.ac.uk

2 14 A. Mijaović foreign exchange. A knock-ou double-barrier conrac is nullified if eiher of he wo barriers is breached by he underlying asse price process during he life of he opion, and delivers φs T ), for some predefined payoff funcion φ, oherwise. A knock-in opion becomes a European opion wih payoff φ if one of he barriers is hi by he asse price process before ime T, and expires worhless oherwise. Since simple pariy relaions exis for he prices of knock-in and knock-ou conracs, we shall concenrae only on examining he laer. The main resul of his paper is given by he represenaion formula V,S ) = ϕ,s ) 1 2 e rt T Δ )q S,b ) ) T d Δ + )q S,b + ) ) ) d, 1.1) where V,S ) is he curren ime-dependen barrier opion price, ϕ,s ) is he curren price of he corresponding European payoff and he funcion q is closely relaed o he ransiion densiy of he process S ) see formula 2.7)), which is a local volailiy process given by he sochasic differenial equaion SDE) in 2.1). The funcions Δ,Δ + :[,T] R can be inerpreed as he limiing values of he opion s delas V S, S ) as he asse price process S ) approaches eiher of he wo barriers b ), b + ) a ime see Theorem 2.5 for more deails). Noe ha, since he opion s payoff is nonnegaive, he dela a he lower resp. upper) barrier is posiive resp. negaive) making he barrier premium in he above formula negaive, as one would expec. By Theorems 2.5 and 2.7 he pair of funcions Δ +,Δ ) exiss and solves a sysem of wo Volerra inegral equaions of he firs kind given by 2.11). An imporan feaure of he represenaion 1.1) is ha i can be used for hedging ime-dependen barrier opions. Once he sysem of Volerra inegral equaions in 2.11) is solved numerically or oherwise), he barrier opion price can be obained by compuing he one-dimensional inegral in formula 1.1), which, from a numerical poin of view, can be done very efficienly. Therefore an enire spo-ladder of opion prices i.e., a vecor of values V,S ), where S ranges over a discree subse in some inerval) can be obained wih lile numerical effor, for we are only solving he sysem of Volerra inegral equaions once. Moreover, since he funcion q is available in semi-analyic form in mos models used in pracice, spo-ladders of delas and gammas i.e., vecors wih coordinaes V S,S ) and 2 V,S S 2 ), respecively, where S akes values in a discree inerval), can be found by differeniaing formula 1.1), once we have obained he soluion Δ +,Δ ) of he sysem in 2.11). This feaure of our pricing algorihm is criical for he risk managemen of barrier opion porfolios because spo-ladders are one of he mos imporan ools used by raders for undersanding heir exposure o adverse movemens in he underlying marke. Hedging a down-and-ou call, when he barrier level is below he srike, is no dissimilar o hedging he corresponding European opion, because he presence of he barrier does no desroy he convexiy of he payoff funcion, and hence he dela V S, S ) remains bounded hroughou he life of he opion. In he case of a doublebarrier knock-ou call opion, he siuaion is radically differen since he barrier opion price is nonconvex close o he upper barrier a any ime before expiry. As

3 Local ime and he pricing of ime-dependen barrier opions 15 menioned earlier, he value Δ + ), where Δ + is he funcion in 1.1), is a good approximaion for he dela V S, S ), when S is close o he upper barrier, and can hence be used for hedging. We should sress here ha in pracice he pair of funcions Δ +,Δ ) arises as a soluion of a nonsingular sysem of linear equaions see Sec 3.3 and [41, p. 179, 4.1)]) obained by discreising he Volerra inegral equaions from 2.11), raher han from a numerical scheme for parial differenial equaions, where he firs derivaive is approximaed by a difference quoien ha can be unsable close o he boundary since he value funcion in ha region changes a a very rapid pace. Time-dependen barriers arise naurally in financial markes even if he barriers in he opion s conrac are consan. Le S denoe he foreign exchange rae, and le he funcions R d,r f :[,T] R describe he deerminisic erm srucures of ineres raes in domesic and foreign markes, respecively. The assumpion ha erm srucures of ineres raes are deerminisic is very common in he foreign exchange markes as he majoriy of barrier opion conracs are shor daed wih mauriies up o one year) and have lile dependence on he sochasiciy of ineres raes. The forward process F := S exp R d ) R f ))), which mus be a maringale since i is proporional o an asse divided by he domesic bond, is ofen modelled direcly insead of he FX rae S. I is clear ha he original barrier opion s conrac wih consan barriers ranslaes ino a conrac wih ime-dependen barriers for F. Furhermore, by modelling he forward process direcly we can exend he represenaion in 1.1) o he case of ime-dependen ineres raes. Noe ha he process S ) sudied in Sec. 2 see 2.1)) has a consan drif. Bu if he funcions R d,r f are in C 2 [,T]), as hey usually are since marke paricipans do no like o see kinks in heir erm srucures, we can model he forward F by df = F σf )dw and price he equivalen derivaive wih barriers b± F ) := b ±) exp R d ) R f ))) using 1.1). A feaure frequenly encounered in barrier opion markes is he exisence of disconinuous barriers. The barriers are usually sep funcions or simply sop being acive a a cerain ime before expiry. Since formula 1.1) works for disconinuous payoffs, i can be applied o nonconinuous barriers by backward inegraion. Time seps would in his case be deermined by he inervals of coninuiy of he barriers. The procedure sars a he end of he las such inerval where he payoff is known and uses Theorem 2.7 o deermine he payoff funcion a he beginning of ha inerval. This produces an equivalen problem wih a smaller number of ime inervals, so he same procedure can be reapplied unil we obain he opion value a he curren ime. The key idea behind he proof of Theorem 2.5, which yields he represenaion 1.1), is in some sense analogous o ha used for finding he inegral equaion for he opimal exercise boundary in he American pu problem see Theorem 4.1 in [31] for a survey accoun and [21] for one of he original derivaions). The smooh-fi principle in he American opion problem implies ha he value of he firs derivaive of he opion price a he exercise boundary is known, which allows us o obain a nonlinear inegral equaion for he exercise boundary by applying Iô s lemma and aking expecaions. In he case of barrier opions, he boundary of he region is specified in advance, bu he firs derivaive i.e., he opion s dela) a he barrier is clearly unknown. By judiciously applying Peskir s change-of-variable formula see Appendix A and [35] for more deails) and aking expecaions as in he previous case i is

4 16 A. Mijaović possible o obain a Volerra inegral equaion of he firs kind for he firs derivaive. Unlike wih he American pu opion, where he inegral equaion is nonlinear in he exercise boundary, in he case of barrier opions we obain a linear equaion which, when discreised, yields an upper-riangular linear sysem for he unknown funcion ha can be solved direcly see Sec. 3 and [41] for more deails). The lieraure on coninuously moniored barrier opions is vas and varied. I appears ha wo general approaches have been formed. In he firs one, which mainly deals wih consan barriers, one ries o find a pahwise i.e., robus) hedging sraegy wih European derivaives ha eiher uniquely deermines or provides an admissible range for he barrier opion price. A model-independen approach of his kind is exemplified in [3]. In he case of he Black Scholes model, and more generally for models wih symmeric smiles, his approach has been applied o a number of pah-dependen derivaives including consan double-barrier opions see [4] and [5]). The second approach consiss of calculaing direcly he expecaion in a risk-neural measure of he pah-dependen barrier payoff. A probabilisic approach using Laplace ransforms for consan double-barrier call and pu opions in he Black Scholes model is described in [17]. A mehod using he join densiy of he sock, is maximum and is minimum o find he price of ime-dependen barrier opions in he Black Scholes model was pioneered in [27]. Boundary crossing probabiliies for Brownian moion have been used in [32] o price single-barrier opions where he underlying asse price process has deerminisic ime-dependen drif and volailiy. In [4] i is shown ha he ime-dependen double-barrier opion problem for geomeric Brownian moion can be reduced o he consan barriers case by firs ransforming he sae space and hen ime. A saic hedge using calls and pus for a ime-dependen single-barrier opion is described in [1]. The resul applies o linear diffusions wih compound Poisson jumps, bu he hedging sraegy depends on knowing he values of he barrier conrac one is rying o hedge a cerain imes before expiry. This deficiency was also noed in [22] see p. 16), where a simplified derivaion of he main resul from [1] is given in he case of diffusion processes. More recen work on ime-dependen double-barrier opions for he same kind of asse price process using analyic ools such as Fourier ransforms, Green s funcions and complex inegraion can be found in [11, 18], and [19] and [34], respecively. Specral mehods are applied o find consan double-barrier opion prices in he class of CEV models in [8]. Laplace ransforms and Wiener Hopf facorizaion are used in [24] o obain prices and Greeks for consan barrier opions where he logarihm of he underlying asse price process is a generalised hyper-exponenial Lévy process. This class of processes conains VG, NIG, CGMY and oher models ha are of relevance in finance. Chaper 12 in [29] conains a wealh of analyic mehods for pricing a variey of barrier opions ime-dependen double barriers wih and wihou rebae) in specific modelling frameworks GBM, CEV, Heson) using he heory of parial differenial equaions. A local-ime approach has been pursued for he sudy of he saic superhedging of barrier opions see [26]) and he decomposiion of European opions wih convex payoff funcions see [6]). In his paper we address he quesion of pricing ime-dependen single- and double-barrier opions where he underlying asse price process is a linear diffusion wih mild regulariy condiions on is volailiy funcion. Our approach is enirely

5 Local ime and he pricing of ime-dependen barrier opions 17 probabilisic and combines he wo approaches discussed in he previous paragraph. We do no make use of firs passage ime disribuions, which are prohibiively complicaed in he class of models we are considering. Insead we employ a pahwise analysis of he opion price, which yields he represenaion 1.1). The paper is organised as follows. Secion 2 conains saemens and proofs of our main resuls. In Sec. 3 we propose a semi-analyic soluion using Laplace ransforms) of he sysem of Volerra inegral equaions ha arises in Theorems 2.5 and 2.7 in he case of consan double barriers. We also discuss discreisaion mehods for he general imedependen barrier case. Secion 4 considers briefly some open quesions relaed o our resuls and concludes he paper. 2 Inegral equaions for ime-dependen barrier opions In his secion our goal is o find he inegral equaions ha characerise he delas a he barriers and consequenly he price of any ime-dependen barrier opion. Before defining precisely he class of exoic opions we shall consider, le us specify he underlying model ha provides uncerainy in our economy. The dynamics of he underlying risky securiy are given by a possibly weak soluion in he sense of Definiion in [25]) of he one-dimensional sochasic differenial equaion SDE) S = S + μs u du+ S u σs u )dw u, S, ), 2.1) where he funcion σ : R + R + saisfies σx)> for all x, ) and is locally Lipschiz-coninuous in he inerval, ) i.e., for any compac se C, ), here exiss a posiive consan K C such ha σx) σy) <K C x y for all x,y C). These wo assumpions are he only regulariy condiions applied o he funcion σ hroughou he paper. The consan μ := r δ is he risk-neural drif given by he ineres rae r and he dividend yield δ. The assumpions on σ imply ha he volailiy funcion x xσx) is also locally Lipschiz-coninuous in he inerval, ) bu may vanish a he boundary x =. Under hese hypoheses, Theorem in [25] yields a filered probabiliy space Ω, F ) [,T ], Q) wih a filraion F ) [,T ] ha saisfies he usual condiions, and processes S = S ) [,T ] and W = W ) [,T ] defined on Ω such ha W is a sandard one-dimensional Brownian moion wih respec o F ) [,T ] and he process S solves he SDE 2.1) up o an explosion ime. Furhermore Theorem guaranees he uniqueness in law of he soluion S. For some models given by he SDE 2.1), he soluion S can reach he boundary poin zero of he domain, ) in finie ime wih sricly posiive probabiliy e.g., he CEV process, given by σx) = x ρ 1, can reach zero if he parameer ρ is in he inerval, 1), see[1]). In such cases he absorbing boundary condiion for he process S a zero is assumed in Theorem of [25]. Our aim is o use he measure Q as an equivalen local maringale measure for our economy in he sense of [9]. The absorbing boundary condiion a zero is herefore very naural because any oher boundary behaviour would in general inroduce arbirage an arbirage sraegy would be o buy he asse when i is worh zero and hold i).

6 18 A. Mijaović The soluion S of he SDE 2.1) behaves differenly a he oher boundary poin of is domain. Lemma 2.1 The process S does no reach infiniy in finie ime Q-almos surely. For he precise definiion of explosion a infiniy, see [25, p. 343]. Noe ha Lemma 2.1 implies ha he inegrals in 2.1) are defined on he enire probabiliy space Ω for any fixed ime, because he soluion process S is Q-almos surely finie during he ime inerval [,]. For he proof of Lemma 2.1, see Appendix C. A coninuous ime-dependen barrier b :[,T], ) is by definiion a coninuous funcion of finie variaion. In his paper we shall mainly be concerned wih double-barrier opions. In order o define hem we need wo funcions b ± :[,T], ) which saisfy b ) < b + ) for all [,T]. For any fixed ime s [,T], le he sopping ime τ s be given by τ s := inf { v [,T s]; S s+v R + \ b s + v),b + s + v) )}, 2.2) where R + := [, ). Noe ha by definiion we have s + τ s T, {s + τ s } F for all [,T], and he propery s + τ s + τ holds for s< T. Le [,T] be he curren ime. By definiion he fundamenal price V of he discouned conrac for he barrier opion wih a nonnegaive measurable payoff funcion φ :[, ) [, ) ha sared a ime is given by V = E[φS τ )I {τ =T } F ], where I {τ =T } is he indicaor funcion on Ω see [23, Definiion 7]). Since our marke is complee, by Theorem 6 in [23] see also Theorem 3.3 in [7]), he fundamenal price of a derivaive securiy is he smalles iniial cos of financing a replicaing porfolio of ha securiy. I was shown in [23] see Sec ) ha he marke price of a derivaive securiy equals is fundamenal price when, in addiion o he sandard NFLVR assumpion of [9], we also sipulae he no-dominance assumpion of Meron [3]. No-dominance inuiively says ha, all hings being equal, marke paricipans prefer more o less, and is only violaed if here exiss an agen who is willing o buy a dominaed securiy a a higher price for he mahemaical formulaion of he no-dominance assumpion, see [23, Assumpion 3]). No-dominance is shown o imply ha here are no bubbles in he price of he underlying asse or in he price of a barrier opion ha is dominaed by a call or a pu see [23, Proposiion 1, and Lemma 8 and Theorem 7]). The assumpion is consisen wih a subclass of models given by SDE 2.1), namely hose ha have an equivalen maringale measure. For example in he CEV framework σx)= x ρ 1, where ρ, 1]), which is known o have a unique equivalen maringale measure see [1]), he no-dominance assumpion can be made, and he fundamenal price given by Theorem 2.7 is he marke price. However he no-dominance assumpion canno be made when he discouned asse price process exp μ)s ) [,T ] is a sric local maringale i.e., here is a bubble in he underlying economy), which has been shown o be he case for some of he models in our framework see [3] and [13]). In his case he marke price of he derivaive can exceed he fundamenal price given by Theorem 2.7. In oher words he marke price of he derivaive is sricly larger han he price of he replicaing porfolio, and lile can be said abou

7 Local ime and he pricing of ime-dependen barrier opions 19 is dynamics see [23, Sec , Example 5]). I is no an easy ask o obain general necessary and sufficien condiions for he exisence of an equivalen maringale measure for he soluion of he SDE 2.1), a opic ha meris furher research. In his paper he process V ) denoes he fundamenal price of he barrier opion in he economy given by 2.1), referred o simply as he price in all ha follows. 1 Our aim is o find he price V a any ime [,T] of a ime-dependen doublebarrier conrac iniiaed a ime zero. In order o do his we consider he process Z := E [ φs +τ )I {+τ =T } F ], 2.3) which equals he discouned value of an equivalen ime-dependen barrier conrac iniiaed a ime. Unlike V ), which is a maringale under he pricing measure Q, he process Z ) is no a discouned price process of a securiy in our economy, since a each ime i represens he price of a differen securiy and hence need no be a maringale see Lemma 2.2). This is somewha similar o he well-known observaion in ineres rae heory ha he shor rae i.e., he rae a which funds can be borrowed for an infiniesimal period of ime also known as he insananeous ineres rae) corresponds o a differen asse a each ime and herefore need no saisfy any noarbirage drif resricions. Unlike in he case of he insananeous ineres rae, he drif of Z ) can be deermined uniquely and, as we shall soon see, conains all he informaion needed o obain he curren price of he barrier opion. Before exploring some basic properies of he process Z in he nex lemma, noe ha definiions 2.2) and 2.3) also apply o single-barrier opions wih obvious modificaions. Lemma 2.2 a) Le he imes s, [,T] saisfy s. Then he inequaliy E[Z F s ] Z s holds almos surely in Ω, Q). If eiher he upper barrier b + is presen or he random variable φs T ) is in L 1 Ω, Q), he process Z ) is a nonnegaive submaringale. b) Assume ha he payoff funcion φ : R + R + is coninuous on he complemen of a finie se of poins where i is righ-coninuous wih lef limis and ha, if b + is no presen, he payoff φs T ) is in L 1 Ω, Q). Le he log-normal volailiy σ be locally Lipschiz-coninuous in he inerval, ) and assume ha i saisfies σs)> for all S, ). Then he process Z ) has a coninuous modificaion of he form Z = Z,S ), where he coninuous funcion Z :[,T] R + R + is given by Z,S) := E,S [φs +τ )I {+τ =T }]. Le C := {, S) [,T) R + ; b ) < S < b + ) }, B + := {, S) [,T) R + ; S>b + ) }, B := {, S) [,T) R + ; S<b ) } 1 Thanks are due o he anonymous referee for raising he issue of bubbles and heir implicaions for he pricing of opions.

8 2 A. Mijaović be open subses of he domain [,T) R +. Then Z vanishes on he se B B +, is of order C 1,2 C) and saisfies he parial differenial equaion Z, S) + μsz S, S) + S2 σ 2 S) Z SS, S) = 2 for all, S) C wih erminal condiion ZT,S) = φs)for S b T ), b + T )) and boundary condiions Z,b ± )) = for all [,T]. The same is rue for a single-barrier opion price wih appropriaely modified boundary condiions. From now on we shall assume ha we are working wih he modificaion of he process Z ) given in b) of Lemma 2.2, i.e., we shall assume ha he process Z ) is a coninuous submaringale. Noe also ha he saemen in a) of Lemma 2.2 is inuiively clear. If he underlying asse price is beween he barriers, he process Z ) is a rue maringale up o he firs ime S ) his a barrier, because before ha sopping ime Z equals he discouned barrier opion price. Since Z ) is nonnegaive, if i were a maringale, i would have o say a zero from ha momen onwards. Bu as soon as he sock reurns o he inerval beween he barriers, he process Z ) assumes again a sricly posiive value. Such behaviour makes is mean drif upwards wih ime. We now give a sraighforward bu rigorous proof of his fac. Proof of Lemma 2.2 Pick s, [,T] such ha s<. Noe ha s + τ s + τ for all pahs in Ω, and herefore we have he inclusion {s + τ s = T } { + τ = T } and he ideniy I {s+τs =T } = I {s+τs >}I {+τ =T }. We can now rewrie Z s, using he ower propery and he fac ha {s + τ s >} F,as Z s = E [ E [ ] ] φs +τ )I {s+τs >}I {+τ =T } F F s = E[Z I {τs > s} F s ] E[Z F s ]. The las inequaliy holds because φ, and hence Z, is nonnegaive. If eiher of he wo inegrabiliy condiions in a) of Lemma 2.2 are saisfied, we ge E[Z ] < for all [,T], which implies ha Z ) is a submaringale. This proves a). Par b) in he lemma is a well-known fac abou barrier opions. I suffices o noe ha saemen b) is a special case of Theorem B.1 in Appendix B. Par b) of Lemma 2.2 implies ha he parial derivaive Z S is a coninuous funcion in he open region C, bu he lemma says nohing abou he behaviour of Z S a he boundary of C. A key sep in obaining he inegral represenaion for he double-barrier opion price see 2.1) in Theorem 2.5) will be he applicaion of Theorem A.1 o he funcion Z :[,T] R + R + given in b) of Lemma 2.2.This sep requires a cerain regulariy of he funcion Z and is firs derivaive Z S close o he boundary of C. In principle he limi of he dela of he double-barrier opion price need no exis as he underlying asse S approaches he boundary of he region C. I does no come as a surprise ha addiional hypoheses on he regulariy of he payoff funcion φ : R + R + as well as of he barriers b ± :[,T] R + are required for he funcion Z o saisfy he assumpions of Theorem A.1. Lemma 2.3 gives sufficien condiions for he funcions φ and b ± ha guaranee ha he firs spaial derivaive

9 Local ime and he pricing of ime-dependen barrier opions 21 of he soluion Z of he PDE problem in b) of Lemma 2.2 does no blow up a he boundary of he region C. Lemma 2.3 Le he coninuous barriers b ± :[,T] R + be wice differeniable and assume ha he payoff funcion φ :[b T ), b + T )] R + saisfies φb T )) = φb + T )) = and is wice differeniable wih he second derivaive φ : b T ), b + T )) R which is Hölder-coninuous of order α, 1). If only he lower barrier b is presen, we addiionally assume ha he random variable φs T ) is in L 1 Ω, Q). Then he following hold: a) The limis Δ + ) := lim ɛ Z S,b+ ) ɛ ) and Δ ) := lim ɛ Z S,b ) + ɛ ) exis for all [,T], and his holds uniformly in he sense ha Z S,b ) ± ɛ) converge uniformly o Δ ± on [,T]. b) For some δ>, we have boh sup <ɛ<δ VZ,b + ) ɛ))t ) < and sup <ɛ<δ VZ,b ) + ɛ))t ) <, where V g)t ) denoes he oal variaion of a funcion g :[,T] R. Properies a) and b) hold in he ime-dependen single-barrier case wih obvious modificaions. Lemma 2.3 is a consequence of Schauder s boundary esimaes for parabolic parial differenial equaions. For he proof, see Appendix C. Noe also ha he uniform convergence in he lemma, ogeher wih Theorem B.1, implies ha he dela a he barrier Δ ± ) is a coninuous funcion of ime for all [,T] if he barrier funcions b ± and he payoff φ saisfy he assumpions in Lemma 2.3. This should be conrased wih he known behaviour of he dela of an up-and-ou call opion which goes o minus infiniy if, close o expiry, he underlying asse approaches he barrier level. The ask now is o undersand he pahwise behaviour of he process Z ) [,T ]. For his we need he imporan concep of local ime. Recall ha he local ime a level a R) of any coninuous semimaringale X = X ) [,T ] on he probabiliy space Ω, Q) can be defined as he limi L a 1 X) := lim I [a,a+ɛ) X u )d X, X u ɛ ɛ almos surely in Q see [38, p. 227, Corollary 1.9]), where X, X is he quadraic variaion process as defined in [38, Chap. IV, Theorem 1.3 and Proposiion 1.18]. Noice ha his definiion can be easily exended o a local ime of X along any coninuous curve b :[,T] R wih finie variaion by L b X) := L X b), since he process X b is sill a coninuous semimaringale and he equaliy X b,x b = X, X holds for all. For imes <v T, we denoe he local ime of X beween and v by L a,v X) := La v X) La X). I is well known ha he map L a X) is almos surely a nondecreasing coninuous funcion. Wih his nondecreasing process, one can associae a random measure dl a X) on he inerval

10 22 A. Mijaović [,T] he suppor of which is conained in he se { [,T]; X = a} see [38, p. 222, Proposiion 1.3]). By Lemma 2.2 we know ha he process Z = Z ) [,T ] is a nonnegaive coninuous semimaringale. Therefore we are a libery o apply he Tanaka formula see [38, p. 222, Theorem 1.2]) a level o he nonnegaive process Z ) [,T ], hus obaining he pahwise represenaion Z v = Z + I {Zu >} dz u L,v Z) for <v T. 2.4) Using he represenaion in 2.4), we can prove he following proposiion, which will play a cenral role in all ha follows. Proposiion 2.4 Assume ha he payoff funcion φ and he barriers b ± saisfy he assumpions of Lemma 2.3, and le,v be wo elemens in he inerval [,T] such ha v. If he upper barrier b + is presen, he process I {Zu >} dz u ) v [,T] is a coninuous maringale, and hence he represenaion 2.4) is he Doob Meyer decomposiion of he submaringale Z v ) v [,T]. Therefore we have almos surely he equaliy Z = E[Z v F ] 1 2 E[ L,v Z) F ]. 2.5) Assuming ha φs T ) is in L 2 Ω, Q), he represenaion 2.5) holds also in he case where only he lower barrier b is presen. If ime v equals expiry T and ime equals he curren ime, he equaliy in Proposiion 2.4 yields a represenaion of he barrier opion price a as a sum of he curren value of he European payoff Z T and he expecaion of he local ime from now unil expiry. The former quaniy is usually available in mos models in a semianalyic closed form, and he laer will be obained in Theorem 2.5 by applying he change-of-variable formula from [35]. Noe also ha inuiively he sochasic inegral I {Zu >} dz u ) is a maringale because he inegraor Z u ) equals on he se {Z u > } a discouned double-barrier opion price, which is a maringale. Proof of Proposiion 2.4 Le C denoe he domain beween he barriers as defined in b)oflemma2.2. Recall ha Z v = Zv,S v ), where he funcion Z : C R + is he soluion of he PDE in b) of Lemma 2.2. By Lemma 2.3 we are a libery o apply Theorem A.1 o he funcion Z. In differenial form we obain dz u = I {b u)<s u <b + u)}z S u, S u )S u σs u )dw u + 1 I{Su =b 2 u)}δ u) dl b u S) I {Su =b + u)}δ + u) dl b + u S) ), where Z S denoes he firs derivaive of Z wih respec o S. Definiion 2.3) implies he inclusion {Z u > } {b u) < S u <b + u)} for all u [,T], and herefore I {Zu >} dz u = I {Zu >}Z S u, S u )S u σs u )dw u. 2.6)

11 Local ime and he pricing of ime-dependen barrier opions 23 The funcion Z S is bounded on he domain C by Lemma 2.3, which implies ha he sochasic inegral on he righ-hand side is a coninuous maringale saring a zero. By aking expecaion on boh sides of 2.4) we conclude he proof in he doublebarrier case. However, he same argumen can be applied if only he upper barrier b + is presen. This is because he inegrand on he righ-hand side of 2.6) is sill bounded, making he sochasic inegral in 2.6) a rue maringale. In he single-barrier case wih only b presen, he argumen above does no work because he inegrand in 2.6) is no longer necessarily bounded. However he same reasoning shows ha he ideniy in 2.5) holds for he sopped process Z v τn = Zv τ n,s v τn ), where he sopping ime τ n is he firs passage ime of he diffusion S ino he inerval [n, ) afer ime, because he inegrand in 2.6) is bounded. Jensen s inequaliy for condiional expecaions and he definiion of he process Z given in 2.3) imply he inequaliy max { E [ Zv 2 ] [ ]} [ F, E Z 2 F v τn E φst ) 2 ] F. Our assumpion on φs T ) implies ha Z v and Z v τn are elemens of he space L 2 Ω, Q) for all large naural numbers n. By Lemma 2.1 we have ha lim n τ n is infinie almos surely in Ω. In oher words we have he almos sure pahwise convergence lim n Z v τn = Z v.the Cauchy Schwarz inequaliy implies ha E [ Z v Z v τn ] [ F = E Iτn <v Z v Z v τn ] F E[I τn <v F ] 1/2 E [ Z v Z v τn 2 F ] 1/2 2E[I τn <v F ] 1/2 E [ φs T ) 2 F ] 1/2. Since he sequence E[I τn <v F ] converges o zero Q-almos surely as n goes o infiniy, we obain E[Z v F ]= lim n E[Z v τ n F ] = Z 1 2 lim n E[ L,v τ n Z) F ] = Z 1 2 E[ L,v Z) F ], where he las equaliy follows by he monoone convergence heorem. This concludes he proof of he proposiion. Before we proceed o our main heorem, recall ha, for any poin x, ) and ime,t], he densiy p; x, ) :, ) R + of he ransiion funcion of he underlying asse price process S, given by he SDE in 2.1), is characerised by he ideniy Q x S A) = A p; x,y)dy, where A is any measurable se in, ).The funcion p; x, ) :, ) R + is nonnegaive bu does no necessarily inegrae o one because he process can reach zero and say here) in finie ime. The exisence

12 24 A. Mijaović of p; x,y) can be deduced from [2, Sec. 4.11], where i is shown ha he ransiion funcion of a diffusion is absoluely coninuous wih respec o he speed measure see [2, p. 17], for hedefiniionofhespeedmeasure). In he caseofheprocess S given by 2.1), he speed measure is absoluely coninuous wih respec o Lebesgue measure on he inerval, ), and hence he exisence follows. Furhermore i is proved in [2, p. 149] ha he funcion p ;,y):, ),T] R + saisfies he parabolic PDE in b) of Lemma 2.2 for any y, ). This fac will play a crucial par in he proof of Theorem 2.5 cf. proof of Lemma 2.6). Noe also ha sufficien condiions for he exisence of densiies of soluions of one-dimensional SDEs, which are joinly smooh in all hree variables, are given in [39]. This sronger resul requires a volailiy funcion ha is uniformly bounded away from zero and is herefore no suied o our purpose. For mos models ha are of relevance in mahemaical finance, he densiies can be obained eiher in semi-analyic closed form see for example 2.8) and 2.9)) or numerically. The kernels of he inegral operaors appearing in Theorems 2.5 and 2.7 are relaed o he ransiion funcion of he asse price process S ) [,T ] and will now be specified precisely. The quadraic variaion S,S ) is a coninuous nondecreasing adaped process and as such defines, for each pah of S ), a measure d S,S on he inerval [,T]. Since he asse price process S ) is a soluion of he SDE in 2.1), his measure is absoluely coninuous wih respec o Lebesgue measure on [,T], and he Radon Nikodým derivaive is given by d S,S = S 2σS ) 2 d. The funcion q x, y) ha appears in he kernel of he inegral operaors in Theorems 2.5 and 2.7 can be defined as q x, y) := p; x,y) d S,S d 2.7) S =y, where p; x,y) is he densiy defined above. In he case of geomeric Brownian moionwehaveheformula q x, y) = yσ exp logy/x) μ σ 2 /2)) 2 ) 2π 2σ 2, 2.8) where he drif equals μ = r δ, and σ 2 is he consan variance. The funcion p; x, ) :, ) R + in he case of GBM is a rue probabiliy densiy funcion because he process canno reach zero. In he case of he CEV model, given by 2.1) wih absorbing boundary condiion a zero and he log-normal volailiy funcion σx) = σ x ρ 1 where ρ, 1) and σ, ), we have for he funcion q he closed form expression q x, y) = 2σ 2 y2ρ 1 ρ)k 1/2 2ρ) XY 1 4ρ) 1/4 4ρ) exp X Y)I 1/2 2ρ) 2 XY ). 2.9) This expression is a consequence of 2.7) and he formula for he ransiion densiy p, which can for example be obained from Theorem 3.5 in [1]. The funcion z I α z) is he modified Bessel funcion of he firs kind of order α and he parameers in 2.9)

13 Local ime and he pricing of ime-dependen barrier opions 25 are given by k := 2μ 2σ 2 1 ρ)exp2μ1 ρ)) 1), X := kx 21 ρ) exp2μ1 ρ)), Y := ky 21 ρ), where μ is he drif in he SDE 2.1). We now sae one of our main heorems. Theorem 2.5 Le S ) be he underlying process given by 2.1), and le Z = Z,S ) be he discouned price of a ime-dependen single- or double-barrier opion conrac, saring a he curren ime, given in 2.3). Assume furher ha he barriers b ± :[,T] R + and he payoff φ : R + R + saisfy he assumpions of Lemma 2.3 and ha he local volailiy funcion x σx), x R +, saisfies he assumpions in b) of Lemma 2.2. In he case where only he lower barrier b is presen, we assume in addiion ha he variable φs T ) is in L 2 Ω, Q). Le ϕ,x) := E,x [φs T )] denoe he discouned curren price of he European conrac saring a ime, condiional upon he asse price S being a level x, and le he funcion q x, y) be as in 2.7). Then we have for he ime-dependen double-barrier opion price he inegral represenaion Z,S ) = ϕ,s ) 1 2 T T Δ )q S,b ) ) d + 1 Δ + )q S,b + ) ) d, 2.1) 2 where Δ ± ) is he limiing value of he dela of he double-barrier opion price a b ± ) as defined in a) of Lemma 2.3. Furhermore he coninuous funcions Δ +,Δ :[,T] R saisfy he following linear sysem of wo Volerra inegral equaions of he firs kind: ) ϕ,b+ )) = 1 T ) Δ+ u) Q, u) du, 2.11) ϕ,b )) 2 Δ u) where he marix Q, u), for <u T, is given by ) qu b + ), b + u)) q u b + ), b u)) Q, u) :=. 2.12) q u b ), b + u)) q u b ), b u)) In a ime-dependen up-and-ou resp. down-and-ou) single-barrier case, he represenaion 2.1) conains a single inegral along b + resp. b ). The inegral equaion ha deermines he funcion Δ + resp. Δ ) in he up-and-ou resp. down-andou) case akes he form of he Volerra equaion of he firs kind wih ± equal o + resp. ), i.e., ϕ,b ± ) ) ± 1 2 T q u b± ), b ± u) ) Δ ± u) du =. 2.13)

14 26 A. Mijaović Theorem 2.5 yields an inegral represenaion for he double-barrier opion price for a wide variey of local volailiy models, any pair of ime-dependen barriers and any payoff funcion ha saisfy he assumpions in Lemma 2.3. Raher surprisingly, knowing he values of he dela a he barriers for all fuure imes and he curren price of he corresponding European derivaive recall ha φb T )) = φb + T )) = for payoffs φ saisfying he assumpions in Lemma 2.3), is enough o obain he curren value of he ime-dependen barrier opion. Noe also ha boh inegrals in 2.1)are negaive since Δ ) > resp. Δ + ) < ), which inuiively follows from he fac ha he barrier opion price is increasing resp. decreasing) as he asse price moves away from resp. approaches) he lower resp. upper) barrier. As expeced, his makes he barrier opion cheaper han is European counerpar. The represenaion 2.1) herefore decomposes he double-barrier opion price ino he European opion price and he barrier premium. In order o include payoff funcions φ ha are of ineres in applicaions e.g., he up-and-ou call opion payoff φs) = S K) + I,b+ T ))S) or he payoff of a double-no-ouch φs) = I b T ),b + T ))S)), we mus relax he smoohness requiremens for he funcion φ sipulaed in Lemma 2.3. This will be done in Theorem 2.7, where we show ha he inegral represenaion for he price 2.1) and he inegral equaion 2.11) for he funcions Δ ± coninue o hold. Before proceeding o he proof of Theorem 2.5, we need he following lemma ha bounds he growh of he funcion q, defined in 2.7), over shor ime inervals. Lemma 2.6 Le K be a compac inerval conained in, ). Then here exiss a posiive consan C K such ha he inequaliy holds for all <u T and x,y K. q u x, y) < C K u The proof of Lemma 2.6 is conained in Appendix B. Noe ha he consan C K in Lemma 2.6 depends only on he compac se, and he inequaliy herefore holds uniformly on K. If he funcion σ in he SDE 2.1) were uniformly bounded away from zero, he esimae in Lemma 2.6 would hold on he enire domain, ). Lemma 2.6 implies ha he inegral equaions 2.11) and 2.13) have weakly singular kernels and ha he inequaliies q u b ± ), b ± u)) < M u hold for all u, T ], where M is a posiive consan ± denoes eiher + or ). The linear operaor in 2.13) resp. 2.11)) is compac on he Banach space of coninuous funcions C[,T]) resp. C[,T]) C[,T])) wih he supremum norm and, as such, has in is specrum. Noe ha by consrucion 2.11) and 2.13) have a coninuous soluion. The uniqueness of his soluion is a much more suble quesion, equivalen o asking wheher in he specrum of he operaor is an eigenvalue. Since 2.11) and 2.13) are of he firs kind and he Fredholm alernaive which provides a general answer o he quesion of uniqueness of soluions for inegral equaions of he second kind) canno be used, i is difficul o answer he quesion in general. However, for a ime-dependen single-barrier case in he Black Scholes model, see Proposiion 2.8. Le us now proceed o he proof of Theorem 2.5.

15 Local ime and he pricing of ime-dependen barrier opions 27 Proof of Theorem 2.5 Le us sar by considering a ime-dependen double-barrier opion. Le C be he domain beween he barriers as defined in b) of Lemma 2.2. We begin by applying Theorems A.1 and B.1 o he process Z = Z,S ), where he funcion Z is he soluion of he PDE from b) of Lemma 2.2. For any pair of imes,v [,T] such ha <v, we herefore obain he pahwise represenaion Z v = Z,S ) + I {b u)<s u <b + u)}z S u, S u )S u σs u )dw u + 1 Δ u) dl b u S) 1 Δ + u) dl b + u S), 2 2 where he funcions Δ + and Δ are defined in Lemma 2.3. The random measures dl b ± u S) are by definiion equal o he well-defined random measures dl u S b ±), and he funcions Δ ± are coninuous by Theorem B.1 and a) of Lemma 2.3 and are hence Borel-measurable. Since he funcion Z S : C R + is bounded, his equaliy yields a Doob Meyer decomposiion of he submaringale Z v ) v [,T]. Since such a decomposiion is unique, Proposiion 2.4 implies for he finie variaion processes he ideniy L,v Z) = 1 Δ u) dl b u S) 1 Δ + u) dl b + u S). 2.14) 2 2 The mainideafor heproofoftheorem2.5 iso use heequaliyin Proposiion2.4 o obain he represenaion of he opion price and he inegral equaions in he heorem. We mus herefore find he expecaion E,S [L,v Z)] using he ideniy 2.14). Le us sar by proving he following: Claim. For any coninuous funcion f :[,T] R of finie variaion and for all,v [,T] such ha <v, he equaliy [ ] E,S fu)dl b ± u S) = fu)q u S,b ± u) ) du holds, where q u x, y) isgivenin2.7). Recall ha, since he process S b ± ) [,T ] is a coninuous semimaringale, here exiss a modificaion of he local ime L a,v S b ±) such ha he map a L a,v S b ±) is righ-coninuous and has lef limis for every v [,T] almos surely in Ω. The funcion is herefore Lebesgue-measurable, and he occupaion imes formula see [38, Chap. VI, Corollary 1.6]) implies I [,ɛ) Su b ± u) ) d S,S u = ɛ L a,v S b ±)da for ɛ>. By aking expecaions and dividing by ɛ on boh sides of his equaliy we obain 1 ɛ E [,S I[,ɛ) Su b ± u) ) Su 2 σs u) 2] du = 1 ɛ [ E,S L a ɛ,v S b ± ) ] da. 2.15)

16 28 A. Mijaović b± u)+ɛ The inegrand on he lef-hand side equals 1 ɛ b ± u) q u S,y)dy and in he limi ɛ we obain q u S,b ± u)) for all u, v]. Since for all small ɛ, wehavehe inequaliy q u S,y)< M u for some consan M and y [b ± u), b ± u) + ɛ] see Lemma 2.6), we can apply he bounded convergence heorem o he lef-hand side of 2.15) o obain q u S,b ± u)) du for all values S including S = b ± )). The righ-hand side of 2.15) will converge o E,S [L,v S b ±)] by he fundamenal heorem of calculus if we can show ha he funcion a E,S [L a,v S b ±)] is coninuous a a =. Tanaka s formula yields for he local ime he represenaion 1 2 La,v S b ±) = Ψ,v a) + + I {Su b ± u)+a}s u σs u )dw u I {Su b ± u)+a} μsu b ± u)) du, where Ψ,v a) := a S v b ± v))) + a S b ± ))) + and x) + := max{x,} for any x R see [38, Chap. VI, Theorem 1.2]). Noe ha he variable Ψ,v a) is Lipschiz-coninuous in a wih a Lipschiz consan equal o 1 for all elemens in Ω. By aking expecaion on boh sides we find [ E,S L a,v S b ± ) ] [ = 2E,S Ψ,v a) ] [ ] + 2E,S I {Su b ± u)+a} μsu b ± u)) du, 2.16) since he inegrand in he sochasic inegral is bounded, and hence he maringale erm vanishes in expecaion. The quaniy E,S [Ψ,v a)] is coninuous in a, while he second expecaion on he righ-hand side can be rewrien, using Fubini s heorem, as Fa,u)du, where he funcion F is given by b± u)+a Fa,u):= C u) + μy b ± u) ) pu ; S,y)dy, he funcion p denoes he densiy of he asse price process S in he inerval, ) and C u) := b ± u)q S S u = ), is a funcion independen of a. The esimae pu ; x,y) C u, for a posiive consan C independen of x,y in a compac subseof, ) see Lemma2.6), implieshahefuncion a Fa, u) possesses a parial derivaive ha is bounded in he sense ha F a a, u) D u for all u [,v]. Here D is some posiive consan independen of S. Lagrange s heorem now implies ha he inegral Fa,u)duis a coninuous funcion of a. We have herefore shown ha 2.16) is coninuous in a and hence proved ha he key ideniy E,S [ L b ±,v S) ] = q u S,b ± u) ) du follows from 2.15) upon aking he limi ɛ. For every pah ω in he probabiliy space Ω, he funcion v L b ±,v S)ω) has finie variaion and is coninuous. Since he same is rue for he funcion f in our

17 Local ime and he pricing of ime-dependen barrier opions 29 claim, we can use he inegraion by pars formula o obain he equaliy fu)dl b ± u S) = fv)l b ±,v S) L b ±,u S) df u, where df u is he Radon measure on he inerval [,v] induced by f. By aking expecaions on boh sides of his ideniy and applying Fubini s heorem o he inegral on he righ, which is jusified since local ime is a nonnegaive funcion, we obain he sequence of equaliies [ ] E,S fu)dl b ± u S) = fv)e,s [ L b ±,v S) ] = fv)e,s [ L b ±,v S) ] = fv)e,s [ L b ±,v S) ] = fs)q s S,b ± s) ) ds. E,S [ L b ±,u S) ] df u u df u q s S,b ± s) ) ds fv) fs) ) qs S,b ± s) ) ds The hird equaliy follows by Fubini s heorem, and he las one is a consequence of he formula for he expecaion of local ime. This proves he claim. In order o apply he claim o he ideniy in 2.14), we need o approximae he coninuous funcions Δ ± on he inerval [,v] by sequences of uniformly bounded coninuous funcions f n ± :[,v] R) n N wih finie oal variaion since he funcions Δ ± are bounded on [,v], we can ake piecewise linear approximaions on a uniform grid in [,v]). For each pah ω Ω, he dominaed convergence heorem implies ha he equaliy Δ ± u) dl b ± u S)ω) = lim n f ± n u) dlb ± u S)ω) holds. Since he funcions f ± n are uniformly bounded by some consan K, he random variables f n ± u) dlb ± u S) are bounded by KL b ± S), which is an inegrable random variable. Anoher applicaion of he dominaed convergence heorem and he above claim herefore yield he equaliies [ ] [ ] E,S Δ ± u) dl b ± u S) = lim E,S n f n ± u) dlb ± u S) = Δ ± u)q u S,b ± u) ) du for any pair of imes,v [,T] ha saisfy <v. We can now apply he las equaliy o 2.14) o find he expecaion of he local ime of he ime-dependen double-barrier opion price. In oher words, by Proposiion 2.4 we have for he expecaion of he double-barrier opion price process he represenaion

18 3 A. Mijaović E,S [Z v ]=Z,S ) Δ u)q u S,b u) ) du Δ + u)q u S,b + u) ) du 2.17) for all,v [,T] saisfying <vand all values of S. The represenaion of he double-barrier opion price 2.1) in he heorem can be obained by aking = and v = T in 2.17). The sysem of inegral equaions 2.11) forδ +,Δ ) also follows from formula 2.17) by aking v = T, S = b + ) and S = b ) and observing ha Z,b )) = Z,b + )) = for all [,T], since he double-barrier conrac ha is a he barrier is worh zero by definiion. This complees he proof of he double-barrier case. The single-barrier case can be obained by making sraighforward modificaions o he preceding proof. Our nex ask is o relax he assumpions on he smoohness of he payoff funcion φ : R + R + made in Theorem 2.5. This is crucial, as we should like o be able o apply our mehodology o payoffs ha arise in pracice, such as he up-and-ou call opion payoff φs) = S K) + I,b+ T ))S) or he payoff of a double-no-ouch φs)= I b T ),b + T ))S). The following heorem allows us o do precisely ha. Theorem 2.7 Le φ : R + R + be a payoff funcion ha is coninuous everywhere excep a a finie se of poins where i is righ-coninuous and has lef limis. Assume furher ha he barriers b ± :[,T] R + saisfy he assumpions of Lemma 2.3 and ha he asse price process S ) [,T ] is given by 2.1). In he case where only he lower barrier b is presen, we assume in addiion ha he variable φs T ) is in L 2 Ω, Q). Le Z = Z,S ) be he discouned price of a ime-dependen singleor double-barrier opion conrac, saring a he curren ime given in 2.3), and le ϕ,x) := E,x [φs T )I b T ),b + T ))S T )] denoe he discouned price of he European conrac a he curren ime condiional upon he asse price S being a level x. Then here exis measurable funcions Δ +,Δ :[,T] R, which are in L 1 [,T],m ± d)) and are no necessarily coninuous or bounded, such ha he double-barrier opion price has he inegral represenaion Z,S ) = ϕ,s ) T T Δ )q S,b ) ) d Δ + )q S,b + ) ) d. 2.18) The measure m ± is absoluely coninuous wih respec o Lebesgue measure, and he Radon Nikodým derivaive is given by dm ± d = q b ± ), b ± )). Furhermore he funcions Δ +,Δ saisfy he linear sysem of Volerra inegral equaions of he firs kind given by 2.11). In a ime-dependen up-and-ou resp. down-and-ou) single-barrier case, here exiss a measurable funcion Δ + resp. Δ ), which is in L 1 [,T],m + d)) resp. L 1 [,T],m d))) and is no necessarily coninuous or bounded, such ha he discouned opion price Z,S ) has he inegral represena-

19 Local ime and he pricing of ime-dependen barrier opions 31 ion T Z,S ) = ϕ,s ) ± 1 Δ ± )q S,b ± ) ) d. 2 The inegral equaion saisfied by he funcion Δ + resp. Δ ) akes he form 2.13). A firs glance Theorems 2.5 and 2.7 look similar. The difference lies in he fac ha Theorem 2.7 applies o a much wider class of payoff funcions φ ha do no saisfy he hypohesis of Lemma 2.3 and furhermore invalidae is conclusions. This makes i impossible o apply he key local ime formula from Theorem A.1, which provided he core of he proof of Theorem 2.5. These analyical difficulies will be circumvened by a careful approximaion argumen yielding he exisence of L 1 funcions Δ,Δ +, which saisfy he inegral equaion 2.11) and give he desired represenaion for he ime-dependen barrier opion price. As an illusraion of he difference beween Theorems 2.7 and 2.5, consider he following. I is well known ha he dela of a shor posiion in an up-and-ou call opion becomes arbirarily large if, close o expiry, he asse price approaches he barrier. In paricular his implies ha Δ + canno be bounded close o expiry. A rader rying o hedge his posiion would have o buy unlimied amouns of he underlying asse. Since he gamma of he shor posiion in he up-and-ou call opion is large and posiive close o he barrier, his dela hedge would be very profiable if he barrier were no ouched. However if he barrier were broken, he large accumulaion of he underlying asse would become a huge problem. This is why, in pracice, such a posiion close o expiry would be lef unhedged. Le us now proceed o he proof of Theorem 2.7. Proof of Theorem 2.7 Le E R + be he finie se of disconinuiies of he payoff funcion x φx)i b T ),b + T ))x), which we also denoe by φ for noaional convenience. We sar by consrucing a sequence of funcions φ n : R + R + ha saisfy he assumpions of Lemma 2.3 and have he following wo properies: 1) φ n x) φ n+1 x) for all x R + and n N, and 2) lim n φ n x) = φx) for all x R + \ E. Le p,r E be wo consecuive poins in E such ha p<r. In oher words he funcion φ is coninuous on he inerval [p,r) and has a limi a r. By he Sone Weiersrass heorem for each n N, here exiss an elemen f n C 3 [p,r]) such ha he inequaliies max{φx) n+1 1, } f nx) max{φx) n 1, } hold for all x [p,r). The consrucion implies ha he sequence f n ) n N saisfies propery 1) for all x [p,r] and propery 2) for all x [p,r). The complemen R + \ E consiss of a finie number of open inervals wih he same properies as p, r). For each poin p E, we can choose a decreasing sequence of open inervals Nn p such ha {p}= n=1 Nn p and E Nn p ={p} for all n N, and N p 1 N 1 r = for any r E \{p}. Noe ha on he componens of R + \ E adjacen o any p E, wehave already consruced sequences f n ) n N and g n ) n N of C 3 funcions ha converge o φ in he required way. In he complemen of he neighbourhood Nn p, we define φ n x) equal o eiher f n x) or g n x), depending on x being larger or smaller han p.

20 32 A. Mijaović We can now easily exend φ n o he inerval Nn p so ha he resuling funcion is C 3 and propery 1) remains rue on any neighbourhood of p. Since {p} = n=1 Nn p, propery 2) is also saisfied. Le Z n = Z n, S ) denoe he process given in 2.3) ha corresponds o he payoff funcion φ n. I is clear ha propery 1) implies he inequaliy Z n, S) Z n+1, S) for all poins, S) C, where he region C is defined in b) of Lemma 2.2, and all n N. Since he se E is finie, properies 1) and 2) and he monoone convergence heorem imply he equaliy lim n Z n, S ) = Z,S ), where Z = Z,S ) is given by 2.3). Since φ n saisfies he hypoheses of Lemma 2.3, expression 2.17) can be rewrien as E,S [ φn S T ) ] = Z n,s ) T T Δ n )q S,b ) ) d Δ n + )q S,b + ) ) d 2.19) for all n N and,v [,T] such ha <v. For every n N, he delas a he barriers exis by Lemma 2.3 and are given by Δ n ± ) := lim k Z n S, b ±) ɛ k ), where ɛ k ) k N is a posiive decreasing sequence accumulaing a zero. Noice ha, since Z n, b ± )) = for all [,T], Lagrange s heorem implies he equaliies Δ n Z n, b + ) ɛ k ) + ) = lim, Δ n Z n, b ) + ɛ k ) ) = lim, k ɛ k k ɛ k for each n N. Since he inequaliy Z n, b + ) ɛ k ) Z n+1, b + ) ɛ k ) resp. Z n, b ) + ɛ k ) Z n+1, b ) + ɛ k )) holds for all n N and k N, ifollows ha Δ n + ) Δn+1 + ) resp. Δn ) Δn+1 )) for all [,T]. In oher words he negaive sequence Δ n + )) n N resp. posiive sequence Δ n )) n N) is decreasing resp. increasing) a any ime and hence converges o is infimum resp. supremum), which is no necessarily finie. We can herefore define measurable funcions Δ + :[,T] [, ], Δ :[,T] [, ] by Δ + ) := lim n Δ n + ), Δ ) := lim n Δ n ). By applying he monoone convergence heorem o all he inegrals in 2.19) we obain formula 2.18) in he heorem. Furhermore, formula 2.18) implies ha he inegrals T Δ ±)q S,b ± )) d are finie. Since he funcions Δ +,Δ do no change sign, hey clearly define elemens in L 1 [,T],m + d)), L 1 [,T],m d)), respecively. The sysem of Volerra inegral equaions 2.11) for he funcions Δ +,Δ ) follows in he same way as in he proof of Theorem 2.5. The ime-dependen single-barrier case can be reaed in an analogous way. This complees he proof. We conclude Sec. 2 by considering he uniqueness of he soluion of he Volerra inegral equaion in 2.13) for a ime-dependen single barrier in he Black Scholes model. A much more general resul esablishing he uniqueness of he soluion of he sysem of Volerra inegral equaions of he firs kind given in 2.11), which requires a deailed analysis of he corresponding compac operaors, will be discussed in a subsequen paper.