Option pricing for a partially observed pure jump price process

Transcription

1 Opion pricing for a parially observed pure jump price process Paola Tardelli Deparmen of Elecrical and Informaion Engineering Universiy of L Aquila Via Giovanni Gronchi n. 18, 674 Poggio di Roio (AQ), Ialy ardelli@ing.univaq.i Absrac. Financial asse price movemens are considered, where he risky asse price is a marked poin process. Le is dynamics depend on an underlying even arrivals process which is assumed o be a marked poin process unobserved by he marke agens. Taking ino accoun he presence of caasrophic evens, he possibiliy of common jump imes beween he risky asse price process and he arrivals process is allowed. The equivalen maringale measures are characerized. The arbirage-free pricing of a European coningen claim is idenified as he condiional expecaion wih respec o he observaions under he minimal maringale measure. Keywords. Marked poin processes, Minimal maringale measure, Filering, Pricing under resriced informaion. M.S.C. classificaion. 91B28, 91B7; 6J75; 93E11 1 Inroducion This paper sudies he problem of he arbirage-free pricing of a European coningen claim under he minimal maringale measure in a financial marke where he asses prices are modeled by marke poin processes. In he economic lieraure, sochasic models of financial markes mainly consider coninuous pahs processes, quoe 18 among ohers. Anyway, he form of he real daa, suggess, as in 17, ha he prices are piecewise consan and jump a irregularly spaced random imes in reacion o rades or o significan new informaion. Mahemaical Mehods in Economics and Finance m 2 ef Vol. 5/6, No. 1, 21/11

2 2 Paola Tardelli Inraday informaion on financial asse price quoes and he increasing amoun of sudies on marke microsrucure lead many auhors o believe ha pure jump processes may be more suiable for modeling of he observed price or quaniies relaed o he price, 11, 1, 9, 17, and references herein. Ofen, hey believe ha models ha consider coninuous rajecory processes, even if he presence of jumps are allowed, does no ake ino accoun he discreeness in he daa and could lead o wrong conclusions. In order o describe he amoun of informaion received by he raders relaed o inraday marke aciviy, he aciviy of oher markes, macroeconomics facors or microsrucure rules, some of hese references inroduce exogenous sochasic facors. In all hese cases, he local characerisics of he price process, such as he jump-inensiy and he jump-size disribuion, depend on a laen process whose behaviour is described in differen ways by differen auhors. This is he framework of his paper ha considers a sylized financial marke wih a single risky asse and a bond. All over he paper he price of he risky asse is assumed o be discouned wih respec o he price of he bond and for he sake of simpliciy i will be denoe by price process. In order o link informaions released and he behaviour of rading aciviy, as suggesed by he economic lieraure (as in 4 and 13), his paper assumes ha he local characerisics of he price process depend on he whole hisory of an exogenous marked poin process. Moreover, as in 13, he markovianiy of his processes is assumed, bu as an addiional generalizaion, herein, hey may have common jump imes. This means ha rading aciviy may affec he law of he exogenous process and he possibiliy of caasrophic evens is also allowed. The main ineres of his paper relies on he fac ha he price process has a jump behavior which implies incompleeness of he marke. As a consequence a se may exiss wih an infinie number of risk-neural measures. Chosen a risk-neural probabiliy measure he no-arbirage price of a coningen claim can be defined by he condiional expecaion under his measure, as in he case of complee markes. Many choices have been discussed in he lieraure. Le us quoe he minimal maringale measure, 1, 8, 17, and 18, he mean-variance maringale measure 5 and 19, he minimal enropy maringale measure, 9, 12, and 14. This paper will focus, in paricular, on he minimal maringale measure, inroduced in 8. However, in his noe, recalling ha he risky asse price has a pure jump behavior hen some of he resuls proven in 8 canno be applied. Anyway, sufficien condiions for exisence and uniqueness of he minimal maringale measure are given following 1 and some of is properies are also derived. Finally, he paper deals wih he problem of he valuaion of a coningen claim. Assuming ha he exogenous process canno be observed by he agens,

3 Opion pricing for a parially observed pure jump price process 21 he arbirage-free price of a coningen claim has o be defined. To his end aking ino accoun he discussion performed in 16 and 3, he arbirage-free price of a European coningen claim under resriced informaion is idenified as is condiional expecaion wih respec o he observaions under he minimal maringale measure. Observing ha under suiable assumpions he minimal maringale measure preserves he Markovianiy of he model, following he innovaion mehod he Kushner-Sraonovich equaion for he filer is wrien down. Exisence and uniqueness for is soluion is proven and wih a classical procedure he compuaion of his condiional expecaion wih respec o he observaions reduces o he compuaion of an ordinary expeced value. The paper is organized as follows. In Secion 2 he model and some of is properies are described. Secion 3 is devoed o he characerizaion of he riskneural measures. Recall ha, since he marke is incomplee, his discussion is no rivial. Even if he resul given in 8 canno be used as a consequence of he jump behavior of he price process, in Secion 4 he minimal maringale measure is deermined explicily and some of is properies are derived. Secion 5 is devoed o he evaluaion of a coningen claim, assuming ha he exogenous process is no observable. Las secion is devoed o some conclusions. 2 The Model On a filered probabiliy space, (Ω, F, {F }, P ), where {F } saisfies he usual condiions, le us consider a marke model wih a single risky asse S and a non-risky asse. The price of he risky asse in unis of he numeraire is a process S having he form S = S exp {Y }, wih S IR +. The logreurn price Y is supposed o be an IR-valued marked poin process, wih Y =, characerized by he sequence {τn Y, Y τ Y n Y τ Y n } n, where {τn Y } n, is a nondecreasing nonexplosive sequence of sopping imes. This means ha he jump imes of Y, {τn Y } n, are posiive random variables, such ha {τn Y } F,, and τ Y =, τ Y n < + τ Y n < τ Y n+1, lim n + τ Y n = +. The dynamics of he logreurn price depend on an exogenous marked poin process X, describing arrivals of news o he marke, characerized by he sequence {τ n, X τn X τn } n where again {τ n } n is a nondecreasing nonexplosive sequence of sopping imes. The process X akes values in a finie se X IR, wih iniial condiion X =, and le us assume ha i admis only non-negaive jump sizes. In order o describe he join dynamics of he processes X and Y, le us define N = n 1 {τ Y n }, (1)

4 22 Paola Tardelli he poin process couning he jump imes of Y up o ime. Le N admi a (P, F )-inensiy λ, whose srucure similar o ha given in 4 and 13, is λ = a() + be k Z := λ(, Z ). Thus, λ is a deerminisic measurable funcion of he ime and of he process Z := z + e ks dx s, which is a non homogeneous pure jump process, aking values in a suiable Z IR +, and having he same jump imes of X. The jump sizes of Z are given by Z Z = e k (X X ). The consans b and k are real posiive parameers, z is assumed o be sricly posiive and a( ) is a measurable IR + -valued deerminisic funcion, verifying a() a < +. A more explici expression for λ is given by λ = a() + bz e k + b i (X τi X τi 1 )e k( τ i) 1 {τi }. (2) The laer shows ha λ is sricly posiive and in addiion ha λ is bounded, since for Λ suiable posiive consan and, λ a + bz + bx < Λ < +. (3) There is a naural and inuiive inerpreaion of Equaion (2). When some news reaches he marke a sudden increases in he rading aciviy akes place represened by he posiive jump size of X a a random ime τ n. Successively a progressive normalizaion of he marke occurs wih a speed expressed by k. Finally, he funcion a( ) describes he aciviy of he marke in absence of perurbaions. An adequaely choice of a( ) allows us o ake ino accoun deerminisic feaures like seasonaliies. The consan b compares he effec of he arrivals of he news and ha of he seasonaliies. On one hand, he model proposed in 13 has been generalized by allowing he possibiliy of common jump imes beween he laen process X and he logreurn process Y, hus X and Y are sricly correlaed. On he oher hand he model has been simplified by assuming he following srucure. A firs, le us inroduce he couning processes N 1 = 1 {τ Y n } 1 {Yτ Y Y n τ Y n >} and N 2 = 1 {τ Y n } 1 {Yτ Y Y n τ Y n <}, n n so ha N = N 1 + N 2.

5 Opion pricing for a parially observed pure jump price process 23 Moreover, denoing by {τn X } n {τ n } n he sequence of sopping imes a which X τ X n =/ X τ X n and Y τ X n = Y τ X n, we define N = n 1 {τ X n } and we assume ha N admis a (P, F )-inensiy given by λ := λ (, X, Z ), where λ (, x, z) is a bounded non-negaive measurable funcion, λ (, x, z) Λ. Furhermore, for i = 1, 2, he poin process N i admis a (P, F )-inensiy λ p i, where λ := λ(, Z ), and p i := p i (, X, Z ), i = 1, 2, wih p i (, x, z) sricly posiive measurable funcions such ha p 1 (, x, z) + p 2 (, x, z) = 1. Then, le us assume he exisence of he measurable funcions ξ(, x, z), η 1 (), η 2 (), such ha (a) ξ :, T X Z IR + {}, and for any, T, x X, z Z, x + ξ(, x, z) X, (b) for i = 1, 2, η i :, T IR + is such ha, < η η i () η, for some real consans η and η. Finally, seing ξ := ξ(, X, Z ), he processes involved in his model will be represened as X = Y = Z = z + ξ u dn u + dn u, η 1 (u) dn 1 u η 2 (u) dn 2 u, e ku ξ u dn u + dn u. The discussion performed in 7 abou he srucure of he generaor of he pure jump processes allows us o claim ha he join generaor of he process (X, Y, Z), for f(, x, y, z) belonging o a suiable class of real-valued measurable funcions,, x X, y IR and z Z, is given by Lf(, x, y, z) = f(, x, y, z) + (4) + L f(, x, y, z) + L 1 f(, x, y, z) + L 2 f(, x, y, z), where L f(, x, y, z) = λ (, x, z) f(, x + ξ(, x, z), y, z + e k ξ(, x, z)) f(, x, y, z),

6 24 Paola Tardelli and, for i = 1, 2, L i f(, x, y, z) = λ(, z) p i (, x, z) f(, x + ξ(, x, z), y + ( 1) i 1 η i (), z + e k ξ(, x, z)) f(, x, y, z). Proposiion 1. In he framework of his paper, (X, Y, Z) is a Markov process. Las Proposiion follows by Theorem 7.3 in 7, since he generaor L = L + L 1 + L 2, is bounded. Then he Maringale Problem associaed wih he operaor L and iniial condiion (X =, Y =, Z = z ), is well posed. In addiion, is soluion is a Markov process wih rajecories in D {X Y Z}, T. Lemma 1. The sock price process S is a special (P, F )-local semimaringale wih canonical decomposiion S = S + A S + M S, (5) where A S is a locally bounded variaion predicable process and M S is a locally square inegrable (P, F )-local maringale given by A S = λ u S u e η1(u) p 1 u + e η2(u) p 2 u 1 du, (6) M S = respecively, and < M S > = S u (e η 1(u) 1) dn 1 u λ u p 1 u du + (7) + λ u S 2 u S u (e η 2(u) 1) dn 2 u λ u p 2 u du, (e η1(u) 1) 2 p 1 u + (e η2(u) 1) 2 p 2 u Proof. By Iô formula, he sock price process is such ha S = S + = S + + S u (e η 1(u) 1) dn 1 u + S u (e η 1(u) 1) dn 1 u λ u p 1 u du + S u (e η 2(u) 1) dn 2 u λ u p 2 u du + du. (8) S u (e η 2(u) 1) dn 2 u = (9) S u (e η 1(u) 1) λ u p 1 u du + S u (e η 2(u) 1) λ u p 2 u du.

7 Opion pricing for a parially observed pure jump price process 25 Thus, (6) and (7) follows. Moreover, M S is a locally square inegrable (P, F )- local maringale since λ u S 2 u (e η1(u) 1) 2 p 1 u + (e η2(u) 1) 2 p 2 u For f(y) = S e y and recalling ha du < + P a.s.. < M S > = subsiuing we ge he hesis. Lu (f(y u ) 2 ) 2f(Y u )L u f(y u ) du, 3 Equivalen Maringale Measures From now on, le T be a fixed finie horizon and by a lile abuse of noaions le F := σ{x s, Y s, s }, for T. This secion characerizes he equivalen maringale measures, ha is he se of probabiliy measures Q, equivalen o P, under which S is a (Q, F )-local maringale. The main ool for he characerizaion of he risk-neural measures is a suiable version of he Girsanov Theorem. The choice of he inernal filraion allows us o claim ha, for M (P, F )- local maringale, any probabiliy measure Q equivalen o P is a soluion o he exponenial equaion L = 1 + L s dm s, 6 and 15, whose unique soluion is given by L = e M 1 2 <M c > Π s (1 + M s M s ) e (Ms Ms ). Furhermore, since L = (1+M M ) L, if M is a (P, F )-local maringale such ha M M > 1, hen L is a (P, F )-local maringale and a sricly posiive supermaringale, which is a (P, F )-maringale when IEL T = 1. In his las case, he measure Q defined by he Radon-Nykodim derivaive dq = L T dp FT is a probabiliy measure equivalen o P. On he oher hand, seing p s := λ s/λ s, any (P, F )-local maringale M admis he represenaion M = M + i=,1,2 gs i dn i s λ s p i s ds (1) where gs, i for i =, 1, 2, are (P, F s )-predicable processes. Under he assumpion ha gs i λ i s ds < + P a.s. or IE gs i λ i s ds < +, i=,1,2 i=,1,2

8 26 Paola Tardelli M is a (P, F )-local maringale or a (P, F )-maringale, respecively. In his las case, necessarily, M is uniformly inegrable, since we are working on a finie horizon. Remark 1. By (1), M M = ( ) i=,1,2 gi N i N i. Therefore, if does no coincide wih a jump ime of X or of Y, obviously M M > 1. Oherwise, a any jump ime of he process (X, Y, Z), M M > 1 g i > 1 for i =, 1, 2. (11) In his las case, L T = i=,1,2 exp { T } T log (1 + gs)dn i s i gs i λ s p i s ds (12) is he densiy of Q wih respec o P. Nex, we are going o deermine he semimaringale srucure of he price process under he measure Q. Noe ha under he assumpion i=,1,2 T ( g i + 1 ) λ p i d < + P a.s., (13) he process N i, for i =, 1, 2 admis (Q, F )-inensiy given by ( g i + 1 ) λ p i. Proposiion 2. Under he condiion g i > 1 for i =, 1, 2 and under he condiion (13) he price of he risky asse, S, is a special (Q, F )-local semimaringale, such ha S = S + A Q + M Q where A Q = λ u S u (e η1(u) 1)(gu 1 + 1)p 1 u + (e η2(u) 1)(gu 2 + 1)p 2 u du, and M Q = S u (e η 1(u) 1) dn 1 u (g 1 u + 1)λ u p 1 u du + + S u (e η 2(u) 1) dn 2 u (g 2 u + 1)λ u p 2 u du. The proof is similar o ha of Lemma 1, aking ino accoun he (Q, F )- inensiies of he processes N i for i = 1, 2.

9 Opion pricing for a parially observed pure jump price process 27 Remark 2. Recalling Lemma 1, if A S =, P -a.s., he original measure P is a risk-neural measure. Wih he same procedure, Q is risk-neural if and only if A Q =, P a.s., (14) and M Q is a (Q, F )-local maringale. If λ u S u (e η 1(u) 1)(g 1 u + 1)p 1 u + (1 e η 2(u) ) (g 2 u + 1)p 2 u du < + P a.s. hen M Q is a (Q, F )-local maringale which urns o be a (Q, F )-maringale if } IE λ u S u {(e η1(u) 1)(gu 1 + 1)p 1 u + (1 e η2(u) )(gu 2 + 1)p 2 u du < +. Le us noe ha if he price process is sricly increasing, or sricly decreasing, (14) canno be saisfied and he model does no admi any equivalen maringale measure. Nex, in he se of he risk-neural measures, when i is no empy, one can find an elemen preserving he Markovianiy of he process (X, Y, Z) assuming he exisence of he real valued measurable deerminisic funcions g i (, x, y, z), i =, 1, 2, such ha g i = g i (, X, Y, Z ). (15) In his case, defining L Q f(, x, y, z) = f(, x, y, z) + LQ f(, x, y, z), (16) where, seing λ Q i (, x, z) = λ(, z)p i(, x, z)(1 + g i (, x, y, z)) for i =, 1, 2, L Q f(, x, y, z) = = λ Q (, x, z) f (, x + ξ(, x, z), y, z + e k ξ(, x, z) ) f(, x, y, z) + +λ Q 1 (, x, z) f (, x + ξ(, x, z), y + η 1 (, x, z), z + e k ξ(, x, z) ) f(, x, y, z) + +λ Q 2 (, x, z) f (, x + ξ(, x, z), y η 2 (, x, z), z + e k ξ(, x, z) ) f(, x, y, z), for any bounded, real valued, measurable funcion f, under (13) wih g i given by (15), we have ha he process M Q f () = f(, X, Y, Z ) f(, x,, z ) is a (Q, F )-local maringale. L Q f(s, X s, Y s, Z s )ds The Maringale Problem for he operaor L Q given in (16) and iniial condiion (,,, z ) is well posed since such is he Maringale Problem for he operaor L given in (4) wih he same iniial condiions. This implies ha he

10 28 Paola Tardelli process (X, Y, Z ) is Markovian under he measure Q. We observe ha, when (15) holds rue, he condiion (e η 1(,x,z) 1) 1 + g 1 (, x, y, z) p 1 (, x, z) + (17) provides a sufficien condiion for (14). +(e η2(,x,z) 1) 1 + g 2 (, x, y, z) p 2 (, x, z) = 4 Minimal Maringale Measure The minimal maringale measure P, as observed in 17, was inroduced in 8, o obain hedging sraegies, which are opimal in a suiable sense. In 18, he auhor shows ha he value process can be compued as he condiional expecaion wih respec o P and hen a risk-neural approach o opion valuaion is provided. The classical definiion given in 18 is Definiion 1. An equivalen maringale measure P is called minimal, if each (P, F )-local maringale, R, sricly orhogonal o M S, (7), is a ( P, F )-local maringale. For any iniial probabiliy P, here exiss a mos one minimal maringale measure, (Theorem 2.1, 1). Herein, for his model is exisence is proven and is srucure is described below. The main ool is he following Theorem whose proof can be found in 2, Proposiion 2. We recall his Theorem for sake of compleeness. Theorem 1. Assuming ha here exiss a (P, F )-predicable process c u such ha hen, L := E ( (i) A S = (ii) c u d < M S > u, c u 2 d < M S > u < + P a.s., (iii) 1 c (M S M S ) > P a.s., c u dm S u ) is a (P, F )-sricly posiive local maringale. When L is a (P, F )-maringale, he probabiliy measure P defined by L = d P dp F is he minimal maringale measure. On he oher hand, when P ( 1 c (M S M ) S ) >, he minimal maringale measure does no exis.

11 Opion pricing for a parially observed pure jump price process 29 From now on, aking ino accoun he paricular srucure of he model sudied in his noe, resuls sronger han in a more general seing can be obained. Recalling (6), (7) and (8), (i) implies ha (e η1() 1) p 1 + (e η2() 1) p 2 c =. S (e η 1() 1) 2 p 1 + (e η2(u) 1) 2 p 2 Noe ha c is a predicable process as required. Moreover, Condiion (ii) of Theorem 1 is verified, since λ u is bounded and c u 2 d < M S > u λ u du < + P-a.s.. Condiion (iii) of Theorem 1 is rivially saisfied if is no a jump ime of Y, recalling (7) and Remark 1. Oherwise, if is a jump ime of Y, (iii) becomes (e η1() 1) p 1 + (e η2() 1) p 2 S (e η 1() 1) 2 p 1 + (e η 2() 1) 2 p 2 S (e η1() 1)1 {Y Y >} + (e η2() 1)1 {Y Y <} < 1, ha is always verified. Summing up, L maringale. Seing := E ( M := c u dm S u ) is a (P, F )-sricly posiive local c u dm S u he maringale M has he srucure given in (1), wih ĝ =, ĝ 1 = c S (e η 1() 1), ĝ 2 = c S (e η 2() 1). (18) Proposiion 3. The probabiliy measure P is equivalen o P. I is a riskneural measure and coincides wih he minimal maringale measure. Furhermore P preserves he Markovianiy. Proof. For a real consan K >, depending on η and η, ĝ 1 ĝ 2 K, (19) and, for all, L, he soluion o L = 1 + L s d M s = = 1 + L s ĝs 1 dn 1 s λ s p 1 s ds + L s ĝ 2 s dn 2 s λ s p 2 s ds,

12 3 Paola Tardelli is a sricly posiive supermaringale, 1, 6. Consequenly, 2, IE L = 1, since IE L s ĝs 1 λ s p 1 s ds + L s ĝs 2 λ s p 2 s ds KΛ IE Ls ds < +. The las claim is rue because (15) is a conesquence of (18) and (13) follows by (19) since IE T T (1 + ĝ)λ i p i d IE (λ + λ K)d T Λ(1+K) < + i=,1,2 Under P, he generaor of he Markov process (X, Y, Z) is hen given by LF (, x, y, z) = (2) = F (, x, y, z) + L F (, x, y, z) + L 1 F (, x, y, z) + L 2 F (, x, y, z), where, seing, for i = 1, 2, λ (, X, Z ) = λ (, X, Z ) and λ i (, X, Z ) = λ (1 + ĝ i ) p i, we ge L F (, x, y, z) = L F (, x, y, z) = = λ (, x, z) f(, x + ξ(, x, z), y, z + e k ξ(, x, z)) f(, x, y, z) and Li F (, x, y, z) = λ i (, x, z) F (, x + ξ(, x, z), y + ( 1) i 1 η i (), z + e k ξ(, x, z) ) F (, x, y, z). 5 Pricing Le us consider a European coningen claim wih mauriy T whose payoff is given by H(S T ), referred o as he opion and such ha is a random variable belonging o L 2 (Ω, F S, P ). From now on he exogenous process X is assumed o be unobserved by he marke agens, hence his secion deals wih he problem of pricing, ha is o deermine he value of he payoff a each ime, T in order o avoid arbirage opporuniies, in a parially observed model. Le us define he price of he claim as he expecaion condiioned o he observaions, under a suiable maringale measure. As suggesed by he consideraions developed in 16, here he minimal maringale measure is chosen.

13 Opion pricing for a parially observed pure jump price process 31 In order o compue IE P H(S T ) F Y = IE P IE P H(S T ) F F Y, since P preserves he Markovianiy of he process (X, Y, Z), here exiss a measurable funcion h(, x, y, z) such ha IE P H(S T ) F = h(, X, Y, Z ), (21) where h solves he sysem Lh(, x, y, z) = h(, x, y, z) + L h(, x, y, z) = h(t, x, y, z) = H(S e y ). (22) Nex proposiion shows ha he laer is a linear sysem wih final daa which can be solved by classical recursive mehods. Proposiion 4. Assuming ha H( ) H, he problem (22) admis a unique measurable bounded soluion which is absoluely coninuous wih respec o. Then, for any (x, y, z) and for a.a. here exiss h(, x, y, z) and is bounded. Proof. Le and α(, x, z) := λ (, x, z) + λ 1 (, x, z) + λ 2 (, x, z) = = λ (, x, z) + λ (, z)(1 + ĝ 1 (, x, y, z))p 1 (, x, z) + +λ (, z)(1 + ĝ 2 (, x, y, z))p 2 (, x, z) G(, x, y, z, h) := λ (, x, z) h (, x + ξ(, x, z), y, z + e k ξ(, x, z) ) + + λ 1 (, x, z) h (, x + ξ(, x, z), y + η 1 (, x, z), z + e k ξ(, x, z) ) + + λ 2 (, x, z) h (, x + ξ(, x, z), y η 2 (, x, z), z + e k ξ(, x, z) ), subsiuing (22) can be wrien as h(, x, y, z) α(, x, z)h(, x, y, z) + G(, x, y, z, h) = h(t, x, y, z) = H(S e y ). (23) Consequenly, (23) is equivalen o { h(, x, y, z) = H(S e y ) exp T α(u, x, z)du T { + G(s, x, y, z, h) exp s } + (24) } α(u, x, z)du ds, since, differeniaing boh sides of (24) wih respec o, one can obain an equaion ha, join wih (24) reproduces (23).

14 32 Paola Tardelli The equaion (24) has a unique bounded soluion. In fac, if h 1, h 2 are wo bounded soluions, seing h () = sup h 1 (, x, y, z) h 2 (, x, y, z) x,y,z and recalling (19), we ge ha h () T T α(s, x, z) h (s) ds Λ(3 + 2K) h (s) ds and he asserion follows by a sligh modificaion of he Gronwall Lemma. Finally, he exisence of a bounded soluion absoluely coninuous wih respec o is obained by a classical recursive mehod. Defining, recursively, h (, x, y, z) = H(S e y ) e T and, for j, h j+1 (, x, y, z) = H(S e y ) e T T α(s,x,z)ds + T α(u,x,z)du + H(S e y ) e s α(u,x,z)du α(s, x, z) ds G(s, x, y, z, h j ) e s α(u,x,z)du ds, hen and h 1 h T Λ H (3 + 2K) 2 + T Λ (3 + 2K) h j+1 (, x, y, z) h j (, x, y, z) Λj (3 + 2K) j (T ) j h 1 h j! Λj (3 + 2K) j j! and he conclusion by sandard argumens. T j h 1 h of Thus, he pricing problem reduces o a filering problem, i.e. he compuaion IE P H(S T ) F Y = IE P h(, X, Y, Z ) F Y, and one can deal wih his problem by he classical innovaion mehod. The condiional law of he process (X, Z, Y ), given he σ-algebra F S = F Y, is characerized by inroducing he filer, which is he cadlag version of his condiional law. For any bounded measurable F, he filer, π (F ) = IE P F (, X, Z, Y ) F Y, saisfies a sochasic differenial equaion known as he Kushner-Sraonovich equaion 2. Following a procedure analogous o ha presened in 13, he

15 Opion pricing for a parially observed pure jump price process 33 filering equaion is wrien down and srong uniqueness of is soluions is proven. Recall ha N := N 1 + N 2 Y = and ha he process Y can be represened as η 1 (u) dn 1 u η 2 (u) dn 2 u, hen F Y = F N 1 F N 2. The main resul of his secion is given by Theorem 2 below. Theorem 2. The ( P, F )-inensiy of N i, is given by he process λ i = λ i (, X, Z ), i = 1, 2. For any bounded measurable funcion F defined on X IR Z, he Kushner- Sraonovich equaion can be wrien as ) π (F ) = π (F ) + π s ( Ls F (s,, Y s, ) ds + (25) where + i=1,2 π s ( λi (s,, ) ) + Ψ i s (F ) ( ) dns i π s ( λi (s,, ) ds ) Ψ i s (F ) = π s ( λi (s,, )F ) π s ( λi (s,, ) ) π s (F ) + π s ( Li s F (s,, Y s, ) ), and a + := 1 a 1 {a>}. Moreover, he filering equaion has a unique srong soluion. Proof. The firs claim is obained by aking ino accoun he join dynamics of X, Y, Z given in (2). By applying he classical innovaion mehod as described in 2 he Kushner- Sraonovich equaion is wrien down. In paricular, he las erm in Ψ i s (F ), which arises when common jump imes beween he sae and he observaions are allowed, is relaed wih < F (X, Y, Z), N i >, i = 1, 2. As far as he srong uniqueness of he soluions of (25), observe ha a any jump ime = τ Y k, he filer is uniquely deermined by he knowledge of π. In fac, for Y Y >, π (λ 1 (,, ))=/, or for Y Y <, π (λ 2 (,, ))=/, and π (F ) = π (F ) + +π s ( λ1 (s,, ) ) + Ψ 1 (F )1I {Y Y >} + π s ( λ2 (s,, ) ) + Ψ 2 (F )1I {Y Y <} = = π s ( λ1 (s,, )F + L 1 sf (s,, Y s, ) ) π s ( λ1 (s,, ) ) 1I {Y Y >} + + π s ( λ2 (s,, )F + L 2 sf (s,, Y s, ) ) π s ( λ2 (s,, ) ) 1I {Y Y <}.

16 34 Paola Tardelli For τ Y k, τ Y k+1), he behaviour of he filer is defined by he equaion π (F ) = π τ Y (F ) + k τ Y k {π s (L sf ) + π s ( λ(s, ))π s (F ) π s ( λ(s, )F )}ds, (26) where λ(, X, Z ) = λ(, Z ) 1 + ĝ 1 p 1 + ĝ 2 p 2. For any wo soluions π 1 and π 2 of (26), such ha π 1 (F ) = π 2 (F ), here exiss τ Y τ Y k k a suiable posiive consan C depending on F = sup,x,y,z F (, x, y, z), such ha π 1 (F ) π 2 (F ) C πs 1 πs 2 ds, τ Y k where denoes he bounded variaion norm of he signed measure π 1 s π 2 s. The las inequaliy guaranees uniqueness for τ Y k, τ Y k+1) since (26) is Lipschiz wih respec o he bounded variaion norm and he hesis follows by inducion. Finally a represenaion for he filer via a classical linearized mehod can be performed, as for example in 13, showing ha he compuaion of he filer beween wo consecuive jump imes can be reduced o he evaluaion of an ordinary expecaion. More precisely, for τ Y k, τ Y k+1), π (F ) = X Z { IE s,x,z F (X.Z ) exp X Z { IE s,x,z exp s s } λ(u, X u, Z u ) du } λ(u, X u, Z u ) du s=τ Y k s=τ Y k π τ Y (dx, dz) k π τ Y (dx, dz) k where P s,x,z denoes he law of he process (X, Z), whose dynamics is defined by he operaor L wih iniial condiion (s, x, z). Technical deails on his procedure can be found in 13 and references herein. 6 Conclusions This paper sudies a financial model in which he asse prices have a jump behavior depending on an exogenous sochasic facor, supposed unobservable. The marke is incomplee and an infinie number of risk-neural measures may exis. The conribuion of his paper is wofold. Firs aim is he characerizaion of he equivalen maringale measures and in paricular of he minimal maringale measure. Then, given an European coningen claim, he value of he payoff a

17 Opion pricing for a parially observed pure jump price process 35 each ime is deermined in order o avoid arbirage opporuniies, under he minimal maringale measure. The firs aim requires a change of probabiliy measure obained by a Girsanov ype change of measure. This procedure characerizes he class of he risk-neural measures. The dynamics of he price process guaranee ha his class is no empy. By Proposiion 2 in 2, he minimal maringale measure is given and in our seing we ge an explici expression of i. For he second aim, following he exising lieraure, he problem reduces o evaluae he expecaion condiioned o he observaions, under a suiable maringale measure, which is chosen o be he minimal maringale measure. This bring us o a filering problem for which an explici soluion is given. References 1. Ansel, J.P., Sricker, C.: Unicié e exisence de la loi minimale. Séminaire de Probabiliés, Srasbourg 27 (1993) Brémaud, P.: Poin Processes and Queues, Springer-Verlag, New York (198) 3. Ceci, C., Gerardi, A.: Pricing for geomeric marked poin processes under parial informaion: enropy approach. Technical Repor R Diparimeno di Scienze of he Universià di Chiei-Pescara, Pescara (26) Downloadable from hp:// 4. Cenanni, S., Minozzo, M.: Esimaion and filering by reversible jump MCMC for a doubly sochasic Poisson model for ulra-high-frequency financial daa. Proceedings of he second Workshop on Correlaed Daa Modeling. Franco Angeli, Milano (24) To appear in: Saisical Modeling 5. Delbaen, F., Schachermayer, M.: The variance-opimal maringale measure for coninuous processes. Bernoulli 2 (1996) Doléans-Dade, C.: Quelques applicaions de la formule de changemen de variables pour le semimaringales. Zeischrif fur Wahrscheinlichkeisheorie und verwande Gebiee 16 (197) Ehier, S.N., Kurz, T.G.: Markov Processes: Characerizaion and Convergence. J. Wiley and Sons, Hoboken (1986) 8. Foellmer, H., Schweizer, M.: Hedging of coningen claims under incomplee informaions. In: Davis, M.H.A., Ellio, R.J. (eds.): Applied Sochasic Analysis, Gordon & Breach, New York (1991) Frey R.: Risk minimizaion wih incomplee informaion in a model for high frequency daa. Mahemaical Finance 1 (2) Frey R., Runggaldier W.J.: A nonlinear filering approach o volailiy esimaion wih a view owards high frequency daa. Inernaional Journal of Theoreical and Applied Finance 4 (21) Frey, R., Runggaldier, W.J.: Risk-minimizing hedging sraegies under resriced informaion: he case of sochasic volailiy models observable only a discree random imes. Mahemaical Mehods of Operaion Research 5 (1999) Frielli, M.: The minimal enropy maringale measure and he valuaion problem in incomplee marke. Mahemaical Finance 1 (2) Gerardi, A., Tardelli, P.: Filering on a parially observed ulra-high-frequency daa model. Aca Applicandae Mahemaicae 91 (26)

18 36 Paola Tardelli 14. Grandis, P., Rheinlander, T.: On he minimal enropy maringale measures. The Annals of Probabiliy 3 (22) Jacod, J.: Mulivariae poin processes: predicable projecion, Radon-Nikodym derivaives, represenaion of maringales. Zeischrif fur Wahrscheinlichkeisheorie und verwande Gebiee 31 (1975) Musiela, M., Rukowski, M.: Maringale Mehods in Financial Modeling. Sochasic Modelling and Applied Probabiliy, Vol. 36. Springer, Berlin (1997) 17. Prigen, J.L.: Opion pricing wih a general marked poin process. Mahemaics of Operaion Research 26 (21) Schweizer, M.: A guided our hrough quadraic hedging approaches. In: Jouini, E., Civianovic, J., Musiela, M. (eds.): Opion Pricing, Ineres Raes and Risk Managemen, Cambridge Universiy Press, Cambridge (21) Schweizer, M.: Approximaion pricing and he variance-opimal maringale measure. Annals of Probabiliy 24 (1996) Schweizer, M.: On he minimal maringale measure and he Follmer-Schweizer decomposiion. Sochasic Analysis and Applicaions 13 (1995) Mahemaical Mehods in Economics and Finance m 2 ef Vol. 5/6, No. 1, 21/11 ISSN prin ediion: ISSN online ediion: Web page: hp:// m2ef@unive.i