FIRST PASSAGE TIMES OF A JUMP DIFFUSION PROCESS


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1 Adv. Appl. Prob. 35, Prined in Norhern Ireland Applied Probabiliy Trus 23 FIRST PASSAGE TIMES OF A JUMP DIFFUSION PROCESS S. G. KOU, Columbia Universiy HUI WANG, Brown Universiy Absrac This paper sudies he firs passage imes o fla boundaries for a double exponenial jump diffusion process, which consiss of a coninuous par driven by a Brownian moion and a jump par wih jump sizes having a double exponenial disribuion. Explici soluions of he Laplace ransforms, of boh he disribuion of he firs passage imes and he join disribuion of he process and is running maxima, are obained. Because of he overshoo problems associaed wih general jump diffusion processes, he double exponenial jump diffusion process offers a rare case in which analyical soluions for he firs passage imes are feasible. In addiion, i leads o several ineresing probabilisic resuls. Numerical examples are also given. The finance applicaions include pricing barrier and lookback opions. Keywords: Renewal heory; maringale; differenial equaion; inegral equaion; infiniesimal generaor; markedpoin process; Lévy process; Gaver Sehfes algorihm AMS 2 Subjec Classificaion: Primary 6J75; 44A1 Secondary 6J27 1. Inroducion 1.1. Background Jump diffusion processes are processes of he form N X = σw + µ + Y i ; X. 1.1 i=1 Here {W ; } is a sandard Brownian moion wih W =, {N ; } is a Poisson process wih rae λ, consans µ and σ > are he drif and volailiy of he diffusion par respecively, andhe jump sizes {Y 1,Y 2,...} are independen and idenically disribued i.i.d. random variables. We also assume ha he random processes {W ; }, {N ; }, and random variables {Y 1,Y 2,...} are independen. Noe ha he jump par, N i=1 Y i, is a special case of he socalledmarkedpoin processes; furher backgroundon markedpoin processes can be found, for example, in [9], [15]. The processes in 1.1 have been given differen names in he lieraure, andhey are indeedspecial cases of Lévy processes; see e.g. [6], [26] for backgroundon Lévy processes. The jump diffusion processes are widely used, for example, in finance o model asse sock, bond, currency, ec. prices. Two examples are he normal jump diffusion process where Y Received17 April 22; revision received2 November 22. Posal address: Deparmen of IEOR, Columbia Universiy, New York, NY 127, USA. address: Posal address: Division of Applied Mahemaics, Brown Universiy, Box F, Providence, RI 2912, USA. 54
2 Firs passage imes ofa jump diffusion process 55 has a normal disribuion e.g. [2] and he double exponenial jump diffusion process where Y has a double exponenial disribuion e.g. [18], i.e. he common densiy of Y is given by f Y y = p η 1 e η 1y 1 {y } + q η 2 e η 2y 1 {y<}, where p, q are consans, p + q = 1, and η 1,η 2 >. Noe ha he means of he wo exponenial disribuions are 1/η 1 and1/η 2 respecively. This paper focuses on he firs passage imes of he double exponenial jump diffusion process: τ b := inf{ ; X b}, b >, where X τb := lim sup X on he se {τ b = }. The main problems sudied include he disribuion of he firs passage ime Pτ b = P max X s b 1.2 s for all >, he join disribuion beween he firs passage ime and he erminal value Pτ b, X a, 1.3 andoher relaedquaniies. There are hree reasons why hese problems are ineresing. Firs, from a purely probabilisic poin of view, he double exponenial jump diffusion process offers a rare case in which analyical soluions of he firs passage imes are feasible. Because of he jump par, when a jump diffusion process crosses he boundary level b, someimes i may incur an overshoo over he boundary. In general, he disribuion of he overshoo is no known analyically, hus making i impossible o ge closedform soluions of he disribuion of he firs passage imes. However, if he jump sizes, he Y i, have an exponenialype disribuion, hen he overshoo problems can be solved analyically, hanks o he memoryless propery associaedwih he exponenial disribuion. See [27, Chaper 8] and [29] for some deailed discussions on overshoo problems, including he ladder variables for wosided disribuions which make he overshoo problems more complicaed. Second, he sudy leads o several ineresing probabilisic resuls. i Alhough he exponenial random variables have memoryless properies, he firs passage imes and he overshoo are dependen, despie he fac ha he wo are condiionally independen given ha he overshoo is bigger han. ii The renewalype inegral equaions, which are usedfrequenly in sudying firs passage imes, may no leado unique soluions for he problems, because he boundary condiions are difficul o deermine; see Secion 3.3. Insead, our approach based on differenial equaions andmaringales can circumven his problem of uniqueness. Third, from he applied probabiliy poin of view, he resuls of his paper are useful in opion pricing. Brownian moion and he normal disribuion have been widely used, for example, in he Black Scholes opion pricing framework, o sudy he reurn of asses. However, wo empirical puzzles have recenly receiveda grea deal of aenion, namely he lepokuric feaure, meaning ha he reurn disribuion of asses may have a higher peak and wo asymmeric heavier ails han hose of he normal disribuion, andan abnormaliy called volailiy smile in opion pricing. Many sudies have been conduced o modify he Black Scholes models in order o explain he wo puzzles; see he exbooks [1], [13] for more deails. An immediae problem wih hese differen models is ha i may be difficul o obain analyical soluions for he prices of opions, especially for hose of he popular pahdependen opions, such as
3 56 S. G. KOU AND H. WANG barrier andlookback opions. To ge analyically racable models, ando incorporae boh he lepokuric feaure and he volailiy smile, he double exponenial jump diffusion model is proposed in [18]; see also [12] for pricing of ineres rae derivaives under such a model and more background abou general jump diffusion models. The explici calculaion of 1.2 and1.3 or, more precisely, heir Laplace ransforms, can be used o ge closedform soluions for pricing barrier and lookback opions under he double exponenial jump diffusion model. The deails of is finance applicaions, being oo long o be includedhere, are reporedin [19]. Using a Laplace inverse algorihm he Gaver Sehfes algorihm, boh 1.2 and1.3 can hen be compuedvery quickly Inuiion Wihou he jump par, he process simply becomes a Brownian moion wih drif µ. These disribuions of he firs passage imes can be obained eiher by a combinaion of a change of measure Girsanov heorem andhe reflecion principle, or by calculaing he Laplace ransforms via some appropriae maringales andhe opional sampling heorem. Deails of boh mehods can be found in many classical exbooks on sochasic analysis, e.g. [16], [17]. Wih he jump par, however, i is very difficul o sudy he firs passage imes for general jump diffusion processes. When a jump diffusion process crosses boundary level b, someimes i his he boundary exacly and someimes i incurs an overshoo, X τb b, over he boundary. The overshoo presens several problems if we wan o compue he disribuion of he firs passage imes analyically. i We needo ge he exac disribuion of he overshoo, X τb b; paricularly, PX τb b = andpx τb b>xfor x>. This is possible if he jump size Y has an exponenialype disribuion, hanks o he memoryless propery of he exponenial disribuion. ii We need o know he dependen srucure beween he overshoo, X τb b, andhe firs passage ime τ b. Given ha he overshoo is bigger han, he wo random variables are condiionally independen and X τb b is condiionally memoryless if he jump size Y has an exponenialype disribuion. This condiionally independen and condiionally memoryless srucure seems o be peculiar o he exponenial disribuion, and does no holdfor general disribuions. iii If we wan o use he reflecion principle o sudy he firs passage imes, he dependen srucure beween he overshoo andhe erminal value X is also needed. To he bes of he auhors knowledge, his is no known even for he double exponenial jump diffusion process. Consequenly, we can derive closedform soluions for he Laplace ransforms of he firs passage imes for he double exponenial jump diffusion process, ye canno give more explici calculaions beyondha, as he correlaion beween X and X τb b is no available. However, for oher jump diffusion processes, even analyical forms of he Laplace ransforms seem o be quie difficul, if no impossible, o obain. To compue he Laplace ransform of Pτ b, we use boh maringale anddifferenial equaions. There are wo oher possible approaches: renewalype inegral equaions and Wiener Hopf facorizaion. Renewalype inegral equaions are frequenly usedin he acuarial science lieraure see, for example, [11] andreferences herein o sudy firs passage imes. However, he renewal equaion does no lead o a unique soluion see Secion 3.3 for deails, andhus wouldno solve he problem in our case.
4 Firs passage imes ofa jump diffusion process 57 Wiener Hopf facorizaion andrelaedflucuaion ideniies [6], [7], [21], [25], [26] have also been widely used o sudy he firs passage imes for Lévy processes noe ha he double exponenial jump diffusion process is a special case of Lévy processes. Many such sudies focus on onesided jumps; e.g. [24]. However, because of he onesided jumps, he overshoo problems are avoided, as eiher he jumps are in he opposie direcion o he barrier crossing or here is no ladder variable problem for he onesided jumps. The Wiener Hopf facorizaion for general jump diffusion processes wih wosided jumps is discussed in [8]. In general, explici calculaion of he Wiener Hopf facorizaion is difficul. Because of he special srucure of he exponenial disribuion, especially due o is memoryless propery, we can solve he firs passage ime problems explicily; in some sense his also suggess, hough indirecly, ha he Wiener Hopf facorizaion couldbe performedexplicily in he case of double exponenial jump diffusion processes. An ouline of he paper is as follows. Secion 2 gives some preliminary resuls. Secion 3 presens he compuaion of he Laplace ransform of he firs passage imes, is immediae corollaries, andis connecion wih he inegral equaion approach. The join disribuion of he jump diffusion process and is running maxima is considered in Secion 4. Inversion of Laplace ransforms andnumerical examples are given in Secion 5. Some proofs andechnical deails are deferred o appendices o ease exposiion. 2. Preliminary resuls The infiniesimal generaor of he jump diffusion process 1.1 is given by Lux = 1 2 σ 2 u x + µu x + λ [ux + y ux]f Y y dy for all wice coninuously differeniable funcions ux. In addiion, suppose ha θ η 2,η 1. The momen generaing funcion of jump size Y is given by E[e θy ]= pη 1 η 1 θ + qη 2 η 2 + θ, from which he momen generaing funcion of X can be obainedas φθ, := E[e θx ]=exp{gθ}, where he funcion G is defined as Gx := xµ x2 σ 2 pη1 + λ η 1 x + qη 2 η 2 + x 1. Lemma 2.1. The equaion Gx = α for all α> has exacly four roos: β 1,α, β 2,α, β 3,α, β 4,α, where <β 1,α <η 1 <β 2,α <, <β 3,α <η 2 <β 4,α <. In addiion, le he overall drif of he jump diffusion process be p ū := µ + λ q. η 1 η 2
5 58 S. G. KOU AND H. WANG Then, as α, β 1,α { if ū, β1 if ū<, and β 2,α β 2, where β1 and β 2 are defined as he unique roos Gβ 1 =, Gβ 2 =, <β 1 <η 1 <β 2 <. Proof. Since Gβ is a convex funcion on he inerval η 2,η 1 wih G = λp+q 1 = and Gη 1 =+, G η 2 + =+, here is exacly one roo β 1,α for Gx = α on he inerval,η 1, andanoher one on he inerval η 2,. Furhermore, since Gη 1 + = and G+ =, here is a leas one roo on η 1,. Similarly, here is a leas one roo on, η 2,asG = and G η 2 =. Bu he equaion Gβ = α is acually a polynomial equaion wih degree four; herefore, i can have a mos four real roos. I follows ha, on each inerval,, η 2 and η 1,, here is exacly one roo. The limiing resuls when α follow easily once we noe ha G =ū. The following resul shows ha he memoryless propery of he random walk of exponenial random variables leads o he condiional memoryless propery of he jump diffusion process. Proposiion 2.1. Condiional memorylessness and condiional independence. x>, For any Pτ b,x τb b x = e η1x Pτ b,x τb b>, 2.1 PX τb b x X τb b> = e η1x. 2.2 Furhermore, condiional on X τb b>, he sopping ime τ b and he overshoo X τb b are independen; more precisely, for any x>, Pτ b,x τb b x X τb b> = Pτ b X τb b> PX τb b x X τb b>. 2.3 Proof. We only need o show ha 2.1 holds. The equaliy 2.2 follows readily by leing andobserving ha, on he se {X τb >b}, he hiing ime τ b is finie by definiion; and 2.3 also holds since Pτ b,x τb b x X τb b> = Pτ b,x τb b x PX τb b> = e η 1x Pτ b,x τb b> PX τb b> = PX τb b x X τb b> Pτ b X τb b>. Denoe by T 1,T 2,... he arrival imes of he Poisson process N. I follows ha Pτ b,x τb b x = PT n = τ b,x Tn b x =: P n, n=1 n=1
6 Firs passage imes ofa jump diffusion process 59 as he overshoo inside he probabiliy can only occur during he arrival imes of he Poisson process because x>. However, wih X = σw + µ, wehave P n = P max X s <b,x Tn b + x,t n s<t n I follows ha = E{PX Tn b + x F Tn,T n 1 {max s<tn X s <b,t n }} = E{p exp{ η 1 b + x X T n Y 1 Y n 1 } 1 {max s<tn X s <b,t n }} = e η1x E{p exp{ η 1 b X T n Y 1 Y n 1 } 1 {max s<tn X s <b,t n }} = e η1x E{PX Tn >b F Tn,T n 1 {max s<tn X s <b,t n }} = e η1x P max X s <b,x Tn >b,t n s<t n = e η 1x PX Tn b>,t n = τ b. Pτ b,x τb b x = e η1x PT n = τ b,x τb b> n=1 = e η 1x Pτ b,x τb b>. This complees he proof. Remark 2.1. The condiional independence hough no he condiional memorylessness in Proposiion 2.1 acually holds wih greaer generaliy; see, for example, [14], [28] and heir furher generalizaion [22]. I is easy o verify from Proposiion 2.1 ha, for any x>, he following equaliies hold: PX τb b x = e η1x PX τb b>, Ee ατ b 1 {Xτb b+x} = e η1x Ee ατ b 1 {Xτb b>}. 3. Disribuion of he firs passage imes 3.1. The Laplace ransforms Theorem 3.1. For any α,, le β 1,α and β 2,α be he only wo posiive roos ofhe equaion α = Gβ, where <β 1,α <η 1 <β 2,α <. Then we have he following resuls concerning he Laplace ransforms of τ b and X τb : E[e ατ b ]= η 1 β 1,α β 2,α e bβ 1,α + β 2,α η 1 β 1,α e bβ 2,α, 3.1 η 1 β 2,α β 1,α η 1 β 2,α β 1,α E[e ατ b 1 {Xτb b>y}] =e η 1y η 1 β 1,α β 2,α η 1 [e bβ 1,α e bβ 2,α ] for all y, η 1 β 2,α β 1,α 3.2 E[e ατ b 1 {Xτb =b}] = η 1 β 1,α β 2,α β 1,α e bβ 1,α + β 2,α η 1 β 2,α β 1,α e bβ 2,α. 3.3
7 51 S. G. KOU AND H. WANG Proof. Here we focus on he proof for 3.1 and3.2 since 3.3 follows immediaely by aking he difference of 3.1 and3.2 andby leing y =. For noaional simpliciy, we shall wrie β i β i,α, i = 1, 2. We firs prove 3.1. For any fixedlevel b>, define he funcion u o be { 1, x b, ux := A 1 e β1b x + B 1 e β2b x, x < b, where A 1 and B 1 are defined o be he wo coefficiens in fron of he exponenial erms in 3.1. Clearly, ux 1 for all x,, because β 1, β 2. Noe ha, on he se {τ b < }, ux τb = 1 since A 1 + B 1 = 1. Furhermore, he funcion u is coninuous. Subsiuing his form of u anddoing he inegraion in wo regions, = b x + b x, we have, afer some algebra, ha, for all x<b, αu + Lu is equal o A 1 e b xβ 1 fβ 1 +B 1 e b xβ 2 fβ 2 λpe η 1b x wherefβ:= Gβ α. Since A 1 η 1 η 1 β 1 +B 1 η 1 η 1 β 2 1 fβ 1 = fβ 2 =, A 1 η 1 η 1 β 1 + B 1 η 1 η 1 β 2 1 =,, 3.4 we have αux + Lux = for all x<b. 3.5 Because he funcion ux is coninuous, bu no C 1 a x = b, we canno apply Iô s formula direcly o he process {e α ux ; }. However, i is no difficul o see ha here exiss a sequence of funcions {u n x; n = 1, 2,...} such ha: i u n x is smooh everywhere, and in paricular i belongs o C 2 ; ii u n x = ux for all x b; iii u n x = 1 = ux for all x b + 1/n;iv u n x 2 for all x and n. Clearly, u n x ux for all x. I follows from a sraighforwardcalculaion ha, for x<b, Lu n x = 1 2 σ 2 u n x + µu n x + λ [u n x + y u n x]f Y y dy = 1 2 σ 2 u n x + µu n x λu nx + λ = 1 2 σ 2 u x + µu x λux + λ + λ b x+1/n b x = αux + λ b x u n x + yf Y y dy + λ b x+1/n b x u n x + yf Y y dy λ u n x + yf Y y dy ux + yf Y y dy b x+1/n b x+1/n b x hanks o 3.5. Since u n u 1 by consrucion, i follows ha αu n x + Lu n x λp λpη 1 n b x+1/n b x ux + yf Y y dy ux + yf Y y dy, u n x + y ux + y η 1 dy for all x<b 3.6
8 Firs passage imes ofa jump diffusion process 511 uniformly in x, asn. Applying he Iô formula for jump processes see e.g. [23] o he process {e α u n X ; }, we obain ha he process M n := e α τ b u n X τb τb e αs αu n X s + Lu n X s ds,, is a local maringale saring from M n = u n = u. However, M n 2 + λpη 1 for all, n hanks o 3.6. I follows from he dominaed convergence heorem ha {M n ; } is acually a maringale. In paricular, [ τb ] E M n = E e α τb u n X τb e αs αu n X s + Lu n X s ds = u for all. Leing n, i follows from he dominaed convergence heorem ha lim n E[e α τb u n X τb ]=E[e α τb ux τb ] and, hanks o he uniform convergence in 3.6, ] lim n E Therefore, for any, [ τb = lim n E =. e αs αu n X s + Lu n X s ds [ τb ] e αs αu n X s + Lu n X s ds u = E[e α τb ux τb ] = E[e α τb ux τb 1 {τb < }]+E[e α ux 1 {τb = }]. Now leing, we have, hanks o he boundedness of u, u = E[e ατ b ux τb 1 {τb < }] =E[e ατ b 1 {τb < }] =E[e ατ b ], as ux τb = 1 on he se {τ b < }, from which he resul follows. We now prove 3.2; his is very similar o he previous proof andwe only give an ouline. I suffices o consider he case where y>, as he case for y = follows by leing y. Leing ux := E x [e ατ b 1 {Xτb b>y}], we expec ha u saisfies he equaion αux + Lux = for all x<b, and ux = 1ifx b + y while ux = ifx [b, b + y. This equaion can be explicily solved. Indeed, consider a soluion aking he form 1, x > b+ y, ux =, b < x b + y, A 2 e b xβ 1 + B 2 e b xβ 2, x b,
9 512 S. G. KOU AND H. WANG where he consans A 2 and B 2 are ye o be deermined. Subsiue o obain ha αu + Lux = A 2 e b xβ 1 fβ 1 + B 2 e b xβ 2 fβ 2 λpe η A2 η 1b x 1 + B 2η 1 e η 1y η 1 β 1 η 1 β 2 = for all x<b. Since fβ 1 = fβ 2 =, i suffices o choose A 2 and B 2 so ha η 1 η 1 A 2 + B 2 = e η1y. η 1 β 1 η 1 β 2 Furhermore, he coninuiy of u a x = b implies ha A 2 + B 2 =. Solve he equaions o obain A 2 and B 2 A 2 = B 2, which are exacly he coefficiens in 3.2. A similar argumen as before yields ha u = E[e ατ b ux τb 1 {τb < }] = E[e ατ b 1 {Xτb >b+y} 1 {τb < }] = E[e ατ b 1 {Xτb b>y}], as ux τb = 1 {Xτb >b+y} on he se {τ b < }, from which he proof is finished. Noe he following Laplace ransform, which is convenien for numerical Laplace inversion: e α Pτ b d = 1 e α dpτ b = 1 α α Ee ατ b. Remark 3.1. The special form of double exponenial densiy funcions enables us o explicily solve he differenial inegral equaions 3.5 associaed wih he Laplace ransforms, hanks o 3.4. For general jump diffusion processes, however, such explici soluions will be very difficul, if no impossible, o obain Properies Corollary 3.1. We have Pτ b < = 1 ifand only ifū. Furhermore, if ū, hen PX τb b>y= e η 1y β 2 η 1 β2 [1 e bβ 2 ] for all y, If ū<, hen PX τb = b = η 1 β 2 + β 2 η 1 β2 e bβ 2. Pτ b < = η 1 β1 β2 η 1 β2 e bβ 1 + β 2 η 1 β1 β 1 η 1 β2 e bβ 2 < 1, β 1 Pτ b <,X τb b>y= e η 1y η 1 β1 β 2 η 1 η 1 β2 β 1 [e bβ 1 e bβ 2 ] for all y, Pτ b <,X τb = b = η 1 β 1 β 2 β 1 Here β1 and β 2 are defined as in Lemma 2.1. e bβ 1 + β 2 η 1 β2 e bβ 2. β 1
10 Firs passage imes ofa jump diffusion process 513 Proof. By Lemma 2.1, if ū, hen β 1,α and β 2,α β2 as α. Thus, Pτ b < = lim E[e ατ b ]=1. α If ū<, hen β 1,α β1 and β 2,α β2 as α. The resul follows by leing α in 3.1, 3.2, and3.3. Remark 3.2. I is surprising o see from Theorem 3.1 andcorollary 3.1 ha he firs passage ime τ b andhe overshoo X τb b are dependen, alhough Proposiion 2.1 shows ha hey are condiionally independen. Corollary 3.2. The expecaion ofhe firs passage ime is finie, i.e. E[τ b ] <, ifand only if ū>. Indeed, [ 1 b + β 2 η ] 1 E[τ b ]= ū η 1 β 1 e bβ 2, if ū>, 2 +, if ū. Furhermore, for ū<, we have E[τ b 1 {τb < }] =C 1 e bβ 1 + C 2 e bβ 2 <, where [ 1 β C 1 := 2 β η 1 β2 2 η 1 + bβ2 η 1 β1 β 2 β 1 β 1 2 G β1 + β 1 η 1 β1 ] G β2, [ 1 β C 2 := 1 β η 1 β2 1 η 1 + bβ1 η 1 β2 β 1 β 2 β 1 2 G β2 + β 2 η 1 β2 ] G β1. See Lemma 2.1 for he definiion of β1,β 2. Proof. To ease exposiion, we will use β i o denoe β i,α. Since he funcion 1/x1 e x is decreasing for x [, +, i follows ha, wih probabiliy 1, 1 e ατ b 1 {τb < } τ b 1 {τb < } as α. α By monoone convergence, [ 1 e ατ b ] E[τ b 1 {τb < }] =lim E 1 {τb < } α α Pτ b < Ee ατb = lim α α d = lim α dα Ee ατ b, where he las equaliy follows from L Hôpial s rule. However, i follows from he implici funcion heorem ha lim α d dα β 1 i = lim α G β i = 1 G βi. For ū, we have Pτ b < = 1 ande[τ b 1 {τb < }] =E[τ b ]. Moreover, in his case, we have β 1, β 2 β2 as α, and G =ū, according o Lemma 2.1. For ū<, i is rivial ha E[τ b ]=. Moreover, in his case, β 1 β1, β 2 β2 as α, where β1 and β 2 are boh posiive. The res of he proof is a sraighforwardcalculaion, andis hus omied.
11 514 S. G. KOU AND H. WANG Corollary 3.3. For any α> and θ<η 1, E[e ατ b+θx τb 1 {τb < }] =e θb [c 1 e bβ 1,α + c 2 e bβ 2,α ], where c 1 = η 1 β 1,α β 2,α θ β 2,α β 1,α η 1 θ, c 2 = β 2,α η 1 β 1,α θ β 2,α β 1,α η 1 θ. Proof. I follows ha E[e ατ b+θx τb 1 {τb < }] = E[e ατ b+θx τb 1 {Xτb =b,τ b < }]+e θb E[e ατ b+θx τb b 1 {Xτb >b,τ b < }] = e θb E[e ατ b 1 {Xτb =b,τ b < }]+e θb η 1 η 1 θ E[e ατ b 1 {Xτb >b,τ b < }] = e θb E[e ατ b 1 {Xτb =b}]+e θb η 1 η 1 θ E[e ατ b 1 {Xτb >b}], where we have usedhe condiional memoryless propery. The claim follows from Theorem 3.1. Noe ha, if ū, hen Pτ b < = 1 andcorollary 3.3 implies ha E[e ατ b+β 1,α X τb ]=1, which can be verifiedalernaively by applying he opional sampling heorem o he exponenial maringale e β 1,αX Gβ 1,α = e β 1,αX α,. Remark 3.3. For general Lévy processes X, under appropriae condiions some represenaions for he join disribuion of τ D,X τd,x τd can be obainedin erms of he Lévy characerisics, where D is a general se and τ D is he firs passage ime o D c ; see [14], [28], [22] for more deails Connecion wih renewalype inegral equaions We have used maringale and differenial equaions o derive closedform soluions of he Laplace ransforms for he firspassageime probabiliies. Anoher possible andpopular approach o solving he problems, now invesigaedin his secion, is o se up some inegral equaions by using renewal argumens. For simpliciy, we shall only consider he case where overall drif is nonnegaive, i.e. ū, in which τ b < almos surely. For any x>, define Pxas he probabiliy ha no overshoo occurs for he firs passage ime τ x wih X, ha is, Px := PX τx = x. Proposiion 3.1. The funcion Pxsaisfies he following renewalype inegral equaion: Px + y = PyPx+ 1 Px y Py zη 1 e η 1z dz. However, he soluion o his renewal equaion is no unique. Indeed, for every ξ, he funcion P ξ x = η 1 η 1 + ξ + ξ η 1 + ξ e η 1+ξx saisfies he inegral equaion wih he boundary condiion P ξ = 1.
12 Firs passage imes ofa jump diffusion process 515 Proof. We have Px + y = PX τx+y = x + y = PX τx+y = x + y X τx dz PX τx dz. [x, However, Proposiion 2.1 assers ha PX τx dz = Pxδ x z + 1 P xη 1 e η 1z x dz, z x; here δ x sands for he Dirac measure a poin {x}. Therefore, Px + y = PxPX τx+y = x + y X τx = x + 1 Px = PxPy+ 1 Px = PxPy+ 1 Px x PX τx+y = x + y X τx dzη 1 e η 1z x dz x+y x y Px + y zη 1 e η 1z x dz Py zη 1 e η 1z dz, hanks o he srong Markov propery andhe fac ha τ b is finie almos surely. Now i remains o check ha P ξ x saisfies he inegral equaion for every ξ. To his end, noe ha y P ξ y zη 1 e η1z dz = η 1 η 1 + ξ 1 e η1y + η 1 η 1 + ξ e η1y η 1 η 1 + ξ e η1+ξy. I is hen very easy o check ha and Thus, P ξ xp ξ y = 1 P ξ x = y P ξ xp ξ y + 1 P ξ x P ξ y zη 1 e η 1z dz ξη 1 η 1 + ξ 2 [1 e η 1+ξx e η 1+ξy + e η 1+ξx+y ] 1 η 1 + ξ 2 {η2 1 + η 1ξe η 1+ξx + η 1 ξe η 1+ξy + ξ 2 e η 1+ξx+y }. y P ξ y zη 1 e η1z dz = η 1 η 1 + ξ + ξ η 1 + ξ e η 1+ξx+y = P ξ x + y, andhe proof is complee. Remark 3.4. Proposiion 3.1 shows ha, in he presence of wosided jumps, he renewalype inegral equaions may no have unique soluions, mainly because of he difficuly of deermining enough boundary condiions based on renewal argumens alone. I is easy o see ha ξ = P ξ. Indeed, as we have shown in Corollary 3.1, i is possible o use he infiniesimal generaor and differenial equaions o deermine ξ. The poin here is, however, ha he renewalype inegral equaions canno do he job by hemselves.
13 516 S. G. KOU AND H. WANG 4. Join disribuion of jump diffusion and is running maxima The probabiliy P X a, max X s b = PX a,τ b, s for some fixednumbers a b and b>, is useful, for example, in pricing barrier opions while he logarihm of he underlying asse price is modelled by a jump diffusion process. In his secion, we evaluae he Laplace ransform e α PX a,τ b d for all α>. I urns ou ha he above Laplace ransform has an explici expression, in erms of Hh funcions. We shall firs give a brief accoun of he Hh funcions Hh funcions The Hh funcions are defined as Hh n x := x Hh n 1 y dy = 1 n! x x n e 2 /2 d, n =, 1, 2,... ; 4.1 Hh 1 x := e x2 /2, Hh x := 2π x, where x is he cumulaive disribuion funcion of he sandard normal densiy. The Hh funcions are nonincreasing, andhave a hreeerm recursion, which is very useful in numerical calculaion: for more deails, see [2, p. 691]. Inroduce he following funcion: Hh n x = 1 n Hh n 2x x n Hh n 1x, n 1; 4.2 H i a,b,c; n := 1 2π e 1/2c2 b n+i/2 Hh i c + a d. 4.3 Here i 1,n are boh inegers andwe make he following assumpion. Assumpion 4.1. The parameers a,b,c are arbirary consans such ha b> and c> 2b. For i 1, i follows from 4.2 ha H i a,b,c; n = 1 i H i 2a,b,c; n + 1 c i H i 1a,b,c; n + 1 a i H i 1a,b,c; n. This recursive formula can be usedo deermine all he values of he H i, saring from H 1 a,b,c; n and H a,b,c; n. See Appendix A for deails.
14 Firs passage imes ofa jump diffusion process Laplace ransform Proposiion 4.1. The Laplace ransform of he join disribuion is given by e α PX a,τ b d = A e α PX a b d + B e α PX + ξ + a b d. Here ξ + is an independen exponenial random variable wih rae η 1 > and A := E[e ατ b 1 {Xτb =b}] = η 1 β 1,α β 2,α β 1,α e bβ 1,α + β 2,α η 1 β 2,α β 1,α e bβ 2,α, 4.4 B := E[e ατ b 1 {Xτb >b}] = η 1 β 1,α β 2,α η 1 [e bβ 1,α e bβ 2,α ]. 4.5 η 1 β 2,α β 1,α Proof. We needo compue wo inegrals: For he firs one, I 1 = I 2 = e α PX a,x τb = b, τ b d, e α PX a,x τb >b,τ b d. I 1 = = = e α PX a,x τb = b, τ b dsd e α PX τb = b, τ b dspx s a b d e αs PX τb = b, τ b ds = E[e ατ b 1 {Xτb =b}] e α PX a b d, e αu PX u a b du where he secondequaliy follows from he srong Markov propery, andhe hirdequaliy follows from he fac ha he Laplace ransform of a convoluion is he produc of Laplace ransforms. As for he secondinegral, observe ha, for any s [,], PX a,x τb >b,τ b ds = PX τb >b,τ b dspx s + ξ + a b, by he condiional memoryless propery and he condiional independence see Proposiion 2.1, as well as he srong Markov propery; here ξ + is some independen exponenial random variable wih rae η 1. I follows exacly as for I 1 ha I 2 = E[e ατ b 1 {Xτb >b}] from which he proof is compleed. e α PX + ξ + a b d,
15 518 S. G. KOU AND H. WANG Theorem 4.1. The Laplace ransform of he join disribuion can be furher wrien as Here e α PX a,τ b d = A + B j=i + e hσ η 1 e hσ η 2 + e hσ η 1 B n= n! H h, ϒ α, µ σ ; n j 1 n! AP n,j + BP n,j n=1 i= i= j 1 n! AQ n,j + B Q n,j σ η 1 i H i h, ϒ α,c + ; n i= σ η 2 i H i h, ϒ α,c ; n λp n σ η 1 i H i h, ϒ α,c + ; n + e hσ η 1 BH h, ϒ α,c + ;. n! n 1 j i n j n P n,i := p j j q n j n i 1 η1 η2, 1 i n 1, j i η 1 + η 2 η 1 + η 2 n 1 j i n j n Q n,i := q j j p n j n i 1 η2 η1, 1 i n 1, j i η 1 + η 2 η 1 + η 2 j=i while P n,n := p n and Q n,n := q n ; P n,1 := Q n,i := Q n,i i=1 j=i η2 η 1 + η 2 n q j j p n j n i j i i, P n,i = P n,i 1, 2 i n + 1, η2 η 1 + η 2 j i η1 η 1 + η 2 c + := ση 1 + µ σ, c := ση 2 µ σ, ϒ α := α + λ + µ2 2σ 2, and A and B are given by 4.4 and 4.5. The proof of his heorem is long andis given in Appendix B. n j+1, 1 i n; h := b a σ, 4.6 Remark 4.1. All he parameers involvedin he funcions H i in Theorem 4.1 saisfy Assumpion 4.1. Remark 4.2. I is easy o derive he corresponding resul for PX a, τ b, a b, b>, where τ b := inf{ : X b}. More precisely, we only needo make he following changes in Theorem 4.1: p q, q p, β 1,α β 3,α, η 1 η 2, η 2 η 1, and β 2,α β 4,α.
16 Firs passage imes ofa jump diffusion process Laplace inversion and numerical examples Since he disribuions of he firs passage imes are given in erms of Laplace ransforms, numerical inversion of Laplace ransforms becomes necessary. To do his, we shall use he Gaver Sehfes algorihm. The reason is ha, among all he Laplace inversion algorihms, o he bes of he auhors knowledge, he Gaver Sehfes is he only one ha does he inversion on he real line; all ohers perform he calculaion in he complex domain, and so are unsuiable for our purpose as he Laplace ransforms in our case involve finding he roos β 1,α and β 2,α. See [1] for a survey of Laplace inversion algorihms. The algorihm is very simple. For any boundedrealvaluedfuncion f defined on [, ha is coninuous a, f= lim f n, where f n = ln2 2n! n! n 1! k= n 1 k n fˆ n + k ln2 k 5.1 and fˆ is he Laplace ransform of f, i.e. fα= ˆ e α fd. To speedup he convergence, an npoin Richardson exrapolaion can be used. More precisely, f can be approximaed by fn for large n, where fn = wk, n f k, k=1 andhe exrapolaion weighs wk, n are given by k n wk, n = 1 n k k! n k!. 5.2 Numerically, we findha i is beer o ignore he firs few iniial calculaions of f k.asa resul, he algorihm approximaes fby fn, where f n = k=1 wk, n f k+b, wih f and w given by 5.1 and5.2, andb is he iniial burningou number ypically equal o 2 or 3. The main advanages of he Gaver Sehfes algorihm are: a i is very easy o program several lines of code will do he job; b i converges very quickly; as we will see, he algorihm ypically converges nicely even for n beween 5 and1; c i is sable i.e. a small perurbaion of iniial inpus will no leado a dramaic change of final resuls if highaccuracy compuaion is used. The main disadvanage of he algorihm is ha i needs high accuracy, as boh f n andhe weighs wk, n involve facorials andalernaive +/ signs. The ineresedreader may refer o [1] for more discussions on he algorihm. In our numerical examples, an accuracy of 3 8 digis is ypically needed. However, in many sofware packages e.g. MATHEMATICA an arbirary accuracy can be specified, and in sandard programming languages e.g. C++ subrouines for highprecision calculaion are available. So his is no a big problem. I is easy o compue he marginal andjoin disribuions of he firs passage imes for he double exponenial jump diffusion process by using he Laplace ransform formulae given in
17 52 S. G. KOU AND H. WANG Table 1: The cases of posiive overall drifs ū>µ =.1. The Mone Carlo resuls are basedon 16 simulaion runs. Pτ b Pτ b, X a n λ =.1 λ = 3 λ =.1 λ = Toal CPU ime 1.26 sec 1.76 sec 4.53 min 4.61 min Brownian moion case.2661 N.A N.A. Mone Carlo simulaion 2 poins CPU ime: 15 min Poin es. and95% C.I..241, , , , poins CPU ime: 1 hr 2 min Poin es. and95% C.I..247, , , ,.226 Secions 3 and4, in conjuncion wih he Gaver Sehfes algorihm. As a numerical illusraion, we shall presen wo examples; one is o compue Pτ b andhe oher Pτ b, X a for b =.3, a =.2, and = 1. The resuls are presenedin Tables 1 and2. The parameers, which are chosen o reflec hose in ypical finance applicaions, for he double exponenial jump diffusion are µ =±.1, σ =.2, p =.5, η 1 = 1/.2, η 2 = 1/.3, and λ = 3. To make a comparison wih he Mone Carlo simulaion, we also use λ =.1, so ha he resuls may be comparedwih he limiing Brownian moion case λ = ; he formulae for he firs passage imes of Brownian moion can be foundin many exbooks, e.g. [17]. All he compuaions are done on a Penium 4 MHz PC. The iniial burningou number usedin all calculaions is B = 2. Also, in calculaing Pτ b, X a, we runcae he Poisson sum afer he enh erm, as addiional numerical calculaions sugges ha he error involvedin he runcaion is less han 1 6. The reason why he calculaion of Pτ b, X a akes a longer ime is ha i requires ha he funcions H i are compuedrecursively andmathematica is slow a recursive calculaion. To speedup he simulaion, binomial approximaion is usedo simulae he Poisson processes. Noe ha he Mone Carlo simulaion is biasedandslow, due o wo sources of errors: random sampling error and sysemaic discreizaion bias. I is quie possible o significanly reduce he random sampling error here and, hus, he widh of he confidence inervals by using some variance reducion echniques, such as conrol variaes and imporance sampling suiable for he case of ū<. The sysemaic discreizaion bias, resuling from approximaing a coninuousime process by a discreeime process in simulaion, is, however, very difficul o reduce; in
18 Firs passage imes ofa jump diffusion process 521 Table 2: The cases of negaive overall drifs ū<µ =.1. The Mone Carlo resuls are basedon 16 simulaion runs. Pτ b Pτ b, X a n λ =.1 λ = 3 λ =.1 λ = Toal CPU ime 1.2 sec 1.81 sec 4.49 min 4.67 min Brownian moion case.5815 N.A N.A. Mone Carlo simulaion 2 poins CPU ime: 15 min Poin es. and95% C.I..51,.59.52,.6.38,.46.4,.46 2 poins CPU ime: 1 hr 2 min Poin es. and95% C.I..53,.61.55,.63.4,.46.41,.47 he examples given above, i makes he calculaion from he simulaion biasedlow. Even in he Brownian moion case, because of he presence of a boundary, he discreizaion bias is significan, resuling in a surprisingly slow rae of convergence for simulaing he firs passage ime, boh heoreically andnumerically; e.g. in [3] i is shown ha he discreizaion error has an order 2 1, which is much slower han he order1 convergence for simulaion wihou he boundary; 16 poins are suggesed for a Brownian moion wih µ = 1, σ = 1, and ime = 8. In he presence of jumps, he discreizaion bias is even more serious, especially for large or λ. This explains he large bias in our simulaion resuls. Appendix A. Compuaion of H i We defined in 4.3 he following funcion: H i a,b,c; n := 1 e 1/2c2 b n+i/2 Hh i c + a d 2π for inegers i 1,n. We assume ha Assumpion 4.1 holds hroughou his secion, ha is, b> and c> 2b. The following recursion formula holds: H i a,b,c; n = 1 i H i 2a,b,c; n + 1 c i H i 1a,b,c; n + 1 a i H i 1a,b,c; n. Therefore, i suffices o evaluae H 1 a,b,c; n and H a,b,c; n, boh of which can be calculaedexplicily.
19 522 S. G. KOU AND H. WANG Lemma A.1. If a =, hen, for all inegers n, H 1 a,b,c; n = e ac 2a 2 b 1 a 2 n n j n + 1 j 2b 2b j= j! 2 2a 2 b, j and, for all inegers n 1, H 1 a,b,c; n = e ac 2a 2 b 1 a 2 n n 1 n j n + 1 j 2b 2b j= j! 2 2a 2 b, j where n j := nn + 1 n + j 1 for all inegers n wih he convenion ha n 1. If a =, hen, for all inegers n, H 1,b,c; n = 2n! 1 n! 4b n 2b, and, for all inegers n 1, H 1,b,c; n =+. Proof. We shall prove he case of a = firs. Since Hh 1 x = e x2 /2, by definiion H 1 a,b,c; n = 1 e 1/2c2 b n 1/2 e 1/2c +a/ 2 d 2π = 1 e ac e b+a2 /2 n 1/2 d. 2π Recall he socalledmodifiedbessel funcion of he hirdkind[4, p. 5], K ν x, which has an inegral represenaion [5, p. 146] as α ν 2 e z/2+α2 / 1 ν+1 d = K ναz for arbirary consans ν and α>,z>. I is easy o show ha H 1 a,b,c; n = 2 a 2 n+1/2 π e ac K n+1/2 2a 2b 2 b. However, he modified Bessel funcion K ν x has he propery ha K ν x = K ν x for all ν, π n j n + 1 j K n+1/2 x = 2x e x j! 2x j for all n ; see [4, pp. 5, 1]. The resul follows. j=
20 Firs passage imes ofa jump diffusion process 523 Now consider he case of a =. By definiion, H 1,b,c; n = 1 e b n 1/2 d. 2π For n 1, he inegral is obviously +. Forn, his inegral is he Laplace ransform of n 1/2, which can be foundin many inegral ables, andwe have This complees he proof. H 1,b,c; n = 1 2π 2n! π n! 4b n b = The following lemma gives he value of H in erms of H 1. Lemma A.2. Suppose ha n is an ineger. 1. If b = 1 2 c2, hen H a,b,c; n = 2. If b = 1 2 c2, a>, hen n! H a,b,c; n = b 2 1 c2 n+1 b 2 1 c2 i i! 3. If b = 1 2 c2, a<, hen 2n! 1 n! 4b n 2b. c 2n + 1 H a 1a,b,c; n + 1 2n + 1 H 1a,b,c; n. i= n! H a,b,c; n = b 2 1 c2 n+1 4. If b = 1 2 c2, a =, hen H,b,c; n = n! + b 2 1 c2 n+1 b 2 1 c2 i i! i= n! 2b 2 1 c2 n! n+1 b 2 1 c2 n+1 Proof. I follows from he definiion 4.1 of Hh ha a 2 H 1a,b,c; i 1 c 2 H 1a,b,c; i. a 2 H 1a,b,c; i 1 c 2 H 1a,b,c; i. b 2 1 c2 i c i! 2 H 1,b,c; i. i= d dx Hh nx = Hh n 1 x, n =, 1, 2,....
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