FIRST PASSAGE TIMES OF A JUMP DIFFUSION PROCESS


 Emmeline Bates
 3 years ago
 Views:
Transcription
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,....
MTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationAn Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price
An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationOn the paper Is Itô calculus oversold? by A. Izmailov and B. Shay
On he paper Is Iô calculus oversold? by A. Izmailov and B. Shay M. Rukowski and W. Szazschneider March, 1999 The main message of he paper Is Iô calculus oversold? by A. Izmailov and B. Shay is, we quoe:
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More informationT ϕ t ds t + ψ t db t,
16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationLECTURE 7 Interest Rate Models I: Short Rate Models
LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk
More informationPricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates
Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationNew Pricing Framework: Options and Bonds
arxiv:1407.445v [qfin.pr] 14 Oc 014 New Pricing Framework: Opions and Bonds Nick Laskin TopQuark Inc. Torono, ON, M6P P Absrac A unified analyical pricing framework wih involvemen of he sho noise random
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationA Probability Density Function for Google s stocks
A Probabiliy Densiy Funcion for Google s socks V.Dorobanu Physics Deparmen, Poliehnica Universiy of Timisoara, Romania Absrac. I is an approach o inroduce he Fokker Planck equaion as an ineresing naural
More informationStochastic Calculus and Option Pricing
Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 BlackScholes
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationFourier series. Learning outcomes
Fourier series 23 Conens. Periodic funcions 2. Represening ic funcions by Fourier Series 3. Even and odd funcions 4. Convergence 5. Halfrange series 6. The complex form 7. Applicaion of Fourier series
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationLife insurance cash flows with policyholder behaviour
Life insurance cash flows wih policyholder behaviour Krisian Buchard,,1 & Thomas Møller, Deparmen of Mahemaical Sciences, Universiy of Copenhagen Universiesparken 5, DK2100 Copenhagen Ø, Denmark PFA Pension,
More informationValuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate
Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his
More informationOn Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations
On Galerkin Approximaions for he Zakai Equaion wih Diffusive and Poin Process Observaions An der Fakulä für Mahemaik und Informaik der Universiä Leipzig angenommene DISSERTATION zur Erlangung des akademischen
More informationDOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR
Invesmen Managemen and Financial Innovaions, Volume 4, Issue 3, 7 33 DOES TRADING VOLUME INFLUENCE GARCH EFFECTS? SOME EVIDENCE FROM THE GREEK MARKET WITH SPECIAL REFERENCE TO BANKING SECTOR Ahanasios
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationKeldysh Formalism: Nonequilibrium Green s Function
Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium
More informationJumpDiffusion Option Valuation Without a Representative Investor: a Stochastic Dominance Approach
umpdiffusion Opion Valuaion Wihou a Represenaive Invesor: a Sochasic Doance Approach By Ioan Mihai Oancea and Sylianos Perrakis This version February 00 Naional Bank of Canada, 30 King Sree Wes, Torono,
More informationA Note on Renewal Theory for T iid Random Fuzzy Variables
Applied Mahemaical Sciences, Vol, 6, no 6, 97979 HIKARI Ld, wwwmhikaricom hp://dxdoiorg/988/ams6686 A Noe on Renewal Theory for T iid Rom Fuzzy Variables Dug Hun Hong Deparmen of Mahemaics, Myongji
More informationDynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract
Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationCommunication Networks II Contents
3 / 1  Communicaion Neworks II (Görg)  www.comnes.unibremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP
More informationSEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, Email: toronj333@yahoo.
SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.
More informationConditional Default Probability and Density
Condiional Defaul Probabiliy and Densiy N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari Absrac This paper proposes differen mehods o consruc condiional survival processes, i.e, families of maringales decreasing
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationTime Consistency in Portfolio Management
1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen
More informationPart 1: White Noise and Moving Average Models
Chaper 3: Forecasing From Time Series Models Par 1: Whie Noise and Moving Average Models Saionariy In his chaper, we sudy models for saionary ime series. A ime series is saionary if is underlying saisical
More informationAn accurate analytical approximation for the price of a Europeanstyle arithmetic Asian option
An accurae analyical approximaion for he price of a Europeansyle arihmeic Asian opion David Vyncke 1, Marc Goovaers 2, Jan Dhaene 2 Absrac For discree arihmeic Asian opions he payoff depends on he price
More informationTimeinhomogeneous Lévy Processes in CrossCurrency Market Models
Timeinhomogeneous Lévy Processes in CrossCurrency Marke Models Disseraion zur Erlangung des Dokorgrades der Mahemaischen Fakulä der AlberLudwigsUniversiä Freiburg i. Brsg. vorgeleg von Naaliya Koval
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationUNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.
UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
More informationStochastic Calculus, Week 10. Definitions and Notation. TermStructure Models & Interest Rate Derivatives
Sochasic Calculus, Week 10 TermSrucure Models & Ineres Rae Derivaives Topics: 1. Definiions and noaion for he ineres rae marke 2. Termsrucure models 3. Ineres rae derivaives Definiions and Noaion Zerocoupon
More informationSinglemachine Scheduling with Periodic Maintenance and both Preemptive and. Nonpreemptive jobs in Remanufacturing System 1
Absrac number: 050407 Singlemachine Scheduling wih Periodic Mainenance and boh Preempive and Nonpreempive jobs in Remanufacuring Sysem Liu Biyu hen Weida (School of Economics and Managemen Souheas Universiy
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationLecture III: Finish Discounted Value Formulation
Lecure III: Finish Discouned Value Formulaion I. Inernal Rae of Reurn A. Formally defined: Inernal Rae of Reurn is ha ineres rae which reduces he ne presen value of an invesmen o zero.. Finding he inernal
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationUNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES. Nadine Gatzert
UNDERSTANDING THE DEATH BENEFIT SWITCH OPTION IN UNIVERSAL LIFE POLICIES Nadine Gazer Conac (has changed since iniial submission): Chair for Insurance Managemen Universiy of ErlangenNuremberg Lange Gasse
More informationOptimalCompensationwithHiddenAction and LumpSum Payment in a ContinuousTime Model
Appl Mah Opim (9) 59: 99 46 DOI.7/s45895 OpimalCompensaionwihHiddenAcion and LumpSum Paymen in a ConinuousTime Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationPricing American currency options in a jump diffusion model
Pricing American currency opions in a jump diffusion model Marc Chesney and M. Jeanblanc July 1, 23 This paper has benefied from he helpful commens of N. Bellamy, J. Beroin, P.Collin Dufresne, R. Ellio,
More information4.8 Exponential Growth and Decay; Newton s Law; Logistic Growth and Decay
324 CHAPTER 4 Exponenial and Logarihmic Funcions 4.8 Exponenial Growh and Decay; Newon s Law; Logisic Growh and Decay OBJECTIVES 1 Find Equaions of Populaions Tha Obey he Law of Uninhibied Growh 2 Find
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationSensitivity Analysis for Averaged Asset Price Dynamics with Gamma Processes
Sensiiviy Analysis for Averaged Asse Price Dynamics wih Gamma Processes REIICHIRO KAWAI AND ASUSHI AKEUCHI Absrac he main purpose of his paper is o derive unbiased Mone Carlo esimaors of various sensiiviy
More informationThe Path Integral Approach to Financial Modeling and Options Pricing?
Compuaional Economics : 9 63, 998. 9 c 998 Kluwer Academic Publishers. Prined in he Neherlands. The Pah Inegral Approach o Financial Modeling and Opions Pricing? VADIM LINETSKY Financial Engineering Program,
More informationSPEC model selection algorithm for ARCH models: an options pricing evaluation framework
Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,
More information#A81 INTEGERS 13 (2013) THE AVERAGE LARGEST PRIME FACTOR
#A8 INTEGERS 3 (03) THE AVERAGE LARGEST PRIME FACTOR Eric Naslund Dearmen of Mahemaics, Princeon Universiy, Princeon, New Jersey naslund@mahrinceonedu Received: /8/3, Revised: 7/7/3, Acceed:/5/3, Published:
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationAn empirical analysis about forecasting Tmall airconditioning sales using time series model Yan Xia
An empirical analysis abou forecasing Tmall aircondiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More informationABSTRACT KEYWORDS. Markov chain, Regulation of payments, Linear regulator, Bellman equations, Constraints. 1. INTRODUCTION
QUADRATIC OPTIMIZATION OF LIFE AND PENSION INSURANCE PAYMENTS BY MOGENS STEFFENSEN ABSTRACT Quadraic opimizaion is he classical approach o opimal conrol of pension funds. Usually he paymen sream is approximaed
More informationVariance Swap. by Fabrice Douglas Rouah
Variance wap by Fabrice Douglas Rouah www.frouah.com www.volopa.com In his Noe we presen a deailed derivaion of he fair value of variance ha is used in pricing a variance swap. We describe he approach
More informationIntroduction to Option Pricing with Fourier Transform: Option Pricing with Exponential Lévy Models
Inroducion o Opion Pricing wih Fourier ransform: Opion Pricing wih Exponenial Lévy Models Kazuhisa Masuda Deparmen of Economics he Graduae Cener, he Ciy Universiy of New York, 365 Fifh Avenue, New York,
More informationInventory Planning with Forecast Updates: Approximate Solutions and Cost Error Bounds
OPERATIONS RESEARCH Vol. 54, No. 6, November December 2006, pp. 1079 1097 issn 0030364X eissn 15265463 06 5406 1079 informs doi 10.1287/opre.1060.0338 2006 INFORMS Invenory Planning wih Forecas Updaes:
More information