Brownian excursions outside a corridor and two-sided Parisian options

Transcription

1 Brownian excursions ouside a corridor and wo-sided Parisian opions Angelos Dassios, hanle Wu Deparmen of aisics, London chool of conomics Houghon ree, London WCA A adassios@lseacuk swu3@lseacuk Absrac In his paper, we sudy he excursion ime of a Brownian moion wih drif ouside a corridor by using a four saes semi-markov model In mahemaical finance, hese resuls have an imporan applicaion in he valuaion of double barrier Parisian opions In his paper, we obain an explici expression for he Laplace ransform of is price Keywords: excursion ime, four saes emi-markov model, double barrier Parisian opions, Laplace ransform 1 Inroducion The concep of Parisian opions was firs inroduced by Chesney, Jeanblanc- Picque and or [6] I is a special case of pah dependen opions The owner of a Parisian opion will eiher gain he righ or lose he righ o exercise he opion upon he price reaching a predeermined barrier level L and saying above or below he level for a predeermined ime d before he mauriy dae T More precisely, he owner of a Parisian down-and-ou opion loses he opion if he underlying asse price reaches he level L and remains consanly below his level for a ime inerval longer han d For a Parisian down-and-in opion he same even gives he owner he righ o exercise he opion For deails on he pricing of Parisian opions see [6], [13], [15] and [1] The double barrier Parisian opions are a version wih wo barriers of he sandard Parisian opions inroduced by Chesney, Jeanblanc-Picque and or [6] In conras o he Parisian opions menioned above, we consider he excursions boh below he lower barrier and above he upper barrier, ie ouside a corridor formed by hese wo barriers Le us look a wo examples, depending on wheher he condiion is ha he required excursions above he upper barrier and below he lower barrier have o boh happen before he mauriy dae or ha eiher one of hem happens before he mauriy In one example, he owner 1

2 of a double barrier Parisian max-ou opion loses he opion if he underlying asse process has boh an excursion above he upper barrier for longer han a coninuous period d 1 and below lower he barrier for longer han d before he mauriy of he opion In he oher example, he owner of a double barrier Parisian min-ou opion loses he righ o exercise he opion if eiher one of hese wo evens happens before he mauriy Laer on, we will derive he Laplace ransforms which can be used o price his ype of opions In his paper, we are going o use he same definiion for he excursion as in [6] and [7] Le be a sochasic process and l 1, l, l 1 > l be he levels of hese wo barriers As in [6], we define g l i, sups s l i, d l i, infs s l i, i 1,, 1 wih he usual convenions, sup and inf Assuming d 1 >, d >,we now define τ τ 3 τ 1 inf > 1 >l 1 g l 1, d 1, inf > 1 l < <l 1 1 n g gl 1, >g l l, 1, d, 3 inf > 1 l < <l 1 1 n g gl 1, <g l, d 3, l, 4 τ 4 inf > 1 <l g l, d 4, 5 τ τ 1 τ 4 6 We can see ha τ1 is he firs ime ha he lengh of he excursion of process above he barrier l 1 reaches a given level d 1 ; τ4 corresponds o he one below l wih required lengh d 4 ; and τ is he smaller of τ1 and τ4 We also see ha τ is he firs ime ha he lengh of he excursion in he corridor reaches given level d, given ha he excursion sars from he upper barrier l 1 ; τ3 corresponds o he one in he corridor saring from he lower barrier l Our aim is o sudy he excursion ouside he corridor, herefore τ and τ3 are no of ineres here However we need o use hese wo sopping imes o define our four saes semi-markov model ha will be he main ool used for calculaion Now assume r is he risk-free rae, T is he erm of he opion, is he price of is underlying asse, K is he srike price and Q is he risk neural measure If we have a double barrier Parisian min-ou call opion wih he barrier l 1 and l, is price can be expressed as: DP min ou call e rt Q 1 τ >T T K + ; and he price of a double barrier Parisian min-in pu opion is: DP min in pu e rt Q 1 τ <T K T +

3 In his paper, we are going o sudy he excursion ime ouside he corridor using a semi-markov model consising of four saes By applying he model o a Brownian moion, we can ge he explici form of he Laplace ransform for he price of double barrier opions One can hen inver using echniques as in [13] In ecion we inroduce he four saes semi-markov model as well as a new process, he doubly perurbed Brownian moion, which has he same behavior as a Brownian moion excep ha each ime i his one of he wo barriers, i moves owards he oher side of he barrier by a jump of size ɛ In ecion 3 we obain he maringale o which we can apply he opional sampling heorem and ge he Laplace ransform ha we can use for pricing laer We give our main resuls applied o Brownian moion in ecion 4, including he Laplace ransforms for he sopping imes we defined by-6 for boh a Brownian moion wih drif, ie W µ, and a sandard Brownian moion, ie W In ecion 5, we focus on pricing he double barrier Parisian opions Definiions From he descripion above, i is clear ha we are acually considering four saes, he sae when he sochasic process is above he barrier l 1 he sae when i is below l and wo saes when i is beween l 1 and l depending on wheher i comes ino he corridor hrough l 1 or l For each sae, we are ineresed in he ime he process spends in i We inroduce a new process 1, if > l 1 Z, if l 1 > > l and gl > 1, g l, 3, if l 1 > > l and gl 1, < g l, 4, if < l We can now express he variables defined above see definiions 1-5 in erms of Z : g l i, sup s Z s Z, 7 We hen define τ1 τ τ3 τ4 d l i, inf s Zs Z, 8 inf > 1 Z 1 g l1, d1, 9 inf > 1 Z g l1, d, 1 inf > 1 Z 3 g l, d3, 11 inf > 1 Z 4 g l, d4 1 V max g l 1,, g l,, 13 3

4 he ime Z has spen in he curren sae I is easy o see ha Z, V is a Markov process Z is herefore a semi-markov process wih he sae space 1,, 3, 4, where 1 sands for he sae when he sochasic process is above he barrier l 1 ; 4 corresponds o he sae below he barrier l ; and 3 represen he sae when is in he corridor given ha i comes ino i hrough l 1 and l respecively For Z, define he ransiion inensiies λ ij u by P Z + j, i j Z i, V u λ ij u + o, 14 P Z + i Z i, V u 1 i j λ ij u + o 15 Define P i µ exp Noice ha µ λ ij vdv, p ijµ λ ij µ P i µ i j P i µ 1 P i µ is he disribuion funcion of he excursion ime in sae i, which is a random variable U i defined as U i inf Z s i Z i, V s> Noe ha because he process is ime homogeneous his has he same disribuion as inf Z +s i Z i, V s> for any ime We have herefore p ij µ lim µ P U i µ, µ + µ, Z U i j Moreover, in he definiion of Z, we deliberaely ignored he siuaion when l i, i 1, The reason is ha we only consider he processes, which µ 1 ul idu, i 1,, as Also, when l 1 and l are he regular poins of he process see [5] for definiion, we have o deal wih he degeneraion of p ij Le us ake a Brownian Moion as an example Assume W µ µ + W wih µ, where W is a sandard Brownian Moion eing x o be is saring poin, we know is densiy for he firs hiing ime of level l i, i 1, is p x l i x exp l i x µ π 3 4

5 The Original Brownian Moion W_ δ_ σ_ δ_1 σ_ Figure 1: A ample Pah of W see [4] According o he definiion of he ransiion densiy, p 1 p 1 p l1 and p 34 p 43 p l, for > In [9] in order o solve he similar problem, we inroduced he perurbed Brownian moion X ɛ wih he respec o he barrier we are ineresed in We apply he same idea here, and consruc a new process he doubly perurbed Brownian moion, ɛ, l 1 l > ɛ >, wih he respec o barriers l 1 and l Assume W µ l 1 + ɛ Define a sequence of sopping imes δ, σ n inf > δ n W µ l 1, δ n+1 inf > σ n W µ l 1 + ɛ, where n, 1, see Figure 1 Now define X ɛ W µ if δ n < σ n X ɛ W µ ɛ if σ n < δ n+1 imilarly, we hen define anoher sequence of sopping imes wih he respec o process X ɛ and barrier l ζ, η n inf > ζ n X ɛ l, ζ n+1 inf > η n X ɛ l + ɛ, 5

6 Process X_ X_ ζ_ η_ ζ_ Figure : A ample Pah of X ɛ where n, 1, see Figure Then define ɛ ɛ X ɛ if ζ n < η n X ɛ ɛ if η n < ζ n+1 I is acually a process which sars from l 1 + ɛ and has he same behavior as he relaed Brownian Moion expec ha each ime i his he barrier l 1 or l, i will jump owards he opposie side of he barrier wih size ɛ see Figure 3 From he definiion, i is clear ha l 1 and l become irregular poins for ɛ Also ɛ converges o W µ wih W µ l 1 almos surely for all Therefore as we saw in [9], he Laplace ransforms of he variables defined based on ɛ converge o hose based on W µ As a resul, we can obain he resuls for he Brownian Moion by carrying ou he calculaion for ɛ and ake he limi as ɛ For ɛ, we can define Z, τ1, τ and τ as above we suppress ɛ on he 6

7 Process Figure 3: A ample Pah of ɛ superscrip For Z, we have he ransiion densiies see [4] ɛ p 1 exp ɛ + µ, 16 π 3 where p 1 exp µɛ µ p 4 exp µ l 1 l ɛ µ p 31 exp µ l 1 l ɛ µ ss l 1 l ɛ, l 1 l, 17 ss ɛ, l 1 l, 18 ss ɛ, l 1 l, 19 p 34 exp µɛ µ ss l 1 l ɛ, l 1 l, p 43 ss x, y Also we know ha ɛ exp π 3 k k + 1y x π 3 ɛ µ exp, 1 k + 1y x p 3 p 3 p 14 p 41 7

8 Clearly, all he argumens above apply o he sandard Brownian moion, which is a special case of W µ when µ 3 Resuls for he semi-markov model In we have inroduced he Markov process Z, V Now we apply he same definiion o he doubly perurbed Brownian moion ; herefore we have Z, V, where Z is he curren sae of, aking value from sae space 1,, 3, 4 and V is he ime has spen in curren sae V is also a sochasic process Now we consider a funcion of he form f V, Z, f Z V,, where f i, i 1,, 3, 4 are funcions from R o R The generaor A is defined as an operaor such ha f V, Z, s A f Vs, Zs, s ds is a maringale see [1], chaper Therefore solving A f subjec o cerain condiions will provide us wih maringales of he form f V, Z, o which we can apply he opional sopping heorem o obain he Laplace ransform we are ineresed in More precisely, we will have A f 1 u, f 1u, + f 1u, u + λ 1 uf, f 1 u, A f u, f u, A f 3 u, f 3u, + f u, u + f 3u, u + λ 1 uf 1, f u, + λ 4 uf 4, f u, + λ 31 uf 1, f 3 u, + λ 34 uf 4, f 3 u,, A f 4 u, f 4u, Assume f i has he form + f 4u, u f i u, e β g i u + λ 43 uf 4, f 3 u, 8

9 A f 1 A f By solving he equaion A f, ie A f 3 A f 4 we can ge di g i u α i exp β + λ ij v dv j i + j i u di g j u λ ij s exp s u subjec o g 1 d 1 α 1 g d α g 3 d α 3 g 4 d α 4 β + λ ij v dv ds j i In our case, we are only ineresed in he excursion ouside he corridor Hence, we se d and d 3 o be Also lim d g d lim d3 g 3 d 3 gives α α 3 Therefore, we have g 1 α 1 e βd1 P 1 d 1 + g 1 ˆP 1 β + g 4 ˆP 4 β P1 β, 4 g 4 α 4 e βd4 P 4 d 4 + g 1 ˆP 31 β + g 4 ˆP 34 β P43 β 5 olving 4 and 5 gives 3 where g 1 6 α 1 e βd1 P 1 d 1 1 ˆP 34 β P 43 β + α 4 e βd4 P 4 d 4 ˆP 4 β P 1 β 1 ˆP 1 β P 1 β ˆP 34 β P 43 β + ˆP 1 β P 1 β ˆP 34 β P 43 β ˆP 31 β P 43 β ˆP 4 β P 1 β, g 4 7 α 4 e βd 4 P4 d 4 1 ˆP 1 β P 1 β + α 1 e βd 1 P1 d 1 ˆP 31 β P 43 β 1 ˆP 1 β P 1 β ˆP 34 β P 43 β + ˆP 1 β P 1 β ˆP 34 β P 43 β ˆP 31 β P 43 β ˆP 4 β P 1 β ˆP ij β e βs p ij sds, 8 P ij β di As a resul, we have obained he maringale e βs p ij sds 9 M f V, e β g Z V, i 1,, 3, 4 3 We now can apply he opional sopping heorem o M wih he sopping ime τ, where τ is he sopping ime defined by 6: M τ M 31 9

10 The righ hand side of 31 is M τ M τ 1 τ < + M 1 τ > Furhermore, M τ 1 τ < M τ 1 τ 1 <τ 4 1 τ 1 < e βτ g 1 d 1 1 τ 1 <τ 4 1 τ 1 < α 1 e βτ 1 τ 1 <τ 4 1 τ 1 < + α 4 + M τ 1 τ 1 >τ 4 1 τ 4 < + e βτ g 4 d 4 1 τ 1 >τ 4 1 τ 4 < e βτ 1 τ 1 >τ 4 1 τ 4 < We also have M 1 τ > e β g Z V 1τ >, where Z can ake values 1,, 3 or 4 When Z 1 or 4, since τ >, we have V < d 1 d 4 According o he definiion of g i µ in 3, we have g 1 V and g4 V are bounded When Z or 3, since lim d g d lim d3 g 3 d 3 and looking a 3 wih d and d 3 replaced by we have ha g V and g3 V are bounded Therefore Hence we have lim M τ α 1 lim M 1 τ > e βτ 1 τ 1 <τ4 + α 4 e βτ 1 τ 1 >τ4 3 The lef hand side of 31 gives lim M M g 1, g 4, ɛ l 1 + ɛ ɛ l ɛ By aking he proper α 1 and α 4, we will have when ɛ l 1 + ɛ e βτ 1 τ 1 <τ4 33 e βd 1 P1 d 1 1 ˆP 34 β P 43 β 1 ˆP 1 β P 1 β ˆP 34 β P 43 β + ˆP 1 β P 1 β ˆP 34 β P 43 β ˆP 31 β P 43 β ˆP 4 β P 1 β, e βτ 1 τ 1 >τ4 34 e βd 4 P43 d 4 ˆP 4 β P 1 β 1 ˆP 1 β P 1 β ˆP 34 β P 43 β + ˆP 1 β P 1 β ˆP 34 β P 43 β ˆP 31 β P 43 β ˆP 4 β P 1 β ; 1

11 when ɛ l ɛ e βτ 1 τ 1 <τ4 35 e βd1 P 1 d 1 ˆP 31 β P 43 β 1 ˆP 1 β P 1 β ˆP 34 β P 43 β + ˆP 1 β P 1 β ˆP 34 β P 43 β ˆP 31 β P 43 β ˆP 4 β P 1 β, e βτ 1 τ 1 >τ4 36 e βd 4 P43 d 4 1 ˆP 1 β P 1 β 1 ˆP 1 β P 1 β ˆP 34 β P 43 β + ˆP 1 β P 1 β ˆP 34 β P 43 β ˆP 31 β P 43 β ˆP 4 β P 1 β 4 Main Resuls In we have saed ha he main difficuly wih he Brownian Moion is ha is origin poin is regular, ie he probabiliy ha W µ will reurn o he origin a arbirarily small ime is 1 We have herefore inroduced he new processes ɛ and Z, V wih ransiion densiies for Z defined in 16 o In order o simplify he expressions, we define Ψx πxn x πx + e x, where N is he cumulaive disribuion funcion for he sandard Normal Disribuion Theorem 1 For a Brownian Moion W µ, τ W µ 1, τ W µ 4, τ W µ defined as in, 5 and 6 wih W µ, we have he following Laplace ransforms: when W µ l 1, e βτ W µ 1 τ W µ 1 <τ W µ 4 e βτ W µ 1 τ W µ 1 >τ W µ 4 e βτ W µ G 1d 1, d 4, µ Gd 1, d 4, µ ; 37 G d 4, d 1, µ Gd 1, d 4, µ ; 38 G 1d 1, d 4, µ + G d 4, d 1, µ ; 39 Gd 1, d 4, µ when W µ l, e βτ W µ 1 τ W µ 1 <τ W µ 4 e βτ W µ 1 τ W µ 1 >τ W µ 4 e βτ W µ G d 1, d 4, µ Gd 1, d 4, µ ; 4 G 1d 4, d 1, µ Gd 1, d 4, µ ; 41 G 1d 4, d 1, µ + G d 1, d 4, µ ; 4 Gd 1, d 4, µ 11

12 where G 1 x, y, z e l1 l β+z βx yψ x πxy z + z 43 1 e l 1 l β+z e βx x πx + Ψ z + z β + z β + π Ψ z y + β + z y, G x, y, z e l 1 l β+z z βx yψ z x + z πxy, 44 Gx, y, z e l1 l β+z β + z x yψ + β + z y xψ 1 e l 1 l β+z β + + z x β + z πx Ψ + β + z β + π Ψ z y + β + z y 45 Proof: We apply he ransiion densiies in 16 o o he resuls in 33 o 36 and ake he limi as ɛ According o he definiion of ɛ, we know ha ɛ as as W µ, for all As we saw in [9], since ɛ W µ, for all, by aking he limi ɛ, he quaniies defined based on ɛ converge o hose based on Brownian moion wih drif Therefore we will ge he resuls shown by 37, 38, 4 and 41 We can herefore ge 39 and 4 by e βτ W µ e βτ W µ 1 τ W µ 1 <τ W µ 4 + e βτ W µ 1 τ W µ 1 >τ W µ 4 Corollary 11 For a sandard Brownian Moion µ, we have when W l 1, e βτ W 1 τ W1 <τ W4 e βτ W 1 τ W1 >τ W4 G 1d 1, d 4, Gd 1, d 4, ; 46 G d 4, d 1, Gd 1, d 4, ; 47 e βτ W G 1d 1, d 4, + G d 4, d 1, Gd 1, d 4, 1 ; 48

13 when W l, where e βτ W 1 τ W1 <τ W4 e βτ W 1 τ W1 >τ W4 G d 1, d 4, Gd 1, d 4, ; 49 G 1d 4, d 1, Gd 1, d 4, ; 5 e βτ W G 1d 4, d 1, + G d 1, d 4, Gd 1, d 4, ; 51 G 1 x, y, e l1 l β βx y 5 1 e l 1 l β e βx + β π Ψ βy + βy, G x, y, e l 1 l β βx y, 53 Gx, y, e l1 l β yψ βx + xψ βy 54 1 e l1 l β + Ψ βx + βπx β π Ψ βy + βy Remark 1: By aking he limi l 1 l, we can ge he resul for he single barrier wo-sided excursion case as in [9] Remark : If we only wan o consider he excursion above a barrier, we can le l imilarly, for he one below a barrier, we can le l 1 + These resuls have been shown in [9] Corollary 1 For a Brownian Moion W µ, τ W µ defined as in 6 wih W µ, we have he following Laplace ransforms: when W µ x, x > l 1, e βτ W µ 55 µ+ e β+µ x l 1 β N + µ d 1 x l 1 d1 +e µ β+µ x l 1 N β + µ d 1 x l 1 G1 d 1, d 4, µ + G d 4, d 1, µ d1 G d 1, d 4, µ +e 1 βd1 e µ+ µ x l1 N µ d 1 x l 1 d1 e µ µ x l 1 N µ d 1 x l 1 ; d1 13

14 when W µ x, l x l 1, e βτ W µ 56 e e l1 xµ β+µ x l e β+µ x l G 1 d 1, d 4, µ + G d 4, d 1, µ e β+µ l 1 l e β+µ l 1 l G d 1, d, µ e e l xµ β+µ l 1 x e β+µ l 1 x + e β+µ l 1 l e β+µ l 1 l G d 1, d 4, µ + G 1 d 4, d 1, µ ; G d 1, d, µ when W µ x, x < l, e βτ W µ 57 µ β+µ e l x β N + µ d 4 l x d4 µ+ β+µ l +e x N β + µ d 4 l x G1 d 4, d 1, µ + G d 1, d 4, µ d4 G d 1, d 4, µ +e βd 4 1 e µ µ l x N µ d 4 l x d4 e µ+ µ l x N µ d 4 l x 58 d4 Proof: We will firs prove he case when x > l 1 Define T inf W µ l 1, ie he firs ime W µ his l 1 By definiion, we have τ W µ d 1, if T d 1 ; τ W µ T + τ W µ, if T < d 1, where W µ here sands for a Brownian moion wih drif sared from l 1 As a resul e βτ W µ e βτ W µ 1 T d1 + e βτ W µ 1 T <d1 e βd 1 P T d 1 + e βt 1 T <d1 e βτ W f µ e βτ W f µ has been calculaed in Theorem 1 see 39 The densiy for T is given in [4] as p x l 1 x exp l i x µ π 3 We can herefore calculae P T d 1 1 e µ+ µ x l1 N µ d 1 x l 1 d1 e µ µ x l1 N µ d 1 x l 1, d1 14

15 e βt 1 T <d1 e µ+ β+µ x l 1 β N + µ d 1 x l 1 d1 +e µ β+µ x l 1 N β + µ d 1 x l 1 d1 We herefore ge he resul in 55 For he case when x < l, we can apply he same argumen When l x l 1, we define T inf W µ l, l 1 By definiion, we have τ W µ W T + τ fµ, if W µ T l 1; τ W µ T + τ W µ, if W µ T l, where W µ sands for a Brownian moion wih drif sared from l Consequenly, e βτ W µ e βt e βτ W f µ 1 T l1 + e βt e βτ W µ 1 T l e βt 1 T l1 e βτ W f µ + e βt 1 T l e βτ W µ e βτ W f µ and e βτ W µ have been obained by Theorem 1, 39 and 4 According o [4], we have e βt e e l1 xµ β+µ x l e β+µ x l 1 T l1 e β+µ l 1 l e, β+µ l 1 l e βt e l x µ e β+µ l 1 x e β+µ l 1 x 1 T l e β+µ l 1 l e β+µ l 1 l We have herefore obained 56 Theorem The probabiliy ha W µ wih W µ x, l x l 1, achieves an excursion above l 1 wih lengh as leas d 1 before i achieves an excursion below l wih lengh a leas d 4 is P τ W µ 1 < τ W µ 4 el1 xµ e µ x l e µ x l F 1 d 1, d 4, µ e µ l 1 l e µ l 1 l F d 1, d 4, µ + el x µ e µ l 1 x e µ l 1 x F d 1, d 4, µ e µ l 1 l e µ l1 l, F d 1, d 4, µ 59 P τ W µ 1 > τ W µ 4 el1 xµ e µ x l e µ x l F d 4, d 1, µ e µ l 1 l e 6 µ l 1 l F d 1, d 4, µ + el x µ e µ l 1 x e µ l 1 x F 1 d 4, d 1, µ e µ l 1 l e µ l1 l ; F d 1, d 4, µ 15

16 where F 1 x, y, z e l 1 l z + yψ z 1 e l 1 l z z x πxy + z x πx z + z Ψ yψ x F x, y, z e z l1 l z z + z F x, y, z yψ x e z l1 l z 1 e l 1 l z + Ψ z z + xψ z x + z π Ψ z πxy y πx, 6 π Ψ z Proof: From Theorem 1 and 56 in Corollary 1, we acually know ha, when W µ x, l x l 1, e βτ W µ 1 τ W µ 1 <τ W µ 4 el 1 x µ e µ x l e µ x l G 1 d 1, d 4, µ e µ l 1 l e µ l 1 l Gd 1, d 4, µ 61 y + z y, 63 y + z y 64 + el xµ e µ l1 x e µ l1 x G d 1, d 4, µ e µ l 1 l e µ l 1 l Gd 1, d 4, µ, e βτ W µ 1 τ W µ 1 >τ W µ 4 el 1 x µ e µ x l e µ x l G d 4, d 1, µ e µ l 1 l e µ l 1 l Gd 1, d 4, µ + el xµ e µ l1 x e µ l1 x G 1 d 4, d 1, µ e µ l 1 l e µ l 1 l Gd 1, d 4, µ 65 eing β in 64 and 65 yields he resuls Theorem leads o he following remarkable resul Corollary 1 The probabiliy ha a sandard Brownian moion W wih W x, l x l 1, we have P τ1 W < τ4 W d4 + x l π d1 +, 66 d 4 + l 1 l π P τ W 1 > τ W 4 d1 + l 1 x π d d 4 + l 1 l π 16

17 Remark: When we ake l 1, l, x, we can ge he resuls for he one barrier case as in [9] We will now exen Corollary 1 o obain he join disribuion of W and τ W a an exponenial ime This is an applicaion of 56 and Girsanov s heorem Theorem 3 For a sandard Brownian Moion W wih W x, l x l 1 and τ W defined as in 4 wih W, we have he following resul: For he case x l 1, P W T dx, τ W < T a 1 x f x l 1, d 1 + a x f x l, d 4 + a 1 x hx l 1, d 1 ; 68 For he case l x < l 1, P W T dx, τ W < T a 1 x f x l 1, d 1 + a x f x l, d 4 ; 69 For he case x < l, P W T dx, τ W < T a 1 x f x l 1, d 1 + a x f x l, d 4 + a x hx l, d 4 ; 7 where T is a random variable wih an exponenial disribuion of parameer γ ha is independen of W and γ x fx, y e e γy γ x πyn γy, 71 γ hx, y πye γy e γ x N x γy e γ x N y x γy, 7 y a 1 x a x γ e γx l e γx l b 1 d 1, d 4 + γ e γl 1 x e γl 1 x b d 1, d 4 G e γl 1 l e, 73 γl 1 l γ e γx l e γx l b d 4, d 1 + γ e γl 1 x e γl 1 x b 1 d 4, d 1 G e γl 1 l e, 74 γl 1 l b 1 x, y e l 1 l γ γx y + 1 γ e γ e γx γ π Ψ γy + γy, 75 b x, y e l1 l γ γx y, 76 G e l1 l γ d4 Ψ γd1 + d 1 Ψ γd e l1 l γ + Ψ γd1 + γπd 1 γ π Ψ γd4 + γd 4 Proof: see appendix 17

18 5 Pricing double barrier Parisian Opions We wan o price a double barrier Parisian call opion wih he curren price of is underlying asse o be x, L 1 < x < L, he owner of which will obain he righ o exercise i when eiher he lengh of he excursion above he barrier L 1 reaches d 1, or he lengh of he excursion below he barrier L reaches d before T Is price formula is given by P min in call e rt Q T K + 1 τ <T, where is he underlying sock price, Q denoes he risk neural measure The subscrip min-in-call means i is a call opion which will be riggered when he minimum of wo sopping imes, τ1 and τ4, is less han T, ie τ < T We assume is a geomeric Brownian moion: d r d + σ dw, x, where L 1 < x < L, r is he risk free rae, W wih W is a sandard Brownian moion under Q e m 1 r 1 σ σ, b 1 K σ ln, B m + W, x l 1 1 σ ln L1, l 1 x σ ln L x We have x exp r 1 σ + σw x exp σm + W xe σb By applying Girsanov s Theorem, we have P min in call e r+ 1 m T P [ xe σb T K + e mb T 1 τ B <T where P is a new measure, under which B is a sandard Brownian moion wih B, and τ B is he sopping ime defined wih he respec o barrier l 1, l And we define P min in call e r+ 1 m T P min in call We are going o show ha we can obain he Laplace ransform of P min in call wr T, denoed by L T Firsly, assuming T is a random variable wih an exponenial disribuion ], 18

19 wih parameer γ ha is independen of W, we have [ xe σb + ] P T K e mb T 1τ B < T xe σy K e my P B T dy, τ B < T γ Hence we have b γl T L T 1 γ γe γt xe σy K e my P B T dy, τ B < T dt b e γt P [ xe σb T K + e mb T 1 τ B <T b xe σy K e my P ] dt B T dy, τ B < T By using he resuls in Theorem 3, his Laplace ransform can be calculaed explicily When b l 1, ie K L 1, we have where L T x γ F 1σ + m K γ F 1m, 1 F 1 x a 1 e γd1 πd1 N γ 1 +a e 4 γd πd4 N γ γd 1 e γl 1 +x γb γ x γd 4 e γl +x γb γ x +a 1 xe xl1 rd1+ d 1 x N x d 1 b l1 πd 1 e γd1 d1 γ x e γl 1 +x γb b l N d11 γd 1 e γl 1 +x+ γb N b l1 d1 γd γ x γ + x when l < b < l 1, ie L < K < L 1, we have L T x γ F σ + m K γ F m, ; 19

20 where F x a 1e l 1x γ x a x πd 1 e d 1 x N 1 γ e γd 1 πd1 N 1 +a e γd4 πd4 N γ when b l, ie K L, we have where F x a 1e l1x γ x a 1 x d 1 L T x γ F 3σ + m K γ F 3m, 1 + x πd 1 e d 1 x N 1 γ e γd 1 πd1 N γd 1 e γl1+x+ γb γ x γd 4 e γl +x γb γ x ; x d 1 e γl1 +x+ γb γd 1 γ x x d 4 + a e l x γ x 1 πd 4 γe d 4 x N 1 a e 4 γd πd4 N e γl +x+ γb γd 4 γ γ x +a γe xl rd 4 + d 4 x N x d 4 b l πd 4 e γd d4 4 γ x e γl +x γb N b l d4 γd 4 e γl +x+ γb N b l d4 γd 4 γ x γ + x Remark: The price can be calculaed by numerical inversion of he Laplace ransform For o far, we have shown how o obain he Laplace ransform of P min call in e r+ 1 m T P min call in P min call ou e rt Q T K + 1 τ >T, we can ge he resul from he relaionship ha P min call ou e rt Q T K + P min call in

21 Furhermore, if we se τ L τ 1,L τ,l, we can define anoher ype of Parisian opions by τ L : P max call in e rt Q T K + 1 τ L <T In order o ge is pricing formula, we should use he following relaionship: We have herefore 1 τ L <T 1 τ 1,L <T + 1 τ,l <T 1 τ L <T P max call in P up in call + P down in call P min call in imilarly, from P max call ou e rt Q T K + P max call in, we can work ou P max call ou 6 Appendix: Proof of Theorem 3 Le T be he final ime According o he definiion of Ψx, we have Ψx πxn x πx + e x πx πxrfc x + e x I is no difficul o show ha e βτ W µ By Girsanov s heorem, his is equal o eing γ β + 1 µ gives e βτ W µ βe βt 1 τ W µ <T dt βe β+ 1 µ T µx e µw T 1 τ W <T dt γ 1 µ e γt µx e µw T 1 τ W <T dt γ 1 µ e µx γ e µw T 1τ W < T, where T is a random variable wih an exponenial disribuion of parameer γ ha is independen of W Therefore we have e µw T 1τ W < T γeµx γ 1 e βτ W µ µ 1

22 In order o inverse he above momen generaing funcion, we jus need o inverse he following expressions: µ γ µ e µx e γx dx e µx e γx dx, 1 γ µ 1 e d 1 µ e µx 1 e γx dx + e µx 1 e γx dx, γ γ di di πµe µ di rfc µ e µx 1 exp x πd1 d 1 dx, µx x e e x d i d i dx The inversion of µe d 1 µ γ µ is e γd 1 e γy 1 e γx N e x y d 1 πd1 dy e γy 1 x d1 γd 1 e γx N e x y d 1 πd1 dy x d1 γd 1 q The inversion of 1 di πµe d i µ rfc q «di µ For x, y e d i y 1 d i γ µ γx e γx y dy e γ γ is given below e γd i γx πd i N γd i ; For x <, x y e d i y 1 d i e γx γ e γd i γx πd i N +e γd i+ γx πd i N e γx y y dy + e γ x d i x di γd i γdi N y 1 d i γ e γx y dy x di + γd i Consequenly, we can ge Theorem 3 References [1] Akahori, J ome Formulae for a new ype of pah-dependen opion Ann Appl Prob 5, , 1995

23 [] Azema, J and or, M ude d une maringale remarquable eminaire de probabiliies XXIII Lecure Noes in Mahemaics 137 pringer, Berlin, 1989 [3] Beeman, H Table of Inegral Transforms, Volume I McGraw-Hill book company, INC, 1954 [4] Borodin, A and alminen, P Handbook of Brownian Moion - Facs and Formulae Birkhauser, 1996 [5] Beroin, Jean Levy Processes Cambridge Universiy Press, 1996 [6] Chesney, M, Jeanblanc-Picque, M and or, M Brownian excursions and Parisian barrier opions Adv Appl Prob, 9, , 1997 [7] Chung, KL xcursions in Brownian moion Ark mah 14, , 1976 [8] Dassios, A The disribuion of he quaniles of a Brownian moion wih drif Ann Appl Prob 5, , 1995 [9] Dassios, A Wu, Two-sided Parisian opion wih singl barrier Working paper L, 8 [1] Davis, MHA Markov Models and Opimizaion Chapman and Hall, 1994 [11] Grandel, J Aspec of Risk Theory pringer, 1991 [1] Harley, P Pricing parisian opions by Laplace inversion Decisions in conomics and Finance, [13] Labar, C and Lelong, J Pricing parisian opions Technical repor, NPC, hp:// cermicsenpcfr/repors/crmic-5/crmic-5-94pdf, December 5 [14] Pechl, A ome applicaions of occupaion imes of Brownian moion wih drif in mahemaical finance,journal of Applied Mahemaics and Decision ciences, 3, 63-73, 1999 [15] chröder, M Brownian excursions and Parisian barrier opions: a noe J Appl Prob 44, , 3 3