EXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION

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1 EXPONENTIAL MARTINGALES AND TIME INTEGRALS OF BROWNIAN MOTION VICTOR GOODMAN AND KYOUNGHEE KIM Absrac. We find a simple expression for he probabiliy densiy of R exp(b s s/2ds in erms of is disribuion funcion and he disribuion funcion for he ime inegral of exp(b s + s/2. The relaion is obained wih a change of measure argumen where expecaions over evens deermined by he ime inegral are replaced by expecaions over he enire probabiliy space. We develop precise informaion concerning he lower ail probabiliies for hese random variables as well as for ime inegrals of geomeric Brownian moion wih arbirary consan drif. In paricular, E[ exp`θ/ R exp(b sds ] is finie iff θ<2. We presen a new formula for he price of an Asian call opion. Keywords: Girsanov heorem, Geomeric Brownian Moion, Asian opion. Subjec Classificaion: Primary 6J65, 6H3 Secondary 91B Inroducion Time inegrals of one-dimensional geomeric Brownian moion have appeared in financial models where cerain expeced values are he compued prices of Asian opions. Approximae values for some of hese expecaions were obained in [RS]. Some fundamenal work in [Y] and [GY] focuses on he disribuion and densiy funcions of ime inegrals. We presen new relaionships beween he densiy funcions and disribuion funcions for hese random variables. Our resuls provide a precise descripion of he lower ail of hese disribuions and we sele several momen quesions involving an exponen which is he reciprocal of a ime inegral. We also consider oher momens where he exponen includes furher exponenial erms involving Brownian moion. A surprising resul shows ha cerain of hese double exponenial momens are indeed finie. We le B denoe a one-dimensional Brownian moion. Our noaion for a ime inegral of exponenial Brownian moion is somewha nonsandard. We le denoe he simple exponenial maringale and we define is ime inegral as =exp(b 2 A = M s ds. In [Y] and [BTW] he auhors use A o denoe he ime inegral of exponenial Brownian moion wihou drif, or an inegral wih a drif oher han 1/2. We will obain some disribuional properies of he random variable A.Inparicular 1

2 2 VICTOR GOODMAN AND KYOUNGHEE KIM we presen a formula for is probabiliy densiy and we obain a sharp lower ail esimae for is disribuion. Our mehod of argumen uses a delicae change of measure, where he process (1.1 B := B 2 log(1 y 2 A becomes a sandard Brownian moion. Here, y is a posiive consan. Of course, his pah ranslaion has a singulariy a he random ime τ defined by (1.2 A τ = 2 y Therefore, we will consider sample pahs only for sricly less han τ. 2. The Change of Measure We follow he Girsanov formalism: A sufficien condiion for a process B := B θ s ds o be a Brownian moion over a compac ime inerval [,T] is ha he process (2.1 Λ := exp ( θ s db 1 θ 2 2 sds be a maringale over he ime inerval (see [KS]. B is a sandard Brownian moion w.r.. he measure Q defined by dq (2.2 dp =Λ T In view of equaion (1.1, we consider as our choice of θ he process (2.3 R =2 d d log(1 y 2 A = y A Noice ha R = y and ha R is defined up o he random ime τ. The process R has he following convenien and remarkable propery: Lemma 2.1. The process given in equaion (2.3 saisfies he SDE (2.4 dr = R db 1 2 R2 d for <τ. Proof. This is a simple calculaion using he fac ha d = db. Remark. By wriing equaion (2.4 in is inegral form, R = y + R s db 1 2 R 2 s ds,

3 GEOMETRIC BROWNIAN MOTION 3 we see ha R is essenially he exponen of he Girsanov densiy process i generaes. This unusual propery of R allows us o analyze he behavior of A hrough a change of measure. Definiion 2.2. For each n =1, 2,... le τ n denoe he sopping ime given by τ n =inf{ : R n} Alhough each sopping ime, and τ as well, depends on he choice of y, we will omi menioning heir dependence on his parameer unless we explicily change he value of y. We see from equaion (1.2 and he fac ha R is negaive ha τ n <τ and also lim τ n = τ a. s. n Proposiion 2.3. For each n =1, 2,... he process (2.5 Λ (n =exp[y + R τn ] forms a maringale. Moreover, he process B 2 log(1 y 2 A τ n is a sandard Brownian moion for T w.r.. he probabiliy measure dq =Λ (n T dp Proof. We choose a Girsanov densiy process as in equaion (2.1 by seing θ s = R s 1 {s<τn} Wih his choice, θ s is a bounded adaped process and hence saisfies a Novikov condiion (see Corollary 5.13 of [KS]. The Novikov condiion is sufficen for Λ o be a maringale. In our case, Λ =exp ( τ n R s db 1 τn R 2 sds 2 I follows from Lemma 2.1 ha he exponen above is precisely R τn + y. Thais, Λ =exp(y + R τn This proves he maringale asserion of he proposiion. Moreover, he calculaion in equaion (2.3 shows ha θ s ds = 2 log(1 y 2 A τ n and so he Girsanov heorem implies ha he process B 2 log(1 y 2 A τ n

4 4 VICTOR GOODMAN AND KYOUNGHEE KIM is a sandard Brownian moion on compac ime inervals wih he change of measure given by Λ T. 3. The Correspondence Beween B and B Definiion 3.1. For fixed n =1, 2,... and y> we le B denoe he process appearing in Proposiion 2.3. Tha is (3.1 B := B 2 log(1 y 2 A τ n We also define and =exp( B /2 à = M s ds We noe ha all quaniies in his definiion depend on our choice for n and y. Proposiion 3.2. (3.2 M = and (1 y 2 A τ n 2 ( y 1 2 à τ n = 1 y 2 A τ n and τn (3.4 R τn = y à τ n Proof. From Definiion 3.1 we have Now if <τ n hen =exp(b /2 2 log(1 y 2 A τ n = d d (1 + y 2 à = y 2 = (1 y 2 A τ n 2 y 2(1 y 2 A 2 = d ( 1 d 1 y 2 A

5 GEOMETRIC BROWNIAN MOTION 5 Hence 1+ y 1 2 Ã = 1 y 2 A Finally, for in his same range, R = y A = (1 y 2 A 2 y A = y (1 y 2 A = y 1+ y 2 Ã These equaliies hold up o he ime τ n, and hese are he asserions (3.3 and (3.4 in he proposiion. Proposiion 3.3. If f(x, z is a nonnegaive Borel-measurable funcion and y> hen (3.5 E[f(,R ; A < 2/y] = E[f( (1 + y 2 A 2, y A exp( y A y] Proof. For fixed n and y>weconsider Now E[f(,R ; τ n >] f(,r 1 {τn>} = f(,r exp( R yexp(r + y1 {τn>} Since his funcion vanishes for τ n, each ime parameer may be replaced by τ n. This allows us o apply Proposiion 2.3 where we ake Λ (n 1 {τn>} =exp(r + y1 {τn>} We obain he ideniy (3.6 E[f(,R ; τ n >]=E Q [f(,r exp( R y1 {τn>}] Each erm in he r.h. expeced value can be expressed in erms of he Brownian moion B. We use he ideniies in Proposiion 3.2 o see ha

6 6 VICTOR GOODMAN AND KYOUNGHEE KIM f(,r exp( R y1 {τn>} = f( (1 y 2 A M 2, exp( y1 y à y {τn>} 2à = f( (1 + y 2 Ã, M exp( y1 2 y à y {τn>} 2à Moreover, he even τ n >equals he even whichinurnequalsheeven min R s > n s max s y Ãs The r.h. expeced value in equaion (3.6 is hen M s <n E Q [f( (1 + y 2 Ã, M exp( y1 2 y à y {maxs Ms <n}] 2à y Ãs Bu since he inegrand is nonnegaive we may ake he limi as n and obain he limiing value E Q [f( (1 + y 2 Ã, M exp( y] 2 y à y à In addiion, since lim n τ n = τ as τ is defined in equaion (1.2 we see ha he limi of he l.h. side of equaion (3.6 is E[f(,R ; A < 2/y] This esablishes he ideniy (3.5 of he proposiion. Remark. Proposiion 3.3 is quie similar o Theorem 1 of [WH]. The auhors sudy Girsanov densiy processes and develop necessary and sufficien condiions for a Girsanov process o be a maringale. Theorem 1 shows ha, in grea generaliy, he expeced value of a Girsanov densiy equals he ail probabiliy for a cerain sopping ime. Our argumens proving our proposiion are similar o hose in [WH]. Inour case, we exend heir resul o include expeced values of a funcion of he Brownian moion process muliplied by a Girsanov densiy. The choice f(x, z 1 is a special case of he heorem in [WH]. The reader may see his by making he choice X( = 2 a + A

7 GEOMETRIC BROWNIAN MOTION 7 as required by Theorem 1. To define he correc sopping ime, one should consider he process Y (udu := 2log(1 1 a A One may verify direcly ha Y ( saisfies he funcional equaion menioned in Proposiion 1 of [WH]. 4. The Disribuion of A Theorem 4.1. For a> he disribuion of A is given by (4.1 Pr{A a} = e 2 a E[exp ( 2 a + A ] Moreover, he random variable A has a coninuous, posiive probabiliy densiy funcion g (a which is simply relaed o he disribuion funcions of A and A : (4.2 g (a = 2 a 2 Pr{A a} 2 a 2 Pr{ A a} Proof. The firs asserion of he Theorem follows from Proposiion 3.3 by making he simple choice Ideniy (3.5 becomes in his case f(x, z 1 Pr{A < 2/y} = E[exp( y A y] = E[exp( y A ]e y This ideniy has he raher surprising corollary ha he expeced value above is finie. The inegrand involves a double exponenial of Browian moion. Noiceha he inegrand is a monoone funcion of y. If y varies over some posiive inerval (y,y 1 hen each inegrand is dominaed by he inegrable random variable exp( y A If a sequence {y k } converges o ỹ in his inerval, we apply he dominaed convergence heorem o prove ha he expeced value converges o is value for ỹ. Hence, he expeced value is a coninuous funcion of y. I follows ha he disribuion funcion of A is coninuous and we may wrie he ideniy as

8 8 VICTOR GOODMAN AND KYOUNGHEE KIM (4.3 Pr{A 2/y} = E[exp( y A ]e y This esablishes equaion (4.1. We show he probabiliy densiy exiss by proving ha he righ hand expression in equaion (4.1 is differeniable w.r.. a. To see his we consider he case of Proposiion 3.3 for Ideniy (3.5 becomes f(x, z =x E[ ; A 2/y] =E[ (1 + y 2 A 2 exp( y A ]e y = y 2 E[ (y A exp( 2 y A ]e y As in he previous case, he expeced value E[ (y A exp( 2 y A ] is necessarily finie and again he inegrand is monoone in y. Therefore, he expression is a coninuous funcion of y as we argued in proving (4.1. However, his same expeced value arises by formally differeniaing he expeced value (4.4 E[exp( y A ] on he r.h. side of (4.1 w.r.. y. Now, as y varies over some posiive inerval (y,y 1 eachinegrand y 2 (y A exp( 2 y A is dominaed by he inegrable random variable (y 1 y A exp( 2 y A So, he y inegral from y o y 1, which equals exp( y A exp( y A is dominaed by he produc of y 1 y and an inegrable funcion. Therefore, he difference quoien for he expeced value in (4.4 converges as y y 1 and he limi is he enire expression for (4.5 E[ ; A 2/y]e y The same argumen applies o he case y 1 y so ha expression (4.4 has a derivaive which equals he expression (4.5. Since he disribuion funcion is he produc of e y and expression (4.4 we conclude ha he disribuion funcion of A has a coninuous probabiliy densiy. We differeniae erms in he ideniy (4.3 o obain

9 GEOMETRIC BROWNIAN MOTION 9 (4.6 2y 2 g (2/y = Pr{A 2/y} + E[ ; A 2/y] Now, he expeced value E[ ; A 2/y] is anoher disribuion funcion. Using he change of measure induced by he facor,weseeha B s s = W is a sandard Brownian moion and so he expeced value equals Pr{ exp(w s + s s/2ds 2/y} We subsiue his expression ino equaion (4.6 o obain g (a = 2 ( Pr{A a 2 a} Pr{ exp(b s + s/2ds a} We see ha he expression for g (a is sricly posiive since exp(b s + s/2ds is sricly larger han he random variable A for each sample pah. Finally we noe ha he random variable A has he same disribuion as he ime inegral above. A = has he same disribuion as exp(b s B + /2 s/2ds exp(w s + /2 s/2ds where W s denoes a sandard Brownian moion; we may change variables in he ime inegral o obain he ime inegral of geomeric Brownian moion wih posiive drif. This esablishes asserion (4.2 of he heorem. Remark. We can rewrie he densiy formula (4.2 by combining he wo probabiliies. The densiy equals (4.7 g (a = 2 a 2 Pr{ exp(b s s/2ds a< exp(b s + s/2ds} This expresses he probabiliy densiy for A in erms of a single condiion on Brownian moion sample pahs up o ime. The exisence of a coninuous probabiliy densiy for A is a corollary of Proposiion 2 of Yor [Y]. In he proposiion a condiional densiy for A is given as an inegral ransform of various ranscendenal funcions. No explici connecion is made beween he densiy and he disribuions of A and A /. In Dufresne [D] nice formulas are obained for he densiy of a reciprocal of a ime inegral for some values of a drif parameer in he Brownian moion. Our

10 1 VICTOR GOODMAN AND KYOUNGHEE KIM choice corresponds o he choice of µ = 1, and he densiy is given as an inegral ransform in [D]. Remark. The densiy g (a is a soluion of he PDE derived in [BTW, equaion (26] where i is shown ha ime inegrals of more general geomeric Brownian moions have smooh densiies. In paricular, he random variable A has a smooh densiy since i has he same disribuion as he ime inegral of exp(b s + s/2. The PDE, equaion (26, has a simple form in he case of he densiy of A : g = 2 {a 2 g } g a 2 2 a Bu, equaion (4.2 shows ha a 2 g/2 is he difference of wo disribuion funcions. Therefore, he PDE becomes g = { g a a Pr{ A a} } g a = 2 a 2 Pr{ A a} Tha is, he ime derivaive of he densiy is obained by differeniaing he densiy of A /. Corollary 4.2. For each y> he process Z =exp ( is a supermaringale. In paricular, y A (4.8 E[Z ]=e y Pr{A 2/y} so ha E[Z ] is a sricly decreasing funcion of. Proof. The process Y = y A also saisfies he SDE ha appears in Lemma 2.1. A simple calculaion shows ha (4.9 dy = Y db 1 2 Y 2 d Therefore, he remark following Lemma 2.1 applies o he process Y : Since Y y = Y s db 1 2 Y 2 s ds,

11 GEOMETRIC BROWNIAN MOTION 11 Y y is he exponen of he Girsanov densiy process which Y generaes. A simple sopping ime argumen, similar o one in he proof of Proposiion 2.3, shows ha Z =exp(y is a posiive local maringale. However, he inegral ideniy (4.1 of he Theorem implies ha E[exp(Y ] is decaying funcion of. Consequenly, Z is a local maringale bu i is no a maringale. Remark. The process Y appears implicily in Lemma 2.1 of [BTW]. In order o derive PDE s for cerain expeced values involving A, he auhors compue a diffusion equaion for processes slighly more general han (Y 1,. The PDE ideniies do no apply o equaion (4.8 since Z is no a homogeneous funcion of Y 1. One can derive he corollary from Theorem 1 of [WH], bu he argumen here connecs he resul direcly o he behavior of ime inegrals of geomeric Brownian moion. Corollary 4.3. (4.1 E[exp ( 2 A ]= Proof. For each a> E[exp ( 2 ; A a] exp(2/apr{a a} A and equaion (4.1 of he heorem implies ha he r.h. expression equals E[exp ( 2 ] a + A Bu, his quaniy increases as a and herefore he random variable is no inegrable. exp ( 2 A

12 12 VICTOR GOODMAN AND KYOUNGHEE KIM 5. Geomeric Brownian Moion wih Drif Definiion 5.1. For each ν R we le A (ν denoe he ime inegral (5.1 A (ν = exp(b s + νs s/2ds The random variable A in he previous secions is A ( wih his noaion. Theorem 5.2. For a> he disribuion of A (ν is given by (5.2 Pr{A (ν a} = a 2ν e 2 a E[(a + A (ν 2ν exp ( 2exp(B + ν /2 ] a + A (ν Proof. For y> we apply Proposiion 3.3 o evaluae E[( ν ; A 2/y] Thechoiceoff(x, z =x ν in he proposiion gives he expeced value M ν E[ (1 + y 2 A 2ν exp( y A y] Nex, we muliply hese expeced values by exp(ν/2 ν 2 /2 so ha M ν exp(ν/2 ν2 /2 = exp(νb ν 2 /2 We use his exponenial maringale facor o change measure in each inegral so ha he process B s = B s νs is a sandard Brownian moion for s. Weseeha E[M ν exp(ν/2 ν2 /2 ; A 2/y] =Pr{A (ν while he oher expeced value equals 2/y} E[(1 + y 2 A(ν This esablishes ideniy (5.2. 2ν exp( exp(b + ν /2 y A(ν y] Corollary 5.3. As a he funcion increases. exp(2/a a Pr{A (1/2 a}

13 GEOMETRIC BROWNIAN MOTION 13 Proof. For he case ha ν = 1/2, he formula in (5.2 for he disribuion funcion becomes (5.3 Pr{A (1/2 a} = ae 2 a E[ 1 a + A (1/2 exp ( 2exp(B ] a + A (1/2 The inegrand of he expeced value in (5.3 increases as a. Corollary 5.4. The following expeced value is infinie. (5.4 E[exp ( 2 exp(b sds ]= Proof. The corollary saes ha E[exp ( 2 ]=. Le F (a denoe he disribuion funcion of A (1/2 and A (1/2 consider E[exp ( 2 (1/2 ; A A (1/2 1] We use a sandard argumen ha jusifies inegraion by pars: so ha e 2 a = a 2 x 2 e 2 x dx E[exp ( 2 (1/2 ; A A (1/2 1] = = = Bu, Corollary 5.3 implies ha 2 x 2 e 2 x x 1 1 a df (adx 2 x 2 e 2 x F (x 1dx 2 x 2 e 2 x dxdf (a 1 x 2 e 2 x F (x k x for x<1 where he consan k>. Hence, he inegral is infinie.

14 14 VICTOR GOODMAN AND KYOUNGHEE KIM 6. Finie Exponenial Momens Lemma 6.1. For any > (6.1 E[exp ( 1 2 ] < exp(b sds Proof. The expeced value in (6.1 is E[exp ( 1 2A (1/2 wih he noaion of Secion 5. And, an upper bound for he expeced value is e + e Pr{exp ( 1 2A (1/2 ] x}dx Le x = e s o obain he following form of he inegral above: (6.2 1 Pr{ 1 2A (1/2 s}e s ds = Pr{ 1 1 2s A(1/2 }e s ds I suffices o prove ha he inegral from k o is finie where k is chosen so ha on he inerval of inegraion 4 s For any s in he inerval we have A (1/2 = 4/s exp(b u du 4/s = 4 s s exp(b u du 4 4 s exp( s 4 by Jensen s inequaliy. Since he random variable is normal wih sandard deviaion s 4 4/s 2 3s B u du exp(b u du 4/s B u du we may replace i wih 2 3s Z where Z denoes a sandard normal random variable. The inegrand in equaion (6.2 is dominaed by Pr{ 1 2s 4 s exp ( 2 Z }e s 3s =Pr{ 1 8 exp ( 2 3s Z }e s

15 GEOMETRIC BROWNIAN MOTION 15 < Pr{ 2 2 3s Z}e s since log(1/8 < 2. We may wrie his as Pr{ 3s>Z}e s Hence he inegral in equaion (6.2, is dominaed by 1 Pr{ 1 2s A(1/2 }e s ds, e k + e k + k k Pr{ 3s>Z}e s ds exp( 3s 2 es ds < Theorem 6.2. For any θ<2 and > he process exp ( θ A is a supermaringale for.inparicular, (6.3 E[exp ( θ ] < A and he expeced value is a sricly decreasing funcion of. Proof. We have seen (proof of Theorem 4.1 ha A / has he same disribuion as exp(b s + s/2ds Since his random variable is larger han A (1/2, Lemma 6.1 implies ha Now le E[exp ( 2A ] < (6.4 U = ec 4 for any fixed c>. If > is sufficienly small, so ha e c /4 1/2, we have A We define 1 by E[exp(U ] < e c1 /4=θ

16 16 VICTOR GOODMAN AND KYOUNGHEE KIM and we claim ha E[exp(U 1 ] < The inegrabliiy of exp ( θ A will follow because he choice of c is arbirary. We firs consider he SDE for U : du = cu d + U db ec 4 A 2 d = U db + cu d 4 e c U 2 d = U db + U {c 4e c U }d Nex, we compue he SDE for he process exp(u. d exp(u =U exp(u db +exp(u U {c 4e c U }d exp(u U 2 d In order o compue an expeced value, we inroduce he sopping imes and we obain τ n =inf{ : U n} E[exp(U τn 1 ] E[exp(U ] τn 1 = E[ exp(u U {c U 4e c U } d] E[ τn 1 exp(u U {c U 4e c1 U } d] τn 1 (6.5 = E[ exp(u U {c U θ 1 U } d] Noice ha if U b where b := c(θ hen he inegrand in equaion (6.5 is negaive. I follows ha E[exp(U τn 1 ] E[exp(U ] τn 1 E[ exp(u U {c U θ 1 U }1 {U b}d] Now we ake he limi as n and, using he condiion ha U inegrand, we see ha b in he E[exp(U 1 ] < This esablishes ha he expeced value in equaion (6.3 is finie. I is a sricly decreasing funcion of because he random variable A has he same disribuion as A (1 which is a sricly increasing funcion of. I remains o esablish he supermaringale propery. Corollary 4.2 implies ha for any α<1 he process

17 GEOMETRIC BROWNIAN MOTION 17 { exp ( 2 a + A } α is a supermaringale because i is a concave funcion of a supermaringale. Now as a he poinwise limi of his process is { (2 } α (2α exp =exp A A Tha is, he process in he saemen of he heorem is inegrable and is he limi of non-negaive supermaringales. Hence, i is also a supermaringale. Remark. I follows from Theorem 5.2 ha he disribuion funcion of M A form a 2 exp( 2/aK a has he where he funcion K a increases as a. So, if θ > 2 in equaion (6.3, he expeced value is infinie. I is unclear for he case θ =2if he expeced value is finie. Corollary 6.3. For any θ<2 and > (6.6 E[exp ( θ exp(b sds ] < Proof. Since he expeced value in (6.6 is a decreasing funcion of, i suffices o show he expeced value is finie for arbirarily small. Since M A has he same disribuion as exp(b s + s/2ds, Theorem 6.2 implies ha for any θ <2 E[exp ( θ ] < exp(b s + s/2ds For a given θ<2, choose so small ha θ := θ exp( 2 < 2 so ha E[exp ( θ e /2 ] < exp(b s + s/2ds We see ha he expeced value above is larger han E[exp ( θ ] exp(b sds and so his expeced value is finie.

18 18 VICTOR GOODMAN AND KYOUNGHEE KIM Remark. Some relaed work on exponenial momens of A is menioned in Yor [Y]. Equaion (1.e of he aricle saes ha E[ 1 A exp ( u2 2A ]= 1 (1 + u2 exp ( 1 2 (sinh 1 u 2 However, a difference in noaion requires ha A be expressed wih our noaion as 1 4 A(1/2 /4. By wriing = /4 we see ha he formula in Equaion (1.e is an expression for 2 ( 2u 2 E[ exp ] A (1/2 A (1/2 Equaion (6.6 implies ha he expeced value is finie for all complex values of u such ha Re(u 2 > 1 and is analyic on his region. Since sinh 1 u = log(u + 1+u 2 one sees ha he righ hand expression also has an analyic exension. The formula has a singulariy a he value u = i which corresponds o he infinie exponenial momen of Corollary 5.4. In addiion, Theorem 4.1 of [D] provides an inegral formula for he densiy of A (1/2 and one can show ha he expeced value is finie (for θ<2 usingtheorem 4.1. The corollary provides a probabilisic proof of his fac. The resul ha he momen is infinie for θ =2is new. An expeced value considered in [Y], [RS], and in oher works concerning ime inegrals in financial mahemaics is he price for an Asian call opion. We presen a new expeced value for he simples opion price and indicae how o derive a corresponding formula for arbirary consan drif and volailiy values. Proposiion 6.4. For any a> (6.7 E[ (A a + ]= a + a 2 E[(a + A 1 exp ( 2 2 ] a + A a Proof. The noaion (A a + involves he indicaor funcion of he even A a so i is naural o consider E[ A 2/y ; A < 2/y ] = 2 y E[1 y 2 A ; A < 2/y ] We apply Proposiion 3.3 where we make he choice f(x, z = x z Then 2f(,R = 2 y A = 2 y (1 y 2 A

19 GEOMETRIC BROWNIAN MOTION 19 Equaion (3.5 shows ha he expeced value over A < 2/y is given by On he oher hand, 2 y E[(1 + y 2 A 1 exp( y A y] E[ A 2 y ]= 2 y so we ake he difference of hese expeced values and le 2 y resul. = a o obain he Remark. The proposiion requires he Brownian moion o have drif 1/2. To derive an expecaion formula for arbirary drif one can apply Proposiion 3.3 o he expeced value E[( ν ( A 2/y ; A < 2/y ] The facor of ( ν is relevan for a change of measure (see he proof of Theorem 5.2 so ha he ime inegral will conain an arbirary drif. To incorporae a volailiy facor in he Brownian moion, one can make a simple ime scale change. We le s = σ 2 s so ha 1 σ 2 exp(b s ds = exp(b σ 2 s ds This random variable has he same disribuion as he ime inegral of exp(σb s. References [BTW] R. Bhaacharya, E.Thomann, and E. Waymire: A noe on he disribuion of inegrals of geomeric Brownian moion, Sa. and Prob. Leers 55, (21. [D] D. Dufresne: The inegral of geomeric Brownian moion, Adv. in Appl. Probab. 33, (21. [GY] H. Geman and M. Yor: Asian Opions, Bessel Processes and Perpeuiies, Mah. Finance 2, (1993. [KS] I. Karazas and S.Shreve: Brownian Moion and Sochasic Calculus, Springer-Verlag New York (1991. [K] K. Kim: Momen Generaing funcion of he inverse of inegral of geomeric Brownian Moion, Proc. Amer. Mah. Soc. 132, (24 [RS] L.C.G. Rogers and Z. Shi: The value of an Asian opion, J. Appl Appl. Probab.32, (1995. [WH] B. Wong and C.C. Heyde: On he maringale propery of sochasic exponenials, J. Appl. Probab. 41, (24. [Y] M. Yor: On some exponenial funcionals of Brownian moion, Adv. in Appl. Probab. 24, (1992. Mahemaics Deparmen, Indiana Universiy, Bloomingon, IN address: goodmanv@indiana.edu Mahemaics Deparmen, Indiana Universiy, Bloomingon, IN address: kyoukim@indiana.edu