2013 (Heisei 25) Doctoral Thesis

Transcription

1 3 Heisei 5 Docoal Thesis A sudy on he densiy and sensiiviy analysis concening he maimum of SDs Docoal Pogam in Inegaed Science and ngineeing Gaduae School of Science and ngineeing Tomonoi Nakasu

2 A sudy on he densiy and sensiiviy analysis concening he maimum of SDs Tomonoi Nakasu Depamen of Mahemaical Science Gaduae School of Science and ngineeing Risumeikan Univesiy

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4 Absac In his hesis, we shall give some esuls on he eisence of he densiy funcion and sensiiviy analysis concening he maimum of some sochasic diffeenial equaions SDs, in sho. The Malliavin calculus o sochasic calculus of vaiaions plays an impoan ole o obain he esuls of his hesis. In Chape, we pesen he inoducion of his hesis and he peliminay of Malliavin calculus. In Chape, we conside an m-dimensional SD wih coefficiens which depend on he maimum of he soluion. Fis, we pove he absolue coninuiy of he law of he soluion. Then we pove ha he join law of he maimum of he ih componen of he soluion and he i h componen of he soluion is absoluely coninuous wih espec o he Lebesgue measue in a paicula case. In Chape 3, we give a decomposiion fomula o calculae he vega inde he sensiiviy of an opion conac wih espec o changes in volailiy fo opions depending on he eema maimum o minimum of a geneal one-dimensional model and sudy is behavio. Moeove, we compae he vega inde obained in his one-dimensional model wih he one in he Black-Scholes model. Ou mahemaical and numeical esuls povide mainly hee ineesing popeies of he vega inde fo baie ype opions in he one-dimensional model: Fis, he vega inde can be decomposed ino hee componens which can be called eema sensiiviy, eminal feaue sensiiviy and dif sensiiviy. Second, by using an eample of up-in call opions, we show ha hee is a baie value a which he impoance of eema and eminal sensiiviy ae evesed. Thid, eema sensiiviy is impoan only fo opions wih sho mauiy as fa as he vega inde is concened. The compaison of he vega inde in wo diffeen models claifies ha he behavio of he vega inde in he one-dimensional model consideed in his hesis is fa away fom ha in he Black-Scholes model. In he case of binay baie opions, each componen of he decomposiion fomula fo he vega inde involves he Diac dela funcionals. Kenel mehods ae used in ode o esimae he vega inde in his seing. iii

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6 Acknowledgemens I would like o epess my deepes hanks o Pofesso Auo Kohasu- Higa, fo his guidance and suppo fom he summe of 5 when we me fo he fis ime. He has been aking cae of me in ems of abou sudies and eseaches fom ou fis encoune even when I was a paciione a a company. Needless o say, his consucive ideas and houghful commens lagely conibue o his hesis. I am also indebed o Pofesso Jiô Akahoi who has suppoed my life a Risumeikan Univesiy. He has oganized some symposiums duing my docoal couse a which I have had he oppouniies o give alks. Pofesso Sesuo Fujiie has given me deep kindness houghou my PhD couse. Fuiful discussions wih he following people: Pofesso Asushi Takeuchi, Pofesso Masafumi Hayashi, Pofesso Masaaki Fukasawa and Song Xiaoming, impoved his hesis. I would like o hank hem. I am hankful o Pofesso Taizo Chiyonobu, Pofesso Kazuhio Yasuda, Pofesso Takahio Aoyama, Pofesso Ngô Hoàng Long and Pofesso Azmi Makhlouf fo giving me encouagemens o addess my woks. I would like o hank he membes he of weekly mahemaical finance semina: Pofesso Masaoshi Fujisaki, Pofesso Kenji Yasuomi, Pofesso Takuya Waanabe, Yui Imamua, Nienlin Liu, Gô Yûki, Zhong Jie and Libo Li, fo valuable alks. I have eceived encouagemens hough alking wih seveal paciiones and I mus give my hanks o hem. My colleagues of Risumeikan Univesiy: Hideyuki Tanaka, Hidemi Aihaa and Takafumi Amaba, have given ineesing and inspiing discussions. I gealy appeciae hem. Finally, I would like o hank my paens fo hei suppos. v

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8 Conens Inoducion and peliminay. Poblem of he eisence of he densiy funcion Poblem of he compuaion of Geeks Peliminay of Malliavin calculus Absolue coninuiy of he laws of a muli-dimensional sochasic diffeenial equaion wih coefficiens depending on he maimum 5. Inoducion The eisence, uniqueness and diffeeniabiliy of he soluion o. and he absolue coninuiy of he pobabiliy law of X The absolue coninuiy of he pobabiliy law of X, i M i A concluding emak Volailiy isk fo opions depending on eema and is esimaion using kenel mehods 5 3. Inoducion Main esul: Vega inde fo opions depending on he eema Numeical epeimen : Sucue of he vega inde Peliminay: assumpions of he one-dimensional model and he definiion of he vega in he Black-Scholes model The case of payoff funcions depending on only one componen The case of payoff funcions depending on he eema and he eminal value of he undelying Kenel mehod Numeical epeimen : simaion of he vega inde using he kenel mehod Conclusion and final emaks Appendi A Appendi A Appendi A Appendi A Appendi A Appendi A Appendi B Appendi C vii

9 CONTNTS Appendi D viii

10 Chape Inoducion and peliminay. Poblem of he eisence of he densiy funcion In pobabiliy heoy, we ofen conside an infinie-dimensional pobabiliy space, called he Wiene space. On he Wiene space, compuing he epecaion of a andom vaiable implies inegaing he andom vaiable wih espec o a pobabiliy measue defined on his infinie-dimensional space, called he Wiene measue. If we can pove he eisence of he densiy funcion of he andom vaiable, his inegal wih espec o he Wiene measue can be ansfomed o he inegal wih espec o he Lebesgue measue, namely, a measue on a finie-dimensional space. Meanwhile, in mahemaical finance, we ofen deal wih opions wih non-smooh payoff funcions e.g. uopean call opion o uopean pu opion. The pice of opions is defined by he epecaion of andom vaiables and he isks involved in opions ae defined by he sensiiviies of he pice of opions wih espec o make paamees. Thus, in ode o compue hese sensiiviies, we ae equied o diffeeniae non-smooh payoff funcions. The eisence of he densiy funcion of he andom vaiable guaanees ha we can diffeeniae non-smooh payoff funcions, as long as he Lebesgue measue of he se of all non-smooh poins of he payoff funcions is zeo. Theefoe, o sudy he eisence of he densiy funcion of andom vaiables is one of he mos impoan subjec fom a heoeical and a pacical poin of view. Chape of his hesis is concened wih he poblem of he eisence of he densiy funcions of an SD whose coefficiens ae dependen on he maimum of he soluion. One may inepe a esul obained in his chape as an eension of a esul in 7. Howeve, we shall give some esuls on he join laws which ae no consideed in 7. The esuls of Chape ae aken fom he published pape 5.. Poblem of he compuaion of Geeks In mahemaical finance, he compuaion of he isks involved in opions, called Geeks, is one of he mos impoan poblem since paciiones begin he hedging pocedues fo opions based on he values of Geeks. Thee ae some kinds of Geeks. Fo eample, he sensiiviy of opion pices wih espec o he cuen undelying asse s pice pice is called he dela, and he sensiiviy of he dela wih espec o he cuen asse pice is called he gamma. A make paamee which descibes he vaiance of asse

11 .3. PRLIMINARY OF MALLIAVIN CALCULUS pices is called he volailiy, and he sensiiviy wih espec o he volailiy is called he vega o vega inde. In he Black-Scholes model, he simples financial model, he Geeks can be compued eplicily. Howeve, in he ohe models which may pefom bee han he Black-Scholes model, he Geeks do no have he eplici fomulas, heefoe we ae equied o use some numeical echniques o compue he Geeks, such as he Mone Calo simulaion. Hence, he poblem how we can epess he Geeks is an ineesing and impoan poblem, mahemaically and pacically. In pacice, vaious ypes of opions ae aded by paciiones. A uopean opion may be eecised only a he epiaion dae of he opion e.g. uopean call opion o uopean pu opion. An opion whose payoff is deemined by he aveage undelying pice ove some pe-se peiod of ime is called an Asian opion. A lookback opion is an opion wih he payoff depends on he maimum o minimum undelying asse s pice occuing ove he life of he opion. A baie opion is an opion on he undelying asse whose pice beaching he pe-se baie level eihe spings he opion ino eisence o einguishes an aleady eising opion. In 6, he auhos used he Malliavin calculus o calculae he Geeks fo he fis ime. They obained some epessions o compue he Geeks of some uopean opions and Asian opions, and showed ha hese epessions povide he bee numeical esuls han ones obained by a classical mehod, called he finie diffeence mehod. One can find a fomula o compue he vega inde fo Asian opions, in. In 9, a mehod o compue he dela and gamma of lookback and baie opions is discussed and numeical esuls ae also given. In Chape 3, we focus on he poblem of he compuaions of he vega inde fo lookback and baie opions. We shall give an epession of he vega inde, numeical esuls and a mehod o simulae he vega inde fo some specific opions. The esuls of Chape 3 ae aken fom he submied pape 6..3 Peliminay of Malliavin calculus Recen advances of a diffeenial calculus on he Wiene space, called he Malliavin calculus o sochasic calculus of vaiaions povides many useful ools o us in ode o y he poblems menioned in he pevious subsecions. We inoduce some basic ools of Malliavin calculus ha will be used houghou he hesis. We efe o 7 o inoduce Malliavin calculus. Le Ω, F, P be he canonical Wiene space which suppos a d-dimensional Bownian moion W. The class of eal andom vaiables of he fom F = fw,, W n, f Cb Rnd ; R,,, n is denoed by S. D,p denoes a Banach space which is he compleion of S wih espec o he nom whee F,p = F p p + D j F = n i= DF j d j= f ji W,, W n,i. p p,

12 CHAPTR. INTRODUCTION AND PRLIMINARY D k,p is defined analogously, and is associaed nom is denoed by k,p. Also, we define D k, = p D k,p and D = p k D k,p. Fo F, G D, we define DF, DG H := d j= Dj F DGd j and DF H := d j= Dj F d. Now le us inoduce a localizaion of D k,p. D k,p loc denoes he se of andom vaiables F such ha hee eiss a sequence {Ω n, F n, n } F D k,p wih he following popeies: i Ω n Ω, a.s. ii F = F n, a.s. on Ω n.. The following heoem is well-known and we shall use his heoem of obain he esuls in Chape Theoem. Theoem.. of 7 Le F = F,, F m be a andom veco saisfying he following condiions. i F i belongs o he space D,p loc, p >, fo all i =,, m. ii The mai γ F := DF i, DF j H i,j m is inveible a.s. Then he law of F is absolue coninuous wih espec o he Lebesgue measue on R m. 3

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14 Chape Absolue coninuiy of he laws of a muli-dimensional sochasic diffeenial equaion wih coefficiens depending on he maimum. Inoducion In his chape, we deal wih he following m-dimensional sochasic diffeenial equaion SD: X i = i + A i ls, X s, M s dw l s + l= B i s, X s, M s ds, i m. whee W denoes a d-dimensional Bownian moion, A i l, Bi :, R m R, i m, l d ae measuable funcions and M s = M s,, M m s := ma u s X s,, ma u s X m s. The pupose of his chape is o pove he absolue coninuiy of he join law concening he soluion o. wih Lipschiz coninuous coefficiens using Malliavin calculus. In 7, he auhos poved ha if m = d =, A and B ae Hölde coninuous, fo > he law of X is absoluely coninuous wih espec o he Lebesgue measue on R, whee X is a weak o song soluion o.. The auhos used he mehod o analyze he chaaceisic funcion of X o pove he absolue coninuiy of he law of X in 7. In his chape, fis we pove he absolue coninuiy of he law of X = X,, X m wih espec o he Lebesgue measue on R m. Then we pove he absolue coninuiy of he law of M i, X i, i, i m, wih espec o he Lebesgue measue on R when A i l does no depend on he second space vaiable. To analyze he law of M i, X i may be impoan in he field of applicaions such as finance. Thoughou his chape, we use C o C i, i N o denoe a posiive consan which may depend on consans K, L, d, p, and. 5

15 .. TH XISTNC, UNIQUNSS AND DIFFRNTIABILITY OF TH SOLUTION TO. AND TH ABSOLUT CONTINUITY OF TH PROBABILITY LAW OF X T. The eisence, uniqueness and diffeeniabiliy of he soluion o. and he absolue coninuiy of he pobabiliy law of X In his secion, fisly we pove he eisence, uniqueness and diffeeniabiliy of he soluion o.. Secondly we pove fo >, he absolue coninuiy of he pobabiliy law of X whee X is he soluion o.. We assume he following: A Thee eis K, M, c > such ha A,, A,, + B,, B,, K + fo any,,, R m and, A,, + B,, L, A A,, is coninuous wih espec o,,, A3 hee eiss c > such ha fo any v R m and, R m and. v T A,, c v, Fis, le us sae a lemma on he eisence of a unique soluion o.. Lemma. Assume A, hen. has a unique song soluion fo any iniial value R m. Moeove we have M i p C fo any, i m and p. Poof. Fo s, we define X,i s := i X n+,i s := i + l= A i lu, X u n, M u n dwu l + B i u, X u n, M u n du, i m, n, whee X u n := X u n,,, X u n,m and M u n := ma v u X v n,,, ma v u X v n,m. Fom Hölde s inequaliy and Bukholde-Davis-Gundy s inequaliy, i is easy o see ha ma u s X n+,i u X n,i u C l= A i lu, X n u, M n A i lu, X n u u, M u n du + B i u, X u n, M u n B i u, X u n, M u n du 6.

16 CHAPTR. ABSOLUT CONTINUITY OF TH LAWS OF A MULTI-DIMNSIONAL STOCHASTIC DIFFRNTIAL QUATION WITH COFFICINTS DPNDING ON TH MAXIMUM holds fo s, and i m. By A and a ivial inequaliy ma v u X v n,i ma v u X v n,i X v n,i, we ge heefoe, m i= fo s, and n N. ma u s X n+,i u C s m C X n,i u X n u i= ma u s Xn+,i u X u n+,i We define, fo s, and n N, f n s := f n s C n by.3. Now due o A, we obain i= u f s = X u n + M u n ma v u Xn,i v X v n,i du, m i= s m C i= M u n du ma v u X v n,i ma v u Xn,i v X v n,i du,.3 m i= ma u s X u n+,i X u n+,i, hen we have un f u n du n du, ma u s X,i u C, fo s,, hus we have m ma u s Xn+,i u X u n,i = f n s C s n C..4 n! Relaion.4 and he Čebyšev s inequaliy give m P ma s Xn+,i s X s n,i n+ i= C n C,.5 n! fo n N and he igh hand side of.5 is a convegen seies. Fom he Boel-Canelli s lemma, hee eiss Ω F wih P Ω = such ha fo evey ω Ω hee eiss Nω N wih m i= ma s X s k+,i X s k,i < k+ fo k Nω. Moeove, his implies ha i=,i ma s Xk+m s X s k,i k,.6 fo evey m N, k Nω. We see hen ha he sequence of sample pahs {X s n, s, } is convegen in he supemum nom on coninuous funcions, which concludes he eisence of a coninuous limi {X s, s, } fo all ω Ω. 7

17 .. TH XISTNC, UNIQUNSS AND DIFFRNTIABILITY OF TH SOLUTION TO. AND TH ABSOLUT CONTINUITY OF TH PROBABILITY LAW OF X T Now le us pove ha {X s, s, } saisfies.. Fisly, we shall conside he Lebesgue inegal pa. Due o A, we have fo s, B i u, X n u, M n du u B i u, X u, M u du C M n u M u du, and.6 gives ha ma u s Xu i X u n,i n fo n Nω. Thus, we ge ha B i u, X u n, M u n du B i u, X u, M u du, holds as n, a.s. Ne, we shall conside he sochasic inegal pa. We obseve fom.4 ha fo fied u,, he sequence of andom vaiables {M u n,i } n N is a Cauchy sequence in L Ω, F, P. Indeed, fom.4 we ge M u n,i M n,i u ma v u Xn,i v X n,i v n j=n ma v u Xj+,i v X v j,i, as n, n. Theefoe, hee eiss M u i such ha M u n,i M u i in L Ω, F, P. Since M u n,i Mu, i a.s., we have Mu i M u i lim inf n M u n,i M u = and his implies ha M u n,i Mu i as n. Fom A, we have fo s, l= A i lu, X n u, M u n A i lu, X u, M u dw l u C M u n M u du..7 By A, we have M u n,i C 3 and Faou s lemma gives Mu i lim inf n M u n,i C 3. Fom.7 and he bounded convegence heoem, we have l= A i lu, X n u as n, hus, by aking a subsequence one has l= A i lu, X n u, M u n A i lu, X u, M u, M u n A i lu, X u, M u dw l u dw l u,,.8 as n, a.s. Theefoe, fo >, {X s, s, } saisfies.. Ne, we shall pove he pahwise uniqueness of he equaion.. We assume ha fo fied >, {X s, s, } and { ˇX s, s, } saisfy.. Fom A, i is easy o see ha ma u s Xi u ˇX u i C 8 m i= ma u v Xi u ˇX v i dv,

18 CHAPTR. ABSOLUT CONTINUITY OF TH LAWS OF A MULTI-DIMNSIONAL STOCHASTIC DIFFRNTIAL QUATION WITH COFFICINTS DPNDING ON TH MAXIMUM hus, m i= ma u s Xi u ˇX u i C m i= ma u v Xi u ˇX v i dv holds, hen by defining gs := m i= ma u s Xu i ˇX u i and applying he Gonwall s lemma o gs, we have m i= ma u s Xu i ˇX u i = fo s,. Theefoe, one has he pahwise uniqueness of he soluion {X s, s, } o. fo fied > by he coninuiy of {X s, s, }. Since > is abiay, we have he eisence of a unique song soluion {X s, s, } o.. Moeove, M i p C fo p is a consequence of A. Now, le us pove he popey of he ime when one-dimensional pocess {Xs, i s, } aains is maimum on,. This popey plays an impoan ole o pove he absolue coninuiy of he join law of M i, X i. Lemma. Unde A-A3, fo any and i m, {X i s, s, } aains is maimum on, on a unique poin τ i and < τ i <, a.s. Poof. We define a new pobabiliy measue P by d P dp := ep l= C l s, X s, M s dw l s whee fo, R m and s >, d-dimensional veco Cs,, is defined by Cs, s, := A T AA T Bs,,. Define a d-dimensional pocess W by W s := W l s + C l u, X u M u, l d. Then by he Gisanov s heoem, { W s, s, } is a d-dimensional Bownian moion unde P, heefoe {X s, s, } can be epessed as X i s = i + l=, A i lu, X u, M u d W l u, i m,.9 and fo each i m, {X i s, s, } is a maingale by A. Le {F s, s, } be he augmenaion of he Bownian filaion geneaed by { W s, s, }. Fo s,, we define T s := inf{ > : X i > s} hen he ime-changed pocess B s := X i T s, G s := F T s, s, is a sandad one-dimensional Bownian moion. Moeove, by A3, {X i s, s, } can be wien as fo s,. X i s = i + B Xi s 9

19 .. TH XISTNC, UNIQUNSS AND DIFFRNTIABILITY OF TH SOLUTION TO. AND TH ABSOLUT CONTINUITY OF TH PROBABILITY LAW OF X T Fis, le us pove ha P ma s Xi s = i =. By he law of ieaed logaihm fo Bownian moion, we have lim sup s Xs i i X i s log log X i s = lim sup s B Xi s X i s log log =, X i s P -a.s., hus P ma s Xi s = i P Xs i i, s, =. Then, we shall pove ha P ma s Xi s = X i =. We noe ha fo >, X i is a sopping ime fo he filaion G s, since We define { X i s} = {T s } F T s = G s. ˇB s := { Bs, s, B + Ŵs Ŵ, s,. and ˇF s := σb u, u s σ W u, u s, whee {Ŵs, s, } is a one-dimensional Bownian moion independen of B, hen we find ha { ˇB s, s, } is a one-dimensional Bownian moion, since we can easily check ha { ˇB s, s, } is a ˇF s -maingale and is quadaic vaiaion is given by {s, s, }. Le {B s, s, } be a one-dimensional Bownian moion independen of { ˇB s, s, }. Define ˆB s := { ˇBs, s, B s, s,. hen { ˆB s, s, } is a wo sided Bownian moion. By he definiion of ˇFs, X i is a sopping ime fo he filaion ˇF s, heefoe fom ecise.4 of 4, { ˆB Xi s ˆB Xi, s, } is a onedimensional Bownian moion. Again, by he law of ieaed logaihm fo Bownian moion, we have = lim inf s ˆB X i ˆB X i s s log log s = lim inf s ˆB Xi ˆB Xi s X i X i s log log X i X i s = lim inf s = lim inf s B Xi B Xi s X i X i s log log X i X i s X i X s i, X i X i s log log X i X i s

20 CHAPTR. ABSOLUT CONTINUITY OF TH LAWS OF A MULTI-DIMNSIONAL STOCHASTIC DIFFRNTIAL QUATION WITH COFFICINTS DPNDING ON TH MAXIMUM P -a.s. Theefoe, P ma s Xi s = X i P Xs i X, i s, =. Finally, le us pove he uniqueness of τ i on,. As menioned befoe, we can wie X i s = i + B X i s fo s,. Define and { θ := sup s : B Xi s = sup u B Xi u }, { } θ := inf s : B Xi s = sup B Xi u, u { τ := sup s : B s = sup u B u }, { } τ := inf s : B s = sup B u. u Then by he definiions, we have θ = X i τ X i, θ = X i τ X i. Thus, one has P θ < θ < = P τ Xi < τ Xi < X i = P } {τ X i < < τ X i < < X i =, Q <, Q < P τ Xi < < τ Xi < < X i, whee Q denoes he se of all aional numbes. On {τ Xi < < τ Xi < < X i }, he definiion of τ shows { } τ Xi = sup s : B s = sup B u,. u and he definiion of τ shows τ X i { } = inf s : B s = sup B u..3 u

21 .. TH XISTNC, UNIQUNSS AND DIFFRNTIABILITY OF TH SOLUTION TO. AND TH ABSOLUT CONTINUITY OF TH PROBABILITY LAW OF X T Since B τ X i = B τ X i holds P -a.s.,. and.3 imply ha sup B u = sup B u u u holds P - a.s. Theefoe, we have P τ Xi < < τ Xi < < X i P τ X i,, sup B u = u { } = P sup s : B s = sup B u u sup u B u,, sup B u = sup B u =, u u whee he las equaliy follows fom Poposiion 4 in Secion VI of. This finishes he poof. Le us pove a lemma on he diffeeniabiliy of he maimum of a coninuous pocess which is simila o Poposiion.. of 7 Lemma 3. Fo, le { ˆX s, s, } be a one-dimensional coninuous pocess. Suppose ha i sup s ˆX s <, ii fo any s,, ˆXs D, and sup s D ˆX H <. Then ˆM = sup s ˆXs D, and we have Moeove, if we assume ha D ˆM H sup D ˆX s H s iii { ˆX s, s, } aains is maimum on a unique poin ˆτ,..4 iv fo j d, and almos evey, {D j ˆX s, s, } is coninuous ecep fo s =, and v fo j d, sup s D j ˆX s d <, hen we have whee we have defined D ˆXˆτ := D ˆXs s=ˆτ. Poof. Le { k } k be a dense subse of, and define Define D j ˆM = D ˆXˆτ, a.e.,.5 ˆM n := ma{ ˆX,, ˆX n }. A := { ˆX = ˆM n }, A k := { ˆX ˆM n,, ˆX k ˆM n, ˆX k = ˆM n }, k n.

22 CHAPTR. ABSOLUT CONTINUITY OF TH LAWS OF A MULTI-DIMNSIONAL STOCHASTIC DIFFRNTIAL QUATION WITH COFFICINTS DPNDING ON TH MAXIMUM Then, by he local popey of opeao D we have D ˆM n = n Ak D ˆX k. k= By Poposiion.. of 7, ˆM = sup s ˆXs belongs o D, and D ˆM n D ˆM n in he weak opology of L Ω; L, ; R d unde i and ii. We obain.4 fom D ˆM H lim inf n D ˆM n H. Le us pove.5. Fo ω A k we define ˆτ n := k. Then ˆτ n ˆτ, a.s. due o iii, and we have D ˆM n = n Ak D ˆXˆτ n = D ˆXˆτ n, k= whee we have defined D ˆXˆτ n := D ˆX s s=ˆτ n. Noe ha, if = ˆτ n, hen D ˆXˆτ n is no well defined, due o n he disconinuiy; hus he igoous meaning of he above equaliy is ha D ˆM = D ˆXˆτ n fo almos evey wih pobabiliy. Now le us pove D j ˆXˆτ n u j d D j ˆXˆτ u j d,.6 j= j= fo any u L Ω; L, ; R d. We have D j ˆXˆτ n u j d D j ˆXˆτ u j d = j= j= D j ˆXˆτ n D j ˆXˆτ u j d..7 Fom iv, we have D j ˆXˆτ n D j ˆXˆτ fo ˆτ hen D j ˆXˆτ n D j ˆXˆτ, fo almos evey wih pobabiliy. As D j ˆXˆτ n D j ˆXˆτ sup s D j ˆX s and v, we have j= D j ˆXˆτ n D j ˆXˆτ d n, a.s. j= Due o D j ˆXˆτ n D j ˆXˆτ sup s D j ˆX s and v, we have lim D j ˆXˆτ n D j ˆXˆτ d =. n Then we obain.6. Since D ˆM n we have j= conveges o D ˆM weakly in L Ω; L, ; R d and.6 holds, D j ˆXˆτ D j ˆM u j d =, j= 3

23 .. TH XISTNC, UNIQUNSS AND DIFFRNTIABILITY OF TH SOLUTION TO. AND TH ABSOLUT CONTINUITY OF TH PROBABILITY LAW OF X T fo any u L Ω; L, ; R d. By he fac ha ˆM belongs o D, and v, we have {D ˆXˆτ D ˆM,, } L Ω; L, ; R d. Theefoe we have.5 wih aking u = D ˆXˆτ D ˆM and his finishes he poof. Remak. In Lemma 3, if we assume ha { ˆX s, s, } is adaped, hen we have D ˆXˆτ = fo almos evey such ha > ˆτ. Thus, in his case, we can wie D ˆM =,ˆτ D ˆXˆτ, fo almos evey. Ne, le us pove he diffeeniabiliy of he soluion o. in Malliavin sense. Lemma 4. Assume A-A3. Then, fo s, and i m, Xs, i Ms i belong o D,. Moeove, {DX j s, i s, } saisfies he following equaion: fo s, a.e., and D j X i s = A i j, X, M + + Āi k,lud j X k u + Ãi k,lud j M k u dw l u B i kud j X k u + B i kud j M k u du.8 D i X i s =,.9 fo > s, a.e., whee Āk,lu, Ãk,lu, B k u and B k u ae unifomly bounded and adaped m- dimensional pocesses. Poof. We will use he Picad appoimaion fom Lemma, so X s n, M s n ae he pocesses consuced by ecuence hee. The poof of his lemma uses he poof of Theoem.. of 7. We need o eend he poof o equaion wih coefficiens which depend on he maimum pocess. We sa by poving X s n,i D, fo s,, i m and n. If we assume X s n,i D, and sup u v DX u n,i n,i Hdv < fo s, hen we have M s D, fo s, by Lemma 3 and DA j i lu, X u n, M u n ddu m C DX j u n,k ddu + DM j u n,k ddu = C = C k= m k= C C k= m k= u k= u sup u s sup u s DX j u n,k d j= D j X n,k u DX u n,k Hds DX u n,k Hds du + k=m+ k=m+ d du + k=m+ + k= u sup u s DM j u n,k d du u j= DM u n,k Hds D j M n,k u d du <. 4

24 CHAPTR. ABSOLUT CONTINUITY OF TH LAWS OF A MULTI-DIMNSIONAL STOCHASTIC DIFFRNTIAL QUATION WITH COFFICINTS DPNDING ON TH MAXIMUM by.4. Theefoe, fom Poposiion.3.8 of 7, we have fo s,, X n+,i s D j X n+,i s u s j= = A i j, X n, M n + + Bn,i k udx j u n,k + B n k whee Ān k,l, Ãn n k,l, B k and A and.4, one has u sup = sup j= C C = C = C = C = C u s j= s j= j= j= j= j= k= l= k= u sup u s u sup u s Ān,i k,l udx j u n,k B n,i k ud j M n,k u + Ãn,i k,l du, D, and udm j u n,k dwu l ae unifomly bounded and adaped m-dimensional pocesses. Now, by Ān,i k,l vdx j v n,k l= k= l= l= k= u k= l= k= l= k= l= k= v l= DX n,k v k= u + Ãn,i k,l Ān,i k,l vdx j v n,k Ān,i k,l vdx j v n,k Ān,i k,l vdx j v n,k D j X n,k v D j X n,k v D j X n,k v D j X n,k v H + DM v n,k H and he same compuaion as he above gives u Bn,i k vdx j v n,k + sup u s k= Thus, we have i= sup u s B n,i k DX u n+,i H C + C vdm j n,k + Ãn,i k,l + Ãn,i k,l v + Ãn,i k,l + DM j v n,k dv + DM j v n,k dv + DM j v n,k d + DM j v n,k d C k= dw l v d vdm j v n,k vdm j v n,k dw l v dw l v vdm j v n,k dv d d d dv dv vdm j v n,k dv d C 5 i= sup u v d sup DX u n,k dv, u v k= d sup DX u n,k dv. u v DX u n,i H dv,.

25 .. TH XISTNC, UNIQUNSS AND DIFFRNTIABILITY OF TH SOLUTION TO. AND TH ABSOLUT CONTINUITY OF TH PROBABILITY LAW OF X T and his implies M s n+,i Due o. and Lemma 3, we have sup n DX s n,i D, and sup u v DX u n+,i Hdv < fo s, by Lemma 3. H < and sup n DM n,i sup n sup u s DX u n,i <. By he fac ha X s n,i X i s, M n,i s..3 of 7, Xs i and Ms i belong o D, fo s,. Moeove DX s n,i and DMs i in he weak opology of L Ω; L, ; R d. Le us pove.8. We have DA j i lu, X u, M u ddu C k= s H Ms i in L Ω and Lemma and DM s n,i DX k u H + DM k u H du < convege o DX i s by he same calculaion as., he fac ha DMu k H lim inf n DM u n,k H lim inf n sup v u DX v n,k H holds and.. Theefoe, we have.8 and he poof is compleed. Lemma 5. Assume A-A3. Then, fo {Xs, i s, } and p we have sup D j Xu i p d <,. u s and assumpions i-v of Lemma 3 hold. Moeove, fo s, and p, X i s, M i s D,p. Poof. Fis, le us pove. fo p =. We have sup DX j u i d j= u s s u C + C sup Āi k,l vdx j v k + Ãi k,lvdm j v k u s l= s u + sup Bi k vdx j v k + B kvd i M j v k dv d u s s s C + C Āi k,l vdx j v k + Ãi k,lvdm j v k dv d l= + Bi k vdx j v k + B kvd i M j v k dv d s s C + C D j Xv k + DM j v k dv d = C + C k= D j Xv k + DM j v k ddv k= s v = C + C D j Xv k + DM j v k d dv = C + C m k= k= dw l v d DX k v H + DM k v H dv <,.3 6

26 CHAPTR. ABSOLUT CONTINUITY OF TH LAWS OF A MULTI-DIMNSIONAL STOCHASTIC DIFFRNTIAL QUATION WITH COFFICINTS DPNDING ON TH MAXIMUM by he fac ha DM k v H lim inf n DM n,k v H lim inf n sup u v DX u n,k H is ue,.8 and.. This implies ha v holds fo X i. i follows fom Lemma and we have ii by. fo p =. iii holds due o Lemma and we have iv by.8 and.9. Le us pove. fo p >. I suffices o pove sup i= j= s DX j s i p d C + C u sup i= j= s u Howeve, we ge.4 fom he same compuaion as.3 and an inequaliy DM k u p H C j= u sup s u D j X k u p d, DX j s i p d du..4 which follows fom.8,.9 and.5. Fom. we have X i s, M i s D,p fo s, and p. Now we conside wo m m mai-valued pocess defined by and Z i js = δ i j Y i j s = δ i j + Ā i k,luy k j udw l u + Z i kuāk j,ludw l u B i kuy k j udu, i, j m.5 Z i ku B k j u Āk α,luāα j,ludu, i, j m..6 By he agumen in secion.3 of 7, we have Y s = Zs. Le us epess D j X i s by using Y s and Zs. Lemma 6. Fo s, and i m, j d, D j X i s saisfies DX j s i = Yk i szk k Ak j + Yk i s +Y i k s Zk k Zk k uãk l,ludm j u l dwu l k u B u Āk α,lãα l,ludm j u l du..7 Poof. Fom.8,.5,.6 and Iô s fomula, one has fo i m and j d, k = Z i k sdj X k s = k = + Z i k Ak j, X, M + k = d Z i k, Dj X k u = Zk i Ak j, X, M + + Zk i l k = Z i k uddj X k u + Z i k uāk l,lud j M l dw l u k u B l u Āk α,luãα l,ludm j u l du. 7 k = D j X k u dz i k u

27 .. TH XISTNC, UNIQUNSS AND DIFFRNTIABILITY OF TH SOLUTION TO. AND TH ABSOLUT CONTINUITY OF TH PROBABILITY LAW OF X T By he definiion of.5 and.6, we have heefoe, he esul follows. D j X i s = k,k = Y i k sz k k sdj X k s, Now we pove he absolue coninuiy of he law of X which is he main heoem of his secion. Theoem. Assume A-A3, hen fo >, X has he absoluely coninuous pobabiliy law wih espec o he Lebesgue measue on R m. Poof. Le us pove vt D X d > fo nonzeo veco v R m. By.7 and a ivial inequaliy a + b a b, a, b R, we have v T D X v i Yk i Zk k Ak j =: j= i= v i Yk i Zk k sãk l,lsdm j s l dws l + j= i= v i Yk i Zk k Ak j + A,. j= i= Zk k k s B s Āk α,lãα l,lsdm j l l s ds Then we have A, d ε ε ε + C ε + C ε + Zk k j= i= Zk k ε j= i= v i Yk i Zk k sãk l,lsdm j s l dws l k s B s Āk α,lãα l,lsdm j l j= i= l v i ε Y i k k s B s Āk α,lãα l,lsdm j l l v i ε Y k i ε Zk k Y k i s ds d Zk k sãk l,lsdm j s l dws l s ds d Zk k sãk l,lsdm j l s dw l s d k s B l s Āk α,lãα l,lsdm j s l ds d. 8

28 CHAPTR. ABSOLUT CONTINUITY OF TH LAWS OF A MULTI-DIMNSIONAL STOCHASTIC DIFFRNTIAL QUATION WITH COFFICINTS DPNDING ON TH MAXIMUM Now, one has ε = C C C C C C C εc εc εc Y k i ε k,k,l = k= k,k = k,k,l = k,k,l = k,k,l = k,k,l = k,k,l = k,k,l = k,k,l = k,k,l = ε 3 C m Zk k sãk l,lsdm j l Y k i Y i k k,k,l = ε 3 C m k,k = ε Y i k ε s dw l s d Zk k sãk l,lsdm j l Y i k ε Zk k sãk l,lsdm j l Yk i 4 ε Yk i 4 ε Yk i 4 ε Yk i 4 ε Yk i 4 Yk i 4 s dw l s s dw l s Zk k sãk l,lsdm j l d s dw l s Zk k sãk l,lsdm j l l= l= l= d d s dw l s d Zk k sãk l,lsdm j l s dw l s Zk k sãk l,lsdm j s l ds Zk k sdj Ms l ds d Zk k sdj Ms l 4 ds ε ε Yk i 4 ε d Zk k sdj Ms l 4 ds d Z k k sdj Ms l 4 dsd Z kk s 4 Yk i 4 sup Zk k s 4 s Y i k 4 sup Zk k 4 s 8 s 9 sup ε s l = sup s 4 d DX j s l 4 d ds DX j s l 4 d d sup DX j s l 8 d s 4,

29 .3. TH ABSOLUT CONTINUITY OF TH PROBABILITY LAW OF X I T, M I T and ε C C C Y k i k,k,l = k,k,l = k,k,l = = ε 3 C m Zk k k s B l s Āk α,lsãα l,lsdm j s l ds d Yk i 8 4 sup Zk k 4 s 8 s ε D j M l s ds d Yk i 8 4 sup Zk k 4 s 8 DM j s l dds s ε ε Yk i 8 4 sup Zk k 4 s 8 s ε k,k,l = ε 3 C m k,k,l = ε 3 C m k,k = DM j s l 4 d d ε ε Yk i 8 4 sup Zk k 4 s 8 s Yk i 8 4 sup Zk k 4 s 8 s Y i k 8 4 sup Zk k 4 s 8 s l = ds ε DM j s l 4 d ε DM j s l 4 dds sup DX j s l 4 d s ds. This shows ha ε ε A,d in L Ω as ε ends o. Noe ha we mus choose ε > such ha ε > holds. Theefoe, hee eiss {ε n } n N such ha lim n ε n On he ohe hand, by he coninuiy of A i j, we have lim n ε n ε n j= i= v i Y i k Z k k Ak ε n A, d =, a.s. j, X, M d = v i A i j, X, M fo any nonzeo veco v R m by A3. By Lemma 5 and Theoem he poof is compleed..3 The absolue coninuiy of he pobabiliy law of X i, M i In his secion, we pove he absolue coninuiy of he law of X i, M i, i, i m, in a special case. Tha is: j= i= >,

30 CHAPTR. ABSOLUT CONTINUITY OF TH LAWS OF A MULTI-DIMNSIONAL STOCHASTIC DIFFRNTIAL QUATION WITH COFFICINTS DPNDING ON TH MAXIMUM A4 A i l, i m, l d, do no depend on he second space vaiable, in addiion o A-A3. Remak. Unde A4, Ã k l,l = in.7. The following heoem is he main heoem of his secion. Theoem 3. Assume A-A4. Then, fo > and i, i m, he law of X i, M i is absoluely coninuous wih espec o he Lebesgue measue on R. Poof. Le v, v R\{}. Noe ha, by Lemma 3 and 5, fo >, we have D j M i =,τ i Dj X i. τ i Fis, we assume v, v. By Schwaz s inequaliy and a ivial inequaliy a + b ab, a, b R, we have D v, v X i DX d i DM i DM d i = j= v D j X i d + d τ i v DX j i d j= τ i + v DX j i d j= τ i v DX j i d j= τ i = v v = v v τ i j= τ i v DX j v i DX j i j= j= τ i τ i τ i d + τ i v DX j i d v DX j i d j= τ i v DX j i d τ i τ i j= DX j i d j= τ i DX j i d j= τ i j= τ i τ i v DX j i d τ i DX j i d j= j= j= v DX j i d j= τ i τ i v DX j i d τ i DX j i d + τ i j= DX j i d. DX j i d DX j i d..8 j=

31 .3. TH ABSOLUT CONTINUITY OF TH PROBABILITY LAW OF X I T, M I T Le us pove ha τ i DX j i j= τ i d >, τ i Fom he same compuaion as he poof of Theoem, we ge τ i and τ i τ i C C ε Y i τ i k k,k,l = k,k,l = τ i τ i ε = ε 3 C m k,k,l = ε 3 C m k,k = Theefoe, one has τ i ε D j X i τ i τ i d {τ i ε>} Y i τ i ε k Zk k Ak τ i τ i ε Y i Zk k k s B l sdj M l τ i k s ds DX j i d >, a.s..9 j= j, X d {τ i Zk k k s B l sdj M l d {τ i ε>} Yk i 8 4 sup Zk k 4 s 8 s Yk i 8 4 sup Zk k 4 s 8 s τ i D j M l s ds τ i d τ i τ i ε Yk i 8 4 sup Zk k 4 s 8 s Y i k 8 4 ε τ i τ i ε Y i sup Zk k 4 s 8 s τ i k τ i τ i ε d {τ i l = τ i Zk k k s B l sdj M l τ i ε ε>} s ds τ i ε>} τ i d {τ i ε>} B l k sdj Ms l ds D j M l s ds sup DX j s l 4 d s s ds d {τ i ε>}., d d {τ i {τ i ε>} ε>},.3 as ε ends o. By.3 and he poof of Theoem, hee eiss {ε i n} n N and {ε i n} n N such ha ε i n, ε i n n and lim DX j i d A i n j, X.3 ε i n ε i n j= j=

32 CHAPTR. ABSOLUT CONTINUITY OF TH LAWS OF A MULTI-DIMNSIONAL STOCHASTIC DIFFRNTIAL QUATION WITH COFFICINTS DPNDING ON TH MAXIMUM and lim n ε i n τ i τ i εi n DX j i j= τ i d {τ i εi n >} A i j τ i, X τ i,.3 almos suely, whee we have used he fac ha lim n {τ i εi n >} =, a.s., which is a consequence of Lemma. By Lemma, we have > τ i, hus, hee eiss N N such ha τ i ε i n j= τ i εi D j X i d εi n DX j i j= τ i d εi n ε i n > τ i, j= A i j, X, j= A i j τ i, X τ i, fo any n N, almos suely. This implies ha.9 is ue. Theefoe, he igh hand side of.8 is sicly posiive fo any v R such ha v, v. Second, in he case v = o v = we have Theefoe, we obain D v, v X i DX d i DM i DM d d = i by.3 and.3. This finishes he poof. τ i j= { d j= v DX j i d, if v =, d j= v DM j i d, if v =. D v, v X i DX d i DM i DM d d >, a.s. i Remak 3. The geneal m-dimensional sudy of he law of X,, X m, M,, M m does no follow wih he agumens pesened hee, due o he paicula sucue used in he calculaion of.8. Indeed, in he poof of Theoem 3, we have used an inequaliy a + b ab, a, b R. Coollay. Unde A-A4, by he same calculaion as ha in Theoem 3, fo > and i i m, we can pove he absolue coninuiy of he law of M i, M i condiioned by he se {τ i τ i }. Now we give an eample fo A i l and Ai l ha {τ i τ i } holds, a.s. ample. Fo each k = i, i, le {X k s, s, } saisfies X k s = k + whee A k k is a nonzeo consan hen τ i τ i, a.s. B k u, X u, M u du + A k kw k s, Poof. By Gisanov heoem, he independence of Bownian moions, and he eplici densiy funcion fo τ k, k = i, i Poblem 8.7 in Chape of, we obain he eisence of he densiy funcion fo τ i τ i. Then we have P τ i = τ i =. 3

33 .4. A CONCLUDING RMARK.4 A concluding emak In his chape, we poved he absolue coninuiy of he law of X and X, i M i wih Lipschiz coefficiens unde some addiional assumpions. We end his chape wih some emaks on he law of he maimum of pocesses. Thee ae some heoeical and applicable esuls abou he law of he maimum of coninuous pocesses. In 7 he smoohness of he densiy funcion of he maimum of he Wiene shee is poven. In 9, auhos deived some inegaion by pas fomulae involving he maimum and minimum of a one dimensional diffusion o compue he sensiiviies of he pice of financial poducs wih espec o make paamees called Geeks. Recenly, he smoohness of densiy funcion of he join law of a muli-dimensional diffusion a he ime when a componen aains is maimum ime was poven in. In hese aicles, Gasia-Rodemich-Rumsey s lemma Lemma A.3. of 7 plays an impoan ole o obain he esuls. 4

34 Chape 3 Volailiy isk fo opions depending on eema and is esimaion using kenel mehods 3. Inoducion The Black-Scholes model has been widely used by paciiones due o is simpliciy and he eisence of some eplici pobabiliy densiy funcions concening he model. This model assumes a consan volailiy. Howeve, in he opion make daa, we obseve ha he volailiy can no be a consan. This phenomenon is ofen called volailiy smile afe he shape of obseved daa-implied volailiies see 5 o 8. Fo his eason, i is naual o conside a geneal model which may pefom bee han he Black-Scholes model. On he ohe hand, in a geneal model, usually one knows neihe he associaed eplici densiy funcions no eplici fomulas fo opion pices. Theefoe, he isks involved in opions, called Geeks, can only be compued hough numeical appoimaions. In his chape, we conside he sensiiviy of he model o changes in he volailiy paamee fo opions depending on he eema maimum o minimum. We call his sensiiviy he vega inde and we focus ou discussion on he calculaion of he vega inde. In a geneal model, he volailiy is no a consan and his makes he discussion complicaed mahemaically. We inoduce a peubaion paamee o conside he diecional deivaives fo he diffusion coefficiens o calculae he vega inde. In paicula, his poblem has been discussed by some auhos. In 6, he auhos obained a fomula o calculae he vega inde fo opions whose payoffs depend on he pices of undelying a fied imes hough Malliavin calculus. Ohe Geeks, such as dela and gamma, which ae defined by he sensiiviies wih espec o he cuen pice of he undelying, fo opions depending on he eema ae discussed in 9. In, a fomula o compue he vega inde was obained in he case of opions wih payoffs depending on he undelying smoohly e.g. Asian ype opion by using Malliavin calculus. Howeve, he vega inde fo opions depending on he eema has no been consideed ye, since he eema of a diffusion pocess is no sufficienly smooh and heefoe difficul o ea fom he mahemaical poin of view. In mahemaical finance vaious cedi linked and baie ype poducs have his kind of feaue. Thee ae mainly wo goals in his chape: One is o conside vaious opions which may depend on 5

35 3.. INTRODUCTION he eema of he undelying and obain some financial conclusions abou he popeies of he vega inde in a one-dimensional model. The ohe is o give a mehodology o compue he vega inde fo a specific opion by using so-called kenel mehods. To sudy he sucue of vega inde, we daw he vega isk pofiles in he one-dimensional model and compae he vega inde obained in his one-dimensional model wih he one in he Black-Scholes model see Table 3. and Figue 3.4. Accoding o Table 3. and Figue 3.4, hese diffeen models give diffeen values of he vega inde, even if he payoff funcions ae he same, and his diffeence is cucial fo paciiones, since in pacice hedging pocedues ae done based on he value of vega inde obained in each model. Technically, in his chape, we conside a one-dimensional sochasic diffeenial equaion SD wih ime-independen coefficiens as he dynamics of an asse pice unde he picing measue P. The esuls obained in his chape may be a beakhough o sudy he Geeks in so-called sochasic volailiy models which ae ofen used by paciiones see 4, fo a elaionship beween one-dimensional models and sochasic volailiy models. To deal wih he eema of diffusion pocess, we use he Lampei mehod see ecise 5.. of, fo eample. Tha is, fis we ansfom he SD using Gisanov s heoem o a Saonovich ype SD wihou dif coefficien which can hen be epessed as a monoone ansfomaion of a Wiene pocess. This mehod is diffeen fom he one consideed in 9 whee he Gasia-Rodemich-Rumsey s lemma see Lemma A.3. of 7 plays an impoan ole. Alhough echniques used in 9 ae quie ineesing, he fomulas obained hee have high compuaional compleiy. Howeve, he fomula obained in his chape is much simple. By woking unde a new measue, we can epess he eema of diffusion pocess in a simple fashion and calculae he diecional deivaives. In addiion, we use he dualiy fomula of Malliavin calculus as i appeas in 7, Page 37 o obain a fomula ha may give a bee epession o he vega inde fo some numeical mehods such as Mone Calo simulaion. The fomula of he vega inde obained in his chape allows one o decompose i ino hee componens: he eema and mauiy feaue of opions, and a by-poduc of he Gisanov ansfomaion. The inenion of he cuen eseach is o y o eveal some popeies of he sucue of hese hee componens fo ealisic opions. Though simulaion sudies in Secion 3 of his chape, one can see ha he decomposiion of he vega inde fo baie ype opions has some ineesing popeies. Fo eample, when we conside an up-in call opion, ou Mone Calo analysis shows ha fo he opion wih lowe baie, he vega inde is mosly conveyed by he mauiy feaue of he payoff, while fo he opion wih highe baie, he eema feaue conols mos of he vega inde. We can see he eisence of a baie ha deemines which componen in he decomposiion is of mos impoance see Figue 3.. Moeove, we obseve ha fo he opions wih sho mauiy, we have o pay moe aenion o he change of he value of vega inde wih espec o he mauiy see Figue 3. and 3.3. These esuls seem o be valid among seveal ypes of opions, accoding o ou numeical epeimens. Unfounaely, each componen of he decomposiion fomula obained hee fo binay baie opions involves he Diac dela funcionals, heefoe, we give a mehod o appoimae he dela funcionals called kenel mehods. The kenel mehod is quie effecive o some numeical poblems appeaing in vaious fields such as finance. A basic kenel mehod o esimae pobabiliy densiy funcions is given in 8, Chape -4, and i is applied in 3 o compue he Geeks fo opions wih disconinuous payoffs. To addess his mehod, we shall define an esimao fo he dela funcional by using a so-called kenel funcion and bandwidh paamee. The bandwidh paamee conols he bias and vaiance of he esimao, heefoe, is choice is quie impoan fo using he kenel mehod. In ode o choose he bes bandwidh, we define an asympoic mean squaed eo AMS as an eo fo he esimao, hen we look fo a bandwidh so ha he AMS is as small as possible. If hee eiss a bandwidh 6

36 CHAPTR 3. VOLATILITY RISK FOR OPTIONS DPNDING ON XTRMA AND ITS STIMATION USING KRNL MTHODS which minimizes he AMS, hen we call i he opimal bandwidh. A heoem o epess he opimal bandwidh is saed as he main heoem of Secion 4 and some numeical esuls obained by he kenel mehod ae also given. This chape is oganized as follows. In Secion, we povide he mahemaical esul on he decomposiion of vega inde. In Secion 3, we cay ou Mone Calo simulaions and obain some esuls on he sucue of vega inde as menioned in he pevious paagaph. In Secion 4, we conside a binay baie opion and discuss he kenel mehod. Then we apply i o he compuaion of he vega inde fo his opion. Some numeical esuls obained wih he kenel mehod ae given in Secion 5. In he Appendices, we give some lemmas and poofs of ou esuls. Thoughou he chape, we use Cb k A, B o denoe he space of B-valued k imes coninuously diffeeniable funcions defined on A wih bounded deivaives. Fo a diffeeniable funcion F fom R m o R whee m N, we define i F := F i fo R m and i m. The lees C and C i, i N denoe posiive consans which may depend on f, p, and T ha will appea in his chape, and he values of C and C i may change fom line o line. We define R + :=, and P as he epecaion unde a pobabiliy measue P. 3. Main esul: Vega inde fo opions depending on he eema Le Ω, F, P be a complee pobabiliy space which suppos a one-dimensional Wiene pocess {W,, T }. Fo σ, ˆσ, b : R R and >, we conside he following sochasic diffeenial equaion SD in sho, { ds ε = bs ε d + σ ε S ε dw S ε =, 3. whee σ ε is of he fom σ ε z = σz + εˆσz, ε,. Fo f : R R, we conside he quaniy Π ε := P fma T S ε, ST ε. We assume he following hypoheses. H σ, ˆσ C b R +, R + and b C b R +, R +. H Thee eiss σ > such ha σ ε y σ y fo all y R + and ε,. H3 Thee eiss, and σ > such ha σ y σ ε y σ y, fo all y R + such ha y <. H4 The funcion b σ ε H5 f C b R +, R +. is bounded. Noe ha by H, fo all ε,, 3. has a unique song soluion, and le S ε = {S ε,, T } be he soluion o 3.. In finance, Π ε defines a peubed opion pice wih a payoff funcion f. We conside he quaniy Πε ε ε= and we call his he vega inde of his opion. Ou main esul is he following heoem. I gives he decomposiion fomula fo vega inde. 7

37 3.. MAIN RSULT: VGA INDX FOR OPTIONS DPNDING ON TH XTRMA Theoem 4. Assume he above hypoheses H-H5. Then he following epession fo vega inde is valid. Π ε ma S ε = P f ma S, S T σ ma S T ˆσ T T ε= σ ydy ST + P f ma S ˆσ, S T σs T T σ ydy T + P f ma S, S T Y η I,η + f ma S, S T Y T T T { b σ bσ σ σ S S Y ˆσ σ ydy bˆσ σ + σˆσ S Y } d whee S := S, Y := S and η := ag ma T S. Befoe giving he poof of heoem, le us sae some emaks and pove some pepaaoy lemmas. Remak 4. A. One can pove ha η in Theoem 4 is almos suely unique by using Theoem 5 of his chape and he fac ha he ime a which he maimum of a one-dimensional Wiene pocess ove, T is aained is almos suely unique see Remak.8.6 of. B. We have given he above heoem a geneal mahemaical fom. In finance, we should assume ha P is he equivalen maingale measue, he inees ae is zeo fo convenience and ha S ε is a maingale. Then, in ha paicula case, Π ε has he inepeaion of a peubed opion pice. Fuhemoe, as ou main goal is o descibe he sucue of vega inde, we do no discuss he possible mahemaical eensions o payoff funcions f o avoid cumbesome echnicaliies and long agumens. In Secion 3, we will obain some numeical esuls fo iegula f wih special b, σ and ˆσ. This eension can be done by using he eplici densiy funcions fo ma T S and ma T S, S T. The poof of his eension can be found in Appendi B. C. The fis em of 3. comes fom he diffeeniaion wih espec o he maimum of asse pice, he second em is due o he mauiy pice of he asse and he hid em is a esul of a change of measue. We call hese hee ems eema sensiiviy, eminal sensiiviy and dif sensiiviy, especively. D. When we conside he following measue change dqε dp := ep { T σ ε b σ ε S ε dw whee we have defined σ ε := σ ε, hen unde Qε σ Ŵ ε ε := W b σ ε Sudu ε T σ ε b } σ ε S ε d, is a one-dimensional Wiene pocess. Noe ha due o he boundedness fo b condiion is clealy saisfied. Then unde Qε, S ε can be wien as { ds ε = σε σ ε S ε d + σ ε S ε dŵ ε = σ ε S ε dŵ ε, S ε =, 8 σ ε 3. and σ ε he Novikov

38 CHAPTR 3. VOLATILITY RISK FOR OPTIONS DPNDING ON XTRMA AND ITS STIMATION USING KRNL MTHODS whee dŵ ε denoes Saonovich inegal. Finally, we can wie Π ε as follows, Π ε = P f ma T Sε, ST ε = Qε f ma T Sε, ST ε dp. dqε. Unde Qε, S ε is diven by Ŵ ε and he disibuion of Ŵ ε does no depend on ε unde Qε. Le Ω, F, Q be anohe complee pobabiliy space and W be a one-dimensional Wiene pocess unde Q. Then Π ε is wien as Π ε = Qf ma T Sε, S ε T epx ε T, whee S ε saisfies { ds ε = σ ε S ε d W S ε =, 3.3 and T σ XT ε ε = b σ ε S ε d W T σ ε b σ ε S ε d. We use he SD of he fom 3.3 o wie down S ε wih only W so ha we can epess ma T S ε by ma T W. Fom now on, we use he noaion X T := X T. We inoduce a funcion F ε which is used o epess he soluion o 3.3 in an eplici fom. Definiion. Lampei ansfom Fo ε,, define F ε : R + R as F ε z := z σ ε y dy. Noe ha he invese funcion Fε eiss, since F ε is a coninuous monoone inceasing funcion due o he assumpion H. Fuhemoe, i is clea ha F ε and Fε ae diffeeniable wih espec o z and we have F ε z z = σ ε z, F ε z z = F ε In his se-up one has he following esul. z Fε z = σε F ε z. 3.4 Theoem 5. Unde hypohesis H-H, hee eiss a unique song soluion o 3.3. Fuhemoe, unde Q, he soluion o he SD 3.3 can be wien as follows, Theefoe, one has ma T S ε = F ε F ε + ma T W. S ε = F ε F ε + W

39 3.. MAIN RSULT: VGA INDX FOR OPTIONS DPNDING ON TH XTRMA Poof. By 3.4 and applying Iô s fomula fo Saonovich inegal o Fε F ε + z, i is easy o see ha 3.5 is a soluion o 3.3. On he ohe hand, if hee eiss a soluion o 3.3, hen, again by applying Iô s fomula fo Saonovich inegal o F ε z, he soluion can be epessed by 3.5. Thus, one obains he uniqueness. F ε The equaliy ma T S ε z. = F ε F ε + ma T W is a conclusion of he monooniciy fo Remak 5. Alhough he epesenaion of Fε F ε + W is clealy coninuous in, ε, his does no imply he coninuiy of he soluion o 3.3 in ε, since he ecepional se such ha Iô s fomula does no hold may depend on ε. To ovecome his poblem, we modify he soluion o 3.3 o be coninuous in, ε. This pocedue will be done in Appendi A.. The above epesenaion is he key fomula which allows us o obain Theoem 4. Ou ne sep is o sae some esuls on he egulaiy of S ε and XT ε wih espec o ε and he echange beween Q and ε ε=. The following fou lemmas will be lised and hei poofs can be found in Appendi A. Lemma 7. Le H-H be saisfied. Le S ε be he soluion o 3.3. Then, ma T Sε is diffeeniable wih espec o ε, and he following equaion holds, ma ε T Sε = σ ma T ε= S ma T S ˆσ ydy, a.s. σ Lemma 8. Le H-H be saisfied. Le S ε be he soluion o 3.3. Then, S ε is diffeeniable wih espec o ε, and we have Z := Sε ε = σs ε= S ˆσ ydy,, T, a.s. 3.6 σ Lemma 9. Le H-H3 be saisfied. Then, XT ε is diffeeniable wih espec o ε, and i holds ha XT ε ε = T σ S Z + ˆσ S d ε= W + + T T T b σs Z bs σ S Z + ˆσS σ d S W σ S σ S Z + ˆσ S d b σ σ b σ S b σs Z bs σ S Z + ˆσS σ d, a.s. S Fom he above hee lemmas one can see he coespondence of he deivaives o 3.. In fac, he hee lemmas coespond o he deivaive of he maimum, he undelying a mauiy and he change of measue wih espec o ε, especively. In addiion o he above lemmas, we need he following lemma abou he echange beween Q and ε ε=. 3

40 CHAPTR 3. VOLATILITY RISK FOR OPTIONS DPNDING ON XTRMA AND ITS STIMATION USING KRNL MTHODS Lemma. Le H-H5 be saisfied. Then, we have he following equaion, Qf ma ε T Sε, ST ε epxt ε= ε = Q ε f ma T Sε, S ε T epx ε T ε= Now, le us pove he main heoem. Poof of Theoem 4. As menioned in Remak 4., Π ε is epessed using Q which does no depend on ε, heefoe, we have Π ε ε = Q ε= T Sε, ST ε ma ε T Sε epxt ε i ε= T Sε, ST ε Sε ε T epxε T ii ε= T Sε, ST ε ε epxε T iii. ε= f ma + Q f ma + Q f ma i.by Theoem 5 and Lemma 7 we obain ha i equals he fis em on he igh of 3. afe he applicaion of Gisanov s heoem. ii.by Lemma 8 as in he poof of i, he esul is ivial. iii.by Lemma 9, Xε T ε XT ε T ε ε= = T σ b σ ε= is T σ S Z + ˆσ S d W + S. b σs Z bs σ S Z + ˆσS σ d S W { σ S Z + ˆσ S b σs Z bs σ S Z + ˆσS σ S whee Z is defined by 3.6. By Lemma 8, we can epess Z as follows; whee G is defined as Gz := } d 3.7 Z = σgs. 3.8 z ˆσ σ ydy. Using 3.7, 3.8 and he dualiy fomula in Malliavin calculus we obain { T iii = Q σσ f ma S G + ˆσ, S T epx T σ b G bσσ G + ˆσ T σ S d W T σ b σσ G + ˆσ σ b G bσσ } G + ˆσ σ σ S d { T = ˆP σσ f ma S G + ˆσ, S T σ b G bσσ } G + ˆσ T σ S db T } = ˆP D f ma S, S T {b bσ T σ σσ bˆσ G σ + ˆσ S d, 3

41 3.. MAIN RSULT: VGA INDX FOR OPTIONS DPNDING ON TH XTRMA whee we have used a change of measue d ˆP d Q := epx T and B is a ˆP -Wiene pocess. Moeove, in he above equaliy, D denoes Malliavin deivaive opeao wih espec o B. Fuhemoe, we have D f ma S, S T T Due o Theoem.. of 7 and Lemma 3, we obain whee Y := S Finally we obain 3.. = f ma S, S T D ma S + f ma S, S T D S T. T T T D ma T S and η := ag ma T Sε, and = Y η Y σs I,η, D S T = Y T Y σs. Remak 6. In he calculaion of iii, i is clea ha we can avoid he appeaance of he deivaive of f wihou he dualiy fomula and obain { T iii = ˆP σσ f ma S G + ˆσ, S T σ b G bσσ } G + ˆσ T σ S db. Howeve, we sill pefe o avoid he sochasic inegals in 3.9 in ode o obain he sabiliy of Mone Calo esimaes. Remak 7. We can apply he above echnique o obain he epesenaion of he vega inde fo opions whose payoffs depend on he minimum of he asse. We define Π ε := P f min T Sε, ST ε, hen we have he following fomula fo he vega inde, Π ε ε = P ε= f min T S, S T σ min T S ST + P f min S, S T σs T T + P T min T S ˆσ σ ydy ˆσ σ ydy f min T S, S T Y η I, η + f min T S, S T Y T { b σ bσ σ σ S S Y ˆσ σ ydy bˆσ σ + σˆσ S Y } d, whee η := ag min T S. This fomula povides possibiliies o applicaions o cedi linked poducs