Research Article A Generic Decomposition Formula for Pricing Vanilla Options under Stochastic Volatility Models
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1 Hindawi Pblishing Corporaion Inernaional Jornal of Sochasic Analysis Volme 15, Aricle ID 13647, 11 pages hp://dx.doi.org/1.1155/15/13647 Research Aricle A Generic Decomposiion Formla for Pricing Vanilla Opions nder Sochasic Volailiy Models Raúl Merino 1, and Josep Vives 1 1 Facla de Maemàiqes, Universia de Barcelona, Gran Via 585, 87 Barcelona, Spain VidaCaixa S.A., Invesmen Conrol Deparmen, Jan Gris, -6, 814 Barcelona, Spain Correspondence shold be addressed o Josep Vives; josep.vives@b.ed Received 7 March 15; Acceped 3 May 15 Academic Edior: Jiongmin Yong Copyrigh 15 R. Merino and J. Vives. his is an open access aricle disribed nder he Creaive Commons Aribion License, which permis nresriced se, disribion, and reprodcion in any medim, provided he original work is properly cied. We obain a decomposiion of he call opion price for a very general sochasic volailiy diffsion model, exending a previos decomposiion formla for he Heson model. We realize ha a new erm arises when he sock price does no follow an exponenial model. he echniqes sed for his prpose are nonanicipaive. In pariclar, we also see ha eqivalen resls can be obained by sing Fncional Iô Calcls. Using he same generalizing ideas, we also exend o nonexponenial models he alernaive call opion price decomposiion formla wrien in erms of he Malliavin derivaive of he volailiy process. Finally, we give a general expression for he derivaive of he implied volailiy nder boh he anicipaive and he nonanicipaive cases. 1. Inrodcion Sochasic volailiy models are a naral exension of he Black-Scholes model in order o manage he skew and he smileobservedinrealdaa.iiswellknownhainhese models he average of fre volailiies is a relevan qaniy. See, for example, [1, Chaper ]. Unfornaely, adding a sochasic volailiy srcre makes pricing and calibraion more complicaed, as closed formlas do no always exis. See, for example, [], for a firs reference on his opic. Moreover, even when hese formlas exis, like for he Heson model (see [3]), in general, hey do no allow a fas calibraion of he parameers. Dring las years, differen developmens for finding approximaions o he closed-form opion pricing formlas have been pblished. Malliavin echniqes are narally sed o solve his problem in [4, 5] asheaveragefrevolailiy is an anicipaive qaniy. Oherwise, a nonanicipaive mehod o obain an approximaion of he pricing formla is developed for he Heson model in [6]. he mehod is based on he se of he adaped projecion of he average fre volailiy. As a resl, he model allows obaining a decomposiion of he call opion price in erms of sch fre volailiy. In he presen paper, we generalize [6] o general sochasic volailiy diffsion models. Similarly, following he same kind of ideas, we exend he expansion based on Malliavin calcls obained in [4, 5]. his is imporan becase Heson modelisnoheniqesochasicvolailiymodelcrrenly sed in pracice, and some of hem, like SABR model, are no of exponenial ype. For a general discssion abo sochasic volailiy models in pracice, see [7]. he main ideas developed in his paper are he following: (i) A generic call opion price decomposiion is fond wiho having o specify he volailiy srcre. (ii) A new erm emerges when he sock opion prices do no follow an exponenial model, as, for example, in he SABR case. (iii) he Feynman-Kac formla is a key elemen in he decomposiion. I allows expressing he new erms ha emerge nder he new framework (i.e., sochasic volailiy) as correcions of he Black-Scholes formla. (iv) he decomposiion fond sing Fncional Iô calcls appears o be he same as he decomposiion obained hrogh or echniqes.
2 Inernaional Jornal of Sochasic Analysis (v) A general expression of he derivaive of he implied volailiy, boh for nonanicipaive and anicipaive cases, is given.. Noaions Le S = {S(), [,]}be a sricly posiive price process nder a marke chosen risk neral probabiliy ha follows he model ds () (ii)weseinallhepaperhenoaione [ ] := E[ F ], where {F, } is he naral filraion of S. (iii) In or seing, he call opion price is given by V () =e rτ E [(S () K) + ]. (6) (iv) Recalling ha from he Feynman-Kac formla, he operaor L θ := + 1 θ (, S (),σ()) S +μ(, S ()) S r (7) =μ(, S ()) d +θ(, S (),σ()) (ρdw () + 1 ρ db ()), (1) saisfies L θ BS(, S(), θ(, S(), σ())) =. (v)wewillalsosehefollowingdefiniionsfory : where W and B are independen Brownian moions, ρ ( 1, 1), μ:[,] R + R, θ:[,] R + R +,andσ() is a posiive sqare-inegrable process adaped o he filraion of W. Weassmeonμ and σ sfficien condiions o ensre he exisence and niqeness of he solion of (1).Noiceha we do no assme any concree volailiy srcre. hs, or decomposiions can be adaped o many differen models. In pariclar, we cover he following models: (i) Black-Scholes model: μ(, S()) := rs(), θ(, S(), σ()) := σs(), ρ=, r>, and σ>. (ii) CEV model: μ(, S()) := rs(), θ(, S(), σ()) := σs() β wih β (, 1], ρ=, r>, and σ>. (iii) Heson model: μ(, S()) := rs(), θ(, S(), σ()) := σ()s(), r>, σ>, and dσ () =k(θ σ ()) d + ] σ ()dw (), () where k, θ, and] are posiive consans saisfying he Feller condiion kθ > ]. (iv) SABR model: μ(, S()) := rs(), θ(, S(), σ()) := σ()s() β wih β (, 1], r>, σ>, and dσ () =ασ() dw () (3) wih α>. For exisence and niciy of he solion in he Heson case,see,forexample,[8, Secion.]. For he CEV and SABR models, see [9] and he references herein. he following noaion will be sed in all he paper: (i)wewilldenoebybs(, S, σ) he price of a plain vanillaeropeancallopionnderheclassicalblack- Scholes model wih consan volailiy σ,crrensock price S,imeomariyτ=,srikepriceK,and ineres rae r.inhiscase, BS (, S, σ) =SΦ(d + ) Ke rτ Φ(d ), (4) where Φ( ) denoes he cmlaive probabiliy fncion of he sandard normal law and d ± = ln (S/K) +(r±σ /)τ. (5) σ τ G(,S(),y):=S () SBS (, S (),y), H(,S(),y):=S() S G(,S(),y), K(,S(),y):=S () SG(,S(),y), L(,S(),y):= θ(,s(),y). S () 3. A Decomposiion Formla Using Iô Calcls In his secion, following he ideas in [6], we exend he decomposiion formla o a generic sochasic volailiy diffsion process. We noe ha he new formla can be exended wiho having o specify he nderlying volailiy process, obaining a more flexible decomposiion formla. When he sock price does no follow an exponenial process, a new erm emerges. he formla proved in [6]is a pariclar case. I is well known ha if he sochasic volailiy process is independen from he price process, hen he pricing formla of a plain vanilla Eropean call is given by (8) V () = E [BS (, S (), σ ())], (9) where σ () isheso-calledaveragefrevarianceandiis defined by σ () := 1 σ (s) ds. (1) Narally, σ() is called he average fre volailiy. See [1, page 51]. he idea sed in [6] consiss in considering he adaped projecion of he average fre variance V () := E (σ ()) = 1 E [σ (s)] ds (11) o obain a decomposiion of V() in erms of V(). hisidea swiches an anicipaive problem relaed wih he anicipaive process σ() ino a nonanicipaive one wih he adaped process V().We apply his echniqe o or generic sochasic differenial eqaion (1).
3 Inernaional Jornal of Sochasic Analysis 3 heorem 1 (decomposiion formla). For all [,),we have V () =BS(, S (), V ()) + 1 S BS (, S (), V ()) [L (, S (),σ()) d σ () d] e r (S () E [ e r( ) G (, S (), V ()) (L (, S (),σ()) σ ())d]+ 1 8 E [ e r( ) K (, S (), V ()) d [M, M] ()] + ρ E [ e r( ) L (, S (),σ()) (1) S (S () SBS (, S (), V ()))) d [M, M] () + ρ e r θ (, S (),σ()) ( S (S () S BS (, S (), V ()))) d [W, M] (). (16) aking condiional expecaion and mliplying by e r,we have H(, S (), V ()) d [W, M] ()], where M() := E [σ (s)]ds = σ (s)ds + ( )V(). Proof. Noice ha e r BS(, S(), V()) = e r V(). As e r V() is a maringale, we can wrie e r V () = E (e r V ()) = E (e r BS (, S (), V ())). (13) Or idea is o apply he Iô formlaoheprocesse r BS(, S(), V()). As he derivaives of BS are no bonded, we have o se an approximaion o he ideniy changing BS(, S, σ) by BS n (, S, σ) := BS (, S, σ) ψ n (S), (14) E [e r( ) BS (, S (), V ())] =BS (, S (), V ()) + 1 E [ e r( ) S () SBS (, S (), V ()) (L (, S (),σ()) d σ ())d]+ 1 8 E [ e r( ) (S () S (S () SBS (, S (), V ()))) d [M, M] ()] + ρ E [ e r( ) θ (, S (),σ()) S (S () SBS (, S (), V ()))d[w, M] ()]. (17) where ψ n (S) = φ((1/n)s) for some φ C b sch ha φ(s) = 1 for all S < 1andφ(S) = forall S >, and V() by V ε () = (1/( ))(δ + E[σ (s)ds]), whereε >, and finally apply he dominaed convergence heorem. For simpliciy, we skip his mollifying argmen across he paper. So, applying Iô formla, sing he fac ha σ BS (, S, σ) =S στ SBS (, S, σ) (15) and he Feynman-Kac operaor (7),wededce e r BS (, S (), V ()) e r BS (, S (), V ()) = e r L VS BS (, S (), V ()) d + e r S BS (, S (), V ()) θ (, S (),σ()) (ρdw() + 1 ρ db ())+ 1 e r S () S BS (, S (), V ()) dm () + 1 e r S () Remark. In [6], he following operaors are defined for X() = log S(): (i) G(, X(), σ()) := ( x x)bs(, X(), σ()). (ii) H(, X(), σ()) := ( 3 x x )BS(, X(), σ()). (iii) K(, X(), σ()) := ( 4 x 3 x + x )BS(, X(), σ()). We observe he following: (i) G(, X(), σ()) = G(, S(), σ()). (ii) K(, X(), σ()) = K(, S(), σ()). (iii) H(, X(), σ()) = H(, S(), σ()). Remark 3. We have exended he decomposiion formla in [6] ohegenericsde(1). When we apply Iô Calcls,we realize ha Feynman-Kac formla absorbs some of he erms ha emerge. I is imporan o noe ha his echniqe works foranypayofforanydiffsionmodelsaisfyingfeynman-kac formla.
4 4 Inernaional Jornal of Sochasic Analysis Remark 4. Noe ha when θ(, S(), σ()) = σ()s() (i.e., he sock price follows an exponenial process), hen V () = BS (, S (), V ()) E [ e r( ) K (, S (), V ()) d [M, M] ()] + ρ E [ e r( ) σ () H (, S (), V ()) d [W, M] ()], and he erm 1 E [ e r( ) S () SBS (, S (), V ()) (L (, S (),σ()) σ ())d] (18) (19) vanishes. Indeed, we show ha, de o he se of Feynman-Kac formla, Movemen of he asse + Movemen of he volailiy = 1 θ(,s ) SBS (, S (), V ()) d + σ BS (, S (), V ()) dv () = 1 σ () S () S BS (, S (), V ()) d + 1 S () S BS (, S (), V ()) (dm + V d σ d) = 1 S () S BS (, S (), V ()) (dm + V d), () where (1/)S () S BS(, S(), V())V is sed ino he Feynman-Kac formla and 1 E [ S SBS (, S (), V ()) dm] =. (1) 4. Basic Elemens of Fncional Iô Calcls In his secion, we give he insighs of he Fncional Iô Calcls developed in [1 13]. Le X:[,] Ω R be an Iô process, ha is, a coninos semimaringale defined on a filered probabiliy space (Ω, F,(F ) [,], P) which admis he sochasic inegral represenaion X () =x + μ () d + σ () dw (), () where W is a Brownian moion and μ() and σ() are coninos processes, respecively, in L 1 (Ω [,])and L (Ω [,]). We define D([,],R) he space of cadlag fncions. Given a pah x D([,],R), wewilldenoebyx is resricion o [,].Forh, he horizonal exension x,h is defined as x,h () =x () =x(), [,[ ; (3) x,h () =x(), (, + h] and he verical exension is defined as x h () =x () =x(), [,[ ; x h () =x() +h, ha is xh () =x() +h1 {=}. (4) AprocessY:[,] R R,progressivelymeasrablewih respec o he naral filraion of X, may be represened as Y () =F(, {X (s), s }) =F(,X ) (5) for a cerain measrable fncional F:[,] D([,],R) R. LeF be he space of local Lipschiz fncionals wih respecohenormofhespremmond([,+h],r);ha is, here exiss a consan C> sch ha for any compac K and for any x D([,],K)and y D([,+h],K)we have F(,x ) F(+h,y +h ) C x,h y. (6) +h Under his framework, we have he nex definiions of derivaive. Definiion 5 (horizonal derivaive). he horizonal derivaive of a fncional F F a is defined as F(+h,x,h ) F(,x ) D F(,x )= lim. (7) h + h Definiion 6 (verical derivaive). he verical derivaive of a fncional F F a is defined as h F(,x x F(,x )= lim ) F(,x ). (8) h + h Of corse we can consider ieraed derivaives as xx. We also have he following Iô formla ha works for nonanicipaive fncionals: heorem 7 (Fncional Iô Formla). For any nonanicipaive fncional F F and any [,],wehave F(,X ) F(,X ) = D F(,X )d+ x F(,X )dx() + 1 xx F(,X )d[x, X] (), provided ha D F, x F,and xx F belong o F. Proof. See [11, 1]. (9)
5 Inernaional Jornal of Sochasic Analysis 5 5. A General Decomposiion Using Fncional Iô Calcls In his secion, we apply Fncional Iô Calcls o he problem of finding a decomposiion for he call opion price. he decomposiion problem is an anicipaive pah-dependen problem. Using a smar choice of he volailiy process ino he Black-Scholes formla, we can conver i ino a nonanicipaive one. I is naral o wonder wheher he Fncional Iô Calcls brings some new insides ino he problem. We consider he fncional F(,S(),σ )=e r BS (, S (),f(,σ )), (3) where σ is he pah-dependen process and f F is a nonanicipaive fncional. Under his framework, we calclae he derivaives sing Fncional Iô Calcls wih respec o he variance. hen we wrie hem in erms of he classical Black-Scholes derivaives. We ms realize ha, for simpliciy, he new derivaives are calclaed wih respec o he variance insead of he volailiy of he process. Remark 8. If denoes he classical derivaive, we have (i) Alernaive Vega: σ F=e r f BS (, S (),f(,σ )) σ f(,σ ). (31) (ii) Alernaive Vanna: σ,sf = f,s BS (, S (),f(,σ )) σ f(,σ ). (3) (iii) Alernaive Vomma: σ,σ F (L (, S (),σ ) f (, σ )) d] + 1 E [ e r( ) f(,σ ) τ K(,S(),f(,σ ))d[f(,σ ),f(,σ )]] +ρe [ e r( ) L (, S (),σ()) H(,S(),f(,σ )) f (, σ ) τd[w(),f(,σ )]]. (35) Proof. Noice ha F(, X(), σ ) = e r BS(, S(), f(, σ )) = e r V.Ase r V() is a maringale, we can wrie e r V () = E (e r V ()) = E (e r BS (, S (),f(,σ ))) = E (F (, S (),σ )). (36) Or idea is o apply an approximaion o he ideniy argmen as in heorem 1 andhensehefncionaliô formla o F(,S(),σ )=e r BS (,S,f(,σ )). (37) We dedce ha F(,S(),σ ) F(,S(),σ ) =e r f,f BS (, S (),f(,σ )) ( σ f(,σ )) e r f BS (, S (),f(,σ )) σ f(,σ ). (iv) Alernaive hea: D F= re r BS (, S (),f(,σ )) +e r BS (, S (),f(,σ )) +e r f BS (, S (),f(,σ )) D f(,σ ). (33) (34) heorem 9 (decomposiion formla). For all [,), S(), and f(, σ )>,wehave V () =BS(,S(),f(,σ )) + E [ e r( ) f(,σ )τg(,s(),f(,σ )) D f(,σ )d]+1 E [ e r( ) G(,S(),f(,σ )) = D F(,S(),σ )d + S F(,S(),σ )ds() + σ F(,S(),σ )d + 1 S F(,S(),σ )d[s, S] () + 1 σ F(,S(),σ )d[σ,σ ] + 1 D F(,S(),σ )d + S,σ F(,S(),σ )d[s(),σ ] + f ( S F(,S(),σ ))d[s(),f(,σ )] + f ( σ F(,S(),σ ))d[σ,f(,σ )]. (38)
6 6 Inernaional Jornal of Sochasic Analysis Noe ha (i) as S() is no pah-dependen, we have ha S ( ) = S ( ); (ii) as >and f is a nonanicipaive fncional, hen σ ()f(, σ )=. So, we have F(,S(),σ ) F(,S(),σ ) = D F(,S(),σ )d + S F(,S(),σ )ds() + 1 We dedce ha S F(,S(),σ )d[s, S] () + 1 D F(,S(),σ )d + f,s F(,S(),σ )d[s(),f(,σ )]. F(,S(),σ ) F(,S(),σ )= L f(,σ )BSd + e r f BS (, S (),f(,σ )) D f(,σ )d + 1 e r S BS (, S (),f(,σ )) (θ (, S (),σ()) S f (, σ )) d + S BS (, S (),f(,σ )) θ (, S (),σ()) (ρdw() + 1 ρ db ())+ 1 e r f BS (, S (),f(,σ ))d[f(,σ ),f(,σ )] +ρ e r f,s BS (, S (),f(,σ )) θ(, S (),σ()) d[w(),f(,σ )]. (39) (4) aking now condiional expecaions, sing (15), and mliplying by e r,weobainha e r( ) E [F (, S (),σ )] = BS (, S (),f(,σ )) + E [ e r f(,σ ) ( ) S S BS (, S (),f(,σ )) D f(,σ )] d + 1 E [ e r G(,S(),f(,σ )) (L (, S (),σ()) f (, σ )) d] + 1 E [ e r f(,σ ) τ K(,S(),f(,σ ))d[f(,σ ),f(,σ )]] +ρe [ e r H(,S(),f(,σ )) f (, σ ) τθ(, S (),σ()) d[w(),f(,σ )]]. (41) Remark 1. Noe ha Fncional Iô formla proved in[11] holds for semimaringales b in [1] is also proved for Dirichle process. In boh cases, he hypohesis hold by definiion of f and differeniabiliy of he derivaives of Black- Scholes fncion when τ, S, σ >. herefore, his echniqe can be applied o hese models. Remark 11. Noe ha heorem 9 coincides wih heorem 1 when we choose he volailiy fncion as f(, σ ) = V(). We fond an eqivalence of he ideas developed by [6, 1 13] in he decomposiion problem. Boh formlas come from differen poins of view; he ideas nder [1 13]are based on an exension o fncionals of he work [14], while he main idea of [6] is o change a process by is expecaion. Realize ha sandard Iô CalclsalsocanbeappliedoDirichle processes (for more informaion see [14]). Remark 1. Realize ha heorem 9 holds for any nonanicipaive f(, σ ). I is no rivial o find a differen nonanicipaive process f(, σ ) differen from he one chosen in [6]. 6. Basic Elemens of Malliavin Calcls In he nex secion, we presen a brief inrodcion o he basic facs of Malliavin calcls. For more informaion, see [15]. Le s consider a Brownian moion W = {W(), [, ]} defined on a complee probabiliy space (Ω, F, P).Se H=L ([,]),anddenoebyw(h) he Wiener inegral of afncionh H.LeS be he se of random variables of he form F=f(W(h 1 ),...,W(h n )), wheren 1, f C b, and h 1,...,h n H. Given a random variable F of his form, we define is derivaive as he sochasic process {D W F, [,]}given by D W F= n i=1 xi f(w(h 1 ),...,W(h n )) h i (x), [,]. (4)
7 Inernaional Jornal of Sochasic Analysis 7 he operaor D W and he ieraed operaors D W,n are closable and nbonded from L (Ω) ino L ([,] n Ω),foralln 1. We denoe he closre of S wih respec o he norm k=1 F n, := F L (Ω) + n DW,k F. L ([,] k Ω) (43) We denoe by δ W he adjoin of he derivaive operaor D W. Noe ha δ W is an exension of he Iô inegralinhesense ha he se L a ([,] Ω)of sqare inegrable and adaped processes is inclded in Dom δ and he operaor δ resriced o L a ([,] Ω)coincides wih he Iôsochasicinegral.We se he noaion δ() = ()dw(). WerecallhaLn, W := L ([,];D n, W ) is conained in he domain of δ for all n 1. We will se he nex Iôformlaforanicipaiveprocesses. Proposiion 13. Le s consider he processes X() = x() + (s)dw(s)+ V(s)ds,where, V L a ([,] Ω).Frhermore, consider also a process Y() = θ(s)ds, forsomeθ L 1,.LeF:R 3 R be a wice coninosly differeniable fncion sch ha here exiss a posiive consan C sch ha, for all [,], F and is derivaives evalaed in (, X(), Y()) are bonded by C.henifollowsha F (, X (),Y()) =F(,X(),Y()) + s F (s, X (s),y(s)) ds + x F (s, X (s),y(s)) dx (s) + y F (s, X (s),y(s)) dy (s) + x,y F (s, X (s),y(s)) (D Y) (s) (s) ds + 1 x F (s, X (s),y(s)) (s) ds, where (D Y)(s) := D W s s Y(r)dr. Proof. See [4]. (44) henexproposiionisseflwhenwewanocalclae he Malliavin derivaive. Proposiion 14. Le σ and b be coninosly differenial fncions on R wih bonded derivaives. Consider he solion X= {X, [,]}of he sochasic differenial eqaion: X () =x() + hen, we have D s X () =σ(x (s)) exp ( s σ (X (s)) dw (s) + b (X (s)) ds. (45) σ (X (s)) dw (s) + λ (s) ds) 1 [,] (s), s where λ(s) = [b (1/)(σ ) ](X(s)). (46) Proof. See [15, Secion.]. 7. Decomposiion Formla Using Malliavin Calcls In his secion, we se he Malliavin calcls o exend he call opion price decomposiion in an anicipaive framework. his ime, he decomposiion formla has one erm less han in he Iôformla ssep. We recall he definiion of he fre average volailiy as σ () := 1 σ (s) ds. (47) heorem 15 (decomposiion formla). For all [,),we have V () = E [BS (, S (), σ ())] + 1 E [ e r G (, S (), σ ()) where (L (, S (),σ()) σ ())d]+ ρ E [ e r( ) L (, S (),σ()) H(,S(), σ ) ( D W σ (r) dr) d], G (, S (),σ()) := S () SBS (, S (),σ()), H (, S (),σ()) := S () S G (, S (),σ()), L (, S (),σ()) := θ (, S (),σ()). S () (48) (49) Proof. Noice ha e r BS(, S(), σ()) = e r V. As e r V() is a maringale, we can wrie e r V () = E (e r V ()) = E (e r BS (, S (), σ ())). (5) So, sing he approximaion o he ideniy argmen and applying he Iô formla presened in Proposiion 13 o e r BS (, S (), σ ()), (51)
8 8 Inernaional Jornal of Sochasic Analysis we dedce by sing (15) and (7) ha e r BS (, S (), σ ()) e r BS (, S (), σ ()) = e r L σs BS (, S (), σ ()) d + 1 e r S () SBS (, S (), σ ()) ( θ (, S (),σ()) ) d 1 S () e r S () Remark 18. Noe ha when V() is a deerminisic fncion, we have ha all decomposiion formlas are eqal. Remark 19. When ρ=, we have E [BS (, S (), σ ()) BS (, S, V ())] = 1 E [ e r( ) (G (, S (), σ ()) S BS (, S (), σ ()) σ () d + e r S BS (, S (), σ )θ(, S (),σ()) (5) G(, S (), V ())) L (, S (),σ()) d] 1 E [ e r( ) (G (, S (), σ ()) (55) (ρdw() + 1 ρ db ())+ ρ e r θ (, S (),σ()) S (S () SBS (, S (), σ ())) G(, S (), V ())) σ () d] 1 8 E [ e r( ) K (, S (), V ()) d [M, M] ()]. ( D W σ (r) dr) d. aking condiional expecaion and mliplying by e r,we have E [e r( ) BS (, S (), σ ())] = E [BS (, S (), σ ())] + 1 In pariclar, when θ(, S(), σ()) = σ()s(), E [BS (, S (), σ ()) BS (, S, V ())]= 1 8 E [ e r( ) K (, S (), V ()) d [M, M] ()]. (56) E [ e r( ) G (, S (), σ ()) (L (, S (),σ()) σ ())d]+ ρ E [ e r( ) L (, S (),σ()) H(, S (), σ ()) ( D W σ (r) dr) d]. (53) Remark 16. As i is expeced, a new erm emerges when i is considered (1) like i happens in heorem 1. Remark 17. In pariclar, when θ(, S(), σ()) = σ()s(), V () = BS (, S, σ ())+ ρ E [ e r( ) σ () (54) H(,S(), σ )( D W σ (r) dr) d]. Addiionally,hegammaeffeciscancelledaswehaveseenin he Iôformlasecion. he difference beween he wo approaches is given by he vol-vol of he opion. 8. An Expression for he Derivaive of he Implied Volailiy In his secion, we give a general expression for he derivaive of he implied volailiy nder he framework of IôCalcls and Malliavin calcls. A previos calclaion of his derivaive in he case of exponenial models by sing Malliavin calcls is given in [5]. Le I(S()) denoe he implied volailiy process, which saisfies by definiion V() = BS(, S(), I(S())).Wecalclae hederivaiveofheimpliedvolailiyinhesandardiôcase. Proposiion. Under (1), for every fixed [,) and assming ha (V()) 1 < a.s., we have ha E S I(S [ S F (, S (), V ())d] ()) = σ BS (, S (),I(S ())) (57) E [ (F 1 (, S (), V ())+ S F 3 (, S (), V ())) d] S σ BS (, S (),I(S, ()))
9 Inernaional Jornal of Sochasic Analysis 9 where E [ F 1 (, S (), V ()) d] = 1 E [ e r( ) G (, S (), V ()) (L (, S (),σ()) σ ())d]+ 1 8 E [ e r( ) K (, S (), V ()) d [M, M] ()] + ρ E [ e r( ) θ (, S (),σ()) H (, S (), V ()) d [W, M] S () ()], E [ F (, S (), V ()) d] = ρ E [ e r( ) θ (, S (),σ()) H (, S (), V ()) d [W, M] S () ()], E [ F 3 (, S (), V ()) d] = 1 E [ e r( ) G (, S (), V ()) (L (, S (),σ()) σ ())d]+ 1 8 (58) Using he fac ha (V()) 1 <,wecancheckha S V() is well defined and finie a.s. hs, sing he fac ha S () = K exp(r( )), (59),and(61),weobain S I(S ()) = SBS (, S (), V ()) S BS (, S (),I(S ())) σ BS (, S (),I(S ())) E [ S F 1 (, S (), V ())d] + σ BS (, S. (),I(S ())) (6) From [16], we know ha S I () =, where I () is he implied volailiy in he case ρ=, so So, we have ha S I(S ()) S BS (, S (), V ()) = S BS (, S (),I (S ())) E [ S F 3 (, S (), V ())d]. = SBS (, S (),I ()) S BS (, S (),I(S ())) σ BS (, S (),I(S ())) E [ S F (, S (), V ())d] + σ BS (, S (),I(S. ())) On he oher hand, we have ha S BS (, S (), V ()) =φ(d), BS (, S (), V ()) =S(φ(d) φ( d)), (63) (64) (65) E [ e r( ) K (, S (), V ()) d [M, M] ()]. Proof. aking parial derivaives wih respec o S() on he expression V() = BS(, S(), I(S())),we obain S V () = S BS (, S (),I(S ())) + σ BS (, S (),I(S ())) S I (S ()). On he oher hand, fromheorem 1,wededceha V () = BS (, S (), V ()) which implies ha + E [ F 1 (, S (), V ()) d], S V () = S BS (, S (), V ()) + E [ S F 1 (, S (), V ()) d]. (59) (6) (61) where φ is he sandard Gassian densiy. hen S BS (, S (), V ()) = BS (, S (), V ())+S, S S BS (, S (),I ()) S BS (, S (),I(S ())) = 1 S (BS (, S (),I ()) BS (, S (), I(S ()))) = 1 S E [ (F 1 (, S (), V ()) + S F 3 (, S (), V ())) d]. (66) Now, we derive he implied volailiy sing Malliavin calcls. his is done in [5] for he case θ(, S(), σ()) = σ()s().
10 1 Inernaional Jornal of Sochasic Analysis Proposiion 1. Under (1), for every fixed [,) and assming ha ( σ()) 1 < a.s., we have ha S I(S ()) = where E [ E [ S F (, S (), σ ())d] σ BS (, S (),I(S ())) (67) (F 1 (, S (), σ ())+ S F 3 (, S (), σ ())) d] S σ BS (, S (),I(S, ())) E [ F 1 (, S (), σ ()) d] = 1 E [ e r( ) G (, S (), σ ()) (L (, S (),σ()) σ ())d]+ ρ 9.1. Heson Model. We consider ha he sock price follows he Heson Model (1). Usingheorem 1 or heorem 9, we have V () = BS (, X (), V ()) + ρ E [ e r( ) H (, X (), V ()) ( e k(r s) dr) σ () ] d] E [ e r( ) K (, X (), V ()) ( e k(r s) dr) ] σ () d]. Using heorem 15,we have ha (69) E [ e r( ) L (, S (),σ()) H(, S (), σ ()) ( D W σ (r) dr) d [W, M] ()], E [ F (, S (), σ ()) d] = ρ E [ e r( ) L (, S (),σ()) ( D W σ (r) dr) d [W, M] ()], E [ F 3 (, S (), σ ()) d] = 1 E [ e r( ) G (, S (), σ ()) (L (, S (),σ()) σ ())d]. Proof. See [5] or he previos proof. (68) V () = BS (, S, σ ())+ ρ E [ e r( ) H (, S (), σ ()) ( D W σ (r) dr) σ () d], (7) where D W σ (r) = ]σ()exp((]/) r (1/σ(s))dW(s)+ r [ k ] /8σ (s)]ds). 9.. SABR Model. We consider ha he sock price follows he SABR model (3).Usingheorem 1 or heorem 9,wehave V () = BS (, S (), V ()) + 1 E [ e r( ) G (, S (), V ()) σ () (S (β 1) () 1)d]+ 1 8 E [ e r( ) K (, S (), V ()) d [M, M]] () + ρ E [ e r( ) H (, S (), V ()) σ () d [W, M] ()], (71) Remark. Noe ha his is generalizaion of he formla proved in [5]. In ha case, F 1 =F and F 3 =. 9. Examples In his secion, we provide some applicaions of he decomposiion formla o well-known models in Finance. where d [M, M] = 4α σ 4 () ( e α (s ) ds) d, d [M, W] = ασ () ( e α (s ) ds) d. (7)
11 Inernaional Jornal of Sochasic Analysis 11 Using heorem 15,we have ha V () = E [BS (, S (), σ ())] + 1 E [ e r( ) G (, S (), σ ()) σ () (S (β 1) () 1)d]+ ρ E [ e r( ) H (, S (), σ ()) ( D W σ (r) dr) σ () d], where D W σ (r) = ασ ()1 [,r] (). 1. Conclsion (73) In his paper, we noice ha he idea sed in [6]canbesed for a generic Sochasic Differenial Eqaion (SDE). here is no need o specify he volailiy process. Only exisence and niqeness of he solion of he SDE are needed, allowing mch more flexibiliy in he decomposiion formla. We observe wha is he effec of assming ha he sock price follows an exponenial process and how a new erm arises in a general framework. Addiionally, we have comped he decomposiion sing hree differen mehods: Iô formla, Fncional Iô Calcls, and Malliavin calcls. In he case of callopions,heideasedin[6] is eqivalen o he se of he Fncional Iô formladevelopedin[1 13] bwihohe need of he heory behind he Fncional Iô Calcls.Boh formlas can be applied o Dirichle process, in pariclar, o he fracional Brownian moion wih Hrs parameer eqal or bigger han 1/. Frhermore, we realize ha he Feynman-Kac formla has a key role in he decomposiion process. [5] E. Alòs,J.A.Leòn, and J. Vives, On he shor-ime behavior of he implied volailiy for jmp-diffsion models wih sochasic volailiy, Finance and Sochasics, vol. 11, no. 4, pp , 7. [6] E. Alòs, A decomposiion formla for opion prices in he Heson model and applicaions o opion pricing approximaion, Finance and Sochasics,vol.16,no.3,pp.43 4,1. [7] J. Gaheral, he Volailiy Srface, John Wiley & Sons, New York, NY, USA, 6. [8] A. Glisashvili, Analyically racable sochasic sock price models, Springer Finance, Springer, Berlin, Germany, 1. [9] B. Chen, C. W. Ooserlee, and H. van der Weide, Efficien nbiased simlaion scheme for he SABR sochasic volailiy model, Inernaional Jornal of heoreical and Applied Finance, vol. 15, no., 11. [1] R. Con and D. Fornié, A fncional exension of he Io formla, Compes Rends de l Académie des Sciences,vol.348,no. 1-, pp , 1. [11] R. Con and D.-A. Fornié, Change of variable formlas for non-anicipaive fncionals on pah space, Jornal of Fncional Analysis,vol.59,no.4,pp ,1. [1] R. Con and D.-A. Forniié, Fncional Iô calcls and sochasic inegral represenaion of maringales, he Annals of Probabiliy,vol.41,no.1,pp ,13. [13] B. Dpire, Fncional iô calcls, Bloomberg Porfolio Research Paper 9-4-FRONIERS, 9, hp://papers.ssrn.com/sol3/papers.cfm?absrac id= [14] H. Föllmer, Calcl d io sans probabilies, in Séminaire de Probabiliés XV 1979/8,vol.85ofLecre Noes in Mahemaics, pp , Springer, Berlin, Germany, [15] D. Nalar, he Malliavin Calcls and Relaed opics,springer, nd ediion, 6. [16] E. Renal and N. ozi, Opion hedging and implied volailiies in a sochasic volailiy model, Mahemaical Finance,vol. 6, no. 3, pp. 79 3, Conflic of Ineress he ahors declare ha here is no conflic of ineress regarding he pblicaion of his paper. References [1] J.-P. Foqe, G. Papanicolao, and K. R. Sircar, Derivaives in Financial Markes wih Sochasic Volailiy, Cambridge Universiy Press, Cambridge, UK,. [] J. Hll and A. Whie, he pricing of opions on asses wih sochasic volailiies, he Jornal of Finance, vol. 4, no., pp. 81 3, [3] S. L. Heson, A closed for solion for opions wih sochasic volailiy wih applicaion o bond and crrency opions, Review of Financial Sdies,vol.6,no.,pp ,1993. [4] E. Alòs, A generalizaion of he Hll and Whie formla wih applicaions o opion pricing approximaion, Finance and Sochasics,vol.1,no.3,pp ,6.
12 Advances in Operaions Research Hindawi Pblishing Corporaion hp:// Advances in Decision Sciences Hindawi Pblishing Corporaion hp:// Jornal of Applied Mahemaics Algebra Hindawi Pblishing Corporaion hp:// Hindawi Pblishing Corporaion hp:// Jornal of Probabiliy and Saisics he Scienific World Jornal Hindawi Pblishing Corporaion hp:// Hindawi Pblishing Corporaion hp:// Inernaional Jornal of Differenial Eqaions Hindawi Pblishing Corporaion hp:// Sbmi yor manscrips a hp:// Inernaional Jornal of Advances in Combinaorics Hindawi Pblishing Corporaion hp:// Mahemaical Physics Hindawi Pblishing Corporaion hp:// Jornal of Complex Analysis Hindawi Pblishing Corporaion hp:// Inernaional Jornal of Mahemaics and Mahemaical Sciences Mahemaical Problems in Engineering Jornal of Mahemaics Hindawi Pblishing Corporaion hp:// Hindawi Pblishing Corporaion hp:// Hindawi Pblishing Corporaion hp:// Discree Mahemaics Jornal of Hindawi Pblishing Corporaion hp:// Discree Dynamics in Nare and Sociey Jornal of Fncion Spaces Hindawi Pblishing Corporaion hp:// Absrac and Applied Analysis Hindawi Pblishing Corporaion hp:// Hindawi Pblishing Corporaion hp:// Inernaional Jornal of Jornal of Sochasic Analysis Opimizaion Hindawi Pblishing Corporaion hp:// Hindawi Pblishing Corporaion hp://
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