On the local convexity of the implied volatility curve in uncorrelated stochastic volatility models

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1 On he local conexiy of he implied olailiy cue in uncoelaed sochasic olailiy models Elisa Alòs Dp. d Economia i Empesa and Bacelona Gaduae School of Economics Uniesia Pompeu Faba c/ramon Tias Fagas, Bacelona, Spain Joge A. León y Conol Auomáico CINVESTAV-IPN Apaado Posal -7 7 México, D.F., Mexico Absac In his pape we gie an alenaie poof of he conexiy of he implied olailiy cue as a funcion of he sike, fo sochasic olailiy models in he uncoelaed case. Ou mehod is based on he compuaion of he coesponding s and second deiaies, and on Malliain calculus echniques. We poe ha he implied olailiy is a locally conex funcion of he sike, wih a minimum a he fowad pice of he sock, ecoeing he peious esuls by Renaul and Touzi (996). Moeoe, we obain an expession fo he sho-ime limi of he smile in ems of he Malliain deiaie of he olailiy pocess. Ou analysis only needs some geneal inegabiliy and egulaiy condiions in he Malliain calculus sense and does no need he olailiy o be Makoian no a di usion pocess, as we can see in he examples Inoducion I is well-known ha sochasic olailiy models capue some impoan feaues of he implied olailiy. Fo example, is aiaion wih espec o he Suppoed by he gans ECO-755 and MEC FEDER MTM 9-69 y Paially suppoed by he CONACyT gan 33

2 sike pice, descibed gaphically as a smile o skew, (see Renaul and Touzi (996) o Alòs, León and Vies (7)). This pape is deoed o he analyical sudy of he conexiy of he implied olailiy cue as a funcion of he sike, in he case of uncoelaed sochasic olailiy. As poed in Renaul and Touzi (996) (by inducion on he numbe of posible alues ha he squaed fuue aeage olailiy can ake) his funcion is locally conex wih a minimum a he fowad sock pice. In his pape we pesen an alenaie poof of his conexiy esul, based on he compuaion of he a-he-money s and second deiaies of he implied olailiy. Ou mehod uses implici di eeniaion and Malliain calculus echniques, and gies us explici expessions fo hese deiaies. These expessions allow us o poof he conexiy of he implied olailiy cue, as well as o compue is sho-ime behaiou in ems of he Malliain deiaie of he olailiy pocess. Ou analysis only needs some egulaiy condiions and does no need he olailiy o be a Makoian pocess, as we see in he examples. The pape is oganized as follows. In Secion we inoduce he famewok and he noaion ha we uilize in his pape. In Secion 3 we poe ha he implici olailiy has a saionay poin a he fowad sock pice. In Secion, we obain an expession fo he second deiaie ha allows us o poe he local conexiy of he implied olailiy, as well as o compue is sho-ime a-he-money limi. Finally, some examples ae gien in Secion 5. Saemen of he poblem and noaion In his pape we conside he following model fo he log-pice of a sock unde a isk-neual pobabiliy measue P : X = x + ~ Z sds + Z s dw s ; [; T ]: () Hee, x is he cuen log-pice, ~ is he insananeous inees ae, W is a sandad Bownian moion de ned on a complee pobabiliy space (; G; P ), and is a squae-inegable and igh-coninuous sochasic pocess adaped o he laion geneaed by anohe sandad Bownian moion B independen of W. In he following we denoe by F W and F B he laions geneaed by W and B. Moeoe we de ne F := F W _ F B : I is well-known ha hee is no abiage oppouniy if we pice an Euopean call wih sike pice K by he fomula V = e ~(T ) E [(e X T K) + ]; whee E is he F condiional expecaion wih espec o P (i.e., E (X) = E(XjF )). In he sequel, we make use of he following noaion: R T = T udu: Tha is, epesens he fuue aeage olailiy. R T M = E udu :

3 BS(; x; k; ) denoes he pice of a Euopean call opion unde he classical Black-Scholes model wih consan olailiy, cuen log sock pice x, ime o mauiy T ; sike pice K = exp(k) and inees ae ~: Remembe ha in his case BS(; x; k; ) = e x N(d + ) e k ~(T ) N(d ); whee N denoes he cumulaie pobabiliy funcion of he sandad nomal law and wih k := x + ~(T d := k k p T p T ; ): Noice ha, as is independen o he laion geneaed by W,opion pices ae gien by he so-called Hull and Whie fomula (see fo example Hull and Whie (97)) V = E (BS(; X ; k; )) ; [; T ]: () 3 The a-he-money implied olailiy skew Le us de ne he implied olailiy I = I(; X ; k) as he sochasic pocess such ha V = BS (; X ; k; I) : Someimes we use he conenion I = I(; k) in ode o simplify he noaion. Ou s aim is o sudy he deiaie I k j k=k : Moe pecisely we ge he following esul: Poposiion (he implied olailiy skew) Conside he model (). Then, fo all [; T ] ; I k (; k ) = : Poof. This poof can be deduced following he ideas of he poof of Poposiion 5. in Alòs, León and Vies (7). Namely, by he de niion of I, we know Hence, k V = BS k (; X ; k; I) + BS (; X ; k; I) I (; k); [; T ]: k BS (; X ; k; I) I k = k V Also, fom (), we know V BS k = E k (; X ; k; ) : BS k (; X ; k; I); [; T ]: (3) 3

4 On he ohe hand, since hen we obain Theefoe, (3) yields BS BS k (; x; k ; ) = e x N BS k (; X ; k ; ) p T s = (BS(; x; k ; ) e x ) ; BS k (; X ; k ; I(; k )) = (BS (; X ; k ; ) BS (; X ; k ; I(; k ))) : (; X ; k ; I(; k )) I k (; k ) BS = E k (; X ; k ; ) BS k (; X ; k ; I (; k )) = E (BS (; X ; k ; ) BS (; X ; k ; I (; k ))) = ; whee he las inequaliy is due o () and he de niion of I. Thus, he poof is complee. Remak The aboe heoem poes ha, xed [; T ], he implied olailiy I(; k) has saionay poin a k = k : Noice ha his esul is independen of he sochasic olailiy model. Tha is, we do no need i o be a di usion o a Mako pocess. The a-he-money implied olailiy smile Now ou pupose in his secion is o sudy he a-he-money second deiaie I k (; k ): We poe ha his is posiie. Consequenly, fo eey xed [; T ], he implied olailiy I(; X ; k) is a locally conex funcion of k. Moeoe, we poe ha lim I!T k (; k ) is well-de ned and nie, which is gue i ou explicily. We assume ha he eade is familia wih he elemenay esuls of he Malliain calculus, as gien fo insance in Nuala (6). In he emaining of his pape D ; B denoes he domain of he Malliain deiaie opeao DB wih espec o he Bownian moion B: I is well-known ha D ; B is a dense subse of L () and ha D B is a closed and unbounded opeao fom L () o L ([; T ] ): We will use he noaion L ; B = L ([; T ] ; D ; B ): Fo ou pupose, we inoduce he following hypoheses:

5 (H) belongs o D ; B, and hee exiss an adaped pocess Y = fy ; [; T ]g L ( [; T ]) such ha E D B u Y, fo all u T: (H) Fo eey [; T ] ; hee exiss an F B D + such ha measuable andom aiable lim "T R T E sup ut E D B u D + d T = : (H3) Thee exis wo deeminisic, inegable and igh coninuous funcions ; : [; T ]! R + such ha () j j (); [; T ]: Remak 3 Noice ha unde (H), he Clak-Ocone fomula gies us ha (see, fo insance, Nuala (6)) Z M = M + R T wih M = E sds : E s s! Ds B d db s ; [; T ] ; Befoe saing he main esul of his secion, we esablish he following auxiliay esul. Lemma Le [; T ] and = (BS(; X ; k ; )), hen BS (; X ; k ; ) (T ) exp exp (T ) and BS (; X ; k ; ) 3 E exp 3 3 (T ) A : Poof. We s obsee ha BS (; X ; k ; ) and exp() ae wo conex funcion on R +. Theefoe, Jensen inequaliy implies BS (; X ; k ; ) (T ) exp BS (; X ; k ; (BS(; X ; k ; ))) (T ) = exp BS (; X ; k ; BS(; X ; k ; )) (T ) exp exp (T ) : 5

6 Similaly, using ha x 7! x 3 is a conex funcion on R + and he Taylo expansion fo BS (; X ; k ; ), we hae BS (; X ; k ; ) 3 p! 3 p (BS(; X ; k ; )) e X T p! 3 p BS(; X ; k ; )e X A ; T which implies he esul due o he mean alue heoem. Theoem 5 Assume ha he model (), and Hypoheses (H) and (H3) ae sais ed. Then I k (; k ), fo all [; T ]. Moeoe, if Hypohesis (H) also holds, I lim!t k (; k ) = (D+ ) 5 : Poof. Fom he de niion of he implied olailiy I, we hae k V = BS k + BS (; X ; k; I) By Poposiion, he las equaliy becomes (; X ; k; I) + BS k (; X ; k; I) I k I + BS k (; X ; k; I) I k : BS (; X ; k ; I(; x )) I k (; k ) = k V j k=k Thus () gies BS k (; X ; k ; I(; x )) : BS (; X ; k ; I(; k )) I k (; k ) BS = E k (; X ; k ; ) BS k (; X ; k ; I(; k )) : () Bu he las em on he igh-hand side of () can be wien as BS k (; X ; k ; I(; k )) = BS k (; X ; k ; BS (V )) = BS k (; X ; k ; BS (E (BS (; X ; k ; )))); 6

7 whee, in his case, we denoe BS (; x ; k; ) by BS () in ode o simplify he noaion. Consequenly, using (), we can esablish BS (; X ; k ; I(; k )) I k (; k ) = E BS k (; X ; k ; ) = E BS k BS k (; X ; k ; BS (E (BS (; X ; k ; )))) ; X ; k ; BS (BS (; X ; k ; )) BS k (; X ; k ; BS (E (BS (; X ; k ; )))) : (5) Now he poof is decomposed ino seeal seps. Sep. Le us s poe ha I k (; k ) is posiie. The Clak-Ocone fomula (see Nuala (6)), ogehe wih Hypoheses (H) and (H3), leads o BS (; X ; k ; ) = E (BS (; X ; k ; )) + U db ; whee U : = D B (BS (; X ; k ; )) BS D = (; X ; k B ; ) M T (T ) ; > : (6) Hence, uilizing he conenion (a) := BS k ; X ; k ; BS (a) and equaliy 7

8 (5), we ge BS (; X ; k ; I(; k )) I k (; k ) = E " BS k ; X ; k ; BS E (BS (; X ; k ; )) + BS k (; X ; k ; BS (E (BS (; X ; k; )))) Z u E (BS (; X ; k ; )) + = E " + = E " + = E " a a E (BS (; X ; k ; )) + Z u a (E u (BS (; X ; k ; ))) U u db u a (E u (BS (; X ; k ; ))) U udu a (E u (BS (; X ; k ; ))) U udu U db U u db u U db U udu # # ; # U db!! whee, in he las equaliy, we use he fac ha a (a) = hypoheses (H) and (H3). So, now i is easy o see a (T ) ; and BS (; x ; k; I(; x )) I k (; x ) p 3 (BS exp (E u(bs(;x ;k ;)))) (T s) = E 6 e X (BS (E u (BS (; X ; k ; )))) 3 (T ) U udu 7 3= 5 : Thus I k (; x ) > w.p.. Sep. Hee we show ha E R T a (; ) U BS(; X ; k ; ) (T )D+ (T ) d ; (T ) = whee de ned as in Lemma, coneges o as " T.

9 By Schwaz inequaliy, we can wie U E BS(; X ; k ; ) (T )D+ (T ) Z = BS(; X ; k T ; ) (D B u + D + )du! (T ) Z BS(; X ; k T ; ) (D B u D + )du! (T ) CeX (T ) E exp (T ) (D B u + D + )dua E exp (T ) (D B u D + )dua CeX (T ) E exp (T ) " Z # T A (D B u + D + )du AA " Z # T (D B u D + )du AA C(T ) e X (T ) exp (T ) A sup (D B u D + ) = : ut = = sup (D B u + D + ) = ut 9

10 Hence, Lemma gies E R T a (; ) U Ce X (T ) E sup ut Z Ce X T E (T ) exp E E! ; as " T; BS(; X ; k ; ) (T )D+ (T ) d (T ) = exp (T ) (D B u + D + ) = exp (T ) sup ut (T ) AA C da = exp 3 sup ut (D B u + D + ) d sup E (D B u D + ) d ut (T ) 3 A exp (T ) (D B u D + ) = d exp 3! =! = (T ) 3 due o Hypoheses (H) and (H3). Thus he claim of his pa of he poof is ue. Sep 3. Finally we poe ha lim I!T k (; x ) = (D+ ) : 5 Fom Sep, we obain I lim!t k (; x ) E = lim!t E = lim!t R T a (; ) BS(; X ; k ; ) (T )D+ (T ) d p (T (D + ) R T a (; ) ) = e X BS(; X ; k ; ) A (T ) d p : (T ) 5= e X # A

11 Noe ha he igh coninuiy of and (H3) imply (D + ) R T! (D+ ) 5 a (; ) BS(; X ; k (T ) ; ) d p (T as! T; ) 5= e X w.p.; and (D + ) R T (D+ ) (T ) a (; ) BS(; X ; k (T ) ; ) d p (T ) 5= e X exp (T ) exp 3 (T ) 3 A exp (T ) A d: Theefoe, he esul follows fom he dominaed conegence heoem and he fac ha Z T E exp (T ) exp 3 (T ) A (T ) 3 exp A (T ) A C A d! E ; as! T; which follows fom (H3). Now he poof is nished. Remak 6 Noice ha (H3) can be subsiued by adequae inegabiliy condiions. Remak 7 The aboe heoems poes ha, xed [; T ] ; he implied olailiy I(; k) is a locally conex funcion of he sike wih a minimum a k = k ; accoding o he peious esuls by Renaul and Touzi (996). 5 Examples 5. Di usion sochasic olailiies Assume ha he squaed olailiy can be wien as = f(y ); whee f Cb and Y is he soluion of a sochasic di eenial equaion: dy = a (; Y ) d + b (; Y ) db ; (7)

12 fo some eal funcions a; b Cb : Then, classical agumens (see fo example Nuala, 6) gie us ha Y L ; B and ha Z Z Ds B a Y = s x (u; Y u)ds B b Y u du + b(s; Y s ) + s x (u; Y u)ds B Y u db u : () Taking now ino accoun ha D B s u = f (Y u )D B s Y u i can be easily deduced fom () ha (H), and (H3) hold and ha lim "T R T E sup ut D B u f (Y )b(; Y ) d T = ; which implies (H). Then, Theoem and Theoem gie us ha, fo all [; T ] ; I k (; k ) = and I k (; k ), which poes ha I(; k) is a locally conex funcion wih a minimun a k = k : Moeoe, Theoem say us ha I lim!t k (; k ) = (D+ ) 5 = (f (Y )b(; Y )) 5 : 5. Facional sochasic olailiy models Assume ha he squaed olailiy can be wien as = f(y ); whee f Cb and Y is a pocess of he fom Y = m + (Y m) e ( ) + c p Z exp ( ( s)) db H s ; (9) whee Bs H := R s (s u)h db u : As in Come and Renaul (99), whee his class of models hae been inoduced o capue he long-ime behaiou of he implied olailiy, we assume he olailiy model (9), fo some H > =: Noice ha (see fo example Alòs, Maze and Nuala ()) R exp ( ( s)) dbh s can be wien as H Z Z exp ( ( u)) (u s) H 3 du db s ; s fom whee i follows easily ha sup st E D B s Fs! as! T. In a simila way as in Example 5., Theoem and Theoem gie us ha, fo all [; T ] ; I k (; k ) = and I k (; k ), and hen I(; k) is a locally conex funcion wih a minimun a k = k : Moeoe, Theoem say us ha lim I!T k (; k ) = : This esul indicaes ha, in ode o capue boh he sho-ime and he long ime behaiou of he implied olailiy, a possible appoach could be o conside a olailiy pocess of he fom = f(y ; Y ); whee Y is a soluion of (7) and Y is a soluion of (9), as consideed in a ecen pape by Alòs and Yang ().

13 6 Conclusions We hae seen, by a compuaion of he coesponding s and second deiaies, ha in he case of uncoelaed olailiy, he implied olailiy is a locally conex funcion of he sike, wih a minimum a he fowad pice of he sock. This ecoes he peious esuls by Renaul and Touzi (996). Moeoe, we see ha he sho-ime limi of he a-he-money second deiaie can be explicily compued in ems of he Malliain deiaie of he olailiy pocess. Ou analysis only needs some geneal inegabiliy and egulaiy condiions in he Malliain calculus sense and does no need he olailiy o be a di usion o a Mako pocess Refeences [] Alòs, E., León, J.A. and Vies, J.: On he sho-ime behaio of he implied olailiy fo jump-di usion models wih sochasic olailiy. Finance Soch., 57-59, 7. [] Alòs, E, Maze, O. and Nuala, D: : Sochasic Calculus wih espec o Gaussian pocesses. The Annals of Pobabiliy 9 (), 766-,. [3] Alòs, E. and Yang, Y.: A closed-fom opion picing appoximaion fomula fo a facional Heson model. Woking Pape 6, Uniesia Pompeu Faba,. [] Come, F. and Renaul, E. : Long-memoy in coninuous-ime sochasic olailiy models. Mahemaical Finance, 9-33, 99. [5] Hull, J. C. and Whie, A.: The picing of opions on asses wih sochasic olailiies. Jounal of Finance, -3, 97. [6] Lee, R.: Implied olailiy: saics, dynamics and pobabilisic inepeaion, in Recen adances in applied pobabiliy. Spinge, 5. [7] Nuala, D.: The Malliain Calculus and Relaed Topics. Second Ediion. Pobabiliy and is Applicaions. Spinge-Velag, 6. [] Renaul, E. and Touzi, N: Opion Hedging and Implici Volailiies, Mahemaical Finance 6, 79-3,