A Solution to the Time-Scale Fractional Puzzle in the Implied Volatility

Transcription

1 Arcle A Soluon o he Tme-Scale Fraconal Puzzle n he Impled Volaly Hdeharu Funahash 1, * and Masaak Kjma 1 Mzuho Secures Co. Ld., Tokyo 1-4, Japan Maser of Fnance Program, Tokyo Meropolan Unversy, Tokyo 1-5, Japan; kjma@mu.ac.jp * Correspondence: fr1768@yahoo.co.jp Receved: 8 Ocober 17; Acceped: November 17; Publshed: 5 November 17 Absrac: In he opon prcng leraure, s well known ha ( he decrease n he smle amplude s much slower han he sandard sochasc volaly models and ( he erm srucure of he a-he-money volaly skew s approxmaed by a power-law funcon wh he exponen close o zero. These sylzed facs canno be capured by sandard models, and whle ( has been explaned by usng a fraconal volaly model wh Hurs ndex H > 1/, ( s proven o be sasfed by a rough volaly model wh H < 1/ under a rsk-neural measure. Ths paper provdes a soluon o hs fraconal puzzle n he mpled volaly. Namely, we consruc a wo-facor fraconal volaly model and develop an approxmaon formula for European opon prces. I s shown hrough numercal examples ha our model can resolve he fraconal puzzle, when he correlaons beween he underlyng asse process and he facors of rough volaly and perssence belong o a ceran range. More specfcally, dependng on he hree correlaon values, he mpled volaly surface s classfed no four ypes: (1 he roughness exss, bu he perssence does no; ( he perssence exss, bu he roughness does no; (3 boh he roughness and he perssence exs; and (4 neher he roughness nor he perssence exs. Keywords: fraconal Brownan moon; Hurs ndex; volaly skew; rough volaly; smle amplude; volaly perssence 1. Inroducon In he fnance leraure, here has been a general consensus ha volaly s hghly perssen. There are numerous peces of evdence ha he prce dynamcs of fnancal producs are conssen wh fraconal Brownan moon (fbm volaly models wh Hurs ndex H > 1/, whch mples ha he volaly has a long memory. See, e.g., [1] for he exsence of he long memory feaures n sock marke volales. However, nconssen wh hs sylzed fac, [] recenly fnd ha he log-volaly behaves essenally as an fbm wh H close o zero a any reasonable me scale, by esmang he volaly from hgh frequency daa. Ths puzzle (he word puzzle s used n he conex of usng a fraconal volaly model n opon prcng, bu no used n he conex of fnance n general has no been resolved, alhough one possble explanaon may be he smoohng effec by samplng nervals of daa. Recall ha f H = 1/, he fbm s he sandard Brownan moon (BM, whch s a Markov process wh shor memory. Hence, he fac H < 1/ mples ha volaly s rough. On he oher hand, n he conex of opon prcng, here are also seemngly wo nconssen sylzed facs. Namely, ( he decrease n he smle amplude s much slower han he sandard sochasc volaly models; and ( he erm srucure of he a-he-money volaly skew s well approxmaed by a power-law funcon wh he exponen close o zero. I s well recognzed ha he wo phenomena canno be capured by sandard models. Neverheless, n he opon prcng leraure, hese problems have been suded separaely. Fracal Frac. 17, 1, 14; do:1.339/fracalfrac1114

2 Fracal Frac. 17, 1, 14 of 17 The fac ( seems a resul of long memory feaure of volaly and s ndeed explaned n [3,4] by usng he fraconal volaly model wh H > 1/ under a rsk-neural measure Q. They sar wh a fraconal volaly model wh H > 1/ under he physcal measure P, and assumng ha he marke prce of volaly rsk s zero, hey ransform he model o ha under Q. Whle [3] showed ha, hanks o he long memory feaure of volaly wh H > 1/, he model can explan he slow decay of he smle amplude, [4] developed an approxmaon formula for opon prces and confrm he resul numercally when H > 1/. The fac ( s also well-known from emprcal observaons and has been pad much aenon n he fnance leraure. Whle here have been several aemps o explan hs sylzed fac, (see, e.g., [5 7] for possble models ha explan hs sylzed fac, Alòs e al. [6] and Fukasawa [8] proved ha a fraconal volaly model wh H close o zero under Q can have hs sylzed fac for small T. In oher words, hey show ha he volaly s rough under he rsk-neural measure Q for shor maury opons. The am of hs paper s o provde a soluon o he me-scale fraconal puzzle n he mpled volaly. To hs end, we adop he fraconal volaly model consdered n [4] and exend by nroducng anoher fraconal volaly facor wh Hurs ndex H < 1/. Followng he same dea as n [4], we develop an approxmaon formula for European opon prces and confrm ha our model can ncorporae boh perssence and roughness n he volaly under he rsk-neural measure Q, when he correlaons beween he underlyng asse process and he facors of rough volaly and perssence belong o a ceran range. An mporan fndng n hs paper s ha, dependng on he hree correlaon values, he mpled volaly surface s classfed no four ypes: (1 he roughness exss, bu he perssence does no; ( he perssence exss, bu he roughness does no; (3 boh he roughness and he perssence exs; and (4 neher he roughness nor he perssence exs. Hence, he coexsence of he wo sylzed facs requres he wo-facor fraconal volales of roughness and perssence n a non-rval manner. Ths paper s organzed as follows. In he nex secon, we se up our wo-facor fraconal volaly model. By usng he echnque developed n [4], we derve an approxmaon formula for opon prces n Secon 3. Secon 4 s devoed o numercal examples o nvesgae he mpac of he wo Hurs ndexes on opon prces. Through our exensve numercal expermens, we confrm ha our model can resolve he mpled volaly puzzle menoned above. The defnons of he funcons conaned n our approxmaon formula are gven n Appendx A.. The Seup Inspred by he model proposed n [4], we consder he followng wo-facor fraconal volaly model. Namely, he underlyng asse prce S and s volaly σ = σ(x 1, X are modeled by he sochasc dfferenal equaons (SDEs: ds S = (µ qd σ(x 1, X dw, dx = (θ κ X d γ dw H, = 1,, under he physcal measure P, where µ s he nsananeous mean rae of reurn of he asse, q s he (consan dvdend rae and σ(x, y s some smooh funcon n x and y. Here, whle w s a sandard Brownan moon (BM, w H denoes a fraconal Brownan moon (fbm wh Hurs ndex H under P. The volaly σ = σ(x 1, X s formulaed by usng he wo fraconal mean-reverng processes X. Noe ha θ /κ represens he long-erm average of X, κ s he speed of mean reverson and γ s he volaly of X. Laer, we specfy H 1 > 1/ and H < 1/, because we wan o explan he perssen volaly and rough volaly smulaneously. (1

3 Fracal Frac. 17, 1, 14 3 of Inegral Represenaon The fbm w H can be represened n erms of he sochasc negral wh respec o anoher sandard BM. In hs sudy, we fnd useful for he developmen of our approxmaon o employ he Mandelbro Van Ness represenaon of fbms. Namely, we have: w H = 1 Γ(H 1 { } (( s H 1 ( s H 1 dw s ( s H 1 dw s, >, where Γ(a = a 1 e d denoes he gamma funcon and where w s a sandard BM wh consan correlaons dw 1 dw = ρ 1,d and dw dw = ρ d, = 1,. By defnng: and: g H = 1 ( = 1 Γ(H 1 H 1, >, 1 Γ(H 1 we hen have he followng represenaon: (( s H 1 ( s H 1 dw s, >, w H = 1 ( sdw s g H, >. ( Noe ha, gven w, he fuure behavor of w, >, s ndependen of he pas, because he BM w s a Markov process. Hence, supposng ha he pas w, has been observed, he quany gh s consdered o be a deermnsc funcon of me wh g H =. The fraconal mean-reverng process X gven n (1 can be solved as: X = L ( γ e κ ( s dw H s,, L ( = X e κ (1 e κ θ κ κ γ e κ( s g H s ds. By applyng he negraon-by-pars formula and changng he order of negraon, he volaly can be rewren as: X = L ( ( sdw s,, (3 ( ( = γ Γ(H 1 H 1 1 κ H e κ κ x H 1 e x dx. (4 See [4] for he dealed dervaon. Now, we nroduce he marke prces of rsks η and η o defne he sandard BMs W and W under a marngale measure Q. Namely, defne: dw = dw η d; dw = dw η d, = 1, ;, respecvely. I follows from (1 and he sandard argumen ha he asse prce S follows he SDE: ds S = (r qd σ(x 1, X dw,, (5 under Q, where r denoes he rsk-free spo rae, whch we assume o be consan for smplcy.

4 Fracal Frac. 17, 1, 14 4 of 17 By applyng Io s formula, we oban: S = F(, exp σ(xs 1, Xs dw s 1 σ (Xs 1, Xs ds,, (6 where F(, = S e (r q s he forward prce of he underlyng asse wh delvery dae. On he oher hand, he fraconal mean-reverng process X s gven by: X = L ( ( sdw s,, L ( = L ( ( s η sds. Summarzng, our fraconal volaly model s formulaed as: { S = F(, exp σ(x 1, X dw s 1 X = L ( ( sdw s, = 1,, } σ (X 1, X ds, (7 for, where dw 1 dw = ρ 1, d and dw dw = ρ d, = 1,... Some Specal Cases Recall ha we suppose he pas w, has been observed and he quany gh s a deermnsc funcon of me wh g H =. However, seems dffcul o observe he whole pas of w n he acual marke, and so, we assume ha g H = as n [3] n he res of hs paper. Furhermore, followng mos of he prevous research, we assume ha he volaly rsks are fully dversfed, and so, he marke prces of rsks η are zero,.e., η =, for he sake of smplcy. Under hese specfcaons, he fraconal mean-reverng process X s gven by: X = X e κ θ (1 e κ κ ( sdw s,, (8 where ( s defned n (4. Gven hese processes X, he dynamcs of he asse prce S s deermned by he SDE (5. We observe from (8 ha he process X s a sum of a deermnsc funcon h ( and a sochasc convoluon I (, h ( = X e κ θ (1 e κ, I ( = κ ( sdw s, respecvely. For he purpose of Mone Carlo smulaon, we consder he dscree-me process x n = X (n, n = 1,,..., N, where T = N s he opon maury for suffcenly small >. The sochasc convoluon I ( can be approxmaed by a dscree convoluon such as n k=1 ξ n k W k, where ξ k = λh (k and W k = W (k1 W k. The dscree convoluon can be represened by a marx produc, and so, he mean-reverng process x n, n = 1,,..., N, n dscree me can be expressed compacly n marx form as: x1 x.. = h1 h.. ξ1 ξ ξ W W1.. (9 x N h N ξ N ξ N 1 ξ 1 W N 1

5 Fracal Frac. 17, 1, 14 5 of 17 In he followng, hanks o he above specfcaon, we adop marx Equaon (9 for Mone Carlo smulaon (see, e.g., [9] for Mone Carlo smulaon mehods for general fbms. 3. Approxmaon Formula The call opon prce wren on S wh srke K and maury T s gven by: C(K, T = e rt E [ (S T K ], (1 where (x = max{x, }, and he expecaon s aken over S T under he marngale measure Q. In hs case, all we need o know s he dsrbuon of asse prce S T a he maury T. Le us defne X = 1. From (1, he value of he European call opon s gven by: S F(, [ ( ] C(K, T = S E X T K, K = 1 K F(, T. Wh he densy funcon f T (x of X T a hand, follows ha: C(K, T = S (x K f T (xdx. (11 K In hs secon, we derve an approxmaon formula for opon prces by applyng he echnque used n [4] for he fraconal volaly model (7. To hs end, we expand he underlyng asse no a sum of erave sochasc negrals wh deermnsc negrands. In he followng, we denoe σ ( := σ(l 1 (,, L (, and σ ( := x σ(x1, x x 1 =X 1,x =X The proof of he nex resul s gven n Appendx A. Lemma 1. For X := S /F(, 1, we approxmae by: wh: p 1 (s = σ (s A ( = =1 σ (s σ (s, σ,j ( := x x j σ(x, x j x 1 =X 1,x =X. X A 1 ( A ( A 3 (, A 1 ( = p 1 (sdw s, ρ (s, uσ (udu 1 σ,j,j=1 σ (udw u dw s and A 3 ( = 6 =1 A 3,( wh A 3, ( beng defned by: A 3, ( = 1 A 3,3 ( = 1 A 3,1 ( = =1 (s σ (s ( u σ (s σ (u σ (rdw r dw u dw s, σ,j (s,j=1 σ,j (s,j=1 (s, u ( u λ H j (s, u ( u ρ,j (s, uλh j (s, udu, (s, udw u dw s, λ H j (s, rdwj r dwu dw s, (s, rdw r dwu j dw s,

6 Fracal Frac. 17, 1, 14 6 of 17 A 3,4 ( = A 3,5 ( = A 3,6 ( = σ (s =1 =1 =1 σ (s σ (s σ (u ( u (u, rdw r dw u dw s, ( u σ (u (s, rdw r dw u dw s, ( u (s, u σ (rdw r dwu dw s. Followng he dea gven n [1], he densy funcon f T (x can be approxmaed by f T (x gven n (A16 n Appendx B. By compung (11, he nex resul hen follows. The proof s sraghforward, alhough messy, and omed. Proposon 1. The value of a European call opon wh maury T and srke K under he fraconal volaly model (7 s approxmaed as: C(K, T S [ n( K;, Σ T q3 (T( K 4 Σ 4 6 K Σ T 3Σ T ( K Σ T Σ T (q 4(T q (T { Σ 3 T q 1 (T K q 5 (TΣ T } ] Σ T ( S K 1 Φ( K/ Σ T, where n(x; µ, σ s he densy funcon of he normal dsrbuon wh mean µ and varance σ and Φ(x denoes he cumulave probably funcon of he sandard normal dsrbuon. The defnons of h n (x, q k (T and Σ T are provded n Appendx B. 4. Numercal Examples Ths secon s devoed o numercal sudes of our fraconal volaly model by usng he approxmaon formula gven n Proposon 1. (The accuracy of our approxmaon formula s checked by he Mone Carlo smulaon explaned n Secon.. We noe ha our approxmaon ges gradually worse for he hgh volaly, long maury and deep n-he and ou-of-he money cases. For example, when he percenage volaly defned by η = (γ 1 γ /(X 1 X s bgger han 1.5, our approxmaon seems no enough for praccal uses. For such cases, hgher order approxmaon s requred. Throughou he numercal examples, we use he parameer values lsed n Table 1 as he base case. In parcular, for he Hurs ndexes H 1 and H, Bollerslev and Mkkelsen [1] observed long memory feaures n sock marke volales, and so, we se H 1 =.9 as he perssen volaly. On he oher hand, Bayer e al. [11] clamed (hey repor a good f of her rough volaly model wh H =.7 and he percenage volaly η = 1.9 ha H s of order.1, and so, we se H =.1 as he rough volaly. Fnally, we assume ha he volaly funcon s gven by: T for he sake of smplcy. σ = σ(x 1, X = X 1 X

7 Fracal Frac. 17, 1, 14 7 of 17 Table 1. Base case parameers. (H 1, H S r q X 1 X θ 1 θ κ 1 κ γ 1 γ ρ 1, ρ 1 ρ (.9, In Fgure 1, we depc he ATM (a he money mpled volaly (Fgure 1a and he ATM skew (Fgure 1b wh respec o he opon maury T. I s observed ha he erm srucure of he ATM volaly skew s a power-law funcon of me o maury T. Ths s a ypcal feaure of rough volaly, whch s observed n he S&P ndex opons marke repored by Bayer e al. [11]. The model ATM skew s approxmaed by he power-law funcon wh he order of.449. ddedl ATM Impled Volaly dl ATM Volaly Skew dl dl dl dl dl ddedl /s ddedl dl dl dl dl dl l ddedl ddedl d d d D (a l d d d D (b Fgure 1. ATM mpled volaly (a and ATM skew (b wh respec o he opon maury. The model parameers are lsed n Table 1. Nex, based on our approxmaon formula, we nvesgae he effecs of he model parameers on opon prces Effec of H In Fgure, we plo he skew of he European opons wh respec o H. I s explcly observed ha, as H ncreases, he power-law ndex ncreases o be.449,.1,.44 and.353 for H =.1,.3,.5 and.8, respecvely. Accordng o [5], an emprcal sudy shows ha he ndex s ypcally gven by abou.5, and so, our model s capable of capurng hs sylzed fac by seng he rough volaly H close o zero.,dl,dl,dl,dlde d d d d D Fgure. ATM skew wh respec o H, where H denoes he Hurs ndex of he rough volaly. The model parameers are lsed n Table 1.

8 Fracal Frac. 17, 1, 14 8 of Effec of H 1 We frs check wheher he Hurs ndex H 1 of perssen volaly has he ably o capure he power-law of he ATM skew or no. Fgure 3 shows he ATM skew of he European opons wh respec o H 1. In conras o he rough volaly ndex H gven n Fgure, he effec of H 1 on he ATM skew s very lmed or almos has no effec. On he oher hand, Fgure 4 shows he volaly smle wh respec o he srke and maury.,l,l,l,lde d d d d D Fgure 3. ATM skew wh respec o H 1, he Hurs ndex of volaly perssence. The model parameers are lsed n Table 1. Fgure 4a d show he volaly smle of T =.4, T =.8, T =.1 and T =., respecvely. From Fgure 4, hanks o he long memory feaure of he ndex H 1, he case of H 1 =.9 exhbs a slower decrease of he volaly smle amplude wh respec o me o maury T han he shor memory case H 1 =.5 (he rough volaly case H 1 =.4 shows a much faser decrease. Ths s a preferable feaure, because he observed smle amplude decreases much more slowly wh respec o maury han ha explaned by sandard sochasc volaly models. d,lde,lde d,l,l /s d,l /s,l d d ed ed ddd ddd ddd ed ed ddd ddd ddd ^ ^ (a e (b,lde,lde,l,l /s,l /s,l ed ed ddd ddd ddd ^ (c ed ed ddd ddd ddd (d ^ Fgure 4. Volaly smle wh respec o srke and maury. (a d show he volaly smle of T =.4, T =.8, T =.1, and T =., respecvely. The model parameers are lsed n Table 1.

9 Fracal Frac. 17, 1, 14 9 of 17 Summarzng, an mporan observaon a hs pon s ha, by seng H 1 >.5 and H <.5, we can realze boh he long memory feaure and he roughness of volaly smulaneously by usng our model, whch s he major conrbuon of hs paper Effec of ρ 1, and H 1 We examne he mpac of ρ 1, on opon prces. Recall ha ρ 1, s he correlaon beween he perssen volaly X 1 and he rough volaly X. Noe ha he ATM skew amplude comes from he roughness of σ(x 1, X. From Fgure 5, s observed ha, as he correlaon decreases, he decrease of he smle amplude s deceleraed n he case of H 1.5. Ths means ha he effec of he volaly perssence survves only when he correlaon ρ 1, s negave. d,lde d,lde d,l d d ed d d ed ed ddd ddd ddd ddd (a e,lde e,lde e d,lde e,lde d d,l l d d ed ed ed ed ddd ddd ddd ddd (b ed ed ed ed ddd ddd ddd ddd (c Fgure 5. Volaly smle wh respec o correlaon ρ 1,. (a c show he volaly smle of ρ 1, =.5,, and.5, respecvely. The maury s T =.16 and he oher parameers are lsed n Table Effec of Correlaons Fnally, we examne he effec of he correlaons, (ρ 1, ρ, ρ 1,, on he mpled volaly surface. Fgures 6a c and 7a c, respecvely show he volaly smle (he ATM skew a T =.16 wh respec o ρ 1, ρ and ρ 1,, respecvely. The hree correlaons vary n he range of.5 ρ 1.6,.6 ρ.5 and.9 ρ 1,.9, where he resulng correlaon marx s posve defne. The oher parameers are se o be he same as he base case n Table 1. As we can see form he fgures, dependng on he correlaon values, he volaly surface s classfed no four ypes: (1 he roughness exss, bu he perssence does no; ( he perssence exss, bu he roughness does no; (3 boh he roughness and he perssence exs; and (4 neher he roughness nor he perssence exs. Specfcally, as ρ 1 ncreases, s observed from Fgures 6a and 7a ha he volaly smle s preserved and he skew becomes greaer,.e., he power-law ndex of he ATM skew decreases, respecvely. Form Fgures 6b and 7b, when he absolue value of ρ s small, say ρ <., s observed ha he volaly smle s preserved, bu he ATM skew dsappears, respecvely. On he

10 eeddd dddd deddede de de de de d de de de de Fracal Frac. 17, 1, 14 1 of 17 oher hand, when he absolue value of ρ s hgh, say ρ >.7, we can see ha he volaly smle gradually dsappears, bu he ATM skew becomes greaer. Furhermore, from Fgures 6c and 7c, as ρ 1, ncreases, he ATM skew becomes greaer, bu he volaly smle dsappears, respecvely. (a (b (c Fgure 6. Volaly smle wh respec o he correlaons. (a c show he volaly smle of ρ 1, ρ, and ρ 3, respecvely. The maury s T =.16 and he oher parameers are lsed n Table 1. (a (b (c Fgure 7. ATM skew wh respec o he correlaons. (a c show he ATM skew of ρ 1, ρ, and ρ 3, respecvely. The oher parameers are lsed n Table 1.

11 Fracal Frac. 17, 1, of Conclusons In hs sudy, we exend he fraconal volaly model proposed n [4] by nroducng anoher facor of rough volaly. Through numercal expermens, we demonsrae ha, when one of he Hurs ndexes n fraconal volaly s larger han 1/ (volaly perssence and he oher s smaller han 1/ (rough volaly, our model can explan boh he slower decay of he smle amplude declne and he erm srucure of he a-he-money volaly skew observed n he opons marke, smulaneously. However, he coexsence of he wo sylzed facs seems non-rval. Namely, dependng on he hree correlaon values beween he underlyng asse and he facors of rough volaly and volaly perssence, he mpled volaly surface s classfed no four ypes: (1 he roughness exss, bu he perssence does no; ( he perssence exss, bu he roughness does no; (3 boh he roughness and he perssence exs; and (4 neher he roughness nor he perssence exs. As fuure work, we plan o apply our model o acual markes. Namely, we develop a fas algorhm o calbrae our model o he opons marke, because our model nvolves many parameers self. Furhermore, s of neres o develop a model under he physcal measure P o explan he volaly perssence and he volaly roughness smulaneously. An asymmerc model beween he perssen volaly and he rough volaly may be of grea mporance. Acknowledgmens: Masaak Kjma s graeful for he research grans funded by he Gran-n-Ad (A (#648 from Japan s Mnsry of Educaon, Culure, Spors, Scence and Technology. Auhor Conrbuons: Hdeharu Funahash and Masaak Kjma wroe he paper. Hdeharu Funahash conceved and performed he expermens, and Hdeharu Funahash and Masaak Kjma analyzed he daa. Conflcs of Ineres: The auhors declare no conflc of neres. Appendx A. Proof of Lemma 1 For he proof of Lemma 1, we apply he chaos expanson approach proposed by [1]. Namely, consder he soluon (6,.e., S = F(, exp σ(xs 1, Xs dw s 1 σ (Xs 1, Xs ds,. (A1 Denong J ( f = σ(x1 s, Xs dw s and f = σ(x1 s, Xs dw s, by means of he Herme expanson, we have: S F(, 1 = σ n ( J (σ h n, n! σ n=1 where h n (x denoes he Herme polynomal of order n. See [1] for deals. Le σ ( = f (L 1 (,, L (,, and defne: S (1 I hen follows ha: = F(, exp σ (sdw s 1 σ (sds,. (A I n ( = { σ n n! S S (1 = F(, n=1 ( J (σ h n σ n σ n! I n (, ( } J (σ h n. σ Our approxmaon s o runcae he nfne sum a n. As we shall show laer, hs approxmaon corresponds o neglecng a sum of eraed negrals: I := n=1 n σ 1 ( 1 σ ( σ n ( n dw 1 dw n (A3

12 Fracal Frac. 17, 1, 14 1 of 17 a n = 3. If he volales σ n ( are deermnsc funcons and σ = max n σ n L ([, ] s suffcenly small, hen Proposon. of [1] assures ha he sum of he eraed negrals converges very quckly. Before proceedng, snce σ (s s he deermnsc funcon of s, we can apply he Wener Io chaos expanson o S (1 o oban S (1 F(, = 1 n=1 n Hence, we defne: [ S (1 = F(, 1 σ ( 1 dw 1 σ ( 1 σ ( σ ( n dw 1 dw n. 3 as an approxmaon for S (1. Summarzng, we approxmae he quany S by: σ ( 1 σ ( dw 1 dw σ ( 1 σ ( σ ( 3 dw 1 dw dw 3 ], (A4 S = S (1 F(, (I 1 ( I (, (A5 where S (1 s gven by (A4 and I n ( by (A3. In he followng, we approxmae each I n ( by an eraed negral wh deermnsc volales. In hs paper, we employ Taylor s expanson around S (1 for hs purpose. Recall ha J (σ = f (X1 u, XudW u. I follows ha: J (σ J (σ 1 =1,j=1 σ (s{x s L (, s}dw s σ,j (s{x s L (, s}{x j s L j (, s}dw s (A6 and where we denoe: σ ( := J ( f J (σ J (σ x σ(x1, x x 1 =X 1,x =X =1 σ (s{x s L (, s}dw s,, σ,j ( := x x j σ(x, x j x 1 =X 1,x =X (A7 for he sake of noaonal smplcy. We nex approxmae each I n ( by usng he approxmaons (A6 and (A7. Snce I 1 ( = J ( f J (σ, from (A6 and (7, we ge: I 1 ( =1 1 σ (s σ,j,j=1 (s, udw u dw s (s (s, udw u λ H j (s, udwj u dw s. (A8

13 Fracal Frac. 17, 1, of 17 Moreover, by applyng Io s formula, he second erm on he lef-hand sde of (A8 s wren as: = (s, udw u s s λ H j s (s, u ( u (s, u ( u ρ,j λ H j (s, udwj u λ H j (s, rdwj r (s, uλh j dw u (s, rdw r (s, udu. dw j u (A9 Hence, we have: I 1 ( = σ (s σ,j,j=1 σ,j,j=1 σ,j,j=1 (s, udw u dw s (s (s (s ρ,j (s, uλh j (s, udu dw s (s, u ( u λ H j (s, u ( u λ H j (s, rdwj r dwu dw s (s, rdw r dwu j dw s. (A1 From he defnon, we have: I ( = 1 { ( } J (σ J (σ σ σ. However, from (A7 and (7, we oban: I ( = σ (sdw s =1 σ (s =1 =1 =1 σ (u ( u σ (s (s, udw u dw s 1 {( } σ σ (u, rdw r dw u dw s σ (s σ (udw u (s, udw u dw s σ (sσ (s (s, udw u ds 1 {( } σ σ. Noe ha, by Io s formula, he second and fourh erms n (A11 are wren by: σ (s σ (udw u = σ (s σ (s (s, udw u ( u σ (u (s, rdw r ( u (s, u σ (s ρ (s, uσ (udu dw s dw u dw s σ (rdw r dwu dw s dw s (A11 (A1

14 Fracal Frac. 17, 1, of 17 and: σ σ =1 σ (sσ (X (s s L (, s ds = respecvely. Hence, nserng (A1 and (A13 no (A11, we ge: I ( { =1 σ (s σ (s σ (s σ (s σ (u ( u =1 σ (sσ (s (s, udw u ds, (u, rdw r dw u dw s ( u σ (u (s, rdw r ( u (s, u ρ (s, uσ (udu dw u dw s dw s σ (rdw r dwu } dw s. (A13 (A14 Fnally, nserng (A4, (A1 and (A14 no (A5, we oban he desred resul. Appendx B. Dervaon of he Approxmaed Densy Funcon In hs Appendx, we oban an approxmaon formula for he densy funcon of X = S F(, 1. To hs end, noe ha (7 s a specal case of Equaon ( n [1]. Hence, applyng Proposon 3. n [1], he followng resul can be obaned. Frs, noe ha A 1 ( n Lemma 1 follows a normal dsrbuon wh zero mean and varance Σ = p 1 (sds. Then, by applyng he followng resul, an approxmaon of he densy funcon of X can be obaned. The proof s found n [1] under a general seng. Lemma A1. Le us denoe he densy funcon of X by f X (x. Then, he probably densy funcon of X s approxmaed as: f X (x = n (x;, Σ x {E[a ( a 1 ( = x]n (x;, Σ } x {E[a 3( a 1 ( = x]n (x;, Σ } 1 { } x E[a ( a 1 ( = x]n (x;, Σ. (A15 where n(x; a, b denoes he normal densy funcon wh mean a and varance b. The condonal expecaons n Lemma A1 can be evaluaed explcly by usng he formulas provded n Appendx C. Namely, we oban: E[A (T A 1 (T = x] = q 1 (T E[A 3 (T A 1 (T = x] = q (T E[A (T A 1(T = x] = q 3 (T q 1 (T = ( ( ( x Σ T x 3 Σ 3 T x 4 Σ 4 T σ (p 1 ( σ (sp 1 (sds 1 Σ T 3x Σ T 6x Σ 3 T d,, ( 3 x Σ q 4 (T T Σ 1 q 5 (T, Σ T T =1 σ (p 1( ρ (, sp 1(sds d,

15 Fracal Frac. 17, 1, of 17 q (T = 6 =1 q,(t, q 3 (T = q 1 (T, q 4 (T = 3 =1 q 4,(T and q 5 (T = 3 =1 q 5,(T. Here, we defne: q,1 (T = σ (p 1 ( σ (sp 1 (s σ (up 1 (udu ds d, q, (T = 1 σ,j (p 1( ρ (, sp 1(s ρ j λ H j (, up 1(udu ds d,,j=1 q,3 ( = 1 ( σ,j (p 1( ρ j λ H s j (, sp 1(s ρ (, up 1(udu ds d,,j=1 q,4 (T = q,5 (T = q,6 (T = σ (p 1 ( σ (sp 1(s ρ (s, up 1(udu ds d, =1 σ (p 1( =1 σ (p 1( =1 q 4,1 (T = σ (p 1 ( σ (sp 1 (s ρ (, up 1(udu ds d, ρ (, sp 1(s σ (up 1 (udu ds d, σ (sp 1 (s σ (p 1 ( σ (s σ ( σ (sp 1 (sds d, f (udu ds d σ (up 1 (udu ds d q 4, (T = { σ (p 1( σ j (sp 1(s ρ,j λ H j (s, uλh (, udu ds d,j=1 σ j (p 1( p 1 (sσ (s ρ,j λ H j (, uλh (s, udu ds d σ (p 1( ρ (, sσj (s ρ j λ H j (s, up 1(udu ds d q 4,3 (T = σ j (p 1( ρ j σ (sλh j (, s σ j (σ ( ρ (, sp 1(sds { =1 σ (p 1 ( σ (sp 1(s ρ (s, up 1(udu ρ j λ H j (, sp 1(sds ds d } d, ρ (s, uσ (udu ds d σ (p 1( p 1 (sσ (s ρ (, uσ (udu σ (p 1 ( σ (sσ (s ρ (s, up 1(udu σ (p 1( ρ σ (s (, s σ (up 1 (udu σ (σ ( σ (sp 1 (sds ds d ds d ds d } ρ (, sp 1(sds d,

16 Fracal Frac. 17, 1, of 17 and q 5, (T = q 5,3 (T = q 5,1 (T =,j=1 =1 σ ( σ (udu d, σ j (σ ( ρ,j λ H j (, uλh (, udu d, σ (σ ( ρ (, uσ (udu d. By subsung he condonal expecaons no (A15, he approxmae densy funcon, denoed by f X (x, can be expressed as: f X (x = 1 [ ( n (x;, Σ q3 ( x h 6 (q ( ( q 4 ( x h 4 Σ Σ Σ 3 q ( 1( x ( 3 h 3 q 5( h Σ Σ Σ where h n (x denoes he Herme polynomal of order n: wh h (x = 1. Σ ( x Σ ], h n (x = ( 1 n e x / dn dx n e x /, n = 1,,..., Appendx C. Formulas for Condonal Expecaons (A16 Le W, = 1,..., 5, be he sandard Brownan moons wh correlaon dw dwj = ρ,jd, and le y (x, = 1,..., 5, be deermnsc funcons of me. Moreover, le Σ := y 1 (d, and denoe J T (y 1 = y 1(dW 1. Then, he followng formulas are well known: Frs, [ E y 3 ( y (sdws dw 3 ] ( x J T(y 1 = x = v 1 Σ 1 Σ, Nex, Fnally, v 1 = ρ 1,3 y 3 (y 1 ( ρ 1, y (sy 1 (sds d. [ E y 4 ( y 3 (s y (udwu dws 3 dw 4 v = ] ( x 3 J T(y 1 = x = v Σ 3 3x Σ ρ 1,4 y 4 (y 1 ( ρ 1,3 y 3 (sy 1 (s ρ 1, y (uy 1 (udu ds d. [ ( ( E y 3 ( y (sdws dw 3 = v 3 ( x 4 Σ 4 6x Σ 3 3 Σ v 4 ( x y 5 ( Σ 1 Σ y 4 (sdws v 5,, ] dw 3 J T (y 1 = x

17 Fracal Frac. 17, 1, of 17 ( ( v 3 = ρ 1,3 y 3 (y 1 ( ρ 1, y (y 1 (ds d v 4 = ρ 1,3 y 3 (y 1 ( ρ 1,5 y 5 (sy 1 (s ρ,4 y 4 (uy (udu ρ 1,5 y 5 (y 1 ( ρ 1,3 y 1 (sy 3 (s ρ 1,3 y 3 (y 1 ( ρ,5 y (sy 5 (s ρ 1,5 y 5 (y 1 ( ρ 3,4 y 3 (sy 4 (s { ρ 3,5 y 5 (y 3 ( ρ 1, y (sy 1 (sds v 5 = ρ 3,5 y 5 (y 3 ( ρ,4 y 4 (uy (udu d. ρ 1,5 y 5 (y 1 ( ds d ρ,4 y 4 (uy (udu ds d ρ 1,4 y 4 (uy 1 (udu ds d ρ 1, y (uy 1 (udu ds d } ρ 1,4 y 4 (sy 1 (sds d, ρ 1,4 y 4 (y 1 (ds d, References 1. Bollerslev, T.; Mkkelsen, H.O. Modelng and Prcng Long Memory n Sock Marke Volaly. J. Econom. 1996, 73, Gaheral, J.; Jasson, T.; Rosenbaum, M. Volaly s rough. arxv 14, arxv: Come, F.; Renaul, E. Long memory n connuous-me sochasc volaly models. Mah. Fnanc. 1998, 8, Funahash, H.; Kjma, M. Does he Hurs ndex maer for opon prces under fraconal volaly? Ann. Fnanc. 17, 17, Fouque, J.P.; Papancolaou, G.; Srcar, R.; Solna, K. Maury cycles n mpled volaly. Fnanc. Soch. 4, 8, Alòs, E.; León, J.A.; Vves, J. On he shor-me behavor of he mpled volaly for jump-dffuson models wh sochasc volaly. Fnanc. Soch. 7, 11, Fukasawa, M. Shor-me a-he-money skew and rough fraconal volaly. Quan. Fnanc. 17, 17, Fukasawa, M. Asympoc analyss for sochasc volaly: Marngale expanson. Fnanc. Soch. 11, 15, Kjma, M.; Tam, C.M. Fraconal Brownan moons n fnancal models and her Mone Carlo smulaon. In Theory and Applcaons of Mone Carlo Smulaons; Chan, W.K., Ed.; InTech: Rjeka, Croaa, 13; pp Funahash, H. A chaos expanson approach under hybrd volaly models. Quan. Fnanc. 14, 14, Bayer, C.; Frz, P.; Gaheral, J. Prcng under rough volaly. Quan. Fnanc. 16, 16, Funahash, H.; Kjma, M. A chaos expanson approach for he prcng of conngen clams. J. Compu. Fnanc. 15, 18, c 17 by he auhors. Lcensee MDPI, Basel, Swzerland. Ths arcle s an open access arcle dsrbued under he erms and condons of he Creave Commons Arbuon (CC BY lcense (hp://creavecommons.org/lcenses/by/4./.