Key words. Stochastic volatility, Long-range correlation, Mean reversion, Fractional Ornstein- Uhlenbeck process.

Transcription

1 OPTION PICING UNDE FAST-VAYING LONG-MEMOY STOCHASTIC VOLATILITY JOSSELIN GANIE AND KNUT SØLNA Absrac. ecen empirical sudies sugges ha he volailiy of an underlying price process may have correlaions ha decay slowly under cerain marke condiions. In his paper, he volailiy is modeled as a saionary process wih long-range correlaion properies o capure such a siuaion and we consider European opion pricing. This means ha he volailiy process is neiher a Markov process nor a maringale. However, by exploiing he fac ha he price process sill is a semimaringale and accordingly using he maringale mehod, one can ge an analyical expression for he opion price in he regime where he volailiy process is fas mean revering. The volailiy process is modeled as a smooh and bounded funcion of a fracional Ornsein Uhlenbeck process and we give he expression for he implied volailiy which has a fracional erm srucure. Key words. Sochasic volailiy, Long-range correlaion, Mean reversion, Fracional Ornsein- Uhlenbeck process. AMS subjec classificaions. 9G8, 6H, 6G, 6K37.. Inroducion. Sochasic Volailiy and he Implied Surface. Under many marke scenarios he assumpion ha he volailiy is consan, as in he sandard Black-Scholes model, is no realisic. Pracically, his reflecs iself in an implied volailiy ha depends on he pricing parameers. This means ha, in order o mach observed prices, he volailiy one needs o use in he Black-Scholes opion pricing formula depends on ime o mauriy and log moneyness, wih moneyness being he srike price over he curren price of he underlying. The implied volailiy is a convenien way o parameerize he price of a financial conrac relaive o a paricular underlying. I gives insigh abou how he marke deviaes from he ideal Black-Scholes siuaion and, afer calibraion of an implied volailiy model o liquid conracs, i can be used for pricing less liquid conracs wrien on he same underlying. I is herefore of ineres o idenify a consisen parameerizaion of he implied volailiy ha corresponds o an underlying model for sochasic volailiy flucuaions. As in Garnier and Solna Cenre de Mahémaiques Appliquées, Ecole Polyechnique, 98 Palaiseau Cedex, France josselin.garnier@polyechnique.edu Deparmen of Mahemaics, Universiy of California, Irvine CA 9697 ksolna@mah.uci.edu

2 5 a main objecive of our modeling is o consruc a ime-consisen scheme so ha indeed he volailiy model is chosen as a saionary process and we consider general imes o mauriy. For background on sochasic volailiy models we refer o he books and surveys by Fouque e al. ; Gaheral 6; Ghysels e al. 995; Gulisashvili ; Henry-Labordére 9; ebonao 4 see he references herein. We also refer o our recen paper on fracional sochasic volailiy Garnier and Solna 5 for furher references on he recen lieraure on he class of volailiy models we consider here. Empirical sudies sugges ha volailiy may exhibi a muli scale characer wih long-range correlaions as in Bollerslev e al. 3; Breid e al. 998; Chronopoulou and Viens ; Con, 5; Engle and Paon ; Oh e al. 8. Tha is, correlaions ha decay as a power law in offse raher han as an exponenial funcion as in a Markov process. Here we seek o idenify parameric forms for he implied volailiy consisen wih such long-range correlaions. In our recen paper Garnier and Solna 5 we considered his quesion in he conex where he magniude of he volailiy flucuaions is small. Here, we consider he siuaion where he magniude of he volailiy flucuaions is of he same order as he mean volailiy. Indeed empirical sudies show ha he volailiy flucuaions may be quie large: Breid e al. 998; Con ; Engle and Paon. While in Garnier and Solna 5 he volailiy flucuaions were small leading o a regular perurbaive siuaion, here he siuaion is differen in ha i is he fas mean reversion fas relaive o he diffusion ime of he underlying ha allows us o push hrough an asympoic analysis. However, he presence of long-range correlaions in his conex gives a novel singular perurbaion siuaion. The analysis becomes significanly more involved. In paricular he deailed analysis of he covariaion process is an imporan ingredien. We consider here opion pricing, bu he approach se forh is general and will be useful in oher financial conexs as well. I follows from our analysis ha he form for he implied volailiy surface is similar as in he Markovian case. This confirms he robusness of he implied volailiy parameric model wih respec o he underlying price dynamics. There are, however, cenral differences. In paricular he long-range correlaions produce a volailiy covariance ha is no inegrable which in urn gives an implied volailiy surface ha is a random field, whose saisics can be described in deail. Moreover, in he long-

3 range case he implied volailiy has a fracional behavior as a funcion of ime o mauriy. The empirical sudy in Fouque e al. 3 shows ha, in order o fi well he implied volailiy, i is appropriae o consider a wo-ime scale model wih one slow and one fas volailiy facor. In Garnier and Solna 5 we considered a slow facor, which closely associaes wih a small flucuaions facor. Here, we consider a fas facor wih large flucuaions. Taken ogeher we have a generalizaion of he wo-facor model of Fouque e al. 3, o he case of processes wih longrange correlaions. This leads o a fracional erm srucure of he implied volailiy and i was shown in Fouque e al. 4 ha such a erm srucure may be useful o fi he implied volailiy under cerain marke condiions. Long Memory and Fas Mean eversion. As menioned above he asympoic regime considered in his paper is he siuaion where he volailiy is fas mean revering. We denoe is ime scale by and his is he small parameer in our model. The volailiy hen decorrelaes on he ime scale. Sochasic volailiy models are mos ofen posed wih a volailiy driving process ha has mean zero and mixing properies. This means ha he random values of he volailiy driving process a imes and +, ha are Z and Z+, become rapidly uncorrelaed when, i.e., he auocovariance funcion C = E[Z Z + decays rapidly o zero as. More precisely we say ha he volailiy driving process is mixing if is auocovariance funcion decays fas enough a infiniy so ha i is absoluely inegrable: C d <.. In his case we may associae he process wih he finie correlaion ime c = C d/c, which is of order. Sochasic volailiy models wih long-range correlaion properies have recenly araced a lo of aenion, as more and more daa colleced under various siuaions confirm ha his siuaion can be encounered in many differen markes. Qualiaively, he long-range correlaion propery means ha he random process has long memory in conras wih a mixing process. This means ha he correlaion beween he random values Z and Z+ aken a wo imes separaed by is no compleely negligible even for large. More precisely we say ha he random process Z 3 has

4 he H-long-range correlaion propery if is auocovariance funcion saisfies: C r H H,. where r H > and H /,. We refer o H as he Hurs exponen. Here he correlaion ime is he criical ime scale beyond which he power law behavior. is valid. Noe ha he auocovariance funcion is no inegrable as H,, which means ha a random process wih he H-long-range correlaion propery is no mixing. As we describe in more deail below a common approach for modeling long-range dependence is via using fracional Brownian moion fbm processes as inroduced in Mandelbro and Van Ness 968. Long-memory sochasic volailiy models are indeed easy o pose, however, heir analysis is quie challenging. This is largely due o he fac ha he volailiy process is hen neiher a Markov process nor a semimaringale. I is, however, imporan o noice ha he price process is sill a semimaringale and he problem formulaion does no enail arbirage Mendes e al. 5, as has been argued for some models whose price process iself is driven by fracional processes as in Bjork and Hul 5; ogers 997; Shiryaev 998. A main moivaion for long-memory is o be able o fi observed implied volailiies. One classic challenge regarding fiing of implied volailiy surfaces is o capure a srong moneyness dependence for shor ime o mauriy wihou creaing arificial behavior for long ime o mauriy. Anoher one is o reain a srong parameric dependence for long mauriies despie averaging effecs ha occur in his regime, as discussed in Bollerslev and Mikkelsen 999; Bollerslev e al. 3; Come e al. ; Sundarsen e al.. We remark ha models involving jumps have been promoed as one approach o mee hese challenges by Carr and Wu 3; Mijaovic and Tankov 6. ecen works show ha sochasic volailiy models wih long-range dependence also provide a promising framework for meeing such challenges. Approaches based on using fracional noises in he descripion of he sochasic volailiy process were used by Come and enaul 998; Come e al.. This provides an approach for endowing he volailiy process wih high persisence in he long run long memory wih H > / in order o capure he seepness of long erm volailiy smiles wihou overemphasizing he shor run persisence. In order o ge explici resuls for he implied volailiy a number of asympoic regimes have been considered. Chief among hem has been he regime of shor ime o mauriy. The model presened in Come e al. 4 was

5 recenly revisied in Guennoun e al. 4 where shor and long ime o mauriy asympoics are analyzed using large deviaions principles. In Alòs e al. 7 he auhors use Malliavin calculus o decompose opion prices as he sum of he classic Black-Scholes formula wih volailiy parameer equal o he roo-mean-square fuure average volailiy plus a erm due o he leverage effec i.e., he correlaion beween he underlying reurn and is changes of volailiy and a erm due o he volailiy of he volailiy. Their model is a fracional version of he Baes model Baes 996. They find ha he implied volailiy flaens in he long-range dependen case in he limi of small ime o mauriy. In Forde and Zhang 5 he auhors use large deviaion principles o compue he shor ime o mauriy asympoic form of he implied volailiy. They consider he leverage effec and obain resuls ha are consisen wih hose in Alòs e al. 7. They consider sochasic volailiy models driven by fbms which are analyzed using rough pah heory. They also consider large ime asympoics for some fracional processes. Small ime o mauriy asympoic resuls were recenly also presened in Gulisashvili e al. 5 in a conex of long-range processes. In Fukasawa he auhor discusses he asympoic regime wih small volailiy flucuaions and long-range dependence impac on he implied volailiy as an applicaion of he general heory he ses forh. In his paper as well as in Alòs e al. 7 he auhors use a modeling where he ime plays a special role and hence he modeling is no compleely saisfacory because i leads o a non-saionary volailiy model. This is also he case in Bayer e al. 6 where he auhors consider he so-called rough Bergomi, or rbergomi, model. In his paper and in Garnier and Solna 5 which deals wih small volailiy flucuaions we use a formulaion wih a saionary model. This is also he case in he recen paper by Fukasawa 7 which considers small ime asympoics in he rough volailiy case, wih H < /. This disincion is imporan: If he volailiy facor is a fbm emanaing from he origin, hen he implied volailiy surface is idenified condiioned on he presen value of he volailiy facor only. Below wih a saionary model he implied volailiy surface depends on he pah of he volailiy facor unil he presen, reflecing he non-markovian naure of fbm. We discuss in deail in Secion 6 he consequences of his for he inerpreaion of he implied volaliy surface as a random field. ecenly, pricing approximaions in he regime of small fracional volailiy flucuaions were presened in Alòs and Yang 7. In erms of compuaion of prices for general 5

6 mauriies and fracional volailiy flucuaions, so far mainly numerical approximaions have been available. However, here we presen an asympoic regime based on fas mean reversion which gives explici price approximaions in his conex. Taken ogeher he resuls of Garnier and Solna 5 and his paper allow o consruc a fracional wo-ime scale sochasic volailiy model and flexibiliy o fi boh he shor and long ime o mauriy pars of he implied volailiy surface. We remark ha we here consider he case wih H > / and long-range correlaion only as opposed o he case wih rough volailiy and H < / corresponding o sharp decay of he correlaions a he origin. Indeed boh regimes have been idenified from he empirical perspecive. We refer o for insance Gaheral e al. 6 for observaions of rough volailiy, while in Chronopoulou and Viens cases of long-range volailiy are repored. A persisen or long-range mean revering volailiy siuaion is repored in Jensen 6 in a discree modeling framework. Long-range volailiy siuaions are also repored for currencies in Walher e al. 7, for commodiies in Charfeddine 4, and for equiy index in Chia e al. 5, while analysis of elecriciy markes daa ypically gives H < / as in Simonsen ; ypdal and Lovsleen 3; Bennedsen 5. We believe ha boh he rough and he long-range cases are imporan and can be seen depending on he specific marke and regime. Even hough he rough case wih H < / may be he mos common siuaion, i may be of paricular imporance o undersand he siuaion where H > / and he ramificaion of his for pricing and hedging. In his paper we only consider he analyic aspecs of our model. The fiing wih respec o specific daa is beyond he scope of his paper and will be presened elsewhere. The fracional model we se forh here produces ypical sylized facs, like heavy ails of reurns, volailiy clusering, mean reversion, and long memory or volailiy persisence. Addiionally, we here incorporae he leverage effec. A erm coined by Black e al. 976 referring o sock price movemens which are correlaed ypically negaively wih volailiy, as falling sock prices may imply more uncerainy and hence volailiy. Noe, however, ha he model for he implied volailiy surface derived below is linear in log moneyness. This may seem somewha resricive from he poin of view of fiing because in many cases a srong skew in log moneyness may be observed in cerain markes. This has paricularly been he case for sock markes, bu relaively less so in oher markes like fixed income markes. However, 6

7 if one considers higher order approximaions, hen his generaes also skew effecs. A number of modeling issues no considered here, like ransacion coss, bid-ask spreads and liquidiy, may also affec he skew shape. Noe also ha for simpliciy we do no incorporae a non-zero ineres rae, nor do we incorporae marke price of risk aspecs. apid-clusering, Long-Memory and he Implied Surface. We summarize nex he main resul of he paper from he poin of view of calibraion. Tha is, he form of he implied volailiy surface in he conex of a sochasic volailiy modeled by a fas process wih long-range correlaion properies. We summarize firs some aspecs of he modeling. We consider a coninuous ime sochasic volailiy model ha is a smooh funcion of a Gaussian long-range process. Explicily, we model he fracional sochasic volailiy fsv as a smooh funcion of a fracional Ornsein-Uhlenbeck fou process. The fou process is a classic model for a saionary process wih a fracional long-range correlaion srucure. This process can be expressed in erms of an inegral of a fracional Brownian moion fbm process. The disribuion of a fbm process is characerized in erms of he Hurs exponen H,. The fbm process is locally Hölder coninuous of exponen H for all H < H and his propery is inheried by he fou process. The fbm process, W H, is also self-similar in ha { W H α, } dis. = { α H W H, } for all α >..3 The self-similariy propery is inheried approximaely by he fou process on scales smaller han he mean reversion ime of he fou process which we denoe by below. In his sense we may refer o he fou process as a muliscale process on shor scales. The case H /, ha we address in his paper gives a fou process ha is a long-range process. This regime corresponds o a persisen process where consecuive incremens of he fbm are posiively correlaed. The sronger posiive correlaion for he consecuive incremens of he associaed fbm process wih increasing H values gives a smooher process whose auocovariance funcion decay slowly. For more deails regarding he fbm and fou processes we refer repecively o Biagini e al. 8; Couin 7; Doukhan e al. 3; Mandelbro and Van Ness 968 and Cheridio e al. 3; Kaarakka and Salminen. The volailiy driving process is he -scaled fracional Ornsein-Uhlenbeck pro- 7

8 cess fou defined by: Z = H e s dw H s..4 I is a zero-mean, saionary Gaussian process, ha exhibis long-range correlaions for he Hurs exponen H /,. I is imporan o noe ha his is a process whose naural ime scale is, his in he sense ha he mean reversion ime or ime before he process reaches is equilibrium disribuion scales like. I is also imporan o noe ha he decay of he correlaions on he ime scale is polynomial raher han exponenial as in he sandard Ornsein-Uhlenbeck process. Explicily, he correlaion of he process beween imes and + decays as / H, while he variance of he process is independen of. In his paper we consider a sochasic volailiy model ha is a smooh funcion of he rapidly varying fracional Ornsein-Uhlenbeck process wih Hurs coefficien H /,, i is given by σ = F Z,.5 where F is a smooh, posiive, one-o-one, bounded funcion wih bounded derivaives and wih an addiional echnical condiion ha is given in Eq The process σ inheris he long-range correlaion properies of he fou Z. The main resul we se forh in Secion 5 is an expression for he implied volailiy of he European Call Opion for srike K, mauriy T, and curren ime : [ T I = E σ T s ds [ / H / H 3/ K F + σaf + log τ τ..6 τ τ X Here a F = H σσ ouρ F F τ H 3/ σγh + 3/,.7 τ = T is ime o mauriy, ρ he correlaion beween he Brownian moion driving he fbm and he Brownian moion driving he underlying, and τ = σ.8 is he characerisic diffusion ime. Furhermore, we have wih σou = / sinπh: σ = F = F σ ou z pzdz, σ = F = F σ ou zpzdz, F F = F σ ou zf σ ou zpzdz, 8

9 wih pz he pdf of he sandard normal disribuion. Tha is, we form momens of he volailiy funcion averaged wih respec o he invarian disribuion of he fou process Z. The firs erm in Eq..6 is indeed he expeced effecive volailiy unil mauriy condiioned on he presen. The second erm is a skewness erm ha is non-zero only when he volailiy process and he underlying are correlaed so ha ρ is non-zero. Noe ha he exponen of he fracional erm srucure depends on he Hurs exponen which deermines he smoohness and he decorrelaion rae of he volailiy driving process Z. The smooher he process he larger he implied volailiy for large imes o mauriy. In he fas case presened here wih large and fas volailiy flucuaions he implied volailiy explodes in he regime of shor ime o mauriy. Indeed, shor ime o mauriy means ime o mauriy smaller han he diffusion ime.8, bu larger han he mean reversion ime. Therefore shor ime o mauriy involves large volailiy flucuaions over a shor mauriy horizon resuling in a moneyness correcion ha explodes and dominaes he pure mauriy erm. In he conex of shor or long imes o mauriy he condiional expeced effecive volailiy gives a small conribuion and we have for shor imes o mauriy and K X : [ H 3/ K I a F log,.9 τ τ X and respecively in he regime of long imes o mauriy: I a F τ τ H /.. We remark here ha he fracional scaling in he skewness erm in Eq..6 is exacly he fracional scaling ha corresponds o he case of large ime o mauriy and small volailiy flucuaions given in Garnier and Solna 5. Tha is, wih large imes o mauriy here we have a siuaion reminiscen of he one we have here wih rapid volailiy flucuaions, however, here he volailiy flucuaions are large as compared o he small volailiy flucuaions in Garnier and Solna 5. We remark also ha he case wih a mixing volailiy, and hence inegrable correlaion funcion for he volailiy flucuaions, would correspond o H /. Noe, however, ha our derivaion is valid only for H /,. If we consider he formula 4. for σ φ ha deermines he variance of he firs erm in Eq..6, we can observe ha i vanishes when H /, which shows ha he firs erm in Eq..6 9

10 becomes o leading order deerminisic. In he mixing case he implied volailiy is deerminisic o leading correcion order, while he non-inegrabiliy of he volailiy covariance funcion makes i a sochasic process in he general long-range case wih a variance ha goes o zero as H /. Indeed in he limi case H / we ge a resul as in Fouque e al.,, Secion 5..5 ha deals wih he mixing case. Explicily, consider he mixing case where he volailiy driving process is an ordinary Ornsein-Uhlenbeck process, moreover, he ineres rae and marke price of volailiy risk are zero as we consider here. Then Fouque e al.,, Eq gives he implied volailiy in erms of a coefficien V 3 defined in Fouque e al.,, Secion 5..5: [ I = σ V 3 σ + K σ 3 τ log,. X ha has he same form as he formal limi of.6 as H /. However he averaging expression giving he coefficien V 3 does no correspond o he inerpreaion we arrive a here by he formal limi H /. This is because he singular perurbaion siuaion we consider in fac is singular a H = / and ordering of imporan erms becomes differen. Neverheless i is imporan from he calibraion poin of view ha we have coninuiy of he implied volailiy parameerizaion and is form a H = /, providing robusness o he asympoic framework. In Secion 6 we give he complee saisical descripion of he sochasic correcion coefficien which deermines he random componen of he price correcion and he implied volailiy he firs erm in Eq..6. I is a random funcion of he mauriy T and he curren ime wih Gaussian saisics and wih a covariance funcion ha we describe in deail. This covariance funcion has ineresing and non-rivial selfsimilar properies and i is imporan in order o consruc and characerize esimaors of he implied volailiy surface. Ouline. The ouline of he paper is as follows. In Secion we describe he fracional Ornsein-Uhlenbeck process and derive some fundamenal a priori bounds. In Secion 3 we describe he sochasic volailiy model. In Secion 4 we derive he expression for he price in he fas mean revering fracional case. The derivaion is based on he maringale mehod. Tha is, we make an ansaz for he price as a process ha has he correc payoff and o leading order is a maringale. Then indeed his process is he leading order expression for he price wih an error ha is of he order

11 of he non-maringale par. This approach involves inroducing correcors so ha he non-maringale par is pushed o a small erm and we give he resuling decomposiion in Secion 4. Based on he expression for he price we derive he associaed implied volailiy in Secion 5 and presen finally some concluding remarks in Secion 7. We give a convenien Hermie decomposiion of he volailiy in Appendix A. A number of he echnical lemmas are proved in Appendix B.. The apid Fracional Ornsein-Uhlenbeck Process. We use a rapid fracional Ornsein-Uhlenbeck fou process as he volailiy facor and describe here how his process can be represened in erms of a fracional Brownian moion. Since fracional Brownian moion can be expressed in erms of ordinary Brownian moion we also arrive a an expression for he rapid fou process as a filered version of Brownian moion. A fracional Brownian moion fbm is a zero-mean Gaussian process W H wih he covariance E[W H Ws H = σ H H + s H s H,. where σ H is a posiive consan. We use he following moving-average sochasic inegral represenaion of he fbm Mandelbro and Van Ness 968: W H = ΓH + s H + s H + dw s,. where W is a sandard Brownian moion over. Then indeed W H zero-mean Gaussian process wih he covariance. and where we now have σ H = = [ ΓH + + s H s H ds + H is a ΓH + sinπh..3 We inroduce he -scaled fracional Ornsein-Uhlenbeck process fou as Z = H e s dw H s = H W H H e s W H s ds..4 Thus, he fracional OU process is in fac a fracional Brownian moion wih a resoring force owards zero. I is a zero-mean, saionary Gaussian process, wih variance E[Z = σ ou, wih σ ou = ΓH + σ H = sinπh,.5

12 ha is independen of, and covariance: E[Z Z +s = σ ouc Z s, ha is a funcion of s/ only, wih C Z s = ΓH + [ = sinπh π e v s + v H dv s H cossx x H + x dx. This shows ha is he naural scale of variaion of he fou Z. Noe ha he random process Z is no a maringale, neiher a Markov process. For H /, i possesses long-range correlaion properies: C Z s = ΓH sh + o s H, s..6 This shows ha he correlaion funcion is non-inegrable a infiniy. In his paper we focus on he case H /,. We remark ha if H = /, hen he sandard OU process synhesized wih a sandard Brownian moion is a saionary Gaussian Markov process wih an exponenial correlaion and hence a mixing process. I is possible o simulae pahs of he fracional OU process using he Cholesky mehod see Figure. or oher well-known mehods described in Omre e al. 993; Barde e al. 3. Using Eqs.. and.4 we arrive a he moving-average inegral represenaion of he scaled fou as: Z = σ ou K sdw s,.7 where K = [ K, K = ΓH + H s H e s ds..8 The main properies of he kernel K in our conex are he following ones valid for any H /, : - K is nonnegaive-valued, K L, wih K udu =, bu K L,. - for small imes : K = ΓH + H + O H+,.9

13 - for large imes : K = and in paricular K ΓH H 3 ΓH H 3 + O H 5,. L,. 3. The Sochasic Volailiy Model. The price of he risky asse follows he sochasic differenial equaion: dx = σ X dw, 3. where he sochasic volailiy is σ = F Z, 3. and wih Z being he scaled fou inroduced in he previous secion which is adaped o he Brownian moion W. Moreover, W is a Brownian moion ha is correlaed o he sochasic volailiy hrough W = ρw + ρ B, 3.3 where he Brownian moion B is independen of W. The funcion F is assumed o be one-o-one, posiive-valued, smooh, bounded and wih bounded derivaives. Accordingly, he filraion F generaed by B, W is also he one generaed by X. Indeed, i is equivalen o he one generaed by W, W, or W, Z. Since F is one-o-one, i is equivalen o he one generaed by W, σ. Since F is posiive-valued, i is equivalen o he one generaed by W, σ, or X. We denoe he Hermie coefficiens of he volailiy funcion F wih respec o he invarian disribuion of he fou process by C k : C k = H k zf σ ou zpzdz, H k z = k e z / dk dz k e z /, 3.4 wih pz = exp z // π. We use hese in Appendix A o derive some echnical lemmas. Indeed, for a echnical reason, we require ha F saisfies he following condiion: here exiss some α > such ha k= α k C k k! 3 <. 3.5

14 As we have discussed above, he volailiy driving process Z possesses long-range correlaion properies. As we now show he volailiy process σ propery. Lemma 3.. We denoe, for j =, : F j = F σ ou z j pzdz, where pz is he pdf of he sandard normal disribuion. iself inheris his F j = F σ ou z j pzdz, 3.6. The process σ is a saionary random process wih mean E[σ = F and variance Varσ = F F, independenly of.. The covariance funcion of he process σ is of he form Cov σ, σ+s = F F s C σ, 3.7 where he correlaion funcion C σ saisfies C σ = and σou F C σ s = ΓH F F sh + o s H, for s. 3.8 Consequenly, he process σ possesses long-range correlaion properies i.e. is correlaion funcion is no inegrable a infiniy. Proof. The fac ha σ is a saionary random process wih mean F is sraighforward in view of he definiion 3. of σ. For any, s, he vecor σ ou Z, Z +s is a Gaussian random vecor wih mean, and covariance marix: C = C Z s/. C Z s/ Therefore, denoing F c z = F σ ou z F, he covariance funcion of he process σ is Covσ, σ+s = E [ F c σou Z F c σou Z+s = π F c z F c z exp z, z C z, z T de C s = Ψ C Z, dz dz wih ΨC = π F c z F c z exp C 4 z + z Cz z C dz dz.

15 This shows ha Covσ, σ +s is a funcion of s/ only. Moreover, he funcion Ψ can be expanded in powers of C for small C: ΨC = F c z F c z exp z + z π +C z z F c z F c z exp π dz dz z + z dz dz + OC, C, which gives wih.6 he form 3.8 of he correlaion funcion for σ. 4. The Opion Price. We aim a compuing he opion price defined as he maringale M = E [ hx T F, 4. where h is a smooh funcion. In fac weaker assumpions are possible for h, as we only need o conrol he funcion Q x defined below raher han h. We inroduce he operaor L BS σ = + σ x x, 4. ha is, he sandard Black-Scholes operaor a zero ineres rae and consan volailiy σ. We nex exploi he fac ha he price process is a maringale o obain an approximaion, via consrucing an explici funcion Q x so ha Q T x = hx and so ha Q X is a maringale o firs-order correced erms. Then, indeed Q X gives he approximaion for M o his order. The following proposiion gives he firs-order correcion o he expression for he maringale M in he regime of small. Proposiion 4.. When is small, we have M = Q X + o H, 4.3 where Q x = Q x + x xq x φ + H σρq x, 4.4 Q x is deerminisic and given by he Black-Scholes formula wih consan volailiy σ, L BS σq x =, Q x = hx, T

16 wih σ = F = F σ ou z pzdz, σ = F = F σ ou zpzdz, 4.6 pz he pdf of he sandard normal disribuion, φ is he random componen and Q x is he deerminisic correcion [ T φ = E σ s σ ds F, 4.7 Q x = x x x xq x D, 4.8 wih D defined by D = DT H+, σ ou F F D = ΓH + 3 = σ ou ΓH + 3 F F σ ou zpzdz. 4.9 As shown in Lemma B.3 firs iem, as, he zero-mean random variable H φ has a variance ha converges o σ φ T H, wih σ φ = σ ou F F ΓH + sinπh HΓH +, 4. moreover, i converges in disribuion o a Gaussian random variable wih mean zero and variance σ φ T H. This shows ha he wo correcive erms in 4.4 are of he same order H, bu he firs one is random, zero-mean and approximaely Gaussian disribued, while he second one is deerminisic. Proof. For any smooh funcion q x, we have by Iô s formula dq X = q X d + x x q X σ dw + x xq X σ d = L BS σ q X d + x x q X σ dw, he las erm being a maringale. Therefore, by 4.5, we have wih N dq X = σ σ x x Q X d + dn, 4. a maringale: dn = x x Q X σ dw. Noe also ha in Eq. 4. and below we use he noaion x x x Q X = x x=x Q x. 6

17 Le φ be defined by 4.7. We have φ = ψ where he maringale ψ is defined by [ ψ = E We can wrie T σ σ x x Q X d = x x By Iô s formula: d [ φ x x Q X = x x Q +L BS σ x x + x x x x σ s σ ds, σ s σ ds F. 4. Q X dψ x x Q X dφ. X dφ + x x x x Q X φ d Q X σ d φ, W. Since L BS σ = L BS σ + σ σ x x and LBS σ x x gives d [ φ x x Q X = σ σ x x Q X d + σ σ x x x x Q + x x x x + x x x x Q X σ φ dw Q X φ d X σ d φ, W Q X σ φ dw + x x We have φ, W = ψ, W = ρ ψ, W and herefore where N Therefore: d [ φ x x Q X = σ σ x x Q X d is a maringale, dn = x x x x + σ σ x x x x +ρ x x x x +dn, Q x =, his Q X dψ. Q X φ d Q X σ d ψ, W Q X σ φ dw + x x Q X dψ. d [ Q X + φ x x Q X = σ σ x x x x +ρ x x x x Q X φ d Q X σ d ψ, W +dn + dn

18 The deerminisic funcion Q defined by 4.8 saisfies L BS σq x = x x x xq x θ, Q x =, where θ = dd /d is such ha d ψ, W = H θ + θ d, as shown in Lemmas B.-B. wih θ characerized in Eq. B.9. Applying Iô s formula dq X = L BS σ Q X d + x x Q X σ dw = L BS σq X d + σ σ x x Q X d where N Therefore + x x Q X σ dw = σ σ x x is a maringale, Q X d x x x x Q X θ d + dn, dn = x x Q X σ dw. d [ Q X + φ x x Q X + H ρ σq X = σ σ x x x x + H ρ x x x x Q X φ d + H Q X σ σθ d + ρ x x x x T ρ σ σ σ x x Q X σ θ d Q X d +dn + dn + H ρ σdn. 4.4 We nex show ha he firs four erms of he righ-hand side are of small order H. We inroduce for any [, T :,T = T,T = T 3,T = T 4,T = T x x x x Q s X s σs σ φ sds, 4.5 H ρ σ x x We show ha, for j =,, 3, 4, Q s X s σ s σ ds, 4.6 H ρ x x x x Q s X s θ s σs σds, 4.7 ρ x x x x Q s X s σs θ sds. 4.8 lim sup H [,T E [ j,t / =

19 Sep : Proof of 4.9 for j =. We denoe and so ha Noe ha Y s Y s = x x x x Q s X s γ =,T = T σ s σ φ sds, 4. Y s dγs ds ds. is a bounded semimaringale wih bounded quadraic variaions, so ha is mean square incremens E[Y s Y s are uniformly bounded by K s s. Le N be a posiive ineger. We denoe k = + T k/n. We have N,T =,a,t =,b,t k+ Y s k k= N k+ Y k k k= k+ N = k= dγs ds =,a,t +,b,t ds, N ds ds = dγ s Y s Y k k Noe ha we have by Minkowski s inequaliy: E [,a,t k= dγ s ds ds. Y k γ k+ γ k, / N Y E[γ k / N + Y k= so ha, by Lemma B.4, for any fixed N: On he oher hand lim sup H [,T E [,b,t / F K N k= N k= k+ k k+ E [,a,t / =. E[ Y s Y k k K sup E[φ s 4 /4. N s [,T sup s [,T 4 /4 E[φ s 4 /4 ds s k / ds sup E[φ s 4 /4 s [,T 9 E[γ s /,

20 Therefore, by Lemma B.3 fourh iem, we ge lim sup H sup [,T E [,T / lim sup H Since his is rue for any N, we ge he desired resul. sup [,T E [,b,t / K N. Sep : Proof of 4.9 for j =. We denoe and so ha Noe ha Y s Y s κ = H = ρ σ x x Q s X s,t = T σ s σ ds, 4. Y s dκ s ds ds. is a bounded semimaringale wih bounded quadraic variaions. Le N be a posiive ineger. We denoe as above k = + T k/n. We hen have N,T =,a,t =,b,t Then, on he one hand E [,a,t k+ Y s k k= N k+ Y k k k= k+ N = k= dκ s ds =,a,t +,b,t ds, N ds ds = dκ s Y s Y k k k= dκ s ds ds. Y k κ k+ κ k, / N Y E[κ k / N + Y k= so ha, by Lemma B.6, On he oher hand lim sup H [,T E [,b,t / H F N K H K H N. k= E [,a,t / =. N k= k+ k k+ E[ Y s Y k k s k / ds sup s [,T / ds E[κ s /,

21 Therefore, we ge lim sup H sup [,T E [,T / lim sup H Since his is rue for any N, we ge he desired resul. sup [,T E [,b,t / K N. Sep 3: Proof of 4.9 for j = 3. This proof follows he same lines as he proof of Sep wih η = H σ s σ ds, 4. insead of κ, and using ha θ is bounded. We hen ge he desired resul by Lemma B.5. Sep 4: Proof of 4.9 for j = 4. We have By Lemma B., E [ 4,T / K T lim H [,T E [ θ s / ds K sup sup s [,T E [ 4,T / =. E [ θ s /. We can now complee he proof of Proposiion 4.. In 4.4 we inroduced he approximaion: We hen have Q x = Q x + φ x x Q x + H ρ σq x. because Q T x = hx, φ T By 4.4 we have,t = N = Q T x = hx, =, and Q T x =. Le us denoe,t +,T + 3,T + 4,T, 4.3 dn s + dn s + H ρ σdn s. 4.4 Q T X Q X =,T + N T N.

22 Therefore M = E [ hx T F = E [ Q T X T F = Q X + E [,T F + E [ NT N F = Q X + E [,T F, 4.5 which gives he desired resul because E [,T F is of order o H in L. 5. Call Price Correcion and Implied Volailiy. We denoe he Black- Scholes call price, wih curren ime, mauriy T, srike K, underlying value x, and volailiy σ, by C BS, x; K, T ; σ, so ha Q in Eq. 4.5 is Q x = C BS, x; K, T ; σ. Indeed, C BS gives an explici formula for he price in he case wih consan volailiy. In he siuaion wih a sochasic volailiy as considered here no explici pricing formula exiss. However, as shown in Eq. 4.4 we can ge an asympoic expression for he price in he case wih he sochasic volailiy.5 as a correcion o Q x, he Black-Scholes price evaluaed a he effecive or homogenized volailiy σ. Here, we show ha his correced price akes on a raher simple generic form in he wo parameers, relaive ime o mauriy and moneyness. This represenaion hen leads o a simple represenaion for he implied volailiy as we show below. The long-range characer of he volailiy flucuaions indeed has a srong impac on he form of he implied volailiy and his observaion is imporan in a calibraion conex. We denoe he ime o mauriy by τ = T and we inroduce he characerisic diffusion ime τ = /σ and he dimensionless effecive skewness facor: H H ρ σd τ a F = 3/ σ = σσ ouρ F F τ H H 3/ σγh + 3/, 5. wih σ, σ and D given in Proposiion 4. and he correlaion ρ inroduced in Eq Lemma 5.. The price correcion in Eq. 4.4, normalized by he srike K, can be wrien in he form K φ x x Q e d / x K = π x + H ρ σq {φ τ τ x / + af [ τ τ H + τ τ H log K x }, 5. wih τ [ K d = τ τ τ log x. 5.3

23 Here, he dimensionless random and deerminisic correcion coefficiens are small of order where we used ha φ deviaion H H H φ = O, a F = O, 5.4 τ τ τ τ as defined in Proposiion 4. is cenered and wih sandard Var φ / = τ H τ τ H τσφ + o H, 5.5 wih σ φ defined by Eq. 4. see also Eq. B.4 in Lemma B.3. We commen in more deail abou he saisical srucure of φ in he nex secion. I follows from he above ha he normalized price correcion depends on he wo parameers, he moneyness K/x and he relaive ime o mauriy τ/ τ, and exhibis a erm srucure in fracional powers of relaive ime o mauriy. In Figure 5. we show he relaive price correcion in Eq. 5. as funcion of relaive ime o mauriy τ/ τ for hree values of he moneyness K/x. The solid lines plo he mean relaive price correcion and he dashed lines give he mean plus/minus one sandard deviaion. We use here H =.6, a F =., and / τ H τσ φ =.4. The mean relaive price correcion is larges for a mid range of imes o mauriy. For very shor imes o mauriy relaive o he effecive diffusion ime he effec of he volailiy flucuaions are small, while for large imes he rapid mean reversion averages ou he effec of he flucuaions. Noe, however, ha a he money he random componen of he price correcion decays slowly as τ τ H /, as τ while around he money wih moneyness K/x differen from uniy he decay is like τ τ H / exp τ logk/x. 4τ This reflecs he fac ha he vega is diverging in his limi so ha he sensiiviy o volailiy flucuaions becomes large. We remark ha his would affec calibraion schemes using a he money daa. Moreover, resuls regarding small ime asympoics for he coheren implied volailiy becomes quesionable in his conex as he dominaing conribuion comes from he random componen of he price correcion. 3

24 Noe also ha he parameers chosen are no calibraed o marke daa, his will be considered in anoher publicaion. In Figure 5. we show he price correcion surface as funcion of relaive ime o mauriy τ/ τ and moneyness K/x. We nex presen he proof of Lemma 5.. Proof. For he European call opion wih payoff hx = x K + we have explicily x C BS, x; K, T ; σ = xφ σ T log + σ T K x KΦ σ T log σ T, K where Φ is he cumulaive disribuion funcion of he sandard normal disribuion. We hen have in paricular he Greek relaionships for he call price: σ C BS = T σx xc BS, x x σ C BS = + log K x σ σ C BS. T We hen ge x xq x = x x x xq x = where he Vega is given by σt σc BS, x; K, T ; σ, 5.6 [ σt + log K x σ 3 T σ C BS, x; K, T ; σ, 5.7 σ C BS, x; K, T ; σ = xe d / T π, d = σ T log K x σ. 5.8 T Then, wih Q x given in Eq. 4.8 we can idenify he form of he price correcion as: φ x x Q x + H ρ σq x = φ x x Q x + H ρ σdx x x xq x [ = φ xe d / σ + H xρ σde d / T H πt π σ which in urn gives log K x σ 3 T H,5.9 We nex consider he implied volailiy associaed wih he price correcion. For he sochasic volailiy model in Eq..5 we wan o idenify he implied volailiy I so ha in erms of he correced price in Lemma 4. we have: C BS, x; K, T ; I = Q x + φ x x Q x + H ρ σq x. 5. 4

25 We define he relaive implied volailiy correcion, δi, by I = σ + δi. 5. Lemma 5.. The relaive implied volailiy correcion has he form: δi = φ τ τ + af [ τ τ where φ is defined by 4.7 and a F by 5.. H / + τ τ H 3/ log K X + o H, 5. In Figure 5.3 we show he implied volailiy correcion in Eq. 5. as funcion of relaive ime o mauriy τ/ τ for hree values of he moneyness K/x. We again used H =.6, a F =. and / τ H τσ φ =.4. Noe ha due o he form of he vega, he sensiiviy of he price o he volailiy, he form of he implied volailiy surface is very differen from ha of he price correcion. In Figure 5.4 we show he implied volailiy correcion surface as funcion of relaive ime o mauriy τ/ τ and moneyness K/x. by Proof. We find by using Eqs. 5.9 and 5.8 ha he implied volailiy is given φ [ I = σ + σt + H σρd σt + log K X σ 3 + o H. 5.3 T Since D is deerminisic and given by 4.9, we can hen wrie I = σ + φ σt and he Lemma follows. + H σσ ouρ F F σγh + 3 [ T H + log K X σ T 3 H + o H, 5.4 The firs wo erms in Eq. 5.4 can be combined and rewrien as up o erms of order o H : φ [ σ + σt = E T T Since D is deerminisic and given by 4.9, we can hen wrie [ I = E T +σa F [ τ τ T σ s ds F / σ s ds F / + o H. 5.5 H / + τ τ H 3/ log K X + o H, 5.6 so ha he implied volailiy is he expeced effecive volailiy over he remaining ime horizon condiioned on he presen and wih an added skewness correcion. 5

26 In view of Eq. 5.5, for small ime o mauriy he fourh erm in τ H 3 dominaes in 5.. We remark here ha his is relaed o he fac ha he small parameer in our problem is he mean reversion ime so ha for any order one ime o mauriy in his regime he volailiy has enough ime o flucuae and mean rever giving a price correcion as in Lemma 5.. Then wih he Vega, σ C BS, being mall away from he money, see Eq. 5.8, we ge a srong moneyness dependence and he implied volailiy blows up for small ime o mauriy. Moreover, for large ime o mauriy he hird erm in τ H dominaes in 5.. The long-range dependence gives smooh volailiy flucuaions which gives an implied volailiy ha blows up for large ime o mauriy and wih he curren value for he underlying being less imporan in his large ime o mauriy regime. 6. The -T Process and he Sochasic Implied Surface. We inroduced in Eq. 4.7 he sochasic correcion coefficien φ φ,t which gives he random componen of he price correcion and he implied volailiy and where we here explicily display he dependence on mauriy T. Noe ha if he volailiy process had been a Markovian process hen he correcion would have been deerminisic, as in Fouque e al.. The presence of long-range memory in he volailiy process means ha informaion from he pas volailiy pah mus be carried forward and his makes he price correcion relaive o he price a he homogenized volailiy a sochasic process, and correspondingly for he implied volailiy. We here discuss he saisical srucure of he random field which describes he implied volailiy surface in he scaling regime ha we consider. The implied volailiy is he cenral quaniy in ypical calibraion processes and o design efficien esimaors for boh he coheren and incoheren pars of he implied volailiy, moreover, o characerize he resuling esimaion precision, i is imporan o undersand he saisical flucuaions of he observed implied surface. We give a precise characerizaion of hese flucuaions below. The flucuaions of he implied volailiy for large imes o mauriy relaive o τ become sronger for larger Hurs exponen because he larger Hurs exponen gives sronger emporal coherence and a larger correcion o he anicipaed volailiy. On he oher hand for small imes o mauriy he flucuaions become larger for small Hurs exponen because his gives a rougher process wih large flucuaions even over very small inervals. I is also ineresing o noe ha he correlaion srucure of he implied volailiy surface in fac encodes informa- 6

27 ion abou he long-range characer of he underlying sochasic volailiy. Observing for insance a he money implied volailiy flucuaions as funcion of curren ime for fixed ime o mauriy gives informaion ha makes i possible o esimae he Hurs exponen and check for consisency of he modeling framework. In Livieri e al. 7 observed a he money implied volailiy was used o esimae he Hurs exponen. The auhors found a coefficien ha was slighly larger han he corresponding esimaes using hisorical daa and explained his discrepancy in erms of smoohing effec due o he remaining ime o mauriy. To consruc and inerpre esimaors of his kind a model for he implied surface as a random field relaing i o he underlying volailiy parameers is clearly essenial. In under o undersand he implied volailiy random field noe firs ha i follows from Lemma B.3 ha as, he random process H φ,t /[σ φt H, < T, converges in disribuion in he sense of finie-dimensional disribuions o a Gaussian sochasic process ψ,t, < T, he normalized -T correcion process, wih mean zero, variance one, and covariance E[ψ,T ψ,t = C φ, ; T, T for any [, T, [, T. The four-parameer funcion C φ is given by Eq. B.6. We discuss nex in more deail he -T process ψ,t, a wo-parameer process of curren ime and mauriy T. This process is scaled o have consan uni variance, however, is a non-saionary Gaussian process suppored for < < T. As we see below, close o mauriy T, he process is srongly affeced by he presence of he mauriy boundary. Le us firs consider he case of a fixed mauriy T and inroduce he process ψ ; T = ψ,t, [, T. 6. On shor scales relaive o he ime o mauriy, i.e. for T, i follows from Eq. B.6 ha he process ψ ; T [,T decorrelaes as E [ ψ ; T ψ ; T T, ha is as a Markov process on shor scales. funcion of ψ ; T [,T is More generally, he auocovariance E [ ψ ; T ψ ; T = C, ; T, du [ u + + H [ C = u H u + + H u + H du [ + u H u H, 7

28 wih, ; T = T + 6. which shows ha he correlaion funcion of he process ψ ; T [,T depends only on his relaive separaion so ha we have a siuaion wih a canonical relaive decorrelaion ha depends only on he imes o mauriy τ = T, τ = T. Therefore, we inroduce he process ψ τ; T τ [,T defined by ψ τ; T = ψ T τ,t, τ [, T. 6.3 The process ψ τ; T τ [,T is Gaussian wih mean zero and auocovariance funcion E [ ψ τ; T ψ τ ; T = C τ, τ, wih C as above and τ, τ = τ τ τ + τ. 6.4 For τ τ τ he process decorrelaes on he scale τ so ha he process flucuaions become more rapid close o mauriy. Close o mauriy he price flucuaions become smaller, however, when we magnify hem we see flucuaions on smaller scales for smaller ime o mauriy which reflecs he self similariy of he driving volailiy facor. In Figure 6. we show he correlaion funcion C as funcion of he relaive separaion ime [, and H =.6. The process decorrelaes as a Markov process on shor scales and indeed as one of he imes o mauriy goes o zero relaive o he oher ime he correlaion goes rapidly o zero. Noe ha i follows from he expression 6.4 for ha i is scale invarian in ha aτ, aτ = τ, τ for a >, giving rapid flucuaions for small imes o mauriy. The process has indeed a self-similar propery. We have in disribuion: ψ τ; τ [, ψ τt ; T τ [,, for any T >. In Figure 6. we show wo realizaions of he process ψ τ; as a funcion of ime o mauriy τ. One can also invesigae he srucure of he -T process for a fixed ime o mauriy τ, as a funcion of ime. Thus, if we observe he price for a given ime o mauriy, we would like o know how he price correcion, respecively he implied 8

29 volailiy, would flucuae wih respec o he curren ime, or ime ranslaion. Accordingly we consider he process ψ ; τ = ψ,τ+,, 6.5 for fixed τ >. The process ψ ; τ [, is Gaussian wih mean zero and auocovariance funcion wih E [ ψ ; τψ ; τ = C, ; τ, 6.6 du [ [ u + H u H u + + H u + H C = du [ + u H u H,, ; τ =. 6.7 τ The expression of shows ha he coherence ime of his process scales wih ime o mauriy τ. We see again ha he rescaled implied volailiy surface flucuaions are more rapid closer o mauriy. We also see ha on ransecs parallel o he mauriy boundary in he, T plane hese flucuaions are saionary, his is consisen wih he fac ha we have an underlying consisen model wih a saionary volailiy driving facor. The flucuaions moreover have a self-similar propery. We have in disribuion: ψ ; [, ψ τ; τ [,, for any τ >. The auocovariance funcion of ψ ; [, is ploed in Figure 6.3. In he figure one can see he rapid decay a he origin followed by a long-range behavior. This shows how he implied surface decorrelaes as we move in ime. In Figure 6.4 we show he auocorrelaion funcion in a log-log plo wih he dashed line corresponding o he correlaion decay H. In Figure 6.5 we show wo realizaions of he process ψ ;. Finally, i is of ineres o consider he case where we evaluae he sochasic correcion facor as funcion of ime o mauriy for fixed curren ime : ψ 3 τ; = ψ,+τ, τ. 6.8 The process ψ 3 τ; τ [, is Gaussian wih mean zero and auocovariance funcion E [ ψ 3 τ; ψ 3 τ ; = C 3 3 τ, τ, du [ u + / [ + H u H u + + H u H C 3 = du [ + u H u H, 9

30 wih 3 τ, τ = τ τ τ τ. 6.9 This covariance funcion is ploed in Figure 6.6. Noe ha i follows from he expression 6.9 for 3 ha i is scale invarian in ha 3 aτ, aτ = 3 τ, τ for a >, so ha again he process flucuaes more rapidly for small mauriies. The disribuion of he process ψ 3 τ; τ [, does no depend on and i has a self-similar propery. For any a >, we have in disribuion: ψ3 τ; τ [, ψ 3 aτ; τ [,. In Figure 6.7 we show wo realizaions of he process ψ 3 τ; τ [,. 7. Conclusion. We have considered a coninuous ime sochasic volailiy model wih long-range correlaion properies. We consider he regime of fas mean reversion. This makes i possible o derive an explici expression for he European call opion price and he implied volailiy. Specifically he volailiy is a smooh funcion of a fracional Ornsein-Uhlenbeck process. The analysis of such a non-markovian siuaion is challenging. To he bes of our knowledge we presen he firs analyical expression for he price for general mauriies when he volailiy flucuaions are order one. So far he price compuaions for such siuaions have been based on numerical approximaions. The main resul from he applied view poin is hen he form of he fracional erm srucure we ge for he implied volailiy surface. Indeed we ge an implied volailiy ha grows large wih ime o mauriy while generaing a srong skew for shor imes o mauriy consisenly wih common observaions. We sress ha in our formulaion he mean reversion ime is small compared o any fixed mauriy as we consider a fas mean revering process. Noe finally ha we have considered he case of processes wih long-range correlaion properies wih he Hurs exponen H > / explaining he growh of implied volailiy for large mauriy. A. Hermie Decomposiion of he Sochasic Volailiy Model. We denoe F z = F σ ou z. 3 A.

31 Because E[ F Z < is finie when Z is a sandard normal variable, he funcion F can be expanded in erms of he Hermie polynomials H k z = k e z / dk dz k e z / A. and he series k= C k k! H kz, A.3 wih C k = E [ H k Z F Z = H k z F zpzdz, A.4 converges in L, pzdz o F z. The Hermie polynomials saisfy E[H k ZH j Z = H k zh j zpzdz = δ kj k!, and we have C k k= k! = E[ F Z <. Noe ha C = F. saisfies Lemma A.. If here exiss α > such ha he funcion F defined by A. k= α k C k k! <, A.5 hen he random process I = F Z s F ds A.6 saisfies sup E[I 4 K 4 4H, [,T A.7 for some consan K. Proof. Denoing Z = σ ou Z, which is a zero-mean Gaussian process wih covariance funcion E[ Z Z +s = C Z s/, we have I = F Z s F ds = C m I,m, m= where I,m = m! H m Z sds, m. 3

32 From Taqqu, 978, Lemma. he fourh-order momen of I,m can be expanded as E[I,m 4 = m m! d d d 3 d 4 where he sum is over all indices i, j,..., i m, j m such ha: i i, j,..., i m, j m {,, 3, 4}, ii i j,..., i m j m, m l= il jl C Z, iii each number,, 3, 4 appears exacly m imes in i, j,..., i m, j m. The number N m of erms in his sum is herefore smaller han 4m!/m! 4 i would be exacly his cardinal wihou he second condiion, herefore i is smaller han his number. Since C Z s K s H for some consan K, we have, for any [, T, E[I,m 4 T T m m! d d d 3 d 4 m l= K il jl H. For each erm of he sum, we apply he change of variables s = i, s = j, s 3 = min{,,3,4}\{i,j }, s 4 = max{,,3,4}\{i,j }. In he produc we keep he firs erm: K s s / H, and he firs erm ha has s 3 in i: K s 3 s j / H, so ha we can wrie, for any [, T, E[I,m 4 N mk T T s s [ H s 3 s m ds ds ds 3 ds 4 m! + s 3 s H s 3 s 4 H + K 4m! m m!m! 4 4 4H, H for some consan K ha depends on H and T, because s H is inegrable over [, T. By Sirling s formula, 4m! m m!m! 4 m m! πm. Therefore, by Minkowski s inequaliy, for any [, T, E[I 4 /4 C m E[Im 4 /4 K H m C m m! m= K H m= m= α m Cm / m /, m! α m m= for some consan K, which gives he desired resul. 3 /