SEMIPARAMETRIC INFERENCE FOR INTEGRATED VOLATILITY FUNCTIONALS USING HIGH-FREQUENCY FINANCIAL DATA Yuxiao Liu A dissertatio submitted to the faculty of the Uiversity of North Carolia at Chapel Hill i partial fulfillmet of the requiremets for the degree of Doctor of Philosophy i the Departmet of Statistics ad Operatios Research. Chapel Hill 27 Approved by: George Tauche Chuashu Ji Amarjit Budhiraja Shakar Bhamidi Vladas Pipiras
c 27 Yuxiao Liu ALL RIGHTS RESERVED ii
ABSTRACT YUNXIAO LIU: Essays i High-frequecy Fiacial Ecoometrics Uder the directio of George Tauche ad Chuashu Ji With the advet of itraday high-frequecy data of fiacial assets sice the late 99s, the research of fiacial ecoometrics has etered ito a big data era. New theoretical techiques usig the theory of cotiuous time stochastic processes has bee extesively developed, ad ew empirical evidece has bee documeted. I particular, due to its far-reachig applicatios i various fields such as risk maagemet ad optio pricig, the study of volatility, which quatitatively measures the ucertaity of prices of fiacial assets, has draw substatial attetio from researchers ad there has bee a large amout of literature devoted to this topic, icludig both modellig ad predictio. I this dissertatio, we are firstly cocered with the statistical iferece of the so-called itegrated volatility fuctioals, which is a geeral class of quatities that are computed from volatility. Secodly, we also devise a simulatio method to recover the probability distributio of prices of fiacial assets by takig advatage of the iformatio cotaied i sampled price data. Accordigly, the dissertatio cosists of two parts. I the first part, we focus o the estimatio of itegrated volatility fuctioals, where the volatility process is assumed to be a log memory Itô semimartigale LMIS, which is defied as the sum of a Itô semimartigale ad a process satisfyig certai regularity assumptios that i particular is able to capture the log memory property of fiacial volatility that has bee vastly documeted i literature. We provide cetral limit theorem CLT i such cotext. Furthermore, uder the such LMIS assumptio, we cosider both parametric ad oparametric bootstrap iferece methods of itegrated volatility fuctioals, ad we show the validity of both bootstrap methods by providig CLTs. Furthermore, with the usual assumptio of volatility beig Itô semimartigale, we cosider a empirical-process form of itegrated volatility iii
fuctioals, ad offer fuctioal CLTs whe the idexig parameter is of arbitrary fiite dimesios. We also cosider bootstrap iferece i this empirical-process settig. I the secod part, we cosider Euler method with estimated spot volatility, from which we are able to regeerate ad realize the stochastic dyamics of price of fiacial asset by takig advatage of the iformatio cotaied i the observed prices. We provide both theoretical foudatio ad empirical applicatio of this method. iv
ACKNOWLEDGEMENTS It is my highest hoor ad distict privilege to develop my Ph.D. career i the greatest Departmet of Statistics ad Operatios Research at the Uiversity of North Carolia at Chapel Hill, ad to work with the world-class ecoometricias, statisticias ad probabilists from both UNC Chapel Hill ad Duke Uiversity. I would like to express my most sicere gratitude to all who offered geerous support over the course of my Ph.D. career i the past three ad half years. Firstly, I would like to exted my deepest gratitude to my dissertatio advisor, Professor George Tauche, for his geerous supervisio, guidace, patiece ad ecouragemet from the very begiig of my Ph.D. study. It was Professor Tauche who opeed to me the door to the appealig field of high-frequecy fiacial ecoometrics, ad who guided me through such a challegig but rewardig jourey. His deep uderstadig of the research problems is always visioary which helps me focus o the most importat problems. His critical commets from the weekly semiars have bee sharp ad isightful throughout, which have proved extremely beeficial to my career, both today ad i the future. Also, I wat to thak my coadvisor, Professor Chuashu Ji, for his uderstadig, support ad guidace. It was Professor Ji s support that eables me to exclusively focus o my research work ad make such fast progress. Every coversatio with him was quite iformative, ecouragig ad relievig, i particular whe there was setback durig my secod Ph.D. applicatio. I would also like to express my gratitude to Professor Jia Li for his detailed guidace o my academic research work. Without his geerous support ad ecouragemet, it would be impossible for me to go this far o the road of high-frequecy fiacial ecoometrics. Wheever I got stuck i the log, mysterious proof work, I could always receive illustratig advice from his smartess ad sharp isights ito the research problems. Wheever I fiished a short term task, I could always build up more cofidece ad ehace my work v
from his detailed review ad commets. Furthermore, I have bee ispired by his rigorous attitude to settle dow each piece of proof ad to fialize each academic paper. I would like to ackowledge other committee members, Professor Amarjit Budhiraja, Professor Shakar Bhamidi ad Professor Vladas Pipiras. The probability courses taught by Professor Buhiraja ad Professor Bhamidi offered me rigorous traiig i probability theory which tured out to be crucially helpful to my research work. I also thak them for their help at the start of my research projects, which saved me a lot of time ad eergy. I thak Professor Pipiras for his prompt ad detailed respose to each of my requests o fractioal Browia Motio. His expertise i this area istructively guided me to explore this ew topic without ay burde. I appreciate the geerous support from all other faculty members from STOR departmet for their illustratig lectures ad discussios o various topics over the years of my Ph.D. study. I am also grateful to the care ad support from my frieds ad classmates, from both UNC Chapel Hill ad Duke Uiversity. My special thaks go to Dr. Ruoyu Wu, for his forever geerous ad illumiatig aswers to my aive probability questios. My thaks go to seior graduate studets, icludig but ot limited to, Dr. Tao Wag, Dr. Yu Zhag, Dr. Qig Feg, Dr. Dogqig Yu, Dr. Mighui Liu, Dr. Ququ Yu, Dr. Xi Che, Dr. Hawei Liu, Dr. Yag Yu, Dr. Meilei Jiag ad Dr. Siliag Gog for their geerous suggestios o my both academic ad idustrial careers. My thaks also go to my fellow graduate studets, iclude ot limited to Mrs. Yag Yu, Mrs. Yiche Tu, my former roommate Mr. Zheqi Zhag, Mr. Zheglig Qi, Mr. Jiayu Liu ad Mrs. Huiju Qia for hudreds of meals, jokes ad coservatios together. I am grateful to my frieds from Duke Uiversity. I thak my log-term fried, Mr. Ruibo Ma for stayig costatly coected ad supportive for the past five years ever from the start of my arrival at the Uited States. I also thak the group members o fiacial ecoometrics from Ecoomics Departmet at Duke Uiversity, icludig but ot limited to, Dr. Be Zhao, Mrs. Yua Xue ad Ms. Rui Che, for the valuable discussios ad advice. I wish all the best to all of you i the future. Most importatly, I would like to thak the love ad support from my family. Noe of my graduate life i the U.S. would ever be achieved without the selfless love ad support vi
from my parets ad gradparets. My thaks also go to Ms. Ya Yag, for her support over the darkest period of time durig my graduate life. vii
TABLE OF CONTENTS LIST OF TABLES.................................................................. LIST OF FIGURES................................................................. xi xii Itroductio..................................................................... 2 Prelimiaries.................................................................... 3 2. Itô semimartigale......................................................... 3 2.2 Stable covergece i law................................................... 7 3 Efficiet Estimatio of Itegrated Volatility Fuctioals with Geeral Volatility Dyamics.............................................................. 2 3. Settig..................................................................... 2 3.2 Itegrated volatility fuctioal............................................. 22 3.3 Examples.................................................................. 25 3.3. Fractioal Browia motio........................................ 25 3.3.2 Wieer itegrals w.r.t. fractioal Browia motio.................. 27 3.4 Results..................................................................... 29 3.5 Proofs...................................................................... 32 3.5. Prelimiaries....................................................... 32 3.5.2 Proof of Theorem 3.4.............................................. 35 4 Bootstrap Iferece for Itegrated Volatility Fuctioals......................... 57 4. Settig..................................................................... 57 4.. Itegrated volatility fuctioals...................................... 58 4.2 Parametric Bootstrap....................................................... 59 4.2. Algorithm........................................................... 59 4.2.2 Result.............................................................. 6 viii
4.3 The Local IID Bootstrap................................................... 6 4.3. Algorithm........................................................... 62 4.3.2 Result.............................................................. 63 4.4 Mote Carlo Study........................................................ 63 4.4. The Mote Carlo Set-up............................................. 63 4.4.2 Results.............................................................. 65 4.5 Future Work............................................................... 65 4.6 Proofs...................................................................... 66 4.6. Proof of Theorem 4.2.............................................. 67 4.6.2 Proof of Theorem 4.3.............................................. 7 4.6.2. Elimiatio of jumps ad trucatio....................... 7 4.6.2.2 Proof for the cotiuous case.............................. 73 5 Empirical-process-type CLTs for Estimatig Itegrated Volatility Fuctioals..... 77 5. Settig..................................................................... 77 5.2 A Empirical-process-type Cetral Limit Theorem.......................... 79 5.3 Bootstrap Iferece......................................................... 82 5.3. Parametric Bootstrap............................................... 82 5.3.2 The Local IID Bootstrap Bootstrap.................................. 84 5.4 Future Work............................................................... 85 5.5 Proof....................................................................... 85 5.5. Proof of Theorem 5.2.2.............................................. 85 5.5.. Uiform Covergece for R 3, θ w.r.t. θ................... 86 5.5..2 Uiform Covergece for R 4, θ w.r.t. θ................... 87 5.5.2 Proof of Theorem 5.3............................................... 9 5.5.3 Proof of Theorem 5.3.2.............................................. 96 6 Euler Method with Estimated Volatility.......................................... 97 6. Motivatio................................................................ 97 ix
6.2 Settig..................................................................... 99 6.2. Product Space...................................................... 99 6.2.2 Basic models: o jump or leverage effect............................. 6.3 Euler Approximatio...................................................... 2 6.4 Mai Results............................................................... 5 6.4. Optimal simulatio scheme ad rate of covergece.................. 6 6.4.2 Special case: costat volatility..................................... 8 6.4.3 Closig the gap: couplig ad Wasserstei metric................... 6.4.4 Summary: optimal simulatio scheme............................... 2 6.4.5 Extesio: more tha Hölder cotiuity............................. 3 6.5 Applicatio................................................................ 6 6.5. Estimatio accuracy of diffusive beta............................... 7 6.5.2 Parametric Bootstrap Iferece for Itegrated Volatility Fuctioals. 9 6.5.3 Extesio: resamplig fuctioals of prices.......................... 2 6.6 Future Work............................................................... 24 6.7 Proofs..................................................................... 24 6.7. A Prelimiary result................................................ 24 6.7.2 Proof of Theorem 6.4............................................... 29 6.7.3 Proof of Theorem 6.4.2.............................................. 33 6.7.4 Proof of Propositio 6.4............................................ 47 6.7.5 Proof of Theorem 6.4.3.............................................. 47 6.7.6 Proof of Theorem 6.4.4.............................................. 49 BIBLIOGRAPHY................................................................... 5 x
LIST OF TABLES 4. Mote Carlo coverage probabilities for o-overlappig bootstrap cofidece itervals......................................................... 66 xi
LIST OF FIGURES. A typical price path ad retur............................................. 2 6. Samplig ad Discretizatio Grid whe δ <............................. 4 6.2 Liear Relatio betwee Market ad Idividual Stock....................... 8 6.3 Sample Distributio of Diffusive Beta....................................... 2 6.4 Simulated Returs for Market.............................................. 22 6.5 Simulated Returs for Idividual Stock..................................... 23 xii
CHAPTER Itroductio With the advet of itraday high-frequecy data of fiacial assets sice the late 99s, the research of fiacial ecoometrics has etered ito a big data era. New theoretical techiques usig the theory of cotiuous time stochastic processes has bee extesively developed, ad ew empirical evidece has bee documeted. I particular, due to its farreachig applicatios i various fields such as risk maagemet ad optio pricig, the study of volatility, which quatitatively measures the ucertaity of prices of fiacial assets, has draw substatial attetio from researchers ad there has bee a large amout of literature devoted to this topic. I this Itroductio, we offer a overview of the dissertatio icludig the basic set-up, the research questios we are to explore, the mai results we have obtaied ad a directio of future work. We start with the basic statistical settig. For simplicity, we oly cosider oedimesioal case i the Itroductio ad the multivariate settig will be discussed i the followig chapters. Cosider a filtered probability space Ω, F, F t t, P satisfyig the usual coditios see e.g., Jacod ad Shiryaev, 23, o which are defied a oe-dimesioal Browia motio W ad a Poisso radom measure µ o R + E with determiistic itesity νdt, dz = dt λdz. Here E is a Polish space. As is usual the case e.g. Aït-Sahalia ad Jacod, 24, we model the logarithm of the price process X t of a give stock as a Itô semimartigale i the followig Grigeliois form: X t = x + t b s ds + t σ s dw s + δ { δ } µ ν t + δ { δ >} ν t, where σ is a R valued predictable or simply progressively measurable process o Ω, F, F t t, P, ad δ is a predictable R valued fuctio o Ω R + E. Throughout the dissertatio, all stochastic processes, uless otherwise specified, are assumed to be
Figure.: A typical price path ad retur a The price path ad returs of SPY from /3/27-2/3/25, based o 5-mi data. càdlàg adapted ad hece locally bouded. A typical stock price path ca be see from Figure.. I fiace ad ecoometrics, the process σ t is called the spot or local volatility of X t, ad accordigly the spot local variace process is defied as c t = σt 2. Sice mathematically the sig of σ t caot be idetified ad c t is always oegative, it is more straightforward to cosider c t i study, which, abusig the termiology, we still call volatility. Moreover, sice c t is latet ot observable, it ca oly be recovered by usig the data of X sampled with high-frequecy via certai statistical estimatio procedures. A typical high-frequecy samplig settig goes as follows: give a fixed time spa [, T ], which typically ca be a tradig day, the price process X is discretely sampled with equidistat step size. Hece for ay i =, 2,..., [T/ ], the log-retur of X over 2
iterval [i, i ] is give by i X = X i X i. We cosider ifill asymptotics where the mesh of grid of samplig asymptotically teds to as. Recall that i a traditioal low frequecy settig, the daily risk has bee measured usig daily squared returs, see e.g. Egle, 982 ad Bollerslev, 986, or daily rage differece betwee maximum ad miimum withi oe day, see e.g. Alizadeh et al., 22. With the availability of itraday high-frequecy data sampled as above, however, a ew measure, which is defied as T c s ds ad called itegrated volatility, becomes prevailig. A cosistet ad efficiet estimator for the itergrated volatility is realized volatility, which is defied as the sum of squared itraday returs. Such a method is proposed by Aderse ad Bollerslev, 999 ad popularized by, e.g., Aderse et al., 2a ad Aderse et al., 23a. More geerally, it would be iterestig to study the radom object of the form Sg T gc s ds, for some possibly oliear fuctio g, which is called itegrated volatility fuctioal ad accommodates may quatities that are related to volatility, icludig itegrated volatility as a special case whe gx = x. For other represetative examples of g, see Chapter 3. The first part of this dissertatio, which icludes Chapter 3, 4 ad 5, focuses o the statistical iferece for Sg uder various coditios o the variace process c t ad test fuctio g. More precisely, for give g, the estimator for Sg ca be costructed i two steps: firstly, we oparametrically recover the spot variace c t over the samplig grid by employig a local average of sum of squared trucated returs see Jacod ad Protter, 3
22, Chapter 9 ad 3, that is, for ay i N [T/ ] k, let ĉ i k k j= i+j X 2 { i+j X u}, where k is a sequece of itegers that goes to ifiity represetig the umber of icremets employed i a local widow ad u determies the trucatio threshold for elimiatig jumps i X, see Macii, 2. Next, pluggig the estimated spot volatilities ito a Riema approximatio framework, the estimator of Sg is give by [T/ ] k S g where a higher order bias term is subtracted off. gĉ i g ĉ i ĉ 2 i k,. Assumig that the volatility process follows a Itô semimartigale, Jacod ad Rosebaum, 23a shows a CLT see 4.5 i Chapter 4 below for S g approximatig Sg with rate provided test fuctio g ad its derivatives satisfy a polyomial growth coditio. By a local spatializatio argumet, Li et al., 26a exteds the CLT result to the case of g satisfyig a much weaker coditio give as Assumptio 3.2., ad Li ad Xiu, 26 shows a empirical-process-type CLT i a similar settig. However, all these results assume that volatility process is a Itô semimartigale, which is actually ot able to capture the log memory property of volatility dyamics that has bee widely documeted i literature, see for example, Comte ad Reault, 998. I cotrast, i Chapter 3 we derive the same CLT result for S g approximatig Sg with rate for a larger class of volatility processes: we assume the volatility process follows a log-memory Itô semimartigale LMIS which is give by σ t = σ,t + σ 2,t, where σ,t is a Itô semimartigale ad σ 2,t ca be a fractioal Browia motio or a Weier itegral with respect to fractioal Browia motio. We state the result below, which ca be viewed as oe-dimesioal versio of Theorem 3.4.. 4
Theorem... Uder Assumptios 3..-3.4., it holds that S g Sg L-s MN, V g, where MN, V is a cetered mixed ormal distributio with coditioal variace T V g 2 g c s 2 c 2 s ds. Here L-s deotes stable covergece i law which will be elaborated i Chapter 2. I light of Theorem.., iferece ca be doe for Sg: for example, oe ca costruct cofidece itervals provided that the asymptotic variace V g ca be cosistetly estimated, as Corollary 3.7 i Jacod ad Rosebaum, 23b. O the other had, however, a cosistet estimator for Sg is ot idispesable to obtai cofidece itervals for Sg, as oe may tur to bootstrap method. I Chapter 4, uder the assumptio that the volatility process σ t follows LMIS, we propose algorithms for costructig cofidece itervals for Sg via both parametric bootstrap method ad oparametric bootstrap method. Give bootstrap samples of returs D { i X, i =,..., } geerated either parametrically or oparametrically, the bootstrap estimator for Sg is give by, ot surprisigly, a aalogue form of.: S g; D k [/k ] gĉ,i g ĉ k,iĉ 2,i, where k ik+j X 2 ĉ,i = k j= are bootstrap spot covariace estimators usig D. Here we take T =, = / ad both ĉ,i ad S g; D are costructed over o-overlappig blocks [ik /, i + k /] for i I {,..., [/k ] }. The the bootstrap cofidece iterval of coverage α is formed as [S g; D + S g; D q α/2 S g; D, S g; D + S g; D q α/2 S g; D ], 5
where q α/2 S g; D ad q α/2 S g; D are the α/2 ad α/2 quatiles of S g; D respectively, computed from a large umber of bootstrap repetitios, ad S g; D = k [/k ] gĉ,i is the ucorrected estimator for Sg. Here we use D { i X, i =,..., } to deote the set of origial returs, i cotrast to its bootstrap couterpart D. Theoretically, the asymptotic coverage rate of α is guarateed by Theorem..2. I the sequel, we use L F P Z Z to deote LZ F LZ F for a sequece of radom variables Z ad Z, amely, the coditioal distributio of Z give F coverges to that of Z i probability uder Prokhorov metric. Such a mode of covergece i commoly used i the settig of bootstrap, as well as together with stable covergece i law. Theorem..2. Suppose the Assumptio 3..-3.4., it follows that S g; D S L F g; D MN, V g, where T V g 2 g c s 2 c 2 s ds. Furthermore, we implemet Mote Carlo simulatio to study the coverage rates of both the parametric ad oparametric bootstrap cofidece itervals i fiite sample, the results of which validate our theoretical asymptotic result give i Theorem..2. Chapter 5 cosiders a more geeral form of test fuctio g. We focus o a fuctioal form of the test fuctio g, amely, g : V Θ R, where V R is the rage space of spot volatility, ad Θ R dimθ is the space of some idexig parameter θ. So for each fixed value c t, gc t, is a fuctio over Θ. For example, whe gx = exp ux for u,, Sg is the Laplace trasform of the volatilty occupatio time Todorov ad Tauche, 22b, which summarizes the complete spatial 6
iformatio of the volatility process withi the time spa. See also Li ad Xiu, 26 Sectio 3.3 for other ecoometric applicatios i this cotext. Our goal is to uiformly estimate the quatity of the form Sg; θ T gc s ; θds. Similarly as i Li ad Xiu, 26, the proposed estimator is [T/ ] k S g; θ; D gĉ i ; θ c 2 g ĉ i ; θ ĉ 2 i k. Uder the assumptio that the volatility process σ t is a Itô semimartigale, plus other regularity coditios, we are able to obtai the followig fuctioal cetral limit theorem. Theorem..3. Suppose Assumptio 3.. with σ 2 =, ad Assumptio 3.2.2. Moreover, assume g : V Θ R satisfies Assumptio 3.2. with respect to the first variate ad is cotiuously differetiable with respect to θ Θ, where Θ R dimθ is a compact set, with dimθ <. The the sequece /2 S g; ; D Sg; of processes coverges F stably i law uder the uiform metric to a process ξ which, coditioal o F, is cetered Gaussia with covariace fuctio S g,, where S g, is defied as, for ay θ, θ Θ, T S g θ, θ 2 c gc s ; θ c gc s ; θ c 2 sds. Furthermore, i this fuctioal settig we also develop both parametric ad oparametric bootstrap algorithms to coduct statistical iferece as regard to Sg;. The algorithms are very similar to the oes i Chapter 4 ad we provide empirical-process-type asymptotic results to justify both bootstrap algorithms. From a applicatio poit of view, such asymptotic results could help costructig empirical uiform cofidece regio for Sg;. The secod part of the dissertatio cosists of Chapter 6 aloe, where we develop Euler method with estimated spot volatility. I the field of fiacial ecoometrics, there is always 7
eed to simulate the followig diffusio process dx t = b t dt + σ t dw t, where W is Browia motio. Very ofte X deotes the log-price of fiacial asset, say stock, ad σ is referred to as the volatility process related to X. The most commoly used method to simulate X is the so-called Euler-Maruyama approximatio, which is amed after Leohard Euler ad Gisiro Maruyama, ad is actually a simple geeralizatio of the Euler method for ordiary differetial equatios to stochastic differetial equatios. More precisely, to obtai the value of X at termial time T over a fixed time spa [, T ], oe uses the recursive equatio: X τ+ = X τ + b τ τ + τ + σ τ W τ+ W τ, with give discretizatio grid = τ < τ < < τ N = T. Usually, the equidistat discretizatio scheme is used, amely, τ i+ τ i = δ for some time step < δ < T. For a thorough treatmet o Euler-Maruyama approximatio ad its extesios, see Kloede ad Plate, 992. However, to implemet such procedure, the values of b t t ad σ t t have to be prespecified or simulated beforehad, which might ot replicate the true world as much as possible, eve if the specified values for parameters are claimed to be calibrated to the real world. Alteratively, istead of specifyig particular dyamics for σ t t, we ca use estimated spot volatility based o high-frequecy data i the Euler method. I other words, we would like to desig a data geeratig mechaism, via Euler method, to regeerate data that mimics the real world more realistically, by takig advatage of the iformatio cotaied i the observed real data. As see below, mimic the real world is i the sese that the probability distributio of the simulated data geerated by our Euler method with estimated spot volatility uiformly approximates measured by Wasserstei metric to that of the true data, uder certai assumptios. I fact, we have already used this method i costructig 8
parametric bootstrap cofidece iterval for itegrated volatility fuctioals see Algorithm i Chapter 4. Put it more precisely, we assume that the log-price process follows dx t = c t dw t X = where T is the termial time, c is the variace process ad W is oe-dimesioal Browia Motio itroduced above. I particular, X has either a drift part or a jump part. We cosider a equally spaced time discretizatio grid over [, T ] for Euler approximatio, i.e., for some δ >, let τ =, τ i = iδ. where i {,,..., T δ }. At each discretizatio time poit iδ, the spot volatility estimatio is give by ĉ iδ = k k l= 2 iδ +lx. The for fixed ad δ, the global Euler approximatio with estimated spot volatility is give by the process: Y,δ t = [t/δ] ĉiδ W i+δ W iδ, t T, where W is Browia motio o the simulatio space, which is idepedet of all the iformatio livig i the real world. I particular, for Y,δ to be well-defied, the coditio δ > has to be satisfied. We derive the theoretical results associated with Y,δ. The very first thig oe should otice is that sice i simulatio oly W is available, Y,δ is a cosistet estimator for the simulated log-price defied by X t = t cs d W s, t T, 9
rather tha the true price observed process X t, which has the same distributio as X t uder the o leverage assumptio, i.e., the volatility process c t ad Browia motio W t, both of which are defied from the real world, are idepedet. To derive the covergece rate of Y,δ approximatig X, we have Theorem..4. Suppose Assumptios 6.2. ad 6.2.3. Assume further that {c t : t } has sample paths satisfyig for ay t > s >, E c t c s 2 K t s 2ρ, < ρ, for some costat K. The it holds for ay fixed discretizatio distace δ [, T that E sup Y,δ t X t t T K k + k ρ + δ ρ + δ log 2T 2 δ for some costat K. From Theorem..4 we are able to derive the optimal simulatio scheme i the sese of fastest covergece rate: to make Y,δ coverges to X as fast as possible, oe should first take δ as small as possible, i.e. δ =, which meas takig each data samplig poit as a discretizatio poit; the we strike balace betwee statistical error ad target error arisig from spot volatility estimatio, by requirig or equivaletly k ρ ρ+ 2 k ρ+ 2 ρ β,,. I this fashio, our optimal Euler approximatio becomes: Y t = t/ ĉi i W, t T,
with covergece rate E sup Yt X t t T K k + k ρ + ρ + 2+ ρ. log 2T 2 We show that such a covergece rate is actually exact ot oly a upper boud. As far as applicatios are cocered, we are able to evaluate the accuracy of estimatio of diffusive beta see Reiss et al., 25. Simply speakig, it is doe by obtaiig the samplig distributio of the diffusive beta. More geerally, we ca use the Euler method with estimated volatility to obtai the samplig distributio for ay fuctioal of the path of log-price process.
Future Work: The results obtaied above give us a solid foudatio for future work. As for the project of bootstrap iferece, we have the followig to do Cosider the over-lappig case, besides the o-overlappig case studied already; Prove a similar asymptotic result whe the volatility process is of the mixed form cosidered i Chapter 3. Prove a similar asymptotic result whe g is a fuctioal, characterized by aother idexig parameter θ, as the settig i Chapter 5. As for the project of Euler method with estimated spot volatility, we ca cotiue the study i both theory ad applicatio: Theory: A iterestig directio to geeralize our Euler method with estimated volatility is to take ito accout the so-called leverage effect, which refers to the egative correlatio betwee volatility ad returs. Sice the Browia motio W used i simulatio is idepedet of everythig i the real world, to create egative correlatio betwee the simulated prices ad volatility, we eed to use the same W to regeerate volatility process, which requires to model volatility process as a Itô semimartigale as well ad estimate the volatility of volatility vol. of vol.. As oe may imagie, the covergece rate i this situatio would be eve slower tha /4 as both volatility ad vol. of vol. are latet. Applicatio: The daily rage of a give price process X, defied as the differece betwee max X t ad mi X t withi oe day, had bee a popular measure to quatify daily risk. Obviously, the daily rage depeds o the whole price path over a sigle day, ad hece its sample distributio ca be realized by the Euler method with estimated volatility. Cosequetly, we are able to implemet empirical study usig the Euler method developed here. 2
CHAPTER 2 Prelimiaries We start with otatio that will be used throughout the paper. For a vector B, we use B j to deote its j-th compoet. For a iteger d >, M d deotes the space of d d oegative semidefiite matrices. For a matrix A, we use A ij ad A to deote its i, j elemet ad traspose, respectively. For a matrix valued process A t, the otatios A ij t ad A t are iterpreted similarly. For a matrix A M d ad a differetiable fuctio g defied o M d, the first two partial derivatives of g are deoted as jk ga = ga/ A jk ad 2 jk,lm ga = 2 ga/ A jk A lm respectively. For a set B, B deotes the idicator fuctio of set B. The symbol idicates equality by defiitio. deotes the Frobeius orm. For ay two possibly radom real-valued sequeces a ad b, we write a = O p b if a /b is bouded i probability ad write a = o p b if a /b coverges to i probability. All limits are for. We use, P L ad L-s to deote covergece i probability, covergece i law ad stable covergece i law, respectively. We use K to deote a geeric costat which may vary from lie to lie. I this sectio we give a brief itroduce to two importat otios that will be used frequetly i the rest of dissertatio. Sectio 2. itroduces Itô semimartigale, which is a basic class of stochastic processes commoly used i ecoometrics ad fiace. Sectio 2.2 discusses the so-called stable covergece i law, which is stroger tha the usual covergece i law or weak covergece. 2. Itô semimartigale We begi with the defiitio of geeral semimartigale. For a comprehesive treatmet o this topic, together with other otios i stochastic aalysis, such as theory of Itô itegral, Poisso radom measures ad stochastic itegral with respect to radom 3
measures, see Jacod ad Shiryaev, 23, Jacod ad Protter, 22 ad Aït-Sahalia ad Jacod, 24. We cosider a filtered probability space Ω, F, F t t, P, where the filtratio F t t satisfies the usual coditio as give i Jacod ad Shiryaev, 23 p.2. Throughout the paper, all stochastic processes, uless otherwise specified, are assumed to be càdlàg adapted ad hece locally bouded. Defiitio 2.. a A semimartigale is a process X of the form X = X +M +A where X is fiite-valued ad F measurable, M is a local martigale with M =, ad A is a stochastic process of fiite variatio. b A special semimartigale is a semimartigale X which admits a decompositio X = X + M + A as above, with a process A that is predictable. Give a R d valued process X, the jump measure associated with X is defied as µ X = s>: X s ɛ s, Xs, where ɛ a deotes the Dirac measure sittig at a. The we ca rewrite a semimartigale as X t = X + A t + M t + s t X s { Xs >} where M = A = ad A is of fiite variatio ad M is a local martigale. The the semimartigale A + M by costructio has jumps of size always smaller tha. Hece by Lemma 4.24 i Jacod ad Shiryaev, 23, A + M is special ad we ca write A + M = B + N, where N = B = ad N is a local martigale ad B is a predictable process of fiite variatio. Defiitio 2.2. The characteristic of a R d valued semimartigale X is the followig triple B, C, ν: i B = B i i d is the predictable process of fiite variatio defied above; 4
ii C = C ij i,j d is the quadratic variatio of the cotiuous local martigale part X c of X, that is, C ij = X i,c, X j,c ; iii ν is the predictable compesatig measure of the jump measure µ X of X. Oe should ote that the characteristic triple does NOT characterize the law of the process except for special cases. A importat special case of semimartigale is Levy process, the characteristic triple of which is B t ω = bt, C t ω = ct, νω, dt, dx = dt F dx, which are ot radom actually. For a geeral treatmet of Lévy process, see Bertoi, 998. I fiacial modellig, it is commo to use a special class of semimartigales, but which is also a direct extesio of Lévy process: Defiitio 2.3. A R d valued semimartigale X is a Itô semimartigale if its characteristic B, C, ν are absolutely cotiuous with respect to the Lebesgue measure, i the sese that B t = t b s ds, C t = t c s ds, νdt, dx = dtf t dx, where b = b t is a R d valued process, c = c t is a process with values i M d, ad F t = F t ω, dx is for each ω, t a measure o R d. Now we come to give a fudametal represetatio theorem for Itô semimartigale, which is usually referred to as the Grigeliois form of Itô semimartigale. The followig theorem is Theorem 2..2 i Jacod ad Protter, 22. Let d be a arbitrary iteger with d d, E be a Polish space with a σ fiite measure λ havig o atom, ad qdt, dx = dt λdx. Theorem 2... Let X be a d dimesioal Itô semimartigale o the space Ω, F, F t t, P, with characteristics B, C, ν. There is a very good filtered extesio Ω, F, P of Ω, F, P, o which are defied a d dimesioal Browia motio W ad a 5
Poisso radom measure p o R + E with Lévy measure λ, such that X t = x + t b s ds + t σ s dw s + δ { δ } p q t + δ { δ >} p t, 2. ad where σ is a R d R d valued predictable or simply progressively measurable o Ω, F, F t t, P, ad δ is a predictable R d valued fuctio o Ω R + E. For a more detailed descriptio of extesio of probability space, see Jacod ad Protter, 22 p.36-37. The poit of Theorem 2.. is that ay d dimesioal Itô semimartigale ca be expressed i terms of a Browia motio ad a Poisso radom measure, ad i fact, 2. ca be used as the defiitio for Itô semimartigales, up to extedig the space. Itô semimartigales of form 2. have bee widely used i modellig prices of fiacial assets for various reasos. At first, it has bee widely kow that the prices of fiacial assets, say stocks, have jumps, which for example occurs whe there is sigificat macroecoomic aoucemets. Although we cosider Poisso radom measure with a compesator of product form, dt λdx, which is time-homogeeous, the whole jump part i 2. is actually time-ihomogeeous sice the jump size fuctio δ is radom ad time-varyig. As a cosequece the jump part of stock price is drive by a very geeral class of processes. As for cotiuous part, the drift part captures the persistece i the process, ad also represets the compesatio for risk ad time, while the cotiuous martigale part give as a stochastic itegral models the small moves. I fact, as Back, 99 poits out, special semimartigale appears to be the most geeral cocept of a gais process for which the otio of a local risk premium ca be well-defied. O the other had, Bardorff-Nielse ad Shephard, 24a Remark ad Bardorff-Nielse et al., 26 Remarks 3 demostrate the geerality of the cotiuous local martigale part t σ sdw s i 2.. More precisely, by a represetatio theorem of local martigale as stochastic itegral e.g., Karatzas ad Shreve, 99 p.7-72, all cotiuous local martigales with absolutely cotiuous quadratic variatio ca be writte i the form of t σ sdw s for some process σ t. Usig the Dambis-Dubis-Schwartz theorem, the differece betwee the class of cotiuous local martigale ad the class of stochastic itegrals with respect to Browia motio is the local martigales with cotiuous, but ot 6
absolutely cotiuous quadratic variatio. Thus the form of cotiuous local martigale part we cosider here is oly slightly smaller the class of geeral cotiuous local martigale. 2.2 Stable covergece i law I this subsectio we itroduce the otio of stable covergece i law, which is stroger tha the usual covergece i law or weak covergece. We first review the defiitio of the latter for illustrative purpose. Let E be a Polish space, with Borel σ-field E. Let {Z } be a sequece of E-valued radom variables, allowig each of them defied o its ow probability space Ω, F, P. Defiitio 2.4. We say that Z coverges i law if there is a probability measure µ o E, E such that E fz fxµdx. for all Lipschitz cotiuous bouded fuctios f o E. Usually, oe could realize the limit as a E-valued radom variable Z o some probability space Ω, F, P, the the above covergece reads as E fz E fz. However, the usual covergece i law defied as above may ot be eough i the area of fiacial ecoometrics. As oe ca see, quite ofte we will be i the followig sceario: we eed to estimate some multivariate parameter θ ad we propose a sequece of cosistet estimators ˆθ. We are able to show a cetral limit theorem with certai covergece rate, say ad mixed ormal limitig distributio, amely, ˆθ θ L N, Σ. Very ofte the limitig variace Σ is radom as well, especially i the case of stochastic volatility. I order to do statistical iferece from CLT for example, to costruct cofidece itervals for θ, oe eeds to scale the limitig distributio to multivariate stadard ormal 7
distributio. However, this may ot be achieved eve if Σ ca be cosistetly estimated, as Z L P Z, Y Y do NOT i geeral imply Z, Y L Z, Y, the oly exceptio beig Y is a costat, which is case of the so-called Slutsky Theorem. We hece eed a stroger versio of covergece i law to make sure the joit covergece Z, Y L Z, Y, still holds eve if Y is radom. We require E-valued sequece {Z } of radom variables to be defied o the same probability space Ω, F, P. Defiitio 2.5. We say that Z stably coverges i law if there is a probability measure η o the product space Ω E, F E, such that ηa E = PA for all A F ad E Y fz Y ωfxηdω, dx for all bouded Lipschitz cotiuous fuctios f o E ad all bouded radom variables Y o Ω, F. As before, we ca realize the limit Z o a arbitrary extesio Ω, F, P of Ω, F, P, the the stable covergece i law above ca be writte as E fz Ẽ Y fz, provided PA {Z B} = ηa B for all A F ad B E. The we say Z coverges stably to Z, deoted by Z L s Z. By defiitio, it immediately follows that stable covergece i law implies covergece i law. Moreover, we do have the desired property Z L s Z, Y P Y = Z, Y L s Z, Y. 8
I fact, stable covergece i law is very much like covergece i probability: whe the limitig variable Z is defied o the same space Ω as all Z, it follows that L s P Z Z Z Z. We ed this sectio with a brief literature retrospectio. The otio of stable covergece i law dates back to Réyi, 963, ad is developed by Aldous ad Eagleso, 978, Jacod, 997 ad Jacod ad Protter, 998. A early use of this cocept i ecoometrics is Phillips ad Ouliaris, 99. For a brief summary of stable covergece i law used i a high-frequecy fiacial ecoometrics settig, see Jacod ad Protter, 22, Sectio 2.2., ad a more detailed expositio i geeral cotext of stochastic aalysis is i Jacod ad Shiryaev, 23, Chapter VIII 5c. See also P. ad Heyde, 98 for some differet isights o the topic. 9
CHAPTER 3 Efficiet Estimatio of Itegrated Volatility Fuctioals with Geeral Volatility Dyamics 3. Settig We start with itroducig the formal setup for our aalysis. Cosider a complete filtered probability space Ω, F, F t t, P. Throughout the chapter, all stochastic processes, uless otherwise specified, are assumed to be càdlàg adapted ad hece locally bouded. Our basic assumptios of uderlyig processes are collected i Assumptio 3... Assumptio 3... For some costat r [,, ad a sequece of a sequece τ m m of stoppig times icreasig to, we have i The process X t is a d dimesioal Itô semimartigale with the form X t = x + t b s ds + t σ s dw s + J t, J t = s t X s = t R δ s, z µ ds, dz, 3. where the drift b t is d dimesioal; the spot volatility process σ t is R d R d valued; W t is a d dimesioal Browia motio; µ is a Poisso radom measure o R + E for a auxiliary Polish space E with the determiistic itesity measure νdt, dz = dt λdz for some σ fiite measure λ o E; δ : Ω R + E R d is a predictable fuctio. Moreover, there are a sequece J m m of oegative λ itegrable determiistic fuctios o E such that δω, t, z r J m z for all t τ m ω ad z E. ii The spot volatility process σ t is of the form σ t = σ,t + σ 2,t. 3.2 2
Moreover, both σ,t ad c,t = σ,t σ,t M d are Itô semimartigales of the followig Grigeliois form σ,t = σ, + t + c,t = c, + t + t R t R b σ s ds + t σ σ s dw s + t δ σ s, z { δ σ s,z }µ νds, dz b c s ds + t σ c s dw s + t δ c s, z { δ c s,z }µ νds, dz R R δ σ s, z { δ σ s,z >} µds, dz δ c s, z { δ c s,z >} µds, dz where W ad µ are the same as i 3.; b σ, b c, δ σ, δ c are d d dimesioal ad σ σ, σ c are d d d dimesioal; δ σ, δ c are predictable fuctios such that for a sequece of oegative λ-itegrable fuctios J m o E, δ σ ω, t, z 2 J m z ad δ c ω, t, z 2 J m z for all t τ m ω ad z E. O the other had, σ 2,t is a stochastic process satisfyig, for some ɛ >, E σ 2,t σ 2,s 2 Kt s +2ɛ 3.3 I fiace area, X is usually the logarithm of price of a give stock ad σ is the associated volatility process. For a proper itroductio to Itô semimartigale, see Jacod ad Protter, 22, Chapter 2. I particular, up to expadig dimesios, it is o restrictio to let all Itô semimartigales be drive by the same Browia motio ad Poisso radom measure. We ote that Assumptio 3.. accommodates a large class of models commoly used i fiace ad ecoomics, which allows for jumps i both price ad volatility processes ad for arbitrary depedece structure betwee compoets withi the model. More importatly, the volatility structure 3.2 cosidered i Assumptio 3.. cosists of a geeral Itô semimartigale plus a compoet satisfyig certai regularity coditios, which covers fractioal Browia motio ad related processes that may be used to capture the log-memory property of the volatility process. We refer the readers to the semial works by Comte ad Reault 996,998, which for the first time itroduce the modellig of log-memory property i fiace area. I view of such a mixture of Itô semimartigale 2
ad log-memory process, i the sequel we refer to model 3.2 as the log-memory Itô semimartigale LMIS volatility model. O the techical level, as log as σ is a Itô semimartigale, c is also a Itô semimartigale by Itô s formula. The processes b c, σ c ad δ c ca be expressed as determiistic fuctios of σ, b σ, σ σ ad δ σ, but we do ot eed this here. O the other had, as far as the coditios imposed o the process σ 2 is cocered, sice σ 2 may ot be a martigale ay more e.g., whe σ 2 is fractioal Browia motio, the coditioal expectatio of σ 2,t σ 2,s give F s could be difficult to compute ad hece complicates the proof of Theorem 3.4. below. This is the reaso why more smoothess o the secod momet 3.3 is eeded, which ca be see as compesatio for the loss of martigale property. Now we state the statistical settig i this chapter. At stage, we assume that the process X is sampled at times i for some time step, for i T/, withi the fixed time iterval [, T ]. For ay process Y, the icremets of Y are deoted by i Y Y i Y i, i =,...,. 3.4 Below, we cosider a ifill asymptotic settig, that is, as. 3.2 Itegrated volatility fuctioal With model 3., the spot covariace process of X is give by c = σσ, which is also M d -valued. The radom object of iterest cosidered i this chapter is the itegrated volatility fuctioal of the form Sg T gc s ds, 3.5 for some possibly oliear test fuctio g : M d R, which is assumed to satisfy the followig assumptio. Below, for a compact set K M d ad ɛ >, we deote the ɛ- elargemet about K by K ɛ {M M d : if M A < ɛ}. A K 22
Assumptio 3.2.. There exist a localizig sequece of stoppig times τ m m ad a sequece of covex compact subsets K m M d such that c t K m for t τ m ad g C 3 K ɛ m, the space of three times cotiuously differetiable fuctios o K ɛ m for some ɛ >. Assumptio 3.2. is easily verified i specific settig, which i particular holds, i oedimesioal case, for gc = logc or c, provided that both c t ad /c t are locally bouded with K m beig compact itervals o,. May quatities of iterests i fiace ad ecoometrics ca be writte i the form of 3.5, with Assumptio 3.2. satisfied. For example, whe c is scalar, gx = x correspods to the so-called itegrated volatility Sg = T c tdt, which has bee a popular measure of volatility i high-frequecy settig, see Aderse ad Bollerslev, 999, Aderse et al., 2b ad Aderse et al., 23b. Moreover, gx = x 2 correspods to the itegrated quarticity, which is the half of asymptotic variace whe usig realized volatility to approximate itegrated volatility. The more geerally defied power variatio Sg = T cp t dt for some p > is associated with polyomial test fuctio gx = x p, see for example, Bardorff-Nielse ad Shephard, 23, Bardorff-Nielse ad Shephard, 24b ad Jacod, 28. I bivariate case, the beta for the diffusive movemet of the stock with respect to the market is give by β t c 2,t /c,t, where the market ad the stock are labelled by ad 2 respectively, with the test fuctio beig ga = A 2,t /A,t for A M 2, see Myklad ad Zhag, 29. Moreover, the idiosycratic spot covariace of the stock ca thus be expressed as c 22,t β 2 t c,t = c 22,t c 2 2,t /c,t, with test fuctio ga = A 22,t A 2 2,t /A,t, see Myklad ad Zhag, 26. Other examples iclude: correlatio/leverage effect Kalia ad Xiu, 26, volatility Laplace trasform Todorov ad Tauche, 22b, variace betas Li et al., 26b, eigevalues Aït-Sahalia ad Xiu, 25. Moreover, geeral forms of S g also serve as itegrated momet coditios i specificatio tests ad estimatio problems i ecoomic models Li ad Xiu, 26, followig which we will cosider a fuctioal versio of fuctio g i Chapter 5. We also ote that early discussio o the estimatio of diffusio process ad the samplig frequecy of data goes back to Merto, 98 ad Zhou, 996. I order to give the estimator of Sg for a give fuctio g, we first oparametrically recover the spot variace c i by employig a local average of sum of squared trucated 23
returs see Jacod ad Protter, 22, Chapter 9 ad 3, that is, for ay i N [T/ ] k, let ĉ lm i k k j= i+jx l i+jx m { i+j X u } where l, m d, k is a sequece of itegers that goes to ifiity represetig the umber of icremets employed i a local widow ad u determies the trucatio threshold for elimiatig jumps i X, see Macii, 2 ad Macii, 29. If X is cotiuous, the there is o eed to trucate i formig ĉ by takig u =. The coditios o tuig parameters k ad u are collected i Assumptio 3.2.2. We ote that the study of spot covariace estimatio dates back to Foster ad Nelso, 996, which features a cotiuous settig; oe ca also see Aït-Sahalia ad Jacod, 24 o this topic i a more geeral settig. Assumptio 3.2.2. k γ ad u ϖ for some costats γ ad ϖ satisfyig r 2 3 < γ < 2, γ 2 r ϖ < 2. I particular, k ad k 2. We the defie the estimator for Sg as [T/ ] k S g gĉ i 2k d j,k,l,m= 2 jk,lm gĉ i ĉ jl i ĉ km i + ĉ jm i ĉ kl i 3.6 Assumig that the volatility process is a Itô semimartigale, Jacod ad Rosebaum, 23a shows a CLT for S g approximatig Sg with rate provided test fuctio g ad its derivative satisfy a certai growth coditio. Li et al., 26a exteds the CLT result to the case of g oly satisfyig Assumptio 3.2., ad Li ad Xiu, 26 shows a empirical-process-type CLT i a similar settig, while both papers still assumig volatility process is a Itô semimartigale. I cotrast, i this chapter we wat to derive a associated CLT for S g approximatig Sg with covergece rate ad the same asymptotic 24
variace as i the aforemetioed papers for a larger class of volatility processes give as LMIS 3.2. 3.3 Examples I this sectio we provides some cocrete examples for the process σ 2 satisfyig certai regularity coditios itroduced i Assumptio 3... We begi with fractioal Browia motio i Sectio 3.3., ad the proceed to Wieer itegrals with respect to fractioal Browia motio i Sectio 3.3.2. 3.3. Fractioal Browia motio Fractioal Browia motio fbm B H t t with Hurst idex H, is a cetered Gaussia process with the covariace fuctio EB H t B H s = 2 s2h + t 2H t s 2H, where for simplificatio we assume B H =. The process was itroduced by Kolmogorov, 94, followed by pioeerig works icludig Hurst, 95, Hurst, 956 ad Madelbrot, 983. Fractioal Browia motio has bee widely used i hydrology, egieerig ad fiace. Whe H = 2, the process reduces to the usual stadard Browia motio. For a more comprehesive descriptio of fractioal Browia motio, see, e.g., Duca et al., 2, Nualart, 25, Nualart, 26 ad Mishura, 28. We briefly summarize some importat properties of fractioal Browia motio below:. Self-similarity: for ay a >, {B H au, u R} d = a H {B H u, u R}, where d = deotes the equality i ay fiite-dimesioal distributios. This property ca be regarded as a fractal property i probability. 2. Statioary icremets ad momet estimates: From defiitio it follows that the icremet of B H over a fiite time iterval [s, t] is ormally distributed with mea zero ad variace E B H t B H s 2 = t s 2H. 25