Nonparametric volatility estimation in jump-diffusion models 1

Transcription

1 Nonparametric volatility estimation in jump-diffusion models 1 Roberto Renò University of Siena reno@unisi.it September 17, 27 1 JRC, Ispra, 2-21 september 27

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3 Contents 1 Theory of quadratic variation Quadratic variation Preliminaries Stopping times and subdivisions Martingales Quadratic variation for locally square-integrable martingales Semimartingales Stochastic integral Quadratic Variation Stochastic differential equations Quadratic variation in financial economics Estimating integrated volatility Realized volatility Alternative estimators of integrated volatility for continuous stochastic processes Power and bipower variation Threshold Estimators

4 4 CONTENTS 3 Estimating level-dependent volatility Kernel smoothing The continuous diffusion case The jump-diffusion case Bibliography 47

5 Plan These lecture notes have been written for the Mathematical Finance course at Joint Research Center of Ispra, held on 2-21 September 27. The purpose of the course is to provide an overview of volatility estimation with nonparametric methods. In financial applications, most models have a continuous and a discontinuous component (jumps). Recently, the financial econometrics literature concentrated on the problem of disentangling continuous from discontinuous variation. This course deals with the basic tools to this purpose.the notes are divided as follows. Chapter 1 contains all the relevant theory to define quadratic variation; this is a mathematically oriented chapter which can be used for quick reference. In chapter 2 we focus on the estimation of integrated volatility in jump-diffusion models, that is on estimating volatility over a given time interval, typically a day. In chapter 3 we deal with methods to estimate level-dependent volatility in jump-diffusion models. That is, the aim is to estimate spot volatility but making the assumption that the volatility function depend on the observable state variable. The material does not contain references on two important related topics, that is microstructure noise and multivariate extension. For microstructure noise, popular contributions are Zhang et al. (25); Bandi and Russell (25, 26); Hansen and Lunde (26b); Zhang (26a). For multivariate extensions see Renò (23); Barndorff-Nielsen and Shephard (24a); Hayashi and Yoshida (25); Mancino and Renò (25); Griffin and Oomen (26); Zhang (26b). 5

6 6 CONTENTS

7 Chapter 1 Theory of quadratic variation In this introductory Chapter, we first define quadratic variation and its properties as a tool in stochastic process theory. We subsequently discuss, through a brief review of literature, the importance of the theory of quadratic variation in the financial literature. 1.1 Quadratic variation Preliminaries In what follows, we work in a filtered probability space (Ω,(F t ) t R +,P) satisfying the usual conditions, see Jacod and Shiryaev (1987); Protter (199). We start by the definition of a process, and some other useful definitions. Definition 1.1 A process is a family X = (X t ) t R+ of random variables from Ω to some set E. In applications, the set E will be usually R d, typically with d = 1 (a real-valued stochastic process). A process can be thought as a mapping from Ω R + into E. Definition 1.2 A trajectory of the process X is the mapping t X(ω) for a fixed ω Ω. Definition 1.3 A process X is called càdlàg if all its trajectories are right-continuous and admit left-hand limits. It is called càg if all its trajectories are left-continuous. 7

8 8 CHAPTER 1. THEORY OF QUADRATIC VARIATION If a process is càdlàg, we can naturally define two other processes, X and X as follows: If the trajectory is continuous in t, then X t = X t and X t =. X = X, X t = lim s t X s (1.1) X t = X t X t (1.2) Definition 1.4 A process is said to be adapted to the filtration F if X t is F t -measurable for every t R +. Since our aim is to study quadratic variation, it is important to identify processes of finite quadratic variation for every trajectory. We denote by V the set of all real-valued processes X that are càdlàg, adapted, with X = and whose each trajectory X t (ω) has a finite variation over each finite interval [,t], which implies Var(X) = lim n 1 k n X tk/n (ω) X t(k 1)/n (ω) <. (1.3) We then abbreviate by X V the fact that X is an adapted process with finite variation. We end this subsection with the definitions of increasing and predictable processes. Definition 1.5 A process X is said to be increasing if it is càdlàg, adapted, with X = and such that each trajectory is non-decreasing. Definition 1.6 The predictable σ-field is the σ-field on Ω R + that is generated by all càg adapted processes. A process is said to be predictable if it is measurable with respect to the predictable σ-field Stopping times and subdivisions The concept of stopping time is very useful in econometric analysis, since economic data are recorded at discrete points in time. Definition 1.7 A stopping time is a mapping T : Ω R + such that {ω T(ω) t} F t for all t R +. Given a process X and a stopping time T, we define the stopped process as Xt T things, stopping times are necessary to introduce the localization procedure. = X T t. Among other

9 1.1. QUADRATIC VARIATION 9 Definition 1.8 If C is a class of processes, we define the localized class C loc as follows: a process X belongs to C loc if and only if there exists an increasing sequence T n of stopping times such that lim n T n = a.s. and that each stopped process X T n C. The sequence T n is called a localizing sequence. It is clear that C C loc. We call an adapted subdivision a sequence τ n of stopping times with τ =, supτ n < and τ n < τ n+1 n N on the set {τ n < }. Among subdivisions, we will consider the Riemann sequence. Definition 1.9 A sequence τ n,m,m N of adapted subdivisions is called a Riemann sequence if lim sup n m N [τ n,m+1 τ n,m ] = for all t R Martingales Among processes, a very important role is played by martingales. Definition 1.1 A martingale is an adapted process X whose P-a.s trajectories are càdlàg such that every X t is integrable and such that, for every s t: X s = E[X t F s ] (1.4) Definition 1.11 A martingale X is square-integrable if sup t R + E[X 2 t ] <. In the forthcoming analysis, an important role is played by two special classes of martingales: local martingales and locally square-integrable martingales. Definition 1.12 A locally square-integrable martingale is a process that belongs to the localized class constructed from the space of square integrable martingales. Definition 1.13 A local martingale is a process that belongs to the localized class of uniformly integrable martingales, that is of martingales X such that the family of random variables X t is uniformly integrable. We obviously have that if a martingale X is locally square-integrable, than it is a local martingale. The class of local martingales can be obtained by localization of the class of martingales also. Indeed we have the following:

10 1 CHAPTER 1. THEORY OF QUADRATIC VARIATION Proposition 1.14 Each martingale is a local martingale Proof. Let X be a martingale, and consider the sequence of stopping times T n = n. Then, for every t R +, we have X T n t = E[X n F t ]. Since the class of uniformly integrable martingales is stable under stopping, we have that X T n is uniformly integrable as well. Local martingales, that is martingales, can be decomposed in a continuous and discontinuous part. This concept will be very useful when defining quadratic variation. Definition 1.15 Two local martingales M, N are called orthogonal if their product MN is a local martingale. A local martingale X is called a purely discontinuous local martingale if X = and if it is orthogonal to all continuous local martingales. The following properties help the intuition: Proposition A local martingale X is orthogonal to itself if and only if X is square integrable and X = X up to null sets 2. A purely discontinuous local martingale which is continuous is a.s. equal to. 3. A local martingale X with X = is purely discontinuous if and only if it is orthogonal to all continuous bounded martingales Y with Y =. 4. A local martingale in V is purely discontinuous. Proof. 1. Let X be a local martingale such that X 2 is a local martingale. By localization, we can assume that X,X 2 are uniformly integrable, so that X is square integrable. Thus E[X t ] = E[X ] and E[X 2 t ] = E[X 2 ], and these fact imply X t = X a.s. 2. Is a consequence of point X is orthogonal to Y if and only if it is orthogonal to Y Y. Since Y is continuous, Y Y is locally bounded, then the claim follows from localization. 4. See Jacod and Shiryaev (1987), Lemma I.4.14 (b). The concept of orthogonality, which can be proved to be equivalent to orthogonality in a suitable Hilbert space, allows the following decomposition: Theorem 1.17 Any local martingale X admits a unique (up to null sets) decomposition: X = X + X c + X d (1.5) where X c = X d =, X c is a continuous local martingale and X d is a purely discontinuous local martingale.

11 1.1. QUADRATIC VARIATION 11 Proof. See Jacod and Shiryaev (1987), Theorem I We call X c the continuous part of X and X d its purely discontinuous part. We have also the following: Proposition 1.18 Let X,Y be two purely discontinuous local martingales such that M = N (up to null sets). Then M = N (up to null sets). Proof. Apply Theorem 1.17 to M N Quadratic variation for locally square-integrable martingales We start defining the quadratic variation of two locally square-integrable martingales, see Definition 1.8. We first need the following: Lemma 1.19 Any predictable local martingale which belongs to V is equal to a.s. Proof. See Jacod and Shiryaev (1987), Corollary I Theorem 1.2 For each pair M, N of locally square-integrable martingales there exists a unique, up to null measure sets, predictable process < M,N > V such that MN < M,N > is a local martingale. Proof. The uniqueness comes from Lemma For the existence, see Jacod and Shiryaev (1987), Theorem I.4.2. The process < M,N > is called the predictable quadratic variation of the pair (M,N). Proposition 1.21 The following property holds: < M,N >=< M M,N N > A fundamental example is the Wiener process. Definition 1.22 A Wiener process is a continuous adapted process W such that W = and: 1. E[W 2 t ] <,E[W t ] =, t R +

12 12 CHAPTER 1. THEORY OF QUADRATIC VARIATION 2. W t W s is independent of the σ-field F s, s t. It can be proved that the Wiener process is Gaussian. The function σ 2 (t) = E[W 2 t ] is called the variance function of W t. If σ 2 (t) = t then W is called a Brownian motion. 1 For a proof of the existence of the Wiener process, see Da Prato (1998). The most important properties of the Wiener process can be found in Karatzas and Shreve (1988). We can now prove the following proposition about the quadratic variation of the Wiener process, which is a locally square-integrable martingale. The result is very intuitive: Proposition 1.23 If W is a Wiener process, then < W,W > t = σ 2 (t). Proof. By Theorem 1.2, we have to prove that X t = W 2 t σ 2 (t) is a local martingale. We have: X t X s = W 2 t W 2 s σ 2 t + σ 2 s = (W t W s ) 2 2W 2 s + 2W t W s σ 2 t + σ 2 s. Then E[X t X s F s ] =, hence the result. Note that σ 2 (t) is continuous, null at and increasing Semimartingales Let us denote by L the set of all local martingales M such that M =. Definition 1.24 A semimartingale is a process X of the form X = X + M + A where X is finitevalued and F -measurable, M L and A V (see the discussion of equation 1.3). If there exists a decomposition such that A is predictable, X is called a special semimartingale. From the definition is clear that if X V then it is a semimartingale. Obviously the decomposition X = X + M + A is not unique, but if X is a special semimartingale then there is a unique decomposition with A predictable (Jacod and Shiryaev, 1987). Given that a semimartingale is the sum of a local martingale and a process of finite variation, we can naturally decompose it in a continuous and discontinuous part in the same fashion of Theorem 1.17: 1 In the literature, Wiener process and Brownian motion are often used as synonims, when σ 2 (t) = t.

13 1.1. QUADRATIC VARIATION 13 Proposition 1.25 Let X be a semimartingale. Then there is a unique (up to null sets) continuous local martingale X c such that X c, = and any decomposition X = X + M + A of type 1.24 meets M c = X c up to null sets. Proof. It is enough to use Theorem 1.17 and Proposition 1.16(4). We then follow the above terminology and call X c the continuous martingale part of the semimartingale X. The following Proposition shows that all deterministic processes with finite variation are semimartingales: Proposition 1.26 Let F(t) be a real-valued function on R +, and define the process X t (ω) = F(t). Then X is a semimartingale if and only if F is càdlàg, with finite-variation over each finite interval. Proof. For the sufficiency, it is enough to use the definition of semimartingales. For the converse, see Jacod and Shiryaev (1987), Proposition I Stochastic integral If a process X V, it is easy to define the integral of another process H with respect to X. We define the integral process R HdX by: Z Z t Z t T H s (ω)dx s (ω) i f H s (ω) d[var(x)] s (ω) < h s dx s (ω) = + otherwise This definition stems from the fact that, if X V, then its trajectories are the distribution functions of a signed measure. We want now to define the stochastic integral when X is a semimartingale. In this case, the trajectories do not define a measure; for example, the Wiener process has infinite variation over each finite interval. Now consider a generic process X. The stochastic integral can be naturally defined for processes H such that H = Y 1 [] where Y is bounded and F -measurable, or H = Y 1 ]r,s], where r < s and Y is bounded and F r -measurable. In this case we can define: { Z t i f H = Y 1 H s dx s = [] (1.7) Y(X s t X r t ) i f H = Y 1 ]r,s] (1.6) The distinctive property of semimartingales is that this definition can be extended to any locally bounded predictable process H if and only if X is a semimartingale. The feasibility of the extension for semimartingales is stated in the following theorem.

14 14 CHAPTER 1. THEORY OF QUADRATIC VARIATION Theorem 1.27 Let X be a semimartingale. Then the mapping 1.7 has an extension to the space of all locally bounded predictable processes H, with the following properties: Z t 1. G t = H s dx s is a càdlàg adapted process 2. The mapping H R HdX is linear 3. If a sequence H n of predictable processes converges pointwise to a limit H, and if H n K, where K is a locally bounded predictable process, then R t Hn s dx s converges to R t H sdx s in measure for all t R +. Moreover this extension is unique, up to null measure sets, and in iii) above the convergence is in measure, uniformly on finite intervals: sup s t R s Hn u dx u R s H udx u. A complete proof of the above Theorem can be found in Dellacherie and Meyer (1976). It is important to state the following properties: Proposition 1.28 Let X be a semimartingale and H, K be locally bounded predictable process. Then the following properties hold up to null sets: 1. The mapping X R HdX is linear. 2. R HdX is a semimartingale; if X is a local martingale, then R HdX is a local martingale. 3. If X V then R HdX V and it is given by (1.6). 4. ( R HdX) = and R HdX = R Hd(X X ). 5. ( R HdX) = H X. 6. R Kd( R HdX) = R HKdX. The stochastic integral of a predictable process that is left-continuous can be approximated by Riemann sums. Consider a subdivision τ n. Then the τ-riemann approximant of the stochastic integral R HdX is defined as the process τ( R HdX) defined by We than have the following: τ( R HdX) t = H τn (X τn+1 t X τn t) (1.8) n N

15 1.1. QUADRATIC VARIATION 15 Proposition 1.29 Let X be a semimartingale, H be a càg adapted process and τ n a Riemann sequence of adapted subdivisions. Then the τ n -Riemann approximants converge to R HdX, in measure uniformly on each compact interval. Proof. Consider τ n,m and define H n by H n = H τn,m 1 ]τn,m,τ n,m+1 ] (1.9) m N Then H n is predictable, converges pointwise to H, since H is càg. Now consider K t = sup s t H s. The process K is adapted, càg, locally bounded and H n K. Then the result follow from Theorem 1.27 and from the property τ n ( R HdX) = R H n dx Quadratic Variation We now define the quadratic variation of two semimartingales, and state its most important properties. Definition 1.3 The quadratic variation of two semimartingales X and Y is defined by the following process: Z t Z t [X,Y] t := X t Y t X Y X s dy s Y s dx s (1.1) We will also write [X] t = [X,X] t. From the definition itself, it is straightforward to verify the following properties: Proposition 1.31 The quadratic variation of two semimartingales X,Y has the following properties: 1. [X,Y] = 2. [X,Y] = [X X,Y Y ] 3. [X,Y] = 1 4( [X +Y,X +Y] [X Y,X Y] ) (polarization) The following analysis is crucial for at least two reason. First, the name quadratic variation comes after Theorem Second, it is the basis for realized volatility, a concept which will be illustrated in the following chapters. Indeed, it allows an estimation of quadratic variation.

16 16 CHAPTER 1. THEORY OF QUADRATIC VARIATION Theorem 1.32 Let X and Y be two semimartingales. adapted subdivisions, the process S τn (X,Y) defined by: ( )( S τn (X,Y) t = Xτn,m+1 t X τn,m t Yτn,m+1 t Y τn,m t) m 1 Then for every Riemann sequence τ n,m of converges, for m, to the process [X,Y] t, in measure and uniformly on every compact interval. (1.11) Proof. By polarization, it suffices to prove the claim for X = Y. From equation (1.8) we get: ( Z ) S τn (X,X) = X 2 X 2 2τ n X dx. The last term converges to R X dx by Proposition 1.29, then S τn (X,X) converges to [X,X]. An immediate consequence of Theorem 1.32 is that the quadratic variation of the Wiener process is [W,W] t = σ 2 (t). Let us provide now useful properties of the quadratic variation: Proposition 1.33 Let X and Y be two semimartingales. 1. [X,Y] V. 2. [X, X] is increasing. 3. [X,Y] = X Y. 4. If T is a stopping time, then [X T,Y] = [X,Y T ] = [X T,Y T ] = [X,Y] T. The property 3 implies that if X or Y is continuous, then [X,Y] is continuous as well. Proof. We prove the properties for X = Y, then we can generalize by polarization. 1. [X,X] is càdlàg, adapted and with [X,X] =, thus [X,X] V. 2. Comes directly from Theorem 1.32, since S τn (X,X) is increasing. 3. Using Proposition 1.28 (5) we get [X,X] = (X 2 ) 2X X. Then, since (X 2 ) = ( X) 2 + 2X X, we have [X,X] = ( X) It is a simple consequence of Theorem Proposition 1.34 If X is a special semimartingale and Y V then:

17 1.1. QUADRATIC VARIATION 17 Z t 1. [X,Y] t = X s dy s Z t Z t 2. X t Y t = Y s dx s + X s dy s 3. if Y is predictable, then [X,Y] t = Z t 4. If X or Y is continuous, then [X,Y] =. Y s dx s Proof. See Jacod and Shiryaev (1987), Proposition I We now provide a very useful result. Lemma 1.35 Let X be a purely discontinuous square-integrable martingale. Then [X,X] t = s t ( X s ) 2 Proof. This is Lemma I.4.51 in Jacod and Shiryaev (1987). Theorem 1.36 If X and Y are semimartingales, and if X c,y c denote their continuous martingale parts, then: [X,Y] t =< X c,y c > t + X s Y s (1.12) s t Proof. We prove the theorem in the case X = Y, then polarization yields the result. We can use the decomposition of Proposition 1.25, X = X + X c + M + A, where A V and M is locally squareintegrable and purely discontinuous. By localization we can assume that M is square-integrable. Then: [X,X] = [X c,x c ]+2[X c,m]+2[x c,a]+[m,m]+2[m,a]+[a,a]. (1.13) We have [X c,x c ] =< X c,x c >. Moreover we have [M,M] = ( M s ) 2 from 1.35, while [A,A] = ( A s ) 2 and [M,A] = M s A s from Proposition 1.34(1). Then the sum of the last three terms is ( X s ) 2. From 1.34, 4 we have [X c,a] =. Finally, since X c and M are orthogonal, then < X c,m >=. But [X c,m] is continuous, by Proposition 1.33(3), thus it is equal to < X c,m >=. This ends the proof. Corollary 1.37 Let X,Y be local martingales. Then

18 18 CHAPTER 1. THEORY OF QUADRATIC VARIATION 1. [X,Y] = if X is continuous and Y purely discontinuous. 2. [X,Y] =< X,Y >= if X and Y are continuous and orthogonal. 3. Let H be a locally bounded predictable process. If X is continuous, then R H s dx s is a continuous local martingale. If X is purely discontinuous, then R H s dx s is a purely discontinuous local martingale. Proof. See Jacod and Shiryaev (1987), Corollary I Maybe the most important application of quadratic variation in the field of stochastic processes is Ito s lemma. We state the univariate result, multivariate extension is straightforward. Both proofs can be found in Protter (199), Chapter II. Theorem 1.38 (Ito s lemma) Let X be a semimartingale and f C 2. Then f(x) is a semimartingale and: f(x t ) f(x ) = Z t f (X s )dx s Z t f (X s )d[x,x] c [ s + f(xs ) f(x s ) f ] (X s ) X s <s t (1.14) Applying Ito s lemma to X = W and f(x) = x 2, we immediately get [W,W] t = t. In this respect, the following Theorem provide a characterization of Wiener processes. Theorem 1.39 A stochastic process X is a Brownian motion if and only if it is a continuous local martingale with [X, X] = t. Proof. The fact that, if W is a Brownian motion then [W,W] = t is already known. To show sufficiency, define Z t = exp(iux t + u2 2 t) for some u R. Using Ito s lemma we get: Z t Z Z t = 1+iu Z s dx s + u2 t Z Z s ds u2 t Z t Z s d[x,x] s = 1+iu Z s dx s. 2 2 Then Z is a continuous complex local martingale, as well as any stopping of Z is a martingale. Then we have, u R: hence X is a Brownian motion. ) E[exp(iu(X t X s )) F s ] = exp ( u2 (s t) 2

19 1.1. QUADRATIC VARIATION 19 Theorem 1.4 Let M be a continuous local martingale with M = and such that lim t [M,M] t = a.s. and T s = inf t> [M,M] t > s. Define G s = F Ts and B s = M Ts. Then B s is a Brownian motion with respect to the filtration G s. Moreover [M,M] t are stopping times for G s and M t = B [M,M]t. Proof. See Protter (199), Chapter II, Theorem 41. We finally state the following Theorem which is due to Knight (1971). It allows to transform a vector of orthogonal square-integrable continuous martingales into a vector of independent Brownian motions via a suitable time change. Lemma 1.41 (Knight s Theorem) Let M 1,...,M n be orthogonal square-integrable martingales, and consider the time changes: { inf T i (t) = [B i,b i ] s > t i f this is f inite s (1.15) + otherwise Then the transformed variables: { B i (T i (t)) i f T i (t) < X i (t) = (1.16) B i ( )+W i (t [B i,b i ] ) otherwise where W 1,...W n is an n-dimensional Brownian motion independent of X i, are an n-dimensional Brownian motion relative to their generated filtration Stochastic differential equations In this subsection, we are concerned with the following equation: X(t) = η+ Z t β(s, X(s))ds + Z t σ(s, X(s))dW(s), (1.17) where W(s) is a Brownian motion, as defined in 1.22, and we look for an adapted process X(t) L 2 (Ω). The functions β,σ are applications from [,T] L 2 (Ω) L 2 (Ω), while η is an F measurable process in L 2 (Ω), that is the boundary condition. It is common to write equation (1.17) in the shorthand notation: { dx(t) = β(t,x(t))dt + σ(t,x(t))dw(t) (1.18) X() = η For a review of theory of stochastic differential equation of the kind (1.17), see Da Prato (1998); Karatzas and Shreve (1988). For our purposes, it is sufficient to state the following existence and uniqueness result.

20 2 CHAPTER 1. THEORY OF QUADRATIC VARIATION Theorem 1.42 Assume the following assumptions hold: 1. β and σ are continuous. 2. There exists M > such that: β(t,ζ) 2 + σ(t,ζ) 2 M 2 (1+ ζ 2 ) β(t,ζ 1 ) β(t,ζ 2 ) + G(t,ζ 1 ) G(t,ζ 2 ) M ζ 1 ζ 2 t [,T],ζ L 2 (Ω) t [,T],ζ 1,ζ 2 L 2 (Ω) (1.19) 3. t [,T],ζ L 2 (Ω) such that ζ is F t -measurable, we have that β(t,ζ),σ(t,ζ) L 2 (Ω) and are F t -measurable. Let η L 2 (Ω) and F -measurable. Then there exist a unique (up to null sets) adapted process X(t) L 2 (Ω) fulfilling equation (1.17). Corollary 1.43 Assume the hypothesis of Theorem 1.42 hold. Then the (unique) solution process X of (1.18) is a continuous semimartingale, and [X,X] t = Z t σ 2 (s,x(s))ds (1.2) Proof. The result come from the fact that η is F -measurable and finite-valued, R t β(s,x(s))ds is of finite variation and R t σ(s,x(s))dw(s) is a local martingale, since Wiener process is a local martingale and using For the continuity, see Da Prato (1998). We want now to investigate the link between quadratic variation and the covariance function of the difference process, following Andersen et al. (23). Consider an R d valued semimartingale p(t) in [,T], and its unique decomposition according to Theorem 1.25, p(t) = p()+m(t)+a(t). Let t [, T], h such that t + h < T, and denote the difference process in the interval [t, t + h] by r(t, h) = p(t + h) p(t). In financial economics, if p(t) is the process of logarithmic prices, r(t,h) are called logarithmic returns. We can also define the cumulative difference process r(t) = p(t) p(). It is clear that [r,r] t = [p, p] t. We then have the following: Proposition 1.44 Consider a semimartingale p(t). The conditional difference process covariance matrix at time t over [t, t + h] is given by Cov(r(t,t + h) F t ) = E[[r,r] t+h [r,r] t F t ]+Γ A (t,t + h)+γ AM (t,t + h)+γ AM(t,t + h) (1.21) where Γ A (t,t + h) = Cov(A(t + h) A(t) F t ) and Γ AM (t,t + h) = E[A(t + h)(m(t + h) M(t)) F t.

21 1.2. QUADRATIC VARIATION IN FINANCIAL ECONOMICS 21 Proof. See Proposition 2 of Andersen et al. (23). Proposition 1.44 decomposes the covariance matrix of the difference process in three parts. The first is the contribution of quadratic variation, and it is simply given by its conditional expectation. The second is the contribution of the drift term. The third is the contribution of the covariance between drift and diffusion term. The last two terms disappear if, for instance, the drift term is not stochastic. Even if the drift term is stochastic, so that the last two terms are not null, they are still less relevant when compared to the quadratic variation contribution. For example, we have: Γ i j AM (t,t + h) ( Var[A i (t + h) A i (t) F t ] ) 12 (Var[M j (t + h) M j (t) F t ] ) 1 2 and the latter terms are of order h and h 1 2 respectively, thus Γ AM is at most of order h 3 2. We finally state a proposition on the distribution of returns: Proposition 1.45 Let X be the process satisfying (1.17), and consider the difference process r(t, t +h) of X, and assumptions of Theorem 1.42 hold, and that β,σ are independent of W(s) in the interval [t,t + h]. Then the law of r(t,t + h) conditional to F t is N ( R t+h t β(s)ds, R ) t+h t σ 2 (s)ds. Proof. See Andersen et al. (23), Theorem Quadratic variation in financial economics The importance of quadratic variation in financial economics is widely recognized. The main reason stems from the seminal contribution of Black and Scholes (1973) and Merton (1973), who showed that option prices are a function of asset price volatility. Here, we will circumvent the issue of derivative pricing, since we are more interested in the estimation of quadratic variation from the observation of asset prices, which in the derivative field is called historic volatility. It is well known that the implicit volatility, that is the volatility which prices options, is very different from the historic one, and one very well known example is the smile effect. In particular, we will concentrate on the use of the so-called high-frequency data, whose use became customary in the last decade, see Goodhart and O Hara (1997). Historic volatility was paid a great attention in the financial economics literature. Here we give just few examples of the main problems raised. Christie (1982) analyzes the relation between variance and leverage and variance and interest rates. The leverage effect has been longly studied, since the contributions of Black (1976) and Cox and Ross (1976). The asymmetric link between realized volatility

22 22 CHAPTER 1. THEORY OF QUADRATIC VARIATION and returns is studied in a recent paper by Bekaert and Wu (2), where a model of volatility feedback is introduced, see also Duffee (1995); Wu (21). French and Roll (1986) pose the problem that asset prices variance during trading periods is higher than variance during non-trading periods, and link this finding to the role of private information. The same approach has been followed by Amihud and Mendelson (1987). French et al. (1987), assess the relation between volatility and expected risk premium of stock returns. In the same line, Schwert (1989) studies volatility over more than a century, shows that it is stochastic and tries to explain its movements with regard to macroeconomic variables. In the same spirit, Campbell et al. (21) study the phenomenon of increasing volatility of stocks, explaining this via macroeconomic variables. Intraday volatility has been studied by Lockwood and Linn (199) and in Andersen and Bollerslev (1997), where a link is posed between intraday periodicity and persistence. Maybe the most important stylized fact on volatility is its persistence, or clustering. This idea can be found already in Mandelbrot (1963) or Fama (1965). Poterba and Summers (1986) highlight the importance of persistence on the data used by French et al. (1987). Schwert and Seguin (199) relate the degree of heteroskedasticity to size. Heteroskedasticity leads to modeling persistence in order to get a good picture of asset prices evolution. The result are the ARCH model of Engle (1982) and the GARCH model of Bollerslev (1986), which are very popular nowadays, see Bollerslev et al. (1992) and Bollerslev et al. (1994) for a review. Nelson (1992) assesses the relation between the variance estimated by an ARCH model and the true quadratic variation, showing that the difference between the two converges to zero when the time interval shrinks. The interest in volatility persistence stems from its consequent predictability. Forecasting volatility is probably the main application of the use of the concept of quadratic variation. A quite extensive review of this topic is Poon and Granger (23). On the importance of volatility forecasting for risk management, see also Christoffersen and Diebold (2). Quadratic variation has been used in assessing the informational efficiency of implied volatility, see e.g. Christensen and Prabhala (1998); Blair et al. (21). The financial literature on quadratic variation renewed after the contribution of Andersen and Bollerslev (1998). They show that the low forecasting performance of GARCH(1,1) models, as found e.g. in Jorion (1995), is not due to the poor forecasting ability of these models, but to the poor estimation of integrated volatility. Dating back to an idea of Merton (198), they show, using simulations and FX data, that it is possible to estimate daily volatility using intraday transactions (high-frequency data), and that these estimates are by far more precise than just the daily squared return, and that GARCH forecasting performance is good. They called the measure of volatility via cumulative squared returns realized volatility. This parallels the work of Poterba and Summers (1986); French et al. (1987); Schwert (1989); Schwert and Seguin (199) who compute monthly volatility using daily returns. In some sense, it introduces a new econometric variable, and this leaded to a very large literature.

23 1.2. QUADRATIC VARIATION IN FINANCIAL ECONOMICS 23 Within the same strand, Barndorff-Nielsen and Shephard (22b) study the statistical properties of realized volatility. Hansen and Lunde (26a) compare a large class of autoregressive models using realized volatility measures, concluding that GARCH(1,1) is very difficult to be outperformed. Andersen et al. (21) and Andersen et al. (21) study the statistical properties of realized volatility of stock prices and exchange rates respectively. Andersen et al. (2) study the distribution of standardized returns. The purpose of these studies is to assess the unconditional and conditional properties of volatility, for instance long memory. Similar studies have been conducted for different markets: see Taylor and Xu (1997), Zhou (1996), Areal and Taylor (22) for the FTSE, Andersen et al. (2) for the Nikkei, Bollerslev et al. (2) for an application to interest rates, Bollen and Inder (22), Martens (21), Martens (22), Thomakos and Wang (23) for futures markets and Renò and Rizza (23); Pasquale and Renò (25); Bianco and Renò (26) for the Italian futures market. Using integrated volatility as an observable leads to modeling it directly. The simplest idea to capture persistence is an autoregressive model. Andersen et al. (23) propose an autoregressive model with long memory, end estimate it on foreign exchange rates and stock returns. A similar model is proposed by Deo et al. (23). The HAR-RV model of Corsi (23) is similar, but economically significant restrictions are imposed to the autoregressive structure; long memory is attained using the intuition of Granger (198), that is as the sum of short memory components of different frequencies. Maheu and McCurdy (22) study the importance of non-linear components in the autoregressive structure of volatility dynamics, while Maheu and McCurdy (24) study the impact of jumps on volatility. Fleming et al. (21) show that using a dynamic volatility model instead of a static one, portfolio management can improve substantially. Then, they refine their research using realized volatility as an observable (Fleming et al., 23) and find even better results. Finally, integrated volatility has been used as an observable for estimating stochastic models. One example is Bollerslev and Zhou (22), which estimates a model for exchange rates using realized volatilities and GMM. In the same spirit, Pan (22) uses GMM to estimate a model for stock prices, using as observable the stock prices, option prices and realized volatility. Barndorff-Nielsen and Shephard (22a) suggest a maximum likelihood estimator which uses realized volatilities; Galbraith and Zinde-Walsh (2) use realize volatility to estimate GARCH-like models. Alizadeh et al. (22); Gallant et al. (1999) estimate integrated volatility using the range, that is the squared difference between the high and low of an asset price during a day, and show how to estimate stochastic volatility models including the range as an observable. The range has been used to get more efficient estimates of EGARCH models (Brandt and Jones, 26). Similar studies on the range have been conducted by Brunetti and Lildholdt (22a,b).

24 24 CHAPTER 1. THEORY OF QUADRATIC VARIATION

25 Chapter 2 Estimating integrated volatility Different estimators for the integrated volatility have been proposed. Nowadays, the most popular is realized volatility, which will be discussed thoroughly throughout. The idea behind realized volatility hinges on Theorem Realized volatility Assume that: dx t = µ t dt + σ t dw t + dj t, t [,T] (2.1) that is X t is a Brownian semimartingale plus jumps; µ t and σ t are adapted stochastic processes, and in what follows we will model J t as doubly stochastic Poisson process: J t = N t j=1 c j (2.2) where N t is a Poisson process whose intensity is an adapted stochastic process λ t, the times of the corresponding jumps are (τ j ) j=1,...,nt and c j are adapted random variables measuring the size of the jump at time τ j. The random variables c j are i.i.d. and independent from N t. Here we will not deal with the infinite activity case, see Sato (21) for reference. In this case: Z t [X c ] t = σ 2 s ds 25

26 26 CHAPTER 2. ESTIMATING INTEGRATED VOLATILITY We then have: [X d ] t = N t c 2 j j=1 Z t [X] t = σ 2 s ds+ N t j=1 c 2 j Consider now an interval [,T] with T fixed. Consider a real number δ = T/n. We define the evenly sampled returns as: j X = X jδ X ( j 1)δ, j = 1,...,n (2.3) This definition is used a little bit liberally when δ is generic; in this case n = T/δ is not guaranteed to be integer; when this happens, T is thought to be re-defined as equal to δn. Definition 2.1 Realized variance is defined as: [t/δ] RV δ (X) t = j=1 ( j X) 2 (2.4) Realized volatility is usually the square root of realized variance, however, sometimes the variance is named volatility as well. On the effectiveness of realized volatility as a measure of integrated volatility, see Meddahi (22). As shown by Theorem 1.32, realized variance is an estimator of [X] t, thus it includes jumps. In the case of no jumps, Barndorff-Nielsen and Shephard (22a) proof a central limit theorem regarding RV. Theorem 2.2 As δ, if J =, we have: δ 1 2 (RVδ (X) t [X] t ) L 2 Z t where W is a Brownian motion uncorrelated with W σ 2 s dw s (2.5) Proof. When dj =, it follows from Ito s lemma that: thus Z t Xt 2 = [X] t + 2 X s dx s Z δ j r 2 j = (X δ j X δ( j 1) ) 2 = [X] δ j [X] δ( j 1) + 2 (X s X δ( j 1) )dx s δ( j 1)

27 2.1. REALIZED VOLATILITY 27 which implies: δ 1 2 (RVδ (X) t [X] t ) = 2δ 1 2 = 2δ 1 2 Z δ[t/δ] [t/δ] Z δ j (X s X δ( j 1) )dx s j=1 δ( j 1) (X s X δ[s/δ] )dx s As δ, the right term converges to (Jacod and Protter, 1998): δ 1 2 Z t (X s X δ[s/δ] )dx s 1 Z t 2 σ 2 s dw s, which completes the proof. Intuitively, the latter convergence result is the formalization of the idea that: Z δ j δ( j 1) (X s X δ( j 1) )dx s σ 2 δ( j 1) Z δ j δ( j 1) ( Ws W δ( j 1) ) dws, where the latter term is martingale difference with zero mean and variance 1 2 σ2 δ( j 1). The above Theorem implies that δ 1 2 (RVδ (X) t [X] t ) MN ( Z t ),2 σ 4 s ds Confidence interval for RV can then be constructed using estimators for the integrated quarticity R t σ4 s ds. Barndorff-Nielsen and Shephard (22a) suggest a consistent (but quite unsatisfactory, since it works only in the case J = ) estimator: We can then rewrite: (J = ) QV δ (X) t = 1 3δ (2.6) [t/δ] ( j X) 4 (2.7) j=1 δ 1 2 (RV δ (X) t [X] t ) 2QVδ (X) t N (,1) (2.8) On moderate sample, the performance of the confidence intervals obtained with (2.8) is quite poor; this is mostly due to the non-normality of the distribution of realized volatility, whose distribution is closer to a lognormal one, see e.g. Renò and Rizza (23). For finite δ, better results can be obtained passing to logs, see Meddahi (22): (J = ) δ 2 1 (logrv δ (X) t log[x] t ) N (,1) (2.9) 2 QV δ(x) t (RV δ (X) t ) 2 In the case of jumps, the behavior of realized volatility has been studied by Barndorff-Nielsen et al. (26); Barndorff-Nielsen and Shephard (26) among others, who show that (2.6) still holds even in presence of jumps, with the quadratic variation containing also the jump contribution. Remind that in this case the estimator (2.7) is not robust to the presence of jumps.

28 28 CHAPTER 2. ESTIMATING INTEGRATED VOLATILITY 2.2 Alternative estimators of integrated volatility for continuous stochastic processes Alternative estimators to stochastic volatility have been devised. One example is the Fourier estimator of Malliavin and Mancino (22). Consider, in a filtered probability space, the process (2.1), where σ t,µ t are bounded and J =, so that a single solution of the stochastic differential equation (3.16) exists in the interval [,T = 2π]. Define the Fourier coefficients of dx and σ 2 as follows: a (dx) = 1 2π a k (dx) = 1 π b k (dx) = 1 π Z 2π Z 2π Z 2π dx t a (σ 2 ) = 1 2π cos(kt)dx t sin(kt)dx t a k (σ 2 ) = 1 π b k (σ 2 ) = 1 π Z 2π Z 2π Z 2π σ 2 (X t )dt cos(kt)σ 2 (X t )dt sin(kt)σ 2 (X t )dt. (2.1) It is worth noting that these integrals can be defined on each trajectory for almost every trajectory in Ω; see Föllmer (1979). Then σ 2 t can be computed using its Fourier coefficients: σ 2 (t) = lim M M k= [ ak (σ 2 )cos(kt)+b k (σ 2 )sin(kt) ]. (2.11) Convergence of Fourier sums is in L 2 ([,2π]) norm. The following holds: Proposition 2.3 (Malliavin and Mancino, 22) Consider a process X t satisfying (3.16), and define the Fourier coefficients of dx and σ 2 as in (2.1). Given an integer n >, we almost surely have: a q (σ 2 ) = lim a (σ 2 ) = lim N b q (σ 2 ) = lim N N π N + 1 n π N + 1 n π N + 1 n N 1 ( a 2 k=n 2 k (dx)+b 2 k(dx) ) (2.12) N k=n ( ak (dx)a k+q (dx)+b k (dx)b k+q (dx) ) (2.13) N k=n ( ak (dx)b k+q (dx) b k (dx)a k+q (dx) ) (2.14) Usage of the Fourier estimator is very limited in the literature, and so far it has been employed as an alternative to realized volatility to estimate integrated variance, see Barucci and Renò (22a,b); Kanatani (24); Hansen et al. (23); Barucci et al. (23); Nielsen and Frederiksen (27); Hansen and Lunde (26a).

29 2.3. POWER AND BIPOWER VARIATION 29 Another possibility is given by the range. The range is based on the following observation of Parkinson (198): if p(t) is a one-dimensional solution of of d p(t) = σdw(t), with σ R, and it is observed in [,T], then ( ) ] 2 σ 2 =.361 E[ max p(t) min p(t) t [,T] t [,T] (2.15) This immediately provides an estimate of the variance, which is very popular among finance practitioners, since the maximum and the minimum of the price (so-called high and low) are always recorded. It is simple to show that this idea can be extended to the full variance-covariance matrix, see e.g. Brandt and Diebold (26). This method has been refined using also open and close price, see Garman and Klass (198); Rogers and Satchell (1991); Yang and Zhang (2). For a recent generalization, see Dobrev (27). Other methods have been proposed in the literature. Ball and Torous (1984, 2) regard volatility as a latent variable and estimate it via maximum likelihood; Genon-Catalot et al. (1992) develop a wavelet estimator. Spectral methods have been devised by Thomakos et al. (22) and Curci and Corsi (23). Barndorff-Nielsen and Shephard (27); Andersen et al. (23) are nearly complete reviews of this topic. 2.3 Power and bipower variation An extension of realized variance is the so called realized power variation: Definition 2.4 The realized power variation of order γ is defined as: PV δ (X) t γ [t/δ] = δ 1 γ/2 j X γ (2.16) j=1 It is important to note that the normalization term δ 1 γ/2 disappears when γ = 2, goes to infinity when γ > 2 and goes to zero when γ < 2. When γ = 2, PV coincides with RV. PV is a nice extension of realized volatility. When J = and σ,µ are independent of W, it converges to the integrated powervolatility, see Woerner (26); Barndorff-Nielsen and Shephard (24b): Z t (J = ) p lim PV δ (X) t γ = µ γ σ γ δ sds (2.17)

30 3 CHAPTER 2. ESTIMATING INTEGRATED VOLATILITY where µ γ = E( u γ ) = 2 p+1 γ/2 Γ( 2 ) Γ(1/2) and u N (,1). In the more general case of model (2.1), we have: p lim µ 1 γ PV δ (X) t γ = δ R t σγ sds i f γ < 2 [X] t i f γ = 2 + i f γ > 2 (2.18) Thus, power variation is, in principle, able to disentangle the contribution of jumps according to the power of γ. The robustness of this technique to the case of an infinite number of jumps has been studied by Woerner (26). However, it is impossible to get an estimate of the integrated volatility using power variation. This problem can be solved by using a further generalization: Definition 2.5 We define realized bipower variation of order [γ 1,γ 2 ] as: BPV δ (X) [γ [t/δ] 1,γ 2 ] t = δ 1 2 1(γ 1+γ 2 ) j=2 j 1 X γ1 j X γ 2 (2.19) Bipower variation can be more effective in estimating the continuous part of quadratic variation. In the case of no jumps we have (Barndorff-Nielsen and Shephard, 24b): Z t (J = ) p lim BPV δ (X) [γ,γ] t = µ 2 γ σ 2γ s ds (2.2) δ In the more general case, we have: p lim µ 1 γ δ 1 µ 1 γ 2 BPV δ (X) [γ 1,γ 2 ] t = R t σγ 1+γ 2 s ds i f max(γ 1,γ 2 ) < 2 [X] t i f max(γ 1,γ 2 ) = 2 + i f max(γ 1,γ 2 ) > 2 (2.21) This result means that we can use bipower variation to estimate the integrated variance even in the presence of jumps: that is, choosing γ 1 < 2 and γ 2 = 2 γ 1. Typically, γ 1 = 1 is chosen. The [1,1]-order bipower variation, when it exists, is defined as: BPV(X) [1,1] t [t/δ] = p lim δ In the case µ= and σ independent from W t, we have: BPV(X) [1,1] t j=2 j 1 X j X (2.22) Z t = µ 1 σ 2 s ds = µ 2 1[X c ] t (2.23)

31 2.3. POWER AND BIPOWER VARIATION 31 where µ 1 = 2 π This result can be extended to the case µ, see Barndorff-Nielsen and Shephard (24b). Thus, bipower variation can estimate the continuous quadratic variation of a jump-diffusion process. An intuitive explanation for this feauter of bipower variation is that the counting process N t is finite, thus in the sum (2.19) only a finite number of elements are affected by jumps, and are multiplied by terms (the returns) going to zero as δ. In other words, if the counting process is finite there is always a discrete interval between one jump and the other; when δ is smaller than this finite interval, the jump is only in one return of the product, thus the expected value of the product is null, whatever the nature of the jump. This reasoning poses a serious problem: the result (2.23) holds only asymptotically, that is δ has to be smaller than the typical (unknown) distance between jumps to get a reliable estimate. Remind also that this is infill asymptotics, and that the time span T is fixed. Confidence intervals can be obtained using the results of Barndorff-Nielsen and Shephard (26); Barndorff-Nielsen et al. (26): ( ) δ 2 1 BPV δ (X) [1,1] t BPV(X) [1,1] t L µ 2 Z t 1 2+ϑ where ϑ = π2 4 + π 5. σ 2 s dw s (2.24) Clearly, bipower variation can be used to test for jumps. With mild technical assumptions, including the independence of σ from W t, Barndorff-Nielsen and Shephard (26) show that: ( ) δ 1/2 µ 2 1 BPV δ(x) [1,1] t RV δ (X) t R L N (,ϑ) (2.25) t σ4 s ds (linear jump statistic) and: (ratio jump statistic). ( ) δ 1/2 µ 2 1 BPV δ(x) [1,1] t RV δ (X) t 1 R t σ 4 s ds L N (,ϑ) (2.26) ( R t σ 2 s ds) 2 These tests need the estimation of quarticity, that is R t σ4 s ds, which has to be robust to jumps. It is not difficult to guess that this task can be accomplished using a straightforward extension of bipower variation:

32 32 CHAPTER 2. ESTIMATING INTEGRATED VOLATILITY Definition 2.6 We define the realized multipower variation as: MPV δ (X) [γ 1,...,γ N ] t = δ (γ γ N ) It is clear that, when max(γ 1,...,γ N ) < 2, we have: Examples are quadpower variation: MPV δ (X) [1,1,1,1] t or tripower quarticity: p lim MPV δ (X) [γ 1,...,γ N ] t = δ MPV δ (X) [ 4 3, 4 3, 4 3 ] t = 1 δ = 1 δ ( N k=1 [t/δ] N 1 j=n k= µ γk ) Z t j k X (2.27) σ γ γ N s ds (2.28) [t/δ] Z t j 3 X j 2 X j 1 X j X µ 4 1 σ 4 s ds (2.29) j=4 [t/δ] j=3 Z t j 2 X 3 4 j 1 X 3 4 j X 3 4 µ 3 43 σ 4 s ds (2.3) where µ 4/3 = 2 2/3 Γ(7/6)/Γ(1/2).839. The Monte Carlo evidence of Huang and Tauchen (25) shows that the latter performs better under the selected simulated models, see also the empirical studies of Jiang and Oomen (26); Bollerslev et al. (27). 2.4 Threshold Estimators Threshold estimators are based on the work of Mancini (24, 27) and have been recently studied by Jacod (26). Threshold estimators are based on a property of the Brownian motion established by Lévy. Definition 2.7 A function r(x) is called a modulus of continuity for the function f(x) if, for all sufficiently small δ >, t s δ implies f(t) f(s) < g(δ). The following function is the almost sure modulus of continuity of the Brownian motion: r(δ) = 2δlog 1 δ (2.31) We then have(karatzas and Shreve, 1988): max W(t) W(s) t s δ P lim sup = 1 = 1, (2.32) δ r(δ)

33 2.4. THRESHOLD ESTIMATORS 33 that is, the function in (2.31) defines the speed at which the Brownian motion shrinks to zero. We can use this property to disentangle Brownian paths from discontinuous paths, using an auxiliary function which vanishes slower than the modulus of continuity of the Brownian motion. Definition 2.8 A threshold is a function ϑ(δ) such that: lim ϑ(δ) =, lim δ δlog 1 δ δ ϑ(δ) =. (2.33) The intuition is the following: asymptotically, those variation smaller than the threshold will vanish, since they go to zero faster than the modulus of continuity of the Brownian motion; those variations larger than the threshold can be identified as jumps. This can be formalized as follows: under technical conditions (Mancini, 27), if we denote by N t the non exploding counting process of J in the case of finite activity jumps, we have the following: Theorem 2.9 (Mancini, 24) If J is finite activity with P{c j = } = for all j, and if ϑ( ) is a threshold function, as defined in (2.8) then for P-almost all ω, δ(ω) such that δ < δ(ω) we have i = 1,...,n, I { i N=}(ω) = I {( i X) 2 ϑ(δ)}(ω). (2.34) The above Theorem may be rephrased as: almost surely, for sufficiently small δ, if the squared increments ( i X) 2 in the interval [t,t + δ] are smaller than a threshold ϑ(δ), which vanishes slower than the modulus of continuity of the Brownian motion, then there are no jumps in that interval. On the contrary, jumps occurred only in those intervals in which ( i X) 2 is above the threshold. Definition 2.1 We define the threshold realized variance as: [t/δ] T RV δ (X) t = j=1 ( j X) 2 I { j X 2 ϑ(δ)} (2.35) Using the above result, we can prove that, for every threshold ϑ( ), T RV δ (X) t is a consistent estimator, as δ, of the integrated volatility, and this result is robust under infinite activity jumps. A central limit theorem can be established as well. However, as in (2.24) an estimate of the integrated quarticity should be implemented. This can be accomplished using: T QV δ (X) t = 1 3δ [t/δ] ( j X) 4 I { j X 2 ϑ(δ)} (2.36) j=1