Optimum thresholding using mean and conditional mean square error

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1 Optimum tresolding using mean and conditional mean square error José E. Figueroa-López and Cecilia Mancini August 7, 7 Abstract We consider a univariate semimartingale model for te logaritm of an asset price, containing jumps aving possibly infinite activity IA. Te nonparametric tresold estimator IV ˆ n of te integrated variance IV := T σ sds proposed in 7 is constructed using observations on a discrete time grid, and precisely it sums up te squared increments of te process wen tey are below a tresold, a deterministic function of te observation step and possibly of te coefficients of X. All te tresold functions satisfying given conditions allow asymptotically consistent estimates of IV, owever te finite sample properties of IV ˆ n can depend on te specific coice of te tresold. We aim ere at optimally selecting te tresold by minimizing eiter te estimation mean square error MSE or te conditional mean square error cmse. Te last criterion allows to reac a tresold wic is optimal not in mean but for te specific volatility and jumps pats at and. A parsimonious caracterization of te optimum is establised, wic turns out to be asymptotically proportional to te Lévy s modulus of continuity of te underlying Brownian motion. Moreover, minimizing te cmse enables us to propose a novel implementation sceme for approximating te optimal tresold. Monte Carlo simulations illustrate te superior performance of te proposed metod. Keywords: Tresold estimator, integrated variance, Lévy jumps, mean square error, conditional mean square error, modulus of continuity of te Brownian motion pats, numerical sceme JEL classification codes: C6, C3 Introduction Te importance of including jump components in assets prices models as been extensively igligted. For instance Huang and Taucen in documented empirically tat jumps account for 7% of te S&P5 market price variance, and many different tests for te presence of jumps in asset prices ave been proposed and applied in te literature see 8, Sec. 7.3, for a review of te most used tests. From an economic point of view, jumps may reflect, for instance, reactions of te market to important announcements or events. Tus semimartingale models wit jumps are broadly used in a variety of financial applications, for example for derivative pricing, and also infinite activity jump components ave been considered see e.g. 8, c.5. Separately identifying te contribution of te Brownian part troug te Integrated Variance IV and te one of te jumps to te asset price variations wen we can observe prices discretely is crucial in many respects, for instance, for model assessing and for improving volatility forecasting: e.g. in 5 te proposed test for te presence of jumps is obtained after aving filtered out te jump component; in, te separation allows to construct two tests for recognizing weter te jumps ave finite or infinite variation; in it is sown tat including a separate factor accounting for te jumps in an econometric model for te realized variance substantially improves te out Department of Matematics, Wasington University in St. Louis, MO, 633, USA figueroa@mat.wustl.edu Department of Management and Economics, University of Florence, via delle Pandette 9, 57 cecilia.mancini@unifi.it

2 of sample volatility forecasts. Te correct identification of a model as a significant impact on option pricing and on risk management and tus on assets allocation: for instance Carr and Wu in 7 sow tat te asymptotic beavior of te price of an option as te time-to-maturity approaces zero is substantially different depending on weter te model for te underlying contains jumps or not, and weter te jumps ave finite or infinite variation; Liu, Longstaff, and Pan in 6 find tat incorporating jumps events in te model dramatically affects te optimal investment strategy. Wit discrete non-noisy observations, non parametrically disentangling te jumps from integrated variance IV as mainly been done by using Multipower Variations MPVs and Truncated or Tresold Realized Variance TRV see 8, Sec. 7., for a review of also oter metods. MPV relies on te observation tat, wen te jumps ave finite activity, te probability of aving jumps among subsequent sampling intervals is very small, owever wit infinite activity jumps, tis probability is muc larger. Hence, MPV may not work well in te general case. In contrast, TRV as been sown to be consistent also in te presence of any infinite activity jumps component 7. Furter, it is efficient as soon as te jumps ave finite variation. However te coice of te truncation level tresold as an impact on te estimation performance of IV on finite samples. Te estimation error is large wen eiter te tresold is too small or wen it is too large. In te first case too many increments are discarded, included te increments bearing relevant information about te Brownian part, and TRV underestimates IV. In te second case too many increments are kept witin TRV, included many increments containing jumps, leading to an overestimation of IV. Many different data driven coices of te tresold ave been proposed in te literature, for instance Ait-Saalia and Jacod Sec. 4 terein cose a truncation level of te form α., were is te observation step and α is a multiplier of te standard deviations of te continuous martingale part of te process oter coices are described in 8, p.48. However it is important to control for te estimation error for a given time resolution, and ere we look for an endogenous, teoretically supported, optimal coice. We consider te model dx t = σ t dw t + dj t, were W is a standard Brownian motion, σ is a cádlág process, and J is a pure jump semimartingale SM process. We assume tat we ave at our disposal a record {x, X t,.., X tn } of discrete observations of X spanned on te fixed time interval, T. We also define i Z, or n i Z, te increment Z t i Z ti for any process Z, and a tresold function rσ, any deterministic non-negative function of te observation step, and possibly of a summary measure σ of te realized volatility pat of σ t t, suc tat for any value σ R te following conditions are satisfied We know tat ten TRV, given by lim rσ, =, lim ˆ IV n := rσ, log = +. n i X I { ix rσ ti, i}, were i := t i t i, is a consistent estimator of IV := T σ sds, as sup i i, as soon as σ t t is a.s. bounded away from zero on, T. In te case were te jump process J as finite variation FV and te observations are evenly spaced, te estimator is also asymptotically Gaussian and efficient. For te coice of te tresold TH in finite samples, we consider te following two optimality criteria: minimization of MSE, te expected quadratic error in te estimation of IV; and minimization of cmse, te expected quadratic error conditional on te realized pats of te jump process J and of te volatility process σ s s. Even toug, as mentioned above, many different TH selection procedures ave been proposed, te literature for optimal TH selection is rater scarce. In te TH tat minimizes te expected number of jump misclassifications is considered for a class of additive processes wit finite activity FA jumps and absolutely continuous caracteristics. Even toug it is sown terein tat te proposed criterion is asymptotically equivalent to te minimization of te

3 MSE in te case of Lévy processes wit FA jumps, te latter optimality criterion was not directly analyzed in. Here we go beyond and not only investigate te MSE criterion in te presence of FA jumps but also consider infinite activity jumps and furter introduce te novel cmse criterion. Te last criterion allows to reac a tresold wic is optimal not in mean but for te specific volatility and jumps pats at and, so it is particularly appealing in te cases of non-stationary processes, for wic, even if te MSE was feasible, te deviation of eac realization from te unconditional mean value could be quite large, yielding a poor performance of te unconditional criterion. Moreover, minimizing te cmse is important from a practical point of view, as will be seen in Section 5, were we propose a new TH selection metod in te presence of FA jump processes. Assuming evenly spaced observations, it turns out tat for any semimartingale X, for wic te volatility and te jump processes are independent of te underlying Brownian motion, te two quantities MSE and cmse are explicit functions of te TH and under eac criterion an optimal TH exists, and is a solution of an explicitly given equation, te equation being different under te two criteria. Under certain specific assumptions we also sow uniqueness of te optimal TH: for Lévy processes X, under te first criterion; for constant volatility processes wit general FA jumps, under te second criterion. Te equation caracterizing te optimal tresold depends on te observations time step and so does its solution. Te optimal TH as to tend to as tends to zero and, under eac criterion, an asymptotic expansion wit respect to is possible for some terms witin te equation, wic in turn implies an asymptotic expansion of te optimal TH. Under te MSE criterion, wen X is Lévy and J as eiter finite activity jumps or te activity is infinite but J is symmetric strictly stable, te leading term of te expansion is explicit in, and in bot cases is proportional to te modulus of continuity of te Brownian motion pats and to te spot volatility of X, te proportionality constant being Y, were Y is te jump activity index of X. Tus te iger te jump activity is, te lower te optimal tresold as to be if we want to discard te iger noise represented by te jumps and to catc information about IV. Te leading term of te optimal TH does not satisfy te classical assumptions under wic te truncation metod as been sown in 7 to consistently estimate IV, owever, at least in te finite activity jumps case, we sow erein tat te tresold estimator of IV constructed wit te optimal TH is still consistent. Te assumptions needed for te asymptotic caracterization for te cmse criterion are less restrictive, and also allow for a drift. We find tat, for constant σ and general FA jumps, te leading term of te optimal TH still as to be proportional to te modulus of continuity of te Brownian motion pats and to σ. One of te main motivations for considering te cmse arises from a novel application of tis to tuneup te tresold parameter. Te idea consists in iteratively updating te optimal TH and estimates of te increments of te continuous and jump components Xt c = t σ sdw s and {J t } t of X. We illustrate tis metod on simulated data. Minimization of cmse in te presence of infinite activity jumps in X is a furter topic of ongoing researc. Te constant volatility assumption of some of our results is obviously restrictive. It is possible to allow for stocastic volatility and leverage but, since te proofs are still ongoing, we only discuss ere some ideas and present some simulations experiments tat sow tat also in suc contexts our metods outperform oter popular estimators appearing in te literature. An outline of te paper is as follows. Section deals wit te MSE: te existence of an optimal tresold is establised for a SM X aving volatility and jumps independent on te underlying Brownian motion W ; for a Lévy process X, uniqueness is also establised Subsection. and te asymptotic expansion for te optimal TH is found in Section.3, in bot te cases of a finite jump activity Lévy X and of an infinite activity symmetric strictly stable X. In Section 3, for any finite jump activity SM X, consistency of IVn ˆ is verified even wen te tresold function consists of te leading term of te optimal tresold, wic does not satisfy te classical ypotesis. Section 4 deals wit te cmse in te case were X is a SM wit constant volatility and FA jumps: existence of an optimal TH is establised, its asymptotic expansion is found, ten uniqueness is obtained. In Section 5 te results of Section 4 are used to construct a new metod for iteratively determine te optimal tresold value in finite samples, and a reliability ceck is executed on simulated data. Section 6 presents a Monte Carlo 3

4 study tat sows te superior performance of te new metods over oter metods available in te literature under stocastic volatility and leverage. Section 7 concludes and Section 8 contains te proofs of te presented results. Acknowledgements. José Figueroa-López s researc was supported in part by te National Science Foundation grants: DMS-564 and DMS-636. Cecilia Mancini s work as benefited from support by GNAMPA Italian Group for researc in Analysis, Probability and teir Applications. It is a subunit of te INdAM group, te Itaian Group for researc in Hig Matematics, wit site in Rome and EIF Institut Europlace de Finance, subunit of te Institut Louis Bacelier in Paris. MEAN SQUARE ERROR We compute and optimize te mean square error MSE of IVn ˆ passing troug te conditional expectation wit respect to te pats of σ and J: MSE := E ˆ IV n IV = E E IV ˆ n IV σ, J. Conditioning on σ, as well as assuming no drift in X, is standard in papers were MSE-optimality is looked for, in te absence of jumps see e.g. 4. We also assume evenly spaced observation over a fixed time orizon, T, so tat t i = t i,n = i n, for any i =... n, wit = n = T/n. Denoted by te square root rσ, of a given tresold function, in tis work we focus on te performance of te tresold estimator: ˆ IV n := n i X I { ix }. 3 We indicate te corresponding MSE by MSE. Note tat for we ave IV ˆ n =, so MSE = EIV ; as increases some squared increments i X are included witin IV ˆ n, so IV ˆ n becomes closer to IV and MSE decreases. However, if J, for + te quantity MSE increases again, since IV ˆ n includes all te squared increments i X and tus IV ˆ n estimates te global quadratic variation IV + s T X s of X at time T, and MSE becomes close to E s T X s. We look for a tresold giving MSE = min MSE., In tis section we analyze te first derivative MSE and we find tat an optimal tresold exists, in te general framework were X is a semimartingale satisfying A below, and we furnis an equation to wic is a solution, wile in Section., we find tat is even unique. Te equation as no explicit solution, but is a function of and we can explicitly caracterize te first order term of its asymptotic expansion in, for. Clearly we can always find an approximation of te optimal tresold wit arbitrary precision making use of numerical metods. Let us denote We assume te following i X := i XI { ix }, σ i := ti A. A.s. σ s > for all s; J ; and σ, J are independent on W. t i σ sds, m i := i J. Te independence condition is needed to guarantee tat W remains a Brownian motion conditionally to σ and J. We analyze te leverage case in our simulation study of Sec. 6. Wit te next teorem we compute te first derivative MSE of te mean square error. Te proof is deferred to te Appendix. 4

5 Teorem. Under A and te finiteness of te expectations of te terms below, for fixed and >, we ave tat MSE = G, were wit a i and b i defined as G := n E a i + n j= j i b j IV, 4 m i a i := e σ i + e +m i σ i, σ i π b i := E i X σ, J = e m i σ i + m i + e +m i σ i σi m i + m i + σ i π π mi+ σ i m i σ i e x dx. It clearly follows tat MSE > if and only if G > and, tus, to our aim of finding an optimal tresold, it suffices to study te sign of G as varies. Notation. For brevity we sometimes omit to precise te dependence on of a i and b i. For a function f we sometimes use f+ for lim + f. For two functions fx, gx of a non-negative variable x wic tends to respectively to +, by f g, we mean tat f = og as x respectively x +, by f g we mean tat bot f = Og and g = Of as x respectively x +, wile by f g we mean tat fx/gx as x respectively x +. We denote φx = x e + π, Φx = φsds. x.o.t means iger order terms Remark. Under A and te finiteness of te expectation of te terms in MSE we ave MSE = EIV > and, for small, lim MSE >. + Te next Corollary states te existence of an optimal tresold see te proof in te Appendix. Corollary. Under te same assumptions of Teorem an optimal tresold exists and is solution of te equation G =. To find an optimal tresold to estimate σ we need to find te zeroes of G, wic in turn depends on σ. Also, G depends on te jump process increments m = m,..., m n, wic we don t know. An analogous problem arises wen dealing wit te minimization of te conditional MSE introduced in Section 4, were te optimal tresold as to satisfy te equation F =, wit F := a i + n j i b j IV. However, wen we apply our teory to te case of constant σ and finite activity jumps, as precisely explained in Section 5, we can proceed by estimating σ, m and iteratively. Anoter metod yet to implement is to study te infill asymptotic beavior of in a stationary or deterministic state of σ. In some situations, te leading order terms of will only depend on a few summary measures of te stationary distribution or pat of σ, wic could be estimated separately or jointly wit IV. Remark. In principle MSE could even ave many points were te absolute minimum value MSE of MSE on, + is reaced; also, MSE could ave an infinite number of local not absolute minima. To determine te number of solutions to G =, we need to study te sign of G corresponding to te convexity properties of M SE, but tis is not easy. Define g i := + j i b j IV, 5

6 so tat G = i Ea i g i. We can easily study te functions g i, since we know tat g i = IV <, lim + g i = + and g i = + j i a i > for all >. However witin te joint function G te presence of te terms a i makes it difficult even to know weter a i g i is positive.. Wen X is Lévy Let us assume A. X is a Lévy process. We now ave tat σ > is constant and i X are i.i.d., so te equation caracterizing MSE = is muc simpler to analyze. Indeed, from 4, since witin a i j i b j, te term m i of a i is independent on te terms m j of b j, we ave MSE = G = nea + n Eb IV. Te next result establises uniqueness of te optimal tresold under A. Te proof is in te Appendix. Teorem. If X is Lévy, equation + n Eb IV = 5 as a unique solution and, tus, tere exists a unique optimal tresold, wic is. Te equation in 5 as no explicit solution, owever we can give some important indications to approximate.. Asymptotic beavior of E b i For te rest of Section, in order to empasize te dependence of on, we write := =. We still are under A, so recall tat E b i = E σ n i W + n i J { σ ni W + ni J }, is constant in i. Note tat Eb i is finite for any Lévy process J, regardless of weter J as bounded first moment or not. We consider two cases: te case were J is a finite jump activity process and te one were it is a symmetric strictly stable process. Te asymptotic caracterization of E b i will be used in Subsection.3 to deduce te asymptotic beavior in of te optimal tresold. We anticipate tat in Subsection.3 we will also see tat an optimal tresold as to tend to as and in suc a way tat +... Finite Jump Activity Lévy process Te asymptotic caracterization of E b i in te case were J as finite activity jumps is given in te following Teorem. Its proof is in te Appendix. Teorem 3. Let X be a finite jump activity Lévy process wit jump size density f and wit jump intensity λ. Suppose also tat te restrictions of f on, and, admit C extensions on, and,, respectively. Ten, for any = suc tat and, as, we ave E b = σ σ e π were above Cf := f + + f. σ + λ 3 3 Cf + O + o e σ + o 3, 6

7 .. Strictly stable symmetric Lévy Jump process Let us start by noting tat Eb = E σw + J { σw +J } = σ E W { σw +J } + σe W J { σw +J } + E J { σw +J } =: C + D + E. Te first term above can be written as were C = σ σ E W { σw +J >} = σ σ C + + C, C + = E W {W+σ / J >σ / }, C = E W {W+σ / J < σ }. / By conditioning on J and using te fact tat EW {W>x} = xφx + Φx, for all x R, we ave C ± = E σ J σ φ σ J σ + Φ σ J σ. Te following Lemmas state te asymptotic beavior of te above quantities under te assumption tat. Teir proofs are in te Appendix. Lemma. Suppose tat {J t } t is a symmetric Y -stable process wit Y,. Ten, tere exist constants K < and K suc tat: E φ σ J σ = e σ K Y 3 +.o.t. 6 π E J φ J = K Y +.o.t.. 7 Lemma. Suppose tat {J t } t is a symmetric strictly stable process wit Lévy measure C x Y dx. Ten, te following asymptotics old: E Φ J, 8 σ E J { σw +J } J σ = C Y Y + O Y + O E φ = C Y Y + O Y + O σ σ + O 4 Y Y. 9 As a consequence, te following Teorem states explicitly te asymptotic beavior of E b. It s proof is in te Appendix. Teorem 4. Let X t = σw t + J t, were W is a Wiener process and J is a symmetric strictly stable Lévy process wit Lévy measure C x Y. Ten, for any = suc tat and, as, we ave.3 Asymptotic beavior of We now assume A3. Te support of any jump size J t is R. E b = σ σ e σ + C π Y Y +.o.t.. We firstly see tat an optimal tresold = as to tend to as and in suc a way tat +. Ten we will sow te asymptotic beavior of in more detail. 7

8 Remark 3. Note tat under A3, if minimizes MSE, ten necessarily as. Indeed, if lim inf = c >, ten on a sequence converging to c we would ave IV ˆ n IV s T J s I Js c in probability, rater tan IV ˆ n IV ; since P { s T J s I Js c > } >, te MSE could not be minimized. Lemma 3. Suppose X t = σw t +J t, were W is a Brownian motion and J is a pure-jump Lévy process of bounded variation or, more generally, suc tat, for some Y,, /Y n J n P J, for a real-valued random variable J. Ten, n/ n, as n. Remark. If J as FA jumps, drift d and J t = dt + N t k= γ k, ten we ave P J d and, tus, te assumption in Lemma 3 is satisfied wit Y =. If J is a Lévy process wit Blumental-Getoor index Y, ten Y, and for any η Y, we ave /η a.s. n J n, and again te assumption is satisfied. We are now ready to sow more precisely te asymptotic beavior of. Proposition covers te FA jumps case, wile Proposition tackles te case of symmetric strictly stable jumps. Teir proofs are deferred to te Appendix. Proposition. Let J ave FA jumps and satisfy te assumptions of Teorem 3, let = be te optimal tresold. Ten, σ ln, as. Proposition. Under te conditions of Teorem 4, te optimal tresold = is suc tat Y σ ln, as. As explained in te introduction, te proportionality constant Y of te previous result says tat te iger te jump activity is, te lower te optimal tresold as to be if we want to discard te iger noise represented by te jumps and to catc information about IV. 3 CONSISTENCY WHEN = M log Under te framework described in 7, in te case of equally spaced observations, te tresold criterion allows convergence of n IV ˆ n := i X I { ix rσ ti,} to IV T = T σ sds wen, for all i =,..., n, we ave rσ ti, = r and r is a deterministic function of s.t. r,, as. Here we sow tat, under finite activity jumps, te same estimator is also r log consistent in te case were on any t i, t i we consider a different truncation level r i σ, = M i log, wit suitably cosen random variables M i. Concretely, assume te following A4. Let dx t = a t dt + σ t dw t + dj t, were J t = N t γ i for a non-explosive counting process N and real-valued random variables γ j, a, σ are càdlàg and a.s. σ := inf s,t σ s >. Recall tat a.s. te pats of a and of σ are bounded on, T. Define σ := sup s,t σ s, ten, te following Proposition and Corollary old true. Teir proofs are in te Appendix. Proposition 3. Under A4, if we coose r i = M i log, wit any M iω suc tat M i ω inf s ti,t i σsω, σ, we ave: a.s. η >, for sufficiently small : i =,..., n, I { ix +ηr i} = I { in=}. Corollary. For all η >, we ave n ix P I { ix +ηr i} IV, as. 8

9 4 CONDITIONAL MEAN SQUARE ERROR: FA jumps case We now put ourselves under A. Te quantity of our interest ere, cmse =. E IV ˆ IV σ, J, is suc tat ω, cmse = IV and cmse+ >, because IV ˆ + QV. Furter, from te proof of Teorem, we ave cmse = F, wit F. = n a i g i, g i = + j i b j IV. We analyze te sign of F : for n, fixed, σi and m i also are fixed, and we ave F = IV n a i <, since b j =. Furter we ave F + = + : to see tis, first note tat, from te expression of b i, b i + = m i +σ i, ten g i + j i m j σ i, as +. Moreover, eac a i π / σ i exp σi σ i, tus, for for some sufficiently large, F = n a ig i is a finite sum of n positive terms a i g i Kπ / σ i exp constant K and fixed σ i, so F +, as +. Since F is continuous, it follows tat an optimal tresold exists and solves F =. We now assume also A3. Remark 4. Under A3, as in Remark 3, if = minimizes cmse, ten it as to be true tat, as. In Proposition 4 below we again also find tat under te following A4 ten necessarily +. A4. We assume A4 wit a, constant σ > and n =. Under FA jumps, wen considering, we assume to ave a sufficiently small so tat a.s. te number of jumps occurring during t i, t i is at most ; note tat for any t we ave m i J t, wen selecting i = it suc tat t i < t t i. Tus, wen considering a jump time t, we assume tat is sufficiently small so tat te sign of m it is te same as te one of J t, in particular if J t ten te increments m i approacing it are non-zero. 4. Asymptotic beavior of b i, a i, and F Te following result ensures tat, as previously announced, an optimal tresold as to tend to, as, but at a slower rate tan. Its proof is in te Appendix. Proposition 4. Under A, A3, A4, if = solves F = and =, ten +. We now pass to consider te asymptotic beavior of F for sequences = = satisfying te conditions of Proposition 4. Proposition 5. Under A4, if as in suc a way tat were F := e π e v σ e σ 4σ π σ π. + ten F = F +.o.t., Wit te notation v := and s := σ, we can write F = σ s v 4σs n. Note tat v n, but s, so wic is te leading term between v and ns depends on te coice of v. We also remark tat a solution of F = not necessarily is suc tat F =, owever if a sequence is suc tat F ten te wole F, so it as to be true tat is close in a way tat will become explicit later to one of te solutions of F =. Remark 5. Te asymptotic beavior of F stated in Proposition 5 also olds under te presence of a nonzero drift process {a t } t tat as almost surely locally bounded pats recall tat any cádlág process a satisfies suc a requirement and tat is independent on W. Tis is sown in te Appendix. 9

10 4. Asymptotic beavior of We sow ere tat any cmse optimal tresold as te same asymptotic beavior as te MSE optimal tresold. Te proof of te following result is given in te Appendix. Corollary 3. Under A, A3, A4 we ave tat σ ln, as. Te previous result suggests an approximation for te optimal := of te form = σw, wit w = ln/. It is natural to wonder about oter coices for w. Intuitively, we sould aim at making F to converge to as quickly as possible: in view of 49 witin te proof of Corollary 3, te only possible way is rendering v and ns witin F of te same order, so we coose w suc tat as. For example a function of type w = w +, w, 3 e w w π, ln ln ln ln y, wit any continuous function y tending to π as, satisfies te tree above conditions. However te quickest convergence speed of F to would be reaced by coosing a function w, wic satisfies te following tree more restrictive conditions, as, w + w 3 e w w π, 3 were condition 3 means tat F. In fact suc a w exists, since te following olds true. Teorem 5. Tere exists a unique deterministic function w :,, + suc tat te tree conditions, and 3 above are satisfied. Suc a w turns out to be differentiable and to satisfy also te ODE w = w, +w wic entails tat w w + log. We finally reac te uniqueness of te optimal tresold as a consequence of te following result, wose proof is in te Appendix. We remark tat te asymptotic beavior of described in Corollary 3 is obtained after aving proved just before 4 tat it as to satisfy 4σ s, as. Proposition 6. Te first derivative d d F of F is suc tat, wen evaluated at a function of satisfying, +, and = 4σ s +.o.t., ten, as, F = F +.o.t., as, were F = 4 σ π e Remark 6. Uniqueness of. Since F > for any, we reac tat for sufficiently small we ave d d F > on any sequence as in te above Proposition. Tat entails tat for any sufficiently small te cmse optimal is unique. Indeed, if tere existed two optimal < and i = 4σ s +.o.t., but ten, for small, on suc sequences F i in order to be optimal bot sequences ave to satisfy F i =. σ 3., we would necessarily ave tat i We tank Andrey Sarycev for aving provided suc nice examples. We tank Salvatore Federico for aving provided suc a nice result. Te proof is available upon request. i, +, >, wic is a contradiction, because

11 Remark 7. Te fact tat te asymptotic beavior of te cmse optimal tresold = is te same as te one of te MSE optimal tresold under FA jumps is due to te fact tat solves F =, solves G =, F = F +.o.t., G = G +.o.t., and te leading terms in F are te ones wit m i =, wic do not depend on ω, tus tey are te same as for G. It follows tat, in te case of Lévy FA jumps, we ave F = F +.o.t. = EF +.o.t. = G +.o.t.. n Also, an alternative euristic justification is tat we expect tat F = aigi n n nea i g i, tus te asymptotic beavior of te satisfying G = nea i g i = is te same as any satisfying F =. Remark 8. Comparison wit te results in. In, a process X wit FA jumps is considered, eiter of Lévy type, wit jumps sizes aving distribution density satisfying given conditions, or of Itô SM type, wit deterministic absolutely continuous local caracteristics additive process. Te estimators Ĵ n = n i XI { ix > }, ˆNn = n I { ix > } are considered, and, as, firstly it is sown tat te condition + is necessary and sufficient for te convergence to of bot MSE IV ˆ n IV stronger condition implying consistency of IVn ˆ and MSEĴn J T. Secondly, te autors sow tat σ MSE ˆN n N T e, meaning tat in order to ave L Ω, P convergence to of te estimation error ˆN n N T a stronger condition on is needed, implying. Tirdly, existence and uniqueness of an optimal tresold ˇ minimizing E IV ˆ n IV + ˆN n N T for fixed is obtained, and te asymptotic expansion in of ˇ as leading term 3σ log. Te factor 3 is iger tan te factor of te leading terms of and : tat is due to te fact tat te minimization criterion for ˇ includes also te error on N T, wic requires tat ˇ is iger tan, and tus ˇ > is necessary. 5 A NEW METHOD In tis section, we propose a new metod for tuning te tresold parameter := rσ, of te TRV introduced in. Tis is based on te conditional mean square error cmse = E IV ˆ IV σ, J studied in Section 4. We illustrate te metod for a driftless FA process wit constant volatility σ. As proved terein, te optimal tresold is suc tat n F = b j nσ, a i g i =, g i = + j i were a i and b i are rewritten ere for easy reference: m i a i := a, m i, σ := e σ + e +m i σ σ, π b i := b, m i, σ := σ π σ + m i + e +m i σ m i e m i + m i + σ π It is convenient to set m = m,..., m n and n F ; σ, m := a, m i, σ + b, m j, σ nσ. j i mi+ σ m i σ e x / dx. Te main issue wit te optimal tresold lies on te fact tat tis depends on σ and te increments m = m,..., m n of te jump process, wic we don t know. Note also tat, for small enoug, eac m i will be eiter

12 or one of te jumps of te process and a good proxy of m i is actually n i X { n i X > }. Te idea is ten to iteratively estimating, σ, and m as follows:. Start wit some initial guesses of σ and m, wic we call ˆσ and ˆm. Tere are different possibilities for tese initial values, for instance ˆσ RV defined in item of Section 5. or ˆσ BV defined in item of Section 5. or a truncated ˆσ T RV defined in item of Section 5. wit tresold ˆσ BV log/, for σ, and ˆm =,..., no jumps for m.. Using ˆσ and ˆm, by solving F ; ˆσ, ˆm =, we find an initial estimate for te optimum tat we denote NEW. For instance wit te coice of ˆm =,...,, NEW solves te equation: + n ˆσ e ˆσ + ˆσ ˆσ e x / dx nˆσ =. 4 π π It is easy to see tat, in tat case, NEW is of te form v nˆσ, were vn is te unique solution of te equation: vn + 4n v n e v n vn + e x / dx n =. 5 π π ˆσ Figure sows tat v n ranges from about 3 to 4 wen n ranges from to. vn n, number of observations Figure : Te solution v n of equation 5 as a function of n. 3. Once we ave an initial estimate of, we can update our estimates of σ and m using: ˆσ NEW := T n i X { ix NEW }, ˆm := n X { n X > NEW },..., n nx { n X > NEW } 6 4. We continue tis procedure iteratively: NEW, := NEW, ˆσ NEW, := ˆσ NEW, and for k Find NEW,k s.t. F NEW,k ; ˆσ NEW,k, ˆm k =, 7 set ˆσ NEW,k+ := n i X { ix T NEW,k }, 8 ˆm k+ := n X { n X > NEW,k },..., n nx { n n X > NEW,k }. 9 Te algoritm is stopped wen te sequence of estimates ˆσ NEW,k stabilizes e.g., wen ˆσ NEW,k+ ˆσ NEW,k / ˆσ NEW,k tol, for some desired small tolerance tol.

13 Te previous procedure resembles te one introduced in, wic is based on coosing te tresold so to minimize te expected number of jumps miss-classifications: n Loss := E { n i X >, n i N=} + { n i X, n i N>}. It was proved terein tat, for a Lévy process wit FA jumps, te optimal tresold, ereafter denoted 3mc, is asymptotically equivalent to 3σ ln/, as 3. Using tis information, an iterative metod was proposed, in wic, given an initial estimate ˆσ 3mc, of σ, we set, for k, 3mc,k := 3ˆσ 3mc,k ln, ˆσ 3mc,k := n i X { ix T 3mc,k }. Since, as proved in Section 4, te optimal tresold tat minimizes cmse for given as te asymptotic beavior σ ln/, as, it is natural to consider te following iterative metod to estimate : given an initial guess ˆσ mc, for σ, we set mc,k := ˆσ mc,k ln, ˆσ mc,k := n i X { ix T mc,k }, k. We can go one step furter and consider, as suggested below Corollary 3, a tresold of te form = σw, wit w given as in 3 4. Tis leads us to consider te iterative metod: mc,k := w ˆσ mc,k, ˆσ mc,k := T n i X { ix mc,k }, k. 3 It can be proved tat if we take ˆσ 3mc,, ˆσ mc,, and ˆσ mc, equal ˆσ RV in,, and 3, ten te obtained sequences of estimates {ˆσ mc,k } k, {ˆσ 3mc,k } k, {ˆσ mc,k} k are nonincreasing and, tus, eventually tey reac a constant limiting value. So, for tese two estimators we can and will set te tolerance tol to. Even toug asymptotically w ln/, tere are some differences in finite samples. For instance, for te span of 5 minutes used in our simulations = 5 6.5, we ave w =.98, wile ln/ = 3.4, wic means tat te mc,k will be smaller tan mc,k. 5. Simulation performance: finite actvity jumps and constant volatility We now proceed to assess te metods introduced in tis paper and compare tem against oter popular alternatives. We take a Merton s log-normal model of te form: N t X t = σw t + γ j, 4 were N is a Poisson process wit intensity λ and {γ i } i is an independent sequence of independent normally distributed random variables wit mean and standard deviation µ Jmp and σ Jmp, respectively. We consider te following estimators:. Te Realized quadratic Variation estimator: ˆσ RV := T n n i X ;. Te realized Bipower Variation BV estimator of 5: ˆσ BV := π n i X i+ X ; T 3 mc in te notation 3mc refers to modulus of continuity of te Brownian motion. 4 In order to obtain w, we cange variable, as x = w, in 3 and ten we use a fixed-point algoritm to find te solution x, starting wit x = log. Te algoritm converges very quickly. j= 3

14 3. Te MinRV estimator of 3: 4. Te MedRV estimator of 3: n ˆσ MinRV π n := min{ i X, i+ X } ; T π n n ˆσ MedRV π := T π n median{ i X, i X, i+ X } ; 3 n i= 5. Te TRV given in 3 using a tresold of te form = 4 ω ˆσ BV wit ω =.49. Tis was used in te recent work of Jacod and Todorov 4 and is denoted ˆσ T RV JT ; 6. Te estimator ˆσ 3mc as in wit k =, using te initial tresold 3mc, := 3ˆσ RV log/; 7. Te estimator ˆσ 3mc,k defined by wit k suc tat ˆσ 3mc,l = ˆσ 3mc,l, for all l k; 8. Te estimator ˆσ mc as in wit k =, using te initial tresold mc, := ˆσ RV log/; 9. Te estimator ˆσ mc,k defined by te iterative formulas and wit k suc tat ˆσ mc,l = ˆσ mc,l for all l k;. Te estimator ˆσ mc as in 3 wit k =, using te initial tresold mc, := w ˆσ RV ;. Te estimator ˆσ mc,k defined by te iterative formulas 3 and wit k suc tat ˆσ mc,l = ˆσ mc,l for all l k;. Te estimator ˆσ NEW as defined in 6 were NEW is suc tat F NEW ; ˆσ, ˆm =, wit initial guesses ˆm =,..., and ˆσ T RV := T n n i X { n X as}, wit as := mc, = ˆσ BV log/; 3. ˆσ NEW,k found wit te new metod described by te iterative formulas 7-9, wit initial guesses given as in te previous item and k determined by te stopping rule ˆσ NEW,k ˆσ NEW,k /ˆσ NEW,k tol = 5 ; 4. An Oracle type estimator of te form ˆσ Orc := n i X { ix Orc}, were Orc is suc tat F Orc ; σ, m =, using te true values of te volatility σ and of te jump vector m = m,..., m n = J,..., n J; 5. Te following estimator based on te Tresold Bipower Variation TBV: ˆσ T BV := π n i X i+ X { ix T T BV } { i+x T BV }, using a tresold of te form T BV := 4 ω ˆσ BV wit ω =.49: 6. Te iterated TBV estimator given by: ˆσ T BV, := ˆσ T BV, T BV,k := 4 ω ˆσ T BV,k, ˆσ T BV,k+ := π n i X i+ X { ix T T BV,k } { i+x T BV,k }, k, using ω =.49 and ˆσ T BV as defined in te previous item.5 We stop wen ˆσ T BV,k ˆσ T BV,k /ˆσ T BV,k tol = 5. 5 Te estimators in items 5 and 6 were suggested by an anonymous referee. 4

15 Remark 9. Different variations of te above estimators, tat are not sown ere for sake of brevity, were also analyzed in our simulations. For instance, te 3 alternative tresolds = ω and = ω, wit ω =.495, were implemented; eac one of te estimators in items 5 and 6 was also implemented wit tresolds 3 ω ˆσ BV and 5 ω ˆσ BV. Te results of tese variations were suboptimal to tose sown ere. We also implemented te estimators in te items 6 to starting wit an initial tresold of te form ˆσ BV log/ i.e., using ˆσ BV rater tan ˆσ RV as an initial guess for σ, and te same stopping condition as terein: in tese cases we obtained te same performances for te liming estimators. Te adopted time unit of measure is year 5 days and we consider 5 minute observations over a mont time orizon wit a 6.5 ours per day open market. For our first simulation experiment, we use te following parameters: σ =.4, σ Jmp = 3, µ Jmp =, λ =, = Te dependence of σ Jmp on was done for an easier comparison wit te standard deviation of te increments of te continuous component, wic is.4. So, te standard deviation of te jumps is about 7.5 times te standard deviation of te continuous component increment. Te parameter values in 5 yield an expected annualized volatility of.45, wic is reasonable. Table below sows te sample biases, standard deviations, and MSE s based on 5 simulations. We also sow te sample version of Loss, i.e., te expected number of jump misclassifications as defined by, wit its standard deviation; te sample average of N, i.e. te number of iterations needed to find te estimator s value, wit its standard deviation; and, for te metods using truncation, te average tresold of te last step of te iteration used to obtain te estimate of σ. As expected, te unfeasible oracle estimator, wic is sown as a bencmark for te oter estimators, performs te best, followed by te estimators ˆσ NEW and ˆσ NEW,k based on finding te root of F ; σ, m. Te iterative estimators ˆσ mc,k and ˆσ mc,k, based on te tresolds σ ln/ and σ w, also ave a good performance and significantly improve on te estimator ˆσ 3mc,k number 7 above proposed in and based on 3σ ln/. Te estimator ˆσ T RVJT proposed by Jacod and Todorov 4 also performs quite well in terms of MSE, but te estimation relative error is comparatively large. Te estimators based on TBV namely, te estimators ˆσ T BV and ˆσ T BV,k of items 5 and 6 above as well as te MinRV and MedRV are suboptimal for te considered parameters coice. We now double te intensity of jumps and consider te following parameter setting: σ =.4, σ Jmp = 3, µ Jmp =, λ =, = 5 6.5, 6 wic yields an expected annualized volatility of.5. Te results are sown in Table. We again notice tat te Oracle estimator performs te best followed by te new estimators based on finding te root of F ; σ, m. As before, te estimators based on te MinRV, te MedRV, and ˆσ T BV underperform compared to ˆσ NEW and ˆσ NEW,k ; ˆσ T BV,k as a small relative estimation error, but a comparatively ig MSE. 6 Extensions In tis section we assess our results on models wit stocastic volatility and leverage and on models wit infinite activity jumps. We now mention te main ideas tat we are pursuing in te teoretical ongoing analysis in te presence of stocastic volatility and ten we sow on simulated data tat te performance of our new metods is promising also in suc extended contexts. In te presence of stocastic volatility witout leverage we can deal wit cmse as described in te subsequent paragrap. If also leverage is present, ten we can use a similar approac under MSE or, alternatively, we can work at minimizing cmse by assuming tat dσ t = γ t db t, wit B a Brownian motion correlated wit W, and by 5

16 mean std MSEˆσ mean std mean std mean std Estimator ˆσ σ σ ˆσ σ σ 5 Loss Loss 3 N N ˆσ RV ˆσ BV ˆσ MinRV ˆσ MedRV ˆσ T RVJT ˆσ 3mc ˆσ 3mc,k ˆσ mc ˆσ mc,k ˆσ mc ˆσ mc,k ˆσ NEW ˆσ NEW,k ˆσ Orc ˆσ T BV ˆσ T BV,k Table : Estimation of te volatility σ =.4 for a log-normal Merton model, based on simulated 5-minutes observations of 5 pats over a mont time orizon. Te jump parameters are λ =, σ Jmp = 3 and µ Jmp =. splitting W into a term completely dependent on σ and an independent one 6. Conditioning ten on σ and J, te term independent of σ can be dealt exactly as in tis paper. A popular approac to deal wit te case of stocastic volatility is localization. Assuming continuity of te pats of σ, te idea is tat te volatility is approximately constant in a small time interval. So we can divide te time orizon into k intervals t i, t i and apply our metods tat assume constant volatility to eac interval. More specifically, we want to consider an estimator of te form ˆ IV n := k n i i,l X I { i,l X i}, 7 l= were :=,..., k and, for i =,..., k, eac t i, t i is divided into n i subintervals t i,l, t i,l, l =,.., n i, wit t i = t i, < t i, < < t i,ni = t i, i,l X := X ti,l X ti,l, and te tresold i is uniform on t i, t i. In te case tat n i = for all i, we ave te extreme case of one different tresold for eac subinterval. Wen σ is independent on W, we can consider te cmse of IVn ˆ, denoted by cmse, and use tis to determine te optimal tresolding levels i for te different intervals. We define cmse i i := E ˆ IV i IV i J, σ, were IV i = σ i, wit σ i a random number depending on te pat of σ over te interval t i, t i e.g., t i t i σsds, and IV ˆ i := n i l= i,lx I { i,l X i}. Ten it turns out tat minimizing cmse, as varies wile k and n,.., n k are fixed, is asymptotically equivalent to solve te k problems min i cmse i i,..k, wic can be treated at once and justify wy we tackled te minimization of cmse by assuming constant volatility. 6 We tank Alexei Kolokolov for aving suggested to consider suc an approac 6

17 mean std MSEˆσ mean std mean std mean std Estimator ˆσ σ σ ˆσ σ σ 5 Loss Loss 3 N N ˆσ RV ˆσ BV ˆσ MinRV ˆσ MedRV ˆσ T RVJT ˆσ 3mc ˆσ 3mc,k ˆσ mc ˆσ mc,k ˆσ mc ˆσ mc,k ˆσ NEW ˆσ NEW,k ˆσ Orc ˆσ T BV ˆσ T BV,k Table : Estimation of te volatility σ =.4 for a log-normal Merton model based on simulations of 5-minutes observations of 5 pats over a mont time orizon. Te jump parameters are λ =, σ Jmp = 3 and µ Jmp =. Altoug te teoretical analysis of cmse under stocastic volatility and leverage and under infinite activity jumps are still ongoing, in te rest of tis section, we illustrate on simulated data te beavior of our newly proposed metods. We find in fact tat again tey outperform te metods currently used in te literature, at least in te realistic scenarios tat we considered ere. 6. Simulation performance: stocastic volatility models wit leverage Even toug te new metod presented in Section 5 was originally designed for a model wit constant volatility and tus no leverage, it can still be applied for te more general stocastic volatility model. In tis part, we examine by simulations te performance of te same estimators introduced in Section 5. in te presence of stocastic volatility and leverage. For te continuous part of te process, we take te popular Heston model 3 and consider: dx t =µ t dt + V t db t + dj t, X =, dv t =κθ V t dt + ξ V t dw t, V = θ, 8 were B and W are correlated Wiener processes suc tat EdB t dw t = ρdt wile, in accordance wit our Assumption A, we take te jump component J independent of W, B. For J we adopt te Merton s lognormal model studied in Section 5.. We consider te following settings, were will be set to 5 minutes i.e., = /5 6.5 : Continuous Component Parameters Jump Component µ t κ ξ θ ρ V σ Jmp µ Jmp λ ; Te values of κ and ξ, wic are standard in te literature, are te same as tose used in, were tey also propose ρ =.5 and θ =.4. We adopt ere te value of θ =.6 for easier comparison wit te constant 7

18 volatility case of Section 5., were σ is taken to be.4. We remark owever tat we cecked te performance of all te estimators in te case θ =.4, and tere are no significant canges, except tat it is easier to identify jumps because te variance of te jump part is bigger compared to tat of te continuous component, so te MSEs are smaller. Since te volatility canges from simulation to simulation, to assess te accuracy of te different metods, we compute te relative error, IV E := ˆ n IV, IV for eac simulated pat, were IV is given as below Eq. and IV ˆ n is an estimator of te integrated variance. Te sample mean and standard deviation of te error over 5 simulations for eac of te estimators considered in Section 5. are reported in Table 3. We also sow te sample mean of IV ˆ n IV. As in Section 5., te Oracle is obtained by te formula n ˆσ Orc = i X { ix Orc}, were Orc is suc tat F Orc ; σ Avg, m =, using te true increments of te jump component, m = m,..., m n = J,..., n J, and te true average volatility value σavg := T T σ sds. Te results are consistent wit tose obtained in Section 5.. Te new estimators ˆσ NEW and ˆσ NEW,k based on finding te root of F ; σ, m perform te best. Also, te iterative estimators ˆσ mc,k and ˆσ mc,k based on te tresold σ ln/ and σ w perform quite well and significantly improve on te estimator ˆσ 3mc,k proposed in and based on 3σ ln/. In particular, te leverage factor seems to ave a minor effect on te performance of all te estimators, wile stocastic volatility seems not to ave any adverse effects, compared to te constant volatility case. Tus, for instance, for λ = and a long-run average volatility level of θ =.6 =.4, te sample mean and standard deviation of Eˆσ NEW are.86 and.37, respectively, wic are smaller tat tose attained by Eˆσ NEW in te constant volatility case of σ =.4 namely,.389 and.376 as seeing in Table. 6. Simulation performance: infinite jump activity It is natural to wonder about te robustness of te estimators introduced in tis article against jumps of infinite activity IA. To tis end, in tis section, we consider one of te most popular models of tis kind: te Variance Gamma model VG of 6. Concretely, we assume te model X t = at + σw t + J t := at + σw t + σ Jmp B St + θs t, were W and B are independent Wiener processes and {S t } t is an independent Lévy subordinator suc tat S t is Gamma distributed wit scale parameter β := κ and sape parameter α := t/κ. Note tat, in tat case, ES t = t and VarS t = κt. For te parameter values, we take te following te time unit is one day: σ =. 5 =.6, σ Jmp =., κ =.7; a = θ =. Te values of σ Jmp and κ are consistent wit te empirical results of 9. Te results are sown in Table 4. Basically, te estimators tat use truncation ˆσ 3mc,k, ˆσ mc,k, ˆσ mc, ˆσ NEW,k, ˆσ T BV,k, but not ˆσ T RVJT perform better tan tose witout it ˆσ RV, ˆσ BV, ˆσ MedRV, and ˆσ MinRV. As expected, te Oracle estimator performs te best, followed by te estimators ˆσ mc,k and ˆσ mc,k, wic are based on te respective asymptotic tresolds σ ln/ and σ w. Indeed, teir MSEs are less or equal to a quarter of any oter feasible tresold estimators. Te iterative estimators ˆσ NEW,k based on finding te root of F ɛ; σ, m and ˆσ T BV,k based on truncated bipower variation ave similar performances. 8

19 Estimator ρ = ρ =.5 mean std MSE IV ˆ mean std MSE IV ˆ ˆ IV IV IV ˆ IV IV IV 7 IV ˆ IV IV ˆ IV IV IV 7 ˆσ RV ˆσ BV ˆσ MinRV ˆσ MedRV ˆσ T RVJT ˆσ 3mc ˆσ 3mc,k ˆσ mc ˆσ mc,k ˆσ mc ˆσ mc,k ˆσ NEW ˆσ NEW,k ˆσ Orc ˆσ T BV ˆσ T BV,k Table 3: For eac estimator we report: sample mean and standard deviation of te estimation percentage error E := IV ˆ n IV /IV, and MSE, i.e. te sample mean of IV ˆ n IV. Te means and standard deviations are based on simulations of 5-minutes observations of 5 pats from a jump-diffusion model for X obtained by adding log-normal jumps to te Heston model. Te time orizon of te pats is mont. 7 Conclusions We consider te problem of estimating te integrated variance IV of a semimartingale model X wit jumps for te log price of a financial asset. In view of adopting te truncated realized variance of X, we look for a teoretical and practical way to select an optimal tresold in finite samples. We consider te following two optimality criteria: minimization of MSE, te expected quadratic error in te estimation of IV; and minimization of cmse, te expected quadratic error conditional to te realized pats of te jump process J and of te volatility process σ s s. Under given assumptions, we find tat for eac criterion an optimal TH exists, is unique and is a solution of an explicitly given equation, te equation being different under te two criteria. Also, under eac criterion, an asymptotic expansion wit respect to te step between te observations is possible for te optimal TH. Te leading terms of bot te two expansions turn out to be proportional to te modulus of continuity of te Brownian motion pats and to te spot volatility of X, wit proportionality constant Y, Y being te jump activity index of X. Furter, we sow tat te tresold estimator of IV constructed wit te leading term of te optimal TH is consistent, at least in te finite activity jumps case, even if it does not satisfy te classical assumptions. Te results obtained for te cmse criterion allow for a novel numerical way to tuneup te tresold parameter in finite samples. Based on simulated data, we illustrate te superiority of te new metod on oter broadly used estimators in te literature. Minimization of cmse in te presence of infinite activity jumps in X and in te presence of stocastic volatility and leverage are object of ongoing researc, but te newly proposed estimators are implemented on simulated data under suc frameworks, and again are superior. 9