Testing for Jumps: A Delta-Hedging Perspective

Transcription

1 Testig for Jumps: A Delta-Hedgig Perspective Jia Li Priceto Uiversity Departmet of Ecoomics ad Bedheim Ceter for Fiace Priceto, NJ, 8544 This Versio: Jue 6, 211 Abstract We measure asset price jumps by the hedgig error they iduce o a delta-hedged positio of Europea optios. Based o high frequecy data, we propose a oparametric estimator for this measure ad a test for its positivity. We further costruct a Kolmogorov-type test for the presece of jump hedgig errors for a possibly ifiite-dimesioal family of optios based o the worst-case cotract i this family. Uder regularity coditios, these tests are equivalet to tests for jumps. A empirical applicatio o U.S. stocks i 28 shows that jumps cause statistically sigificat ad ecoomically sizable hedgig errors for short-dated call ad put optios. JEL : C22 K : Hedgig, high frequecy data, semimartigale, testig for jumps. 1 Itroductio The recet availability of observatios o fiacial returs at icreasigly higher frequecies has prompted the developmet of methodologies desiged to test the specificatio of suitable models for these data. Oe of the first issues studied i the literature is testig the presece of jumps. Sice the semial work of Bardorff-Nielse ad Shephard 24), there have bee several tests i the literature. However, these tests are typically based o purely statistical measures of jumps ad thus lack a clear ecoomic iterpretatio. Eve though the literature has documeted that jumps are preset i fiacial data, it is ot clear how the ecoomic relevace of jumps is reflected i the I am grateful to Yacie Aït-Sahalia, Jea Jacod, Ulrich Müller ad Mark Watso for their guidace ad support. I also thak Paolo Colla, Thomas Eisebach, Jakub Jurek, Weicheg Lia, Jose Scheikma, Marti Schmalz, Hyu Shi, Wei Xiog ad Dacheg Xiu for helpful discussios. All errors are mie. jiali@priceto.edu. Webpage: jiali. 1

2 existig testig procedures. The preset paper iteds to fill this gap. We measure jumps by the hedgig error they iduce for Europea optios ad desig tests for jumps based o this measure. To illustrate the idea, let us cosider the classical problem of Black ad Scholes 1973). That is, hedgig a short positio of a Europea call optio with a log positio of the uderlyig stock, where the hedge ratio i.e. the umber of shares of stock required for hedgig oe optio cotract) is the optio delta of the Black-Scholes formula. If the stock price follows a geometric Browia motio, it is well kow that the value of the delta-hedgig portfolio perfectly replicates the model price of the optio. What happes if the stock price jumps? At the jump time, both the model price of the optio, which is a oliear fuctio of the stock price, ad the value of the hedgig portfolio, which is a liear fuctio of the stock price, jump. Because the optio ad its hedge respod to the stock price jump differetly, the jump ecessarily iduces a discrepacy betwee their values, ad thus leads to a hedgig error, heceforth a jump error. We ote that the jump error is ot specific to the call optio or the Black-Scholes formula. Istead, it is a geeric cosequece of the oliearity of the payoff of a derivative security, coupled with the discrete ature of jumps. Qualitatively, it is clear that jumps cause hedgig errors. The remaiig ecoometric questios are: how large is the jump error; is it statistically sigificat? First of all, we ca ot aswer this questio by simply lookig at the actual hedgig error i a tradig book. The reaso is that the observed total hedgig error ot oly icludes the jump error, but also hedgig errors comig from other sources such as discrete implemetatio, trasactio costs, ad volatility misspecificatio. The aswer to this questio would evertheless be immediate if we could observe the cotiuous-time sample path of the stock price. I this case, we could observe the jump time, the pre-jump stock price ad the jump size, which are suffi ciet to determie the hedgig error cotributed by each jump see Sectio 2). However, such iformatio is uavailable i reality because we ca oly observe data sampled at discrete times. The ecoometric challege is to use such data to estimate the jump error of a optio ad the make statistical iferece. We propose estimators ad tests for such purposes. Our method is oparametric i the sese that we oly require the stock price to be a Itô semimartigale without parametrizig the stochastic volatility or the jump compoet i ay way. Sice the jump error is idetified i the cotiuous-time limit, the method relies o the ifill asymptotics with the samplig iterval goig to zero. High frequecy data are ecessary for implemetatio. Our geeral ecoometric result is to joitly test the positivity of jump errors of a possibly ifiite dimesioal) family of optios writte o the same asset, e.g. the cross sectio of call ad put optios with various strikes. This multivariate settig is motivated by our geeral iterest i the whole market istead of a particular cotract. To this ed, we itroduce the otio of jump error profile, which is defied as the collectio of jump errors for a family of optios. We propose a cosistet estimator for the jump error profile. We also propose a Kolmogorov-type test 2

3 to examie whether the profile is zero. The idea behid this test is to look at the jump error of the worst-case cotract i the family ad determie its statistical sigificace. A failure of rejectio would suggest that, uiformly across this family of cotracts, jumps are ulikely to be importat for hedgig. I the effort of costructig this test, we derive a empirical process-type asymptotic theory for Itô semimartigale models. This result seems to be ucovetioal for the theory of empirical processes because our Itô semimartigale model is o-ergodic, o-statioary ad does ot impose weak depedece idepedece, mixig, etc.). We ow discuss the related literature ad some further results. Our test icludes the bipower test of Bardorff-Nielse ad Shephard 24) as a special case correspodig to a quadratic cotract, or equivaletly, a equally weighted portfolio of the complete cross sectio of call optios. By embeddig the bipower test i our framework, we shed ew light o the ecoomic iterpretatio of this test from a hedgig perspective. Aother ispiratio to our work is the paper by Jiag ad Oome 28). Jiag ad Oome measure jumps by the swap variace ad propose a test based o this quatity. The statistical properties of their test are quite differet from ours. The ecoomic iterpretability of their test seems to be arrower tha what we pursue here sice it is restricted to the log cotract, which is a fictitious theoretical device ad rarely, if ever, traded i the market. 1 I cotrast to these papers, we cosider arbitrary Europea optios i a ifiite dimesioal settig, allowig may possibilities for costructig ew tests for jumps. Other tests for jumps based o high-frequecy data have bee proposed by Lee ad Myklad 28), Lee ad Haig 21) based o detectig large returs), ad Huag ad Tauche 25), Aderse, Bollerslev, ad Diebold 27), Aït-Sahalia ad Jacod 29), Podolskij ad Ziggel 21), Aït-Sahalia, Jacod, ad Li 211) based o power variatios of jumps). The tests of Podolskij ad Ziggel 21) ad Aït-Sahalia, Jacod, ad Li 211) are robust to microstructure oise, which is ot cosidered here. Robustifyig the method agaist microstructure oise is crucial for applicatios, but it seems to deserve a separate paper. Our paper is also related to the literature o hedgig derivative securities. I the study of discrete hedgig i diffusio models, Bertsimas, Koga, ad Lo 2) itroduce the otio of temporal graularity as a measure of the magitude of the hedgig error due to discretizatio. This result is further exteded by Hayashi ad Myklad 25) to cotiuous Itô semimartigale models. It turs out that the t-statistic of our test associated with a sigle cotract is proportioal to the estimated ratio of the jump error to the ex-post temporal graularity. Hece, our test is essetially a compariso betwee the jump hedgig error ad the discretizatio hedgig error. A rejectio of the ull hypothesis that the jump error is zero suggests that the jump error is suffi cietly large relative to the discretizatio error. The paper is orgaized as follows. We discuss delta-hedgig ad characterize the jump error 1 The statistical validity of Jiag ad Oome s test does ot deped o the log cotract beig traded. 3

4 i Sectio 2. I Sectio 3, we preset our estimator ad show its cosistecy. I Sectio 4, we preset our test based o a sigle cotract. The testig framework is exteded to a fuctioal settig i Sectio 5, i which we propose a ew test based o the worst-case jump error. Sectio 6 shows simulatio results. We preset a empirical applicatio i Sectio 7. Sectio 8 cocludes. Proofs are collected i Sectio 9. 2 Delta-hedgig ad hedgig errors I this sectio, we characterize jump-iduced delta-hedgig errors of Europea optios. To formalize ideas, let the stock price process X t be a semimartigale with the form X t = X + t b s ds + t σ s dw s + J t, where b t is the istataeous drift, σ 2 t is the istataeous variace hece v t = σ t /X t is the istataeous volatility), ad J t is a purely discotiuous process. Assumig X to be a semimartigale is ecessary to exclude arbitrage opportuities see Harriso ad Pliska 1981), Delbae ad Schachermayer 1994)). We discuss regularity coditios o the price process i Sectio 3. stock. 2 We suppose that a ivestor delta-hedge a optio by dyamically tradig the uderlyig To implemet this hedgig strategy, it is ecessary to specify a optio pricig fuctio, which we deote by V t, X t ). The hedge ratio is the give by the partial derivative V x t, X t ), where X t = lim s t X s is the pre-jump price. The hedgig problem is fully characterized by the pricig fuctio V ) as it specifies both the term of the optio cotract ad the hedgig strategy. I this paper, we do ot wish to eter a discussio about which pricig model should be used for hedgig, as there is a large literature o this topic see e.g. Sigleto 26)). We simply take V ) as a iput ad cosider the correspodig hedgig error. We ote that V ) is ot assumed to be correctly specified, if there ever exists a correctly specified pricig fuctio. We deote the size of a jump at time t by X t = X t X t. For simplicity, we suppose that the bod holdig i the hedgig portfolio ears a costat risk-free rate r f. I a frictioless market, the hedgig error of this strategy from time t to time t, deoted by Π t, is give by t t Π t = V t, X t ) V t, X t ) V x s, X s ) dx s t r f V s, X s ) V x s, X s ) X s )ds. t To simplify otatio, we ormalize t = below. 2 I practice, a ivestor may also gamma- or vega-hedge a optio with the uderlyig ad the other optio. I this case, we ca trasform the gamma- or vega-hedgig problem ito a delta-hedgig problem for a portfolio of optios. 4

5 By Itô s formula, we ca decompose the istataeous hedgig error as dπ t = ϕ t, X t, X t ) + ρ t, X t, σ t ) dt, 1) where ϕ t, X t, X t ) = V t, X t + X t ) V t, X t ) V x t, X t ) X t, 2) ρ t, X t, σ t ) = V t t, X t ) V xx t, X t ) σ 2 t r f V t, X t ) V x t, X t ) X t ), 3) which should be uderstood i itegral form as Π t = s t, X s ϕ s, X s, X s ) + t ρ s, X s, σ s ) ds. 4) The sum i 4) ivolves coutably may ozero terms because there are coutably may jumps ad ϕ s, X s, X s ) oly if X s. The first compoet o the right side of 4) is the hedgig error iduced by jumps, i.e. the jump error, ad the secod compoet is cotributed by the stochastic volatility, heceforth the volatility error. Before elaboratig these iterpretatios, we make a few remarks. First, Π t is the hedgig error uder cotiuous rebalacig i a frictioless market. I practice, discrete implemetatio ad trasactio costs lead to additioal hedgig errors. Such hedgig errors should ot be attributed to jumps. Therefore, they should be isolated from the defiitio of the jump error as cosidered here. Secod, the otatio V t, X t ) idicates that V ) oly depeds o time ad the spot price. I geeral, a pricig fuctio may also deped o parameters goverig the risk-eutral law of X ad/or uobservable state variables such as the stochastic volatility. These uobservable quatities eed to be estimated i order to determie V ). By suppressig the depedece of V ) o these quatities i our otatio, we have implicitly assumed that the estimates of uobservables are ot updated durig [t, t]. This assumptio is ot restrictive sice we are maily iterested i the hedgig error cotributed by itraday jumps accumulated withi a short period, say oe day or oe week. Third, if we could observe the sample path i cotiuous-time, we would have complete kowledge about jumps ad the volatility. Quatifyig each compoet would be a straightforward calculatio. I practice, however, the data are oly sampled at discrete times. The mai focus of our paper is o the estimatio ad iferece for the jump error based o discretely sampled data. Fourth, the jump error is defied through a delta-hedgig strategy. Excludig other hedgig strategies, such as the quadratic hedgig see Schweizer 21)), is restrictive. Elimiatig the delta risk is ecessary i order to obtai a cetral limit theorem ad coductig formal statistical tests. We ow iterpret the jump error ituitively via the istataeous represetatio 1). If a jump occurs at time t with size X t, the model price of the optio jumps by V t, X t + X t ) V t, X t ) 5

6 ad the value of the delta-hedgig portfolio jumps by V x t, X t ) X t. The term ϕ t, X t, X t ), defied as their differece, quatifies the hedgig error iduced by the jump at the istat t. Mathematically, ϕ ) is the differece betwee V t, X t ) ad its first-order Taylor approximatio aroud X t. We ca thus thik of ϕ ) as a local measure of the oliearity of the pricig fuctio. We ote that jumps iduce hedgig errors ot because the pricig fuctio is possibly misspecified, but rather because the value of the optio is oliear i X t. Ideed, eve if the pricig fuctio V ) is correctly specified, ϕ t, X t, X t ) is geerically ozero whe X t ; o the other had, whe X t =, we have ϕ t, X t, X t ) = regardless whether V ) is misspecified or ot. To iterpret the volatility error, we first cosider the basic case i which the pricig fuctio V ) is derived from the Black-Scholes-Merto model with the local volatility fuctio t, x) v BS t, x). It is well kow that V ) should solve the Black-Scholes valuatio equatio: V t t, x) V xx t, x) x 2 vbs 2 t, x) r f V t, x) V x t, x) x) =, 5) subject to a suitable boudary coditio determied by the termial payoff. Combiig 3) ad 5), we have ρ t, X t, σ t ) = 1 2 V xx t, X t ) σ 2 t vbs 2 t, X t ) Xt 2 ). 6) I words, ρ ) is the differece betwee the true istataeous diffusive variace ad its model couterpart, scaled by half of the optio gamma i.e. the secod partial derivative of optio price with respect to the stock price). I our ecoometric framework, we do ot require the pricig fuctio V ) obey the Black-Scholes equatio for ay fuctio v BS ). I this geeral case, ρ t, X t, σ t ) measures the degree to which the pricig fuctio violates the Black-Scholes equatio at the istat t. Our geeral ecoometric framework is based o a family of pricig fuctios V = {V, ; θ) : θ Θ}. The idex θ may iclude the cofiguratio of a cotract, such as the strike price ad the maturity of a call optio. It may also iclude characteristics of the pricig model, such as the volatility parameter i the classical Black-Scholes formula or, more geerally, ay fiite-dimesioal parameter which govers the risk-eutral law of the stock price process. We set for each θ Θ, B T θ) = ϕ s, X s, X s ; θ). s T We refer to the radom) fuctio θ B T θ) as the jump error profile. 6

7 3 Estimatio of hedgig errors 3.1 Assumptios o the stock price process We collect some regularity coditios o the stock price process. Our assumptios are weak eough to icorporate most cotiuous-time models studied i the empirical asset pricig literature. I particular, we allow for stochastic volatility ad jumps ad impose essetially o restrictio o their dyamics. Sice the mathematical presetatio may appear somewhat techical, readers iterested i applicatios may skip this subsectio durig the first readig. We assume that the stock price X = X t ) t is a oe-dimesioal process takig values i a ope set D R, ad is sampled at regularly spaced discrete times i, i, over a fixed time iterval [, T ], with a time lag which asymptotically goes to. The basic assumptio is that X is a Itô semimartigale o a filtered space Ω, F, F t ) t, P), which meas that it ca be writte as X t = X + t b s ds + t σ s dw s + δ1 { δ 1} ) µ ν) t + δ1 { δ >1} ) µ t, where W is a Browia motio, µ is a Poisso radom measure o R + E ad its compesator is νdt, dz) = dt λdz) where E, E) is a auxiliary space ad λ is a σ-fiite measure all these are defied o the filtered space above ad we refer for example to Jacod ad Shiryaev 23) for all uexplaied terms). I particular, whe X is cotiuous, it has the form X t = X + We further assume for some r [, 2] : t b s ds + A H-r): 1) the process b t ) is optioal ad locally bouded; 2) the processes σ t ) is càdlàg ad adapted; t σ s dw s. 7) 3) the fuctio δ is predictable, ad there is a bouded fuctio γ i L r E, E, λ) such that the process sup z E δω, t, z) 1)/γz) is locally bouded; 4) we have almost surely t σ2 s ds > for all t >. A K): We have Assumptio H-2) ad σ t is also a Itô semimartigale which ca be writte as σ t = σ + t t bs ds + σ s dw s + M t + σ s 1 { σs >v}, s t where M is a local martigale orthogoal to W ad with bouded jumps ad M, M t = t a sds, ad the compesator of s t 1 { σ s >v} is t a sds, ad where b t, a t, a t are optioal locally bouded processes, ad σ t is optioal ad càdlàg, as well as b t. 7

8 Assumptio H-r) imposes very mild measurability ad sample-path regularity. The parameter r govers the cocetratio of small jumps. The larger r, the weaker this assumptio. Whe r = 2, we essetially put o restrictio o the cocetratio of small jumps sice jumps of ay semimartigale are square-summable. Whe r = 1 resp. r = ), the jump process has fiite variatio resp. fiite activity). Assumptio K) ca be cosidered as a smoothess coditio for σ t ) t i a stochastic sese. However, this assumptio does ot require sample paths of the volatility process to be smooth. We allow the drift of volatility ad the volatility of volatility to be stochastic i a oparametric maer. We also allow jumps i volatility, which ca have fiite or ifiite activity, or eve ifiite variatio. Although it is fairly urestrictive, Assumptio K) does exclude the process σ t ) beig a fractioal Browia motio. 3.2 Assumptios o the pricig fuctio We collect some regularity coditios o the family V of pricig fuctios. A V): Let Θ, d) be a compact metric space i R q, q 1. For ay bouded subset K D, there exists some fiite costat C >, such that the family V satisfies the followig coditios. 1) For each θ Θ, V t, x; θ) is differetiable i t ad fourth order differetiable i x. The partial derivatives are cotiuous i t, x). 2) For h {V xx, V xxx, V xxxx }, sup t,x,θ) [,T ] K Θ h t, x; θ) <. 3) For h {V xx, V xxx }, sup t,x) [,T ] K h t, x; θ ) h t, x; θ) Cd θ, θ ), θ, θ Θ. 4) For some κ >, N ε, Θ, d) Cε κ for ay ε, 1], where N ε, Θ, d) is the miimum umber of ε-balls eeded for coverig Θ. 5) We have almost surely T V xx s, X s ; θ) 2 ds > for all θ Θ. Assumptios V1-3) are satisfied whe the pricig fuctio V ) is suffi cietly smooth i its state variables ad the parameter θ. I may cases, we ca take the metric d, ) to be Euclidea ad Assumptio V4) automatically holds for κ = q. I geeral, Assumptio V4) restricts the complexity of the set Θ. Assumptio V5) requires that the pricig fuctio has ozero curvature over the horizo. We eed this assumptio to avoid degeerate cetral limit theorems. A basic example which verifies Assumptio V) is the followig. Let V be the collectio of Black-Scholes formulas of a family of Europea call optios idexed by their strike prices. I this case, the idex θ is the strike price. More geerally, θ may iclude the volatility parameter i the Black-Scholes model ad the maturity of the cotract. I this geeral case, V represets a cross sectio of call optios hedged uder various volatility parameters. 8

9 3.3 Cosistet estimators of hedgig errors We propose a estimator for the jump error profile B T ). icremet over the iterval i 1), i ], i 1, by For ay process Y, we deote its i Y = Y i Y i 1). The otatio i Y is ot meat to deote the th differece, but rather highlights that i Y forms a triagular array.) The estimator for the jump error B T θ) is give by BT θ) = A T θ) à T θ), where A T θ) = à T θ) = [T/ ] 1 2m 2 1 ϕ i 1), X i 1), i X; θ ) [T/ ] 1 V xx,i θ) = V xx i, X i ; θ), i, V xx,i 1 θ) i X i+1x ad for w >, m w is the wth absolute momet of a stadard ormal variable, that is, m w = E[ U w ], U N, 1). Followig the covetio i the literature, we refer to BT θ) as the realized jump error ad the fuctioal estimator BT ) as the realized jump error profile. Theorem 1 Suppose that Assumptios H-2) ad V) hold. We have uiformly i θ Θ, A T θ) P 1 2 à T θ) P 1 2 ad cosequetly BT θ) P B T θ). T T V xx s, X s ; θ) σ 2 sds + B T θ), 8) V xx s, X s ; θ) σ 2 sds, 9) Theorem 1 shows the cosistecy of the realized jump error ad the uiform cosistecy of the realized jump error profile. The theorem also clarifies the role of A T ) ad à T ) i our costructio. The estimator A T ) is a ituitive sample aalog to B T ). However, it carries a T bias 1 2 V xx s, X s ; ) σ 2 sds i the first-order asymptotics as show i 8). This bias is corrected by à T ). The correctio term à T ) geeralizes the bipower estimator origially proposed by Bardorff-Nielse ad Shephard 24) with radom weights. We also have a cosistet estimator of the volatility error as a by-product of Theorem 1. For brevity, we oly preset the result i the case whe V ) satisfies the Black-Scholes equatio. We eed this result i our empirical applicatio i which we use the volatility error as a yardstick to gauge the relative magitude of the jump error. 9

10 Corollary 1 Suppose that the same coditios as i Theorem 1 hold. If the fuctio V, ) satisfies the Black-Scholes equatio for some local volatility fuctio t, x) v BS t, x), the the realized volatility error defied by D T = Ã T 2 [T/ ] coverges i probability to the volatility error D T T ρ s, X s, σ s ) ds = 1 2 Vxx,i 1v BS 2 ) i 1), X i 1) X 2 i 1) T 3.4 A sesitivity measure of the jump error V xx s, X s ) σ 2 s v 2 BS s, X s ) X 2 s ) ds. It is helpful to uderstad how sesitive the jump error B T θ) is with respect to chages i θ. Whe θ is oe- or two-dimesioal, we ca visualize the depedece of B T θ) o θ by plottig the fuctioal estimator BT θ) versus θ. However, such presetatio is ifeasible whe θ is of high dimesio. Eve i the low dimesioal case, it is still useful to characterize the sesitivity with a few umbers i additio to presetig a curve or a surface. For this purpose, we propose a local sesitivity measure which is simply defied as the gradiet of B T θ) with respect to θ, i.e., B T θ) / θ. This quatity is uobservable but ca be cosistetly estimated. Corollary 2 Suppose that Assumptio H-2) holds ad the family of fuctios { V, ; θ) / θ : θ Θ} satisfies Assumptio V). The variables BT θ) / θ coverges i probability uiformly i θ Θ to B T θ) / θ. 4 Uivariate tests We costruct tests i order to determie the statistical sigificace of jump errors. I this sectio, we cosider a uivariate problem by fixig oe optio cotract which is delta-hedged accordig to a fixed pricig fuctio t, x) V t, x), so the family V i Assumptio V) is a sigleto. We thus suppress the idex θ for otatioal simplicity. We preset this special case separately to help develop ituitio ad relate our approach to the literature. 4.1 Equivalece to tests for jumps The testig problem ca be formally stated as follows: for a give realizatio ω Ω, oe-sided test { H : ω Ω b+ T H a : ω Ω b+ T, Ω b+ T = {B T > }, 1

11 { H : ω Ω b T two-sided test H a : ω Ω b T, Ω b T = {B T }. The set Ω b+ T is the evet i which the jump error is positive, represetig a ecoomic loss for the optio seller. The oe-sided test is desiged to detect the dow-side risk. O the other had, the sample path of the stock price falls i the evet Ω b T as log as the jump error is ozero, so the jump error may be either a loss or a gai i ecoomic terms. The risk captured by Ω b T is two-sided. We ow show that uder regularity coditios, testig for the presece of jump errors is equivalet to testig for jumps. The latter ca be formally stated as { H : ω Ω j T H a : ω Ω j, T where Ω j T = {ω : t X t ω) is discotiuous o [, T ]}. It is clear that Ω b+ T reverse iclusio is give below. Ωb T Ωj T. The Theorem 2 Suppose that for each t [, T ], the fuctio x V t, x) is twice cotiuously diff eretiable o D. a) If V xx, ) > o [, T ] D, the Ω b+ T = Ωb T = Ωj T. b) Assume that i) the jumps of X have fiite activity with successive arrival times deoted by T q, q N; ii) the coditioal distributio of X Tq give F Tq is absolutely cotiuous for each q N; ad iii) for each t, the fuctio x V t, x) is o-affi e o ay ope iterval i D. The Ω b T = Ωj T almost surely. Theorem 2a) shows that if the pricig fuctio has positive gamma, the jumps ecessarily iduce a positive jump error, ad thus a loss to the hedger who shorts the optio. As the jump error ca ever be egative i this case, it is atural to coduct the oe-sided test. Basic examples of this sort iclude the classical Black-Scholes model Black ad Scholes 1973)) ad Merto s jump-diffusio model Merto 1976)) for vailla call ad put optios. Without the sigle-siged gamma, it is possible that the loss iduced by oe jump is cacelled by the gai from the other. As a result, the presece of jumps does ot ecessarily imply ozero jump error. However, Theorem 2b) shows that such a possibility is egligible. The coditios imposed here are ulikely to be the weakest, but seem to be geeral eough for most applicatios. Coditio i) oly requires that there are fiitely may jumps over a fiite time period. This coditio esures that jumps ca be ordered i time so that coditio ii) is a valid statemet. Coditio ii) is satisfied if the distributio of the jump size, coditioal o the pre-jump iformatio, has a desity. Coditio iii) is geerically true for derivatives with oliear payoffs. 11

12 The sigificace of Theorem 2 is that uder the ull hypotheses, there are o jumps. testig purposes, it is eough to characterize the samplig variability of the realized jump errors uder the ull, as we discuss below. 4.2 Secod order asymptotic properties I this subsectio, we describe the covergece i law of the realized jump error whe the uderlyig process is cotiuous. For a sequece of radom variables ξ, we write ξ L s) MN, Σ) if ξ coverges stably i law to a mixture ormal distributio MN, Σ) which, coditioal o the iformatio set σ-algebra) F, is cetered Gaussia with variace Σ. Stable covergece is a slightly stroger otio tha the usual weak covergece. Its key property which is useful for us is that if there is a sequece of estimators Σ which cosistetly estimate Σ, the ξ / Σ coverges weakly to a stadard ormal distributio. This result ca ot be obtaied from the usual weak covergece because Σ is typically radom i our settig. For a detailed discussio about the stable covergece, see Jacod ad Shiryaev 23). Theorem 3 Suppose that Assumptios K) ad V) hold ad X is cotiuous. We have where T Σ T = k 1/2 BT L s) MN, Σ T ), V xx s, X s ) 2 σ 4 sds, k = π2 + 4π I order to implemet Theorem 3 for statistical iferece, we eed to cosistetly estimate the asymptotic variace Σ T. We propose two estimators: ˆΣ 1) T = k ˆΣ 2) T = where the trucatio level u is give by [T/ ] 2 m 3 4/3 Vxx,i 1 )2 3 j=1 i+j 1 X 4/3 k 3 [T/ ] V xx,i 1 )2 i X)4 1 { i X u } For, 1) u = α ϖ, α >, ϖ, 1/2). 11) These estimators geeralize those proposed by Bardorff-Nielse ad Shephard 26), Bardorff- Nielse, Graverse, Jacod, Podolskij, ad Shephard 26) ad Jacod 28) by accommodatig radom weights. We refer to ˆΣ 1) T as the multipower variace estimator, ad ˆΣ 2) T as the trucatio variace estimator. Both estimators are cosistet, regardless of whether there are jumps or ot. Theorem 4 Suppose that Assumptios H-r) ad V) hold for some r [, 2]. Let ˆΣ T = ˆΣ j) T for j = 1 or 2. Whe j = 2, we further assume that r < 2 ad ϖ [ 1 4 r, 1 2 ). The ˆΣ T = Σ T +o p 1). 12

13 4.3 The test: size ad power The t-statistic of our test is give by B T ST = 1/2 ˆΣ T, where ˆΣ T = ˆΣ j) T for j = 1 or 2. Whe the price of the uderlyig is cotiuous, ST coverges i distributio to a stadard ormal variable Theorems 3 ad 4). Whe there are jumps, ST diverges i probability to + i the restrictio to Ω b+ T, ad S T diverges i probability to + i the restrictio to Ωb T Theorems 1 ad 4). Size ad power properties of tests based o ST readily follow. Corollary 3 For q, 1), let z q be the 1 q quatile of N, 1). Suppose that Assumptio K) ad the same coditios as i Theorem 4 hold. a) Uder the same coditios as i Theorem 2a), the asymptotic rejectio probability of the oe-sided critical regio C T + = {S T z q} is q uder the ull hypothesis B T =, ad oe uder the alterative hypothesis B T >. b) Uder the same coditios as i Theorem 2b), the asymptotic rejectio probability of the two-sided critical regio C T = { S T z q/2} is q uder the ull hypothesis BT =, ad oe uder the alterative hypothesis B T. We ca iterpret the test as a compariso betwee the jump error ad the discretizatio error, i.e., the hedgig error resulted from the discrete implemetatio of the dyamic hedgig strategy. Bertsimas, Koga, ad Lo 2) ad Hayashi ad Myklad 25) show that whe X is cotiuous, the discretizatio error is approximately ormally distributed with mea zero ad stadard deviatio 1/2 dis ΣT /2 k, where Σ T ad k are the same as i Theorem 3 ad dis is the duratio betwee portfolio rebalacig. Followig Bertsimas, Koga, ad Lo 2), we refer to the quatity Σ T /2 k = the t-statistic as 1 2 T V xx s, X s ) 2 σ 4 sds as the ex-post temporal graularity. 3 Rewritig S T = 1 2 k dis ) 1/2 BT, 12) 1/2 dis ˆΣ T /2 k we see that it is proportioal to the ratio of the realized jump error to the discretizatio error measured by the estimate of the ex-post temporal graularity. Up to a multiplicative costat 1/ 2 k, the scalig factor is determied by the relative magitude of the busiess time scale dis with respect to the statistical time scale. Equatio 12) clarifies the ecoomic ituitio 3 Bertsimas, Koga, ad Lo 2) defie the temporal graularity for a derivative cotract with pricig fuctio t, x) V t, x) as [ 1 ] T 2 E Vxxs, Xs)2 σ 4 sds, where E is the coditioal expectatio operator give the iformatio at time. 13

14 behid our test. I practice, a trader ca oly hedge at discrete times ad always suffers from the discretizatio error, regardless of whether there are jumps or ot. Give that the discretizatio error is ievitable, the trader may use it as a yardstick to judge the relative importace of the jump error, which is implicit i the test. As a by-product of our aalysis, we derive cosistet estimators for the ex-post temporal graularity, which might also be useful for other applicatios. We ow discuss the relatio betwee our tests ad the bipower test of Bardorff-Nielse ad Shephard 24). The bipower test is a special case of our oe-sided test with V t, x) = x 2. Igorig the depedece of the optio price o the time variable, this pricig fuctio correspods to a quadratic cotract. I particular, the jump error is reduced to the jump quadratic variatio s T X s) 2. This quatity has received a fair amout of attetio i the recet literature of fiacial ecoometrics, see e.g., Aderse, Bollerslev, ad Diebold 27), Aderse, Bollerslev, ad Dobrev 27) ad Aderse, Bollerslev, ad Huag 21). We ca also iterpret the jump quadratic variatio ad the bipower test based o a argumet of static hedgig. As show by Lelad 198), oe ca replicate the fial payoff of a quadratic cotract with a equally weighted portfolio of the complete cross sectio of call optios, that is, XT 2 = 2 max {X T θ, } dθ. Therefore, the jump error of the quadratic cotract is also the total jump error of this portfolio of call optios. The bipower test ca thus be cosidered as a special way of testig the joit hypothesis that jump errors of the cross sectio of call optios are idetically zero by assigig equal weight to each cotract. 5 The worst-case jump error ad the SUP test I this sectio, we exted the uivariate test to a multivariate settig by developig a fuctioal samplig theory for the realized jump error profile. We begi with a motivatig example. Imagie a trader hedges a few call optios which are deeply i the moey. Sice these optios behave quite similar as the uderlyig stock, they are relatively easy to delta hedge. Cosequetly, the trader is likely to observe a small realized jump error for her portfolio ad coclude that the jump error is isigificat. But we may ask: is this because the trader is lucky eough to maage a tradig book which happes to be jump-eutral or is it because jumps are ot importat for hedgig at all? We address these questios by ivestigatig the etire jump error profile i order to get a global view for a family of cotracts, istead of ay specific portfolio of them. To add some cocreteess, we ote that the family of vailla call optios is of fudametal iterest for two reasos. First, vailla call ad put optios are the backboe of the derivative market. Because of the put-call parity, the jump error of a call optio is idetical to that of a put optio with the same cofiguratio. By studyig call optios, we automatically take care of both types of cotracts. 14

15 Secod, by the result of Lelad 198), the payoff of may optios of which the quadratic cotract is a example ca be statically replicated by holdig the uderlyig stock ad a portfolio of call optios. The study of jump errors of these optios ca thus be reduced to the study of the jump error profile for vailla calls. We cosider call optios i the empirical aalysis, although our ecoometric framework is ot restricted to this special case. I order to examie whether jumps are costly for hedgig i a global maer, we cosider a pessimistic measure defied by B T = sup θ Θ B T θ), i.e., the worst-case jump error for the family of iterest. For call optios, the worst-case cotract is ofte ear the moey, while the exact moeyess depeds o the realizatio of the stock price process, especially whe there are multiple jumps occurrig at various price levels. A aalysis of the worst case sceario tells us what would have happeed if we were hedgig a portfolio which is most sesitive to the jump risk. We also cosider the worst-case absolute jump error defied by B T = sup θ Θ B T θ) which serves as the two-sided couterpart of B T. The realized versios of B T ad B T are give respectively by B T = sup BT θ), θ Θ B T = sup BT θ). θ Θ We ow discuss the ecoometric results. We characterize the samplig variability of the realized jump error profile by extedig Theorem 3 to a fuctioal settig. Agai, we oly cosider the stable covergece i law i the absece of jumps for the testig purpose cosidered here. Theorem 5 Suppose that Assumptios K) ad V) hold ad X is cotiuous. The processes 1/2 BT θ) θ Θ) 1 coverge stably i law to a process U θ) θ Θ defied o a extesio of the probability space Ω, F, P) which, coditioally o F, is a cetered Gaussia process with covariace fuctio Σ T θ, θ ) T = k V xx s, X s ; θ) V xx s, Xs ; θ ) σ 4 sds, θ, θ Θ. The asymptotic properties of B T ad B T follow directly from Theorems 1 ad 5: Corollary 4 a) Uder the same coditios as i Theorem 1, B T = B T + o p 1) ad B T + o p 1). 1/2 b) Uder the same settig as i Theorem 5, 1/2 L s) G T sup θ Θ U θ). B T B T L s) B T = G T sup θ Θ U θ) ad Corollary 4a) shows that B T ad B T are cosistet estimators of BT ad B T respectively. Corollary 4b) characterizes the samplig variability of the realized worst-case jump errors uder the ull hypothesis. To implemet Corollary 4 as a test, we eed to estimate quatiles of G T ad 15

16 G T. For q, 1), let κ q ad κ q be the 1 q quatiles of G T ad G T respectively.4 We first exted the estimators i 1) by settig ˆΣ T, ) = ˆΣ T j;, ), where for j = 1 or 2, ˆΣ T ˆΣ T 1; θ, θ ) k = m 3 4/3 [T/ ] 2 V xx,i 1θ)V xx,i 1θ ) 3 i+j 1X 4/3, 2; θ, θ ) = k [T/ ] V 3 xx,i 1θ)V xx,i 1θ ) i X) 4 1 { i X u }. Similar as i the uivariate settig, we label these two estimators as multipower- ad trucatiobased respectively. Let U θ) θ Θ be a cetered mixture Gaussia process with coditioal covariace fuctio ˆΣ T, ). We set ˆκ q ad ˆκ q to be the 1 q quatiles of sup θ Θ U θ) ad sup θ Θ U θ) respectively. These quatiles, which ca be obtaied via simulatio, serve as cosistet estimators of κ q ad κ q. Below, we oly state results for the test based o the realized worst-case jump error B T. We refer to this test as the SUP test. A test based o B T ca be costructed similarly. Theorem 6 Suppose that Assumptio K) ad the same coditios as i Theorems 4 hold. The ˆΣ T θ, θ ) coverges i probability to Σ T θ, θ ) uiformly i θ, θ Θ. Moreover, for each q,.5], ˆκ q resp. ˆκ q ) coverges i probability to κ q resp. κ q). If we further assume that V xx, ; θ) > for some θ Θ, the the asymptotic rejectio probability of the critical regio { B T ˆκ } q is q uder the ull hypothesis BT =, ad oe uder the alterative hypothesis B T >. j=1 6 Simulatio results 6.1 The settig We ow examie the validity of the asymptotic theory above i a simulatio settig desiged to approximate the costraits faced i a typical real life applicatio. The data geeratig process is give by X t = X c t + J t, dx c t /X c t = v t dw t, v t = c 1/2 t, c t = κ β c t ) dt + γc 1/2 t db t, E [dw t db t ] = ρdt. Here, X c t is the cotiuous part of the price ad J t is a pure jump process to be specified below. The drift part is omitted because it plays little role i the high-frequecy settig. Parameters goverig the dyamics of the stochastic volatility process are calibrated accordig to the estimates 4 For ay real radom variable ξ, the α-quatile of ξ is defied as if{x : Probξ x) α}. 16

17 i Aït-Sahalia ad Kimmel 27): β 1/2 =.4, γ =.5, κ = 5, ρ =.5 i aualized terms. The cotiuous-time process is simulated at 5-secod iterval uder the Euler scheme. We the resample the process at every miute or every 5 miutes. We set the horizo T to be 5 tradig days, with each day cosistig of 6.5 hours. There are 5, simulatios i each experimet. We cosider oe-sided uivariate tests based o weekly vailla call optios, heceforth the VC tests, with strike prices spaig the iterval [2, 3]. For each call optio, we set the pricig fuctio V ) to be the Black-Scholes formula with zero risk-free rate ad divided yield. The volatility parameter i the Black-Scholes formula is take to be β 1/2. Because the pricig fuctio is osmooth at the strike price at maturity, we smooth away this sigularity by addig half a tradig day to the maturity of the optio. We also coduct the SUP test based o the whole family of call optios. For compariso, we coduct the bipower test as a special case of the uivariate test correspodig to V t, x) = x 2. I the power aalysis, we cosider simple jump processes which are egieered to highlight the features of our tests. We cosider sample paths with oe jump occurrig at fixed time τ = 1, 2.5 or 4 days i order to examie how the power of the test depeds o the timig of jumps. We set the jump size to be mx τ β 5 miutes, m = ±2.5, ±5, ±7.5, so the jump size is m times the average stadard deviatio of the diffusive icremet over 5 miutes. To simplify the discussio, we fix the prejump price X τ = 25 so a call optio with strike 25 is at the moey whe the jump occurs. This desig help illustrates how the power depeds o the moeyess of the optio. We refer to VC tests with strikes 22.5, 25 ad 27.5 as i-the-moey ITM), at-the-moey ATM) ad out-of-the-moey OTM) tests, respectively. I the size aalysis, we set the jump process J t to be idetically zero ad X = 25. We fid that tests based o the trucatio variace estimator cotrol size better tha those based o the multipower variace estimator. To save space, we oly preset results based o the trucatio variace estimator. We set the trucatio level u = ᾱ 25 β 1/2.49 with ᾱ = 5. Takig ᾱ = 4 or 6 yields similar results which are omitted for brevity. 6.2 Size Table 1 shows the rejectio rates of 5% ad 1% level tests i the absece of jumps. At the 1-miute samplig frequecy, all tests cotrol size well, suggestig that the asymptotic theory is valid. I the 5-miute case, uivariate tests bipower ad VC) still have good size cotrol. However, the SUP test teds to over-reject: its rejectio rate is 6.5% at the 5% level ad 13.1% at the 1% level. We further ivestigate the sizes of all VC tests with strikes spaig [2, 3]. Figure 1 shows the results for 5% level tests. I most cases, the fiite-sample rejectio rate differs from the omial level by less tha oe percetage poit. Overall, we fid that the tests cotrol size well, as predicted by the asymptotic theory. 17

18 6.3 Power Tables 2 ad 3 show the rejectio rates of tests uder the presece of jumps for 1-miute ad 5- miute data, respectively. For brevity, the omial level is fixed at 5%. A immediate observatio is that the rejectio rate is higher whe data are sampled more frequetly. Below, we discuss the results i more details. Jump size. Not surprisigly, the larger the jump size, the higher the power of each test. For data sampled at 5 miutes, the fiite-sample power is very close to the omial level whe the jump size is m = ±2.5 times the stadard deviatio of the diffusive icremet over 5 miutes. I this case, the jump is too small to be detected because it has little effect o hedgig. O the other had, whe jumps are large m = ±7.5) ad data are sampled at every miute, the bipower, the SUP ad the ATM tests have almost perfect power. Jump time ad moeyess. I Figure 2, we plot the rejectio rate versus the jump time for various jump magitudes rows) ad samplig frequecies colums). The rejectio rate of the bipower test dashed lie) does ot deped the jump time because it correspods to the timehomogeeous fuctio V t, x) = x 2. O the cotrary, whe the jump occurs ear maturity, the SUP test square) ad the ATM test circle) reject more ofte, but the ITM test X) ad the OTM test triagle) reject less. The ituitio is that as the time-to-maturity approaches zero, the pricig fuctio become more oliear resp. liear) aroud resp. far from) the strike price. Therefore, a jump aroud resp. far from) the strike price has a larger effect if it occurs later resp. earlier) durig the life of the optio. I most cases, the ATM test is more powerful tha the ITM ad the OTM tests, reflectig the fact that jumps iduce large hedgig errors for call optios whe they occur at the moey. I Figure 3, we further plot the rejectio rates of all VC tests versus their strikes. The power curve forms a peak aroud the prejump price, i.e. 25, whe the jump occurs late i the sample dot-dashed ad dashed lies); however, the curve is relatively flat across strikes whe the jump occurs early solid lie). This patter suggests that the power of VC tests is more sesitive to the strike price if the jump time is closer to maturity. Bipower, SUP ad ATM. The ATM test is always more powerful tha the SUP test. Ideed, the ATM test directly exploits the fact that the strike price of the worst-case cotract is close to the pre-jump price. The pre-jump price is kow i our experimet by desig but is ukow i practice because jumps are ot directly observable. O the other had, the SUP test has to fid the worst-case cotract. Therefore, the ATM test ca be roughly thought of as a ifeasible versio of the SUP test. As the former exploits extra iformatio, it is ot surprisig it has higher power tha the latter. Except for a few cases, the ATM is also more powerful tha the bipower test. The compariso betwee the SUP test ad the bipower test depeds o the jump time: the former is more powerful tha the latter whe the jump is closer to maturity, ad vice versa. 18

19 Positive ad egative jumps. Rejectio rates of the bipower, the SUP ad the ATM tests do ot seem to deped o the sig of the jump. O the other had, i most cases, the OTM test teds to reject more ofte for positive jumps tha for egative jumps; ad the opposite is true for the ITM test. To illustrate this asymmetry graphically, we plot i Figure 4 the rejectio rates of VC tests versus their strike prices for positive ad egative jumps with various magitudes. Whe the strike is low, positive jumps solid lies) lead to lower rejectio rates tha egative jumps dashed lies), ad vice versa. Ituitively, whe the optio is i the moey, a positive jump makes the pricig fuctio more liear, while a egative jump makes the pricig fuctio more oliear. It is thus easier for a ITM test to pick up egative jumps tha positive oes. To summarize, we illustrates how the power of various tests depeds o features of the jump process, such as the magitude, the directio ad the timig of jumps. The power of a test is high if the correspodig optio is diffi cult to hedge uder the presece of jumps. The purpose of this aalysis is ot to coduct a horse race amog various tests. Istead, we attempt to provide some ecoomic isight o why oe test is more powerful tha the other give certai features of the jump process. Through the les of hedgig, our view is that jumps deserve to be detected oly if they have importat ecoomic cosequeces. This idea might be ucovetioal from a statistical poit of view, but seems atural i a ecoometric cotext. 7 Empirical aalysis 7.1 The settig Our sample cosists of the 3 compoet stocks of Dow Joes Idustrial Average DJIA) i 28; the data source is the TAQ database. Because the compositio of the DJIA chages over time, we use the 3 stocks that are the compoets of the idex as of September, 16th, 21. We use filters to elimiate clear data errors price set to zero, etc.) as is stadard i the empirical market microstructure literature. For each tradig day i 28, we collect all trasactios from 9:3am util 4:pm, ad compute the volume-weighted average of trasactio prices at each time stamp for each oe of these stocks. Followig the commo practice i the literature, we sparsely sample the itraday data at every 5 miutes to reduce the effect of microstructure oise. We perform the tests for itraday returs. We also coduct the same aalysis icludig overight returs withi each week. I so doig, we cosider itraweek stock price jumps i U.S. tradig time uder the assumptio that the hedgig portfolio is locked up after the U.S. stock market closes. Each week ad stock is treated o its ow. The empirical settig is similar as i the simulatio. We cosider vailla call optios with 19

20 moeyess spaig [.85,1.15]. 5 We take the pricig fuctio V ) to be the classical Black- Scholes formula with costat volatility which is updated o a weekly basis. The iterest rate ad the divided yield are set to be zero. Without ay attetio for advocatig the Black-Scholes model as a hedgig device, we choose this simple strategy oly for the purpose of illustratig our method. Nevertheless, we ote that this simple setup might covey a broader message. Ideed, as documeted by Bakshi, Cao, ad Che 1997) ad Cherov ad Ghysels 2), the Black- Scholes strategy has similar hedgig performace as models with stochastic volatility ad jumps. We cojecture that usig more sophisticated pricig models may lead to similar results as ours. A extesive compariso is beyod the scope of this paper. The key tuig parameter i the Black-Scholes hedgig strategy is the volatility parameter, which we deote by v BS. For each stock-week, we calibrate v BS to be the ex-post average volatility 1 T T v2 sds for the week, where we recall that v t = σ t /X t is the volatility. We approximate this quatity by the trucated pre-averagig estimator, which, deoted by ˆv, varies across stocks ad weeks. 6 We are thus cosiderig the bechmark case i which we are correct, up to estimatio errors, about the volatility o average. We perform a sesitivity aalysis with other choices of v BS later. We implemet the tests with the trucatio variace estimator. The trucatio level is set to be u = 5 Xˆv.49, where X is the weekly average stock price. Similarly as i the simulatio, we addig half a tradig day 3.25 hours) to the time-to-maturity i order to smooth away the sigularity of the pricig fuctio at the strike at maturity A illustrative example To help visualize cocepts, we start by illustratig the ecoometric framework via a example based o the stock of Microsoft Corporatio NASDAQ: MSFT). I the top pael of Figure 5, we plot the time series of the stock price for two weeks: week-a begiig August 11th left) ad week-b begiig November 1th right). A visual ispectio suggests that week-a is likely to have a cotiuous sample path, eve though overight returs circles) are icluded, ad week-b is likely to have jumps: oe cadidate is the overight drop at the market s ope o Friday, ad 5 We defie the moeyess of a call optio as the ratio of the strike price to the weekly opeig stock price. For computatioal purpose, we discretize the iterval [.85,1.15] ito grids with mesh size.1. Prelimiary results ot preseted here) suggest that fier mesh size.25) has little effect o the results. 6 The estimate is computed based o data sampled at every 5 secods. This estimator is robust agaist both microstructure oise ad jumps. See Lemma 6 of Aït-Sahalia, Jacod, ad Li 211) for its defiitio ad theoretical justificatio. 7 Amog all 53 weeks i 28, there are two weeks i which Fridays are atioal holidays Good Friday ad Idepedece Day). For these weeks, we cosider a weekly optio with maturity beig 4.5 days. The last day of 28 is Wedesday. The maturity of the optio i this week is set to be 3.5 days. 2

21 aother is the large drop ear the market s close o the same day. For each week, we plot the realized jump error profile with overight returs icluded solid lie) or excluded dashed lie) versus the optio moeyess i the bottom pael of Figure 5. Clearly, the profiles of week-b are more proouced tha those of week-a. I additio, overight returs i week-b have a evidet effect o the jump error, but ot so much i week-a. These observatios are cosistet with what we observe i the time series. We also plot the 5% critical value of each VC test dotted lie). I week-a, the realized jump error is less tha the critical value for every call optio. I week-b, we reject the ull hypothesis of zero itraday jump error whe the moeyess is, roughly speakig, betwee.9 ad.94. To see the ituitio, we ote that the large drop at the ed of the sample occurs whe the stock price is aroud $2.5 which is approximately.94 times the weekly opeig price $ Optios with moeyess betwee.9 ad.94 is ear the moey whe this large jump occurs ad thus leads to a large jump error. Whe we iclude overight returs ito the aalysis, VC tests reject the ull hypothesis for moeyess betwee.89 ad 1.13, suggestig that overight jumps have a sigificat effect o the whole cross sectio of call optios. The SUP test elimiates the ambiguity i the cotract-by-cotract exercise by cocetratig o the worst-case sceario. I the bottom pael of Figure 5, we plot the uiform acceptace regio shaded area) of the SUP test, where the upper boud of the acceptace regio is ˆκ.5 Theorem 6). Not surprisigly, we reject the ull hypothesis that the worst-case jump error is zero for week-b at 5% omial level o matter whether overight returs are icluded or ot. We do ot reject the same ull hypothesis for week-a as the acceptace regio covers the realized jump error profiles. 7.3 Cross-sectioal aalysis We start by performig tests at 5% omial level for each week ad stock. Figure 6 plots the empirical rejectio rate of VC tests versus the optio moeyess solid lie). We first discuss the case whe overight returs are excluded left pael). Whe the moeyess is ear 1, VC tests reject the ull hypothesis at about 41% of the time. The rejectio rate decreases as the moeyess deviates from 1: the rejectio rates at moeyess =.9 ad 1.1 are 26% ad 28% respectively. The patter is similar whe we iclude overight returs i the calculatio right pael). Because overight returs ofte appear to be jumps to the aked eye, it is ot surprisig that the rejectio rate icreases after overight returs are icluded. What is remarkable is the magitude of this icrease the rejectio rate approximately doubles suggestig that overight jumps make a major cotributio to the jump error. We the perform the SUP test for these call optios. We also coduct the bipower test for compariso. Table 4 compares the rejectio rates of the SUP test, the bipower test ad the VC test associated with the ATM optio, which, with some abuse of otatio, will be referred to as the 21