Hybrid scheme for Brownian semistationary processes


 Erin Ball
 1 years ago
 Views:
Transcription
1 Hybrid scheme for Browia semistatioary processes Mikkel Beedse Asger Lude Mikko S. Pakkae September 8, 28 arxiv:57.34v4 [math.pr] 4 May 27 Abstract We itroduce a simulatio scheme for Browia semistatioary processes, which is based o discretizig the stochastic itegral represetatio of the process i the time domai. We assume that the kerel fuctio of the process is regularly varyig at zero. The ovel feature of the scheme is to approximate the kerel fuctio by a power fuctio ear zero ad by a step fuctio elsewhere. The resultig approximatio of the process is a combiatio of Wieer itegrals of the power fuctio ad a Riema sum, which is why we call this method a hybrid scheme. Our mai theoretical result describes the asymptotics of the mea square error of the hybrid scheme ad we observe that the scheme leads to a substatial improvemet of accuracy compared to the ordiary forward Riemasum scheme, while havig the same computatioal complexity. We exemplify the use of the hybrid scheme by two umerical experimets, where we examie the fiitesample properties of a estimator of the roughess parameter of a Browia semistatioary process ad study Mote Carlo optio pricig i the rough Bergomi model of Bayer et al. [], respectively. Keywords: Stochastic simulatio; discretizatio; Browia semistatioary process; stochastic volatility; regular variatio; estimatio; optio pricig; rough volatility; volatility smile. JEL Classificatio: C22, G3, C3 MSC 2 Classificatio: 6G2, 6G22, 65C2, 9G6, 62M9 Itroductio We study simulatio methods for Browia semistatioary BSS processes, first itroduced by BardorffNielse ad Schmiegel [8, 9], which form a flexible class of stochastic processes that are able to capture some commo features of empirical time series, such as stochastic volatility itermittecy, roughess, statioarity ad strog depedece. By ow these processes have bee Departmet of Ecoomics ad Busiess Ecoomics ad CREATES, Aarhus Uiversity, Fuglesags Allé 4, 82 Aarhus V, Demark. Departmet of Ecoomics ad Busiess Ecoomics ad CREATES, Aarhus Uiversity, Fuglesags Allé 4, 82 Aarhus V, Demark. Departmet of Mathematics, Imperial College Lodo, South Kesigto Campus, Lodo SW7 2AZ, UK ad CREATES, Aarhus Uiversity, Demark.
2 applied i various cotexts, most otably i the study of turbulece i physics [7, 6] ad i fiace as models of eergy prices [4, ]. A BSS process X is defied via the itegral represetatio Xt = t gt sσsdw s,. where W is a twosided Browia motio providig the fudametal oise iovatios, the amplitude of which is modulated by a stochastic volatility itermittecy process σ that may deped o W. This drivig oise is the covolved with a determiistic kerel fuctio g that specifies the depedece structure of X. The process X ca also be viewed as a movig average of volatilitymodulated Browia oise ad settig σs =, we see that statioary Browia movig averages are ested i this class of processes. I the applicatios metioed above, the case where X is ot a semimartigale is particularly relevat. This situatio arises whe the kerel fuctio g behaves like a powerlaw ear zero; more specifically, whe for some α 2, 2 \ {}, gx x α for small x >..2 Here we write to idicate proportioality i a iformal sese, aticipatig a rigorous formulatio of this relatioship give i Sectio 2.2 usig the theory of regular variatio [5], which plays a sigificat role i our subsequet argumets. The case α = 6 i.2 is importat i statistical modelig of turbulece [6] as it gives rise to processes that are compatible with Kolmogorov s scalig law for ideal turbulece. Moreover, processes of similar type with α.4 have bee recetly used i the cotext of optio pricig as models of rough volatility [,, 8, 2], see Sectios 2.5 ad 3.3 below. The case α = would roughly speakig lead to a process that is a semimartigale, which is thus excluded. Uder.2, the trajectories of X behave locally like the trajectories of a fractioal Browia motio with Hurst idex H = α + 2, \ { 2 }. While the local behavior ad roughess, measured i terms of Hölder regularity, of X are determied by the parameter α, the global behavior of X e.g., whether the process has log or short memory depeds o the behavior of gx as x, which ca be specified idepedetly of α. This should be cotrasted with fractioal Browia motio ad related selfsimilar models, which ecessarily must coform to a restrictive affie relatioship betwee their Hölder regularity local behavior ad roughess ad Hurst idex global behavior, as elucidated by Geitig ad Schlather [2]. Ideed, i the realm of BSS processes, local ad global behavior are coveietly decoupled, which uderlies the flexibility of these processes as a modelig framework. I coectio with practical applicatios, it is importat to be able to simulate the process X. If the volatility process σ is determiistic ad costat i time, the X will be strictly statioary ad Gaussia. This makes X ameable to exact simulatio usig the Cholesky factorizatio or circulat embeddigs, see, e.g., [2, Chapter XI]. However, it seems difficult, if ot impossible, to develop a exact method that is applicable with a stochastic σ, as the process X is the either Markovia or Gaussia. Thus, i the geeral case oe must resort to approximative methods. To this ed, Beth et al. [3] have recetly proposed a Fourierbased method of simulatig BSS 2
3 processes, ad more geeral Lévy semistatioary LSS processes, which relies o approximatig the kerel fuctio g i the frequecy domai. I this paper, we itroduce a ew discretizatio scheme for BSS processes based o approximatig the kerel fuctio g i the time domai. Our startig poit is the Riemasum discretizatio of.. The Riemasum scheme builds o a approximatio of g usig step fuctios, which has the pitfall of failig to capture appropriately the steepess of g ear zero. I particular, this becomes a serious defect uder.2 whe α 2,. I our ew scheme, we mitigate this problem by approximatig g usig a appropriate power fuctio ear zero ad a step fuctio elsewhere. The resultig discretizatio scheme ca be realized as a liear combiatio of Wieer itegrals with respect to the drivig Browia motio W ad a Riema sum, which is why we call it a hybrid scheme. The hybrid scheme is oly slightly more demadig to implemet tha the Riemasum scheme ad the schemes have the same computatioal complexity as the umber of discretizatio cells teds to ifiity. Our mai theoretical result describes the exact asymptotic behavior of the mea square error MSE of the hybrid scheme ad, as a special case, that of the Riemasum scheme. We observe that switchig from the Riemasum scheme to the hybrid scheme reduces the asymptotic root mea square error RMSE substatially. Usig merely the simplest variat the of hybrid scheme, where a power fuctio is used i a sigle discretizatio cell, the reductio is at least 5% for all α, 2 ad at least 8% for all α 2,. The reductio i RMSE is close to % as α approches 2, which idicates that the hybrid scheme ideed resolves the problem of poor precisio that affects the Riemasum scheme. To assess the accuracy of the hybrid scheme i practice, we perform two umerical experimets. Firstly, we examie the fiitesample performace of a estimator of the roughess idex α, itroduced by BardorffNielse et al. [6] ad Corcuera et al. [6]. This experimet eables us to assess how faithfully the hybrid scheme approximates the fie properties of the BSS process X. Secodly, we study Mote Carlo optio pricig i the rough Bergomi stochastic volatility model of Bayer et al. []. We use the hybrid scheme to simulate the volatility process i this model ad we fid that the resultig implied volatility smiles are idistiguishable from those simulated usig a method that ivolves exact simulatio of the volatility process. Thus we are able propose a solutio to the problem of fidig a efficiet simulatio scheme for the rough Bergomi model, left ope i the paper []. The rest of this paper is orgaized as follows. I Sectio 2 we recall the rigorous defiitio of a BSS process ad itroduce our assumptios. We also itroduce the hybrid scheme, state our mai theoretical result cocerig the asymptotics of the mea square error ad discuss a extesio of the scheme to a class of trucated BSS processes. Sectio 3 briefly discusses the implemetatio of the discretizatio scheme ad presets the umerical experimets metioed above. Fially, Sectio 4 cotais the proofs of the theoretical ad techical results give i the paper. 3
4 2 The model ad theoretical results 2. Browia semistatioary process Let Ω, F, {F t } t R, P be a filtered probability space, satisfyig the usual coditios, supportig a twosided stadard Browia motio W = {W t} t R. We cosider a Browia semistatioary process Xt = t gt sσsdw s, t R, 2. where σ = {σt} t R is a {F t } t R predictable process with locally bouded trajectories, which captures the stochastic volatility itermittecy of X, ad g :, [, is a Borel measurable kerel fuctio. To esure that the itegral 2. is welldefied, we assume that the kerel fuctio g is square itegrable, that is, gx 2 dx <. I fact, we will shortly itroduce some more specific assumptios o g that will imply its square itegrability. Throughout the paper, we also assume that the process σ has fiite secod momets, E[σt 2 ] < for all t R, ad that the process is covariace statioary, amely, E[σs] = E[σt], Covσs, σt = Covσ, σ s t, s, t R. These assumptios imply that also X is covariace statioary, that is, E[Xt] =, CovXs, Xt = E[σ 2 ] gxgx + s t dx, s, t R. However, the process X eed ot be strictly statioary as the depedece betwee the volatility process σ ad the drivig Browia motio W may be timevaryig. 2.2 Kerel fuctio As metioed above, we cosider a kerel fuctio that satisfies gx x α for some α 2, 2 \{} whe x > is ear zero. To make this idea rigorous ad to allow for additioal flexibility, we formulate our assumptios o g usig the theory of regular variatio [5] ad, more specifically, slowly varyig fuctios. To this ed, recall that a measurable fuctio L :, ] [, is slowly varyig at if for ay t >, Ltx lim x Lx =. Moreover, a fuctio fx = x β Lx, x, ], where β R ad L is slowly varyig at, is said to be regularly varyig at, with β beig the idex of regular variatio. Remark 2.. Covetioally, slow ad regular variatio are defied at [5, pp. 6, 7 8]. However, L is slowly varyig resp. regularly varyig at if ad oly if x L/x is slowly varyig resp. regularly varyig at. 4
5 A key feature of slowly varyig fuctios, which will be very importat i the sequel, is that they ca be sadwiched betwee polyomial fuctios as follows. If δ > ad L is slowly varyig at ad bouded away from ad o ay iterval u, ], u,, the there exist costats C δ C δ > such that C δ x δ Lx C δ x δ, x, ]. 2.2 The iequalities above are a immediate cosequece of the socalled Potter bouds for slowly varyig fuctios, see [5, Theorem.5.6ii] ad 4. below. Makig δ very small therei, we see that slowly varyig fuctios are asymptotically egligible i compariso with polyomially growig/decayig fuctios. Thus, by multiplyig power fuctios ad slowly varyig fuctios, regular variatio provides a flexible framework to costruct fuctios that behave asymptotically like power fuctios. Our assumptios cocerig the kerel fuctio g are as follows: A For some α 2, 2 \ {}, gx = x α L g x, x, ], where L g :, ] [, is cotiuously differetiable, slowly varyig at ad bouded away from. Moreover, there exists a costat C > such that the derivative L g of L g satisfies L gx C + x, x, ]. A2 The fuctio g is cotiuously differetiable o,, so that its derivative g is ultimately mootoic ad satisfies g x 2 dx <. A3 For some β, 2, gx = Ox β, x. Here, ad i the sequel, we use fx = Ohx, x a, to idicate that lim sup x a fx hx <. Additioally, aalogous otatio is later used for sequeces ad computatioal complexity. I view of the boud 2.2, these assumptios esure that g is square itegrable. It is worth poitig out that A accommodates fuctios L g with lim x L g x =, e.g., L g x = log x. The assumptio A iflueces the shortterm behavior ad roughess of the process X. A simple way to assess the roughess of X is to study the behavior of its variogram also called the secodorder structure fuctio i turbulece literature V X h := E[ Xh X 2 ], h, as h. Note that, by covariace statioarity, V X s t = E[ Xs Xt 2 ], s, t R. 5
6 Uder our assumptios, we have the followig characterizatio of the behavior of V X ear zero, which geeralizes a result of BardorffNielse [3, p. 9] ad implies that X has a locally Hölder cotiuous modificatio. Therei, ad i what follows, we write ax bx, x y, to idicate ax that lim x y bx =. The proof of this result is carried out i Sectio 4.. Propositio 2.2 Local behavior ad cotiuity. Suppose that A, A2 ad A3 hold. i The variogram of X satisfies V X h E[σ 2 ] 2α + + y + α y α 2 dy h 2α+ L g h 2, h, which implies that V X is regularly varyig at zero with idex 2α +. ii The process X has a modificatio with locally φhölder cotiuous trajectories for ay φ, α + 2. Motivated by Propositio 2.2, we call α the roughess idex of the process X. Igorig the slowly varyig factor L g h 2 i 2.2, we see that the variogram V h behaves like h 2α+ for small values of h, which is remiiscet of the scalig property of the icremets of a fractioal Browia motio fbm with Hurst idex H = α + 2. Thus, the process X behaves locally like such a fbm, at least whe it comes to secod order structure ad roughess. Moreover, the factor 2α+ + y + α y α 2 dy coicides with the ormalizatio coefficiet that appears i the Madelbrot Va Ness represetatio [24, Theorem.3.] of a fbm with H = α + 2. Let us ow look at two examples of a kerel fuctio g that satisfies our assumptios. Example 2.3 The gamma kerel. The socalled gamma kerel gx = x α e λx, x,, with parameters α 2, 2 \ {} ad λ >, has bee used extesively i the literature o BSS processes. It is particularly importat i coectio with statistical modelig of turbulece, see Corcuera et al. [6], but it also provides a way to costruct geeralizatios of Orstei Uhlebeck OU processes with roughess that differs from the usual semimartigale case α =, while mimickig the logterm behavior of a OU process. Moreover, BSS ad LSS processes defied usig the gamma kerel have iterestig probabilistic properties, see [25]. A idepth study of the gamma kerel ca be foud i [3]. Settig L g x := e λx, which is slowly varyig at sice lim x L g x =, it is evidet that A holds. Sice gx decays expoetially fast to as x, it is clear that also A3 holds. To verify A2, ote that g satisfies g x = α x λ gx, g x = α 2 x λ α x 2 gx, x,, where lim x α x λ2 α x 2 = λ 2 >, so g is ultimately icreasig with g x 2 α + λ 2 gx 2, x [,. Thus, g x 2 dx < sice g is square itegrable. 6
7 Example 2.4 Powerlaw kerel. Cosider the kerel fuctio gx = x α + x β α, x,, with parameters α 2, 2 \ {} ad β, 2. The behavior of this kerel fuctio ear zero is similar to that of the gamma kerel, but gx decays to zero polyomially as x, so it ca be used to model log memory. I fact, it ca be show that if β, 2, the the autocorrelatio fuctio of X is ot itegrable. Clearly, A holds with L g x := + x β α, which is slowly varyig at sice lim x L g x =. Moreover, ote that we ca write gx = x β K g x, x,, where K g x := + x β α satisfies lim x K g x =. Thus, also A3 holds. We ca check A2 by computig α + βx α + βx 2 g x = gx, g α 2αx βx2 x = + x + x x + x x 2 + x 2 gx, x,, where α 2αx βx 2 whe x as β < 2, so g is ultimately icreasig. Additioally, we ote that g x 2 α + β 2 gx 2, x [,, implyig g x 2 dx < sice g is square itegrable. 2.3 Hybrid scheme Let t R ad cosider discretizig Xt based o its itegral represetatio 2. o the grid G t := {t, t, t 2,...} for N. To derive our discretizatio scheme, let us first ote that if the volatility process σ does ot vary too much, the it is reasoable to use the approximatio Xt = k= t k + t k gt sσsdw s k= σ t k t k + gt sdw s, 2.3 t k that is, we keep σ costat i each discretizatio cell. Here, ad i the sequel, stads for a iformal approximatio used for purely heuristic purposes. If k is small, the due to A we may approximate [ k k gt s t s α L g, t s, k ] \ {}, 2.4 as the slowly varyig fuctio L g varies less tha the power fuctio y y α ear zero, cf If k is large, or at least k 2, the choosig b k [k, k] provides a adequate approximatio [ bk k gt s g, t s, k ], 2.5 7
8 by A2. Applyig 2.4 to the first κ terms, where κ =, 2,..., ad 2.5 to the remaiig terms i the approximatig series i 2.3 yields k= σ t k t k + gt sdw s t k κ k L g k= + k=κ+ g σ t k t k + t s α dw s t k bk σ t k t k + t k dw s, For completeess, we also allow for κ =, i which case we require that b, ] ad iterpret the first sum o the righthad side of 2.6 as zero. To make umerical implemetatio feasible, we trucate the secod sum o the righthad side of 2.6 so that both sums have N κ + terms i total. Thus, we arrive at a discretizatio scheme for Xt, which we call a hybrid scheme, give by where X t := ˇX t + ˆX t, ˇX t := ˆX t := κ k L g k= N k=κ+ g bk 2.6 σ t k t k + t s α dw s, 2.7 t k σ t k W t k + W t k, 2.8 ad b := {b k } k=κ+ is a sequece of real umbers, evaluatio poits, that must satisfy b k [k, k] \ {} for each k κ +, but otherwise ca be chose freely. As it stads, the discretizatio grid G t depeds o the time t, which may seem cumbersome with regard to samplig X t simultaeously for differet times t. However, ote that wheever times t ad t are separated by a multiple of, the correspodig grids G t ad G t will itersect. I fact the hybrid scheme defied by 2.7 ad 2.8 ca be implemeted efficietly, as we shall see i Sectio 3., below. Sice bk g = g t t b k, the degeerate case κ = with b k = k for all k correspods to the usual Riemasum discretizatio scheme of Xt with Itō type forward sums from 2.8. Heceforth, we deote the associated sequece {k} k=κ+ by b FWD, where the subscript FWD alludes to forward sums. However, icludig terms ivolvig Wieer itegrals of a power fuctio give by 2.7, that is havig κ, improves the accuracy of the discretizatio cosiderably, as we shall see. Havig the leeway to select b k withi the iterval [k, k] \ {}, so that the fuctio gt is evaluated at a poit that does ot ecessarily belog to G t, leads additioally to a moderate improvemet. The tructio i the sum 2.8 etails that the stochastic itegral 2. defiig X is trucated at t N I practice, the value of the parameter N should be large eough to mitigate the. effect of trucatio. To esure that the trucatio poit t N asymptotic results, we itroduce the followig assumptio: 8 teds to as i our
9 A4 For some γ >, N γ+,. 2.4 Asymptotic behavior of mea square error We are ow ready to state our mai theoretical result, which gives a sharp descriptio of the asymptotic behavior of the mea square error MSE of the hybrid scheme as. We defer the proof of this result to Sectio 4.2. Theorem 2.5 Asymptotics of mea square error. Suppose that A, A2, A3 ad A4 hold, so that γ > 2α + 2β +, 2.9 ad that for some δ >, E[ σs σ 2 ] = O s 2α++δ, s. 2. The for all t R, where E[ Xt X t 2 ] Jα, κ, be[σ 2 ] 2α+ L g / 2,, 2. Jα, κ, b := k=κ+ k Remark 2.6. Note that if α 2,, the havig E[ σs σ 2 ] = O s θ, s, y α b α k 2 dy <. 2.2 for all θ,, esures that 2. holds. Take, say, δ := 2 2α+ > ad θ := 2α++δ = α +,. Whe the hybrid scheme is used to simulate the BSS process X o a equidistat grid {,, 2,..., T } for some T > see Sectio 3. o the details of the implemetatio, the followig cosequece of Theorem 2.5 esures that the covariace structure of the simulated process approximates that of the actual process X. Corollary 2.7 Covariace structure. Suppose that the assumptios of Theorem 2.5 hold. The for ay s, t R ad ε >, E[X tx s] E[XtXs] = O α+ 2 +ε,. Proof. Let s, t R. Applyig the Cauchy Schwarz iequality, we get E[X tx s] E[XtXs] E[X t 2 ] /2 E[ Xs X s 2 ] /2 + E[Xs 2 ] /2 E[ Xt X t 2 ] /2. We have sup N E[X t 2 ] /2 < sice E[X t 2 ] E[Xt 2 ] < as, by Theorem 2.5. Moreover, Theorem 2.5 ad the boud 2.2 imply that E[ Xs X s 2 ] /2 = O α+ 2 +ε ad E[ Xt X t 2 ] /2 = O α+ 2 +ε for ay ε >. 9
10 I Theorem 2.5, the asymptotics of the MSE 2. are determied by the behavior of the kerel fuctio g ear zero, as specified i A. The coditio 2.9 esures that error from approximatig g ear zero is asymptotically larger tha the error iduced by the trucatio of the stochastic itegral 2. at t N. I fact, differet kid of asymptotics of the MSE, where trucatio error becomes domiat, could be derived whe 2.9 does ot hold, uder some additioal assumptios, but we do ot pursue this directio i the preset paper. While the rate of covergece i 2. is fully determied by the roughess idex α, which may seem discouragig at first, it turs out that the quatity Jα, κ, b, which we shall call the asymptotic MSE, ca vary a lot, depedig o how we choose κ ad b, ad ca have a substatial impact o the precisio of the approximatio of X. It is immediate from 2.2 that icreasig κ will decrease Jα, κ, b. Moreover, for give α ad κ, it is straightforward to choose b so that Jα, κ, b is miimized, as show i the followig result. Propositio 2.8 Optimal discretizatio. Let α 2, 2 \ {} ad κ. Amog all sequeces b = {b k } k=κ+ with b k [k, k] \ {} for k κ +, the fuctio Jα, κ, b, ad cosequetly the asymptotic MSE iduced by the discretizatio, is miimized by the sequece b give by k b α+ k = k α+ /α, k κ +. α + Proof. Clearly, a sequece b = {b k } k=κ+ miimizes the fuctio Jα, κ, b if ad oly if b k miimizes k yα b α k 2 dy for ay k κ +. By stadard L 2 space theory, c R miimizes the itegral k yα c 2 dy if ad oly if the fuctio y y α c is orthogoal i L 2 to all costat fuctios. This is tatamout to k y α cdy =, ad computig the itegral ad solvig for c yields c = kα+ k α+. α + Settig b k := c/α k, k completes the proof. To uderstad how much icreasig κ ad usig the optimal sequece b from Propositio 2.8 improves the approximatio, we study umerically the asymptotic root mea square error RMSE Jα, κ, b. I particular, we assess how much the asymptotic RMSE decreases relative to RMSE of the forward Riemasum scheme κ = ad b = b FWD usig the quatity Jα, κ, b Jα,, bfwd reductio i asymptotic RMSE = %. 2.3 Jα,, bfwd The results are preseted i Figure. We fid that employig the hybrid scheme with κ leads to a substatial reductio i the asymptotic RMSE relative to the forward Riemasum scheme whe α 2,. Ideed, whe κ, the asymptotic RMSE, as a fuctio of α, does ot blow up as α 2, while with κ = it does. This explais why the reductio i the asymptotic
11 asymptotic RMSE κ = κ = κ = 2 κ = 3 reductio i asymptotic RMSE % κ = κ = κ = 2 κ = α α Figure : Left: The asymptotic RMSE give by Jα, κ, b as a fuctio of α 2, 2 \ {} for κ =,, 2, 3 usig b = b of Propositio 2.8 solid lies ad b = b FWD dashed lies. Right: Reductio i the asymptotic RMSE relative to the forward Riemasum scheme κ = ad b = b FWD give by the formula 2.3, plotted as a fuctio of α 2, 2 \ {} for κ =,, 2, 3 usig b = b solid lies ad for κ =, 2, 3 usig b = b FWD dashed lies. I all computatios, we have used the approximatios outlied i Remark 2.9 with N =. RMSE approaches % as as α 2. Whe α, 2, the improvemet achieved usig the hybrid scheme is more modest, but still cosiderable. Figure also highlights the importace of usig the optimal sequece b, istead of b FWD, as evaluatio poits i the scheme, i particular whe α, 2. Fially, we observe that icreasig κ beyod 2 does ot appear to lead to a sigificat further reductio. Ideed, i our umerical experimets, reported i Sectio 3.2 ad 3.3 below, we observe that usig κ =, 2 already leads to good results. Remark 2.9. It is otrivial to evaluate the quatity Jα, κ, b umerically. itegral i 2.2 explicitly, we ca approximate Jα, κ, b by J N α, κ, b := N k=κ+ k 2α+ k 2α+ 2α + 2bα k k α+ k α+ + b 2α k α + Computig the with some large N N. This approximatio is adequate whe α 2,, but its accuracy deteriorates whe α 2. I particular, the sigularity of the fuctio α Jα, κ, b at 2 is difficult to capture usig J N α, κ, b with umerically feasible values of N. To overcome this umerical problem, we itroduce a correctio term i the case α, 2. The correctio term ca be derived iformally as follows. By the mea value theorem, ad sice b k k 2 for large k, we have y α b α k 2 = α 2 ξ 2α 2 y b k 2 α 2 k 2α 2 y k 2, b = b FWD, α 2 k 2α 2 y k + 2 2, b = b,
12 where ξ = ξy, b k [k, k], for large k. Thus, for large N, we obtai Jα, κ, b J N α, κ, b = k=n+ k y α b α k 2 dy α 2 k=n+ k2α 2 k k y k2 dy, b = b FWD, α 2 k=n+ k2α 2 k k y k dy, b = b, α 2 3 = ζ2 2α, N +, b = b FWD, α 2 2 ζ2 2α, N +, b = b, where ζx, s := k= k+s, x >, s >, is the Hurwitz zeta fuctio, which ca be evaluated x usig accurate umerical algorithms. Remark 2.. Ulike the Fourierbased method of Beth et al. [3], the hybrid scheme does ot require trucatig the sigularity of the kerel fuctio g whe α 2,, which is beeficial to maitaiig the accuracy of the scheme whe α is ear 2. Let us briefly aalyze the effect of trucatig the sigularity of g o the approximatio error, cf. [3, pp ]. Cosider, for ay ε >, the modified BSS process X ε t := t g ε t sσsdw s, t R, defied usig the trucated kerel fuctio gε, x, ε], g ε x := gx, x ε,. Adaptig the proof of Theorem 2.5 i a straightforward maer, it is possible to show that, uder A ad A3, E [ Xt Xε t 2 ] = E[σ 2 ] ε gs gε 2ds 2α + 2 α + + E[σ 2 ]ε 2α+ L g ε 2, ε, }{{} =: Jα for ay t R. While the rate of covergece, as ε, of the MSE that arises from replacig g with g ε is aalogous to the rate of covergece of the hybrid scheme, it is importat to ote that the factor Jα blows up as α 2. I fact, Jα is equal to the first term i the series that defies Jα,, b FWD ad Jα Jα,, b FWD, α 2, which idicates that the effect of trucatig the sigularity, i terms of MSE, is similar to the effect of usig the forward Riemasum scheme to discretize the process whe α is ear 2. I particular, the trucatio threshold ε would the have to be very small i order to keep the trucatio error i check. 2
13 2.5 Extesio to trucated Browia semistatioary processes It is useful to exted the hybrid scheme to a class of ostatioary processes that are closely related to BSS processes. This extesio is importat i coectio with a applicatio to the socalled rough Bergomi model, which we discuss i Sectio 3.3, below. More precisely, we cosider processes of the form Y t = t gt sσsdw s, t, 2.4 where the kerel fuctio g, volatility process σ ad drivig Browia motio W are as before. We call Y a trucated Browia semistatioary T BSS process, as Y is obtaied from the BSS process X by trucatig the stochastic itegral i 2. at. Of the precedig assumptios, oly A ad A2 are eeded to esure that the stochastic itegral i 2.4 exists i fact, of A2, oly the requiremet that g is differetiable o, comes ito play. The T BSS process Y does ot have covariace statioary icremets, so we defie its timedepedet variogram as V Y h, t := E[ Y t + h Y t 2 ], h, t. Extedig Propositio 2.2, we ca describe the behavior of h V Y h, t ear zero as follows. The existece of a locally Hölder cotiuous modificatio is the a straightforward cosequece. We omit the proof of this result, as it would be straightforward adaptatio of the proof of Propositio 2.2. Propositio 2. Local behavior ad cotiuity. Suppose that A ad A2 hold. i The variogram of Y satisfies for ay t, V Y h, t E[σ 2 ] 2α + +, t y + α y α 2 dy h 2α+ L g h 2, h, which implies that h V Y h, t is regularly varyig at zero with idex 2α +. ii The process Y has a modificatio with locally φhölder cotiuous trajectories for ay φ, α + 2. Note that while the icremets of Y are ot covariace statioary, the asymptotic behavior of V Y h, t is the same as that of V X h as h cf. Propositio 2.2 for ay t >. Thus, the icremets of Y apart from icremets startig at time are locally like the icremets of X. We defie the hybrid scheme to discretize Y t, for ay t, as where Y t := ˇY t + Ŷt, 2.5 ˇY t := mi{ t,κ} k= L g k σ t k t k + t s α dw s, t k 3
14 Ŷ t := t k=κ+ g bk σ t k W t k + W t k. I effect, we simply drop the summads i 2.7 ad 2.8 that correspod to itegrals ad icremets o the egative real lie. We make remarks o the implemetatio of this scheme i Sectio 3., below. The MSE of hybrid scheme for the T BSS process Y has the followig asymptotic behavior as, which is, i fact, idetical to the asymptotic behavior of the MSE of the hybrid scheme for BSS processes. We omit the proof of this result, which would be a simple modificatio of the proof of Theorem 2.5. Theorem 2.2 Asymptotics of mea square error. Suppose that A ad A2 hold, ad that for some δ >, E[ σs σ 2 ] = O s 2α++δ, s. The for all t >, E[ Y t Y t 2 ] Jα, κ, be[σ 2 ] 2α+ L g / 2,, where Jα, κ, b is as i Theorem 2.5 Remark 2.3. Uder the assumptios of Theorem 2.2, the coclusio of Corollary 2.7 holds mutatis mutadis. I particular, the covariace structure of the discretized T BSS process approaches that of Y whe. 3 Implemetatio ad umerical experimets 3. Practical implemetatio Simulatig the BSS process X o the equidistat grid {,, 2,..., T } for some T > usig the hybrid scheme etails geeratig i X, i =,,..., T. 3. Provided that we ca simulate the radom variables i+ Wi,j i + j := i W i := σ i i+ i i := σ α s dw s, i = N, N +,..., T, j =,..., κ, 3.2 dw s, i = N, N +,..., T, 3.3, i = N, N +,..., T, 4
15 we ca compute 3. via the formula i κ k X = L g σi k W N b i k,k + g k σi k W i k. 3.4 k= k=κ+ }{{}}{{} = ˇX i = ˆX i I order to simulate 3.2 ad 3.3, it is istrumetal to ote that the κ + dimesioal radom vectors W i := W i, W i,,..., W i,κ, i = N, N +,..., T, are i.i.d. accordig to a multivariate Gaussia distributio with mea zero ad covariace matrix Σ give by Σ, =, Σ,j = Σ j, = j α+ j 2 α+ α + α+, Σ j,j = j 2α+ j 2 2α+ 2α + 2α+, for j = 2,..., κ +, ad Σ j,k = j α + 2α+ α+ k α2f α,, α + 2, j k j 2 α+ k 2 α 2F α,, α + 2, j 2, 3.5 k 2 for j, k = 2,..., κ + such that j < k, where 2 F stads for the Gauss hypergeometric fuctio, see, e.g., [7, p. 56] for the defiitio. Whe k < j, set Σ j,k = Σ k,j. For the coveiece of the reader, we provide a proof of 3.5 i Sectio 4.3. Thus, {Wi } T i= N ca be geerated by takig idepedet draws from the multivariate Gaussia distributio N κ+, Σ. If the volatility process σ is idepedet of W, the {σi } T i= N ca be geerated separately, possibly usig exact methods. Exact methods are available, e.g., for Gaussia processes, as metioed i the itroductio, ad diffusios, see [4]. I the case where σ depeds o W, simulatig {Wi } T i= N ad {σi } T i= N is less straightforward. That said, if σ is drive by a stadard Browia motio Z, correlated with W, say, oe could rely o a factor decompositio Zt := ρw t + ρ 2 W t, t R, 3.6 where ρ [, ] is the correlatio parameter ad {W t} t [,T ] is a stadard Browia motio idepedet of W. The oe would first geerate {Wi } T i= N, use 3.6 to geerate {Z i+ Z i } T i= N } T i= N thereafter. ad employ some appropriate approximate method to produce {σi This approach has, however, the caveat that it iduces a additioal approximatio error, ot quatified i Theorem 2.5. Remark 3.. I the case of the T BSS process Y, itroduced i Sectio 2.5, the observatios Y i, i =,,..., T, give by the hybrid scheme 2.5 ca be computed via i Y = mi{i,κ} k= L g k σ i k W i k,k + i k=κ+ 5 b g k σi k W i k, 3.7
16 usig the radom vectors {Wi } T i= ad radom variables {σi } T i=. I the hybrid scheme, it typically suffices to take κ to be at most 3. Thus, i 3.4, the first sum ˇX i requires oly a egligible computatioal effort. By cotrast, the umber of terms i the secod sum ˆX i icreases as. It is the useful to ote that i N ˆX = Γ k Ξ i k = Γ Ξ i, k= where, k =,..., κ, Γ k := g b k, k = κ +, κ + 2,..., N, Ξ k := σ k W k, k = N, N +,..., T. ad Γ Ξ stads for the discrete covolutio of the sequeces Γ ad Ξ. It is wellkow that the discrete covolutio ca be evaluated efficietly usig a fast Fourier trasform FFT. The computatioal complexity of simultaeously evaluatig Γ Ξ i for all i =,,..., T usig a FFT is ON log N, see [23, pp. 79 8], which uder A4 traslates to O γ+ log. The computatioal complexity of the etire hybrid scheme is the O γ+ log, provided that {σi } T i= N is geerated usig a scheme with complexity ot exceedig O γ+ log. As a compariso, we metio that the complexity of a exact simulatio of a statioary Gaussia process usig circulat embeddigs is O log [2, p. 36], whereas the complexity of the Cholesky factorizatio is O 3 [2, p. 32]. Remark 3.2. With T BSS processes, the computatioal complexity of the hybrid scheme via 3.7 is O log. Figure 2 presets examples of trajectories of the BSS process X usig the hybrid scheme with κ =, 2 ad b = b. We choose the kerel fuctio g to be the gamma kerel Example 2.3 with λ =. We also discretize X usig the Riemasum scheme, κ = with b {b FWD, b } that is, the forward Riemasum scheme ad its couterpart with optimally chose evaluatio poits. We ca make two observatios: Firstly, we see how the roughess parameter α cotrols the regularity properties of the trajectories of X as we decrease α, the trajectories of X become icreasigly rough. Secodly, ad more importatly, we see how the simulated trajectories comig from the Riemasum ad hybrid schemes ca be rather differet, eve though we use the same iovatios for the drivig Browia motio. I fact, the two variats of the hybrid scheme κ =, 2 yield almost idetical trajectories, while the Riemasum scheme κ = produces trajectories that are comparatively smoother, this differece becomig more apparet as α approaches 2. Ideed, i the extreme case with α =.499, both variats of the Riemasum scheme break dow ad yield aomalous trajectories with very little variatio, while the hybrid scheme cotiues to produce accurate results. The fact that the hybrid scheme is able to reproduce the fie properties of rough BSS processes, eve for values of α very close to 2, is backed up by a further experimet reported i the followig sectio. 6
17 3 α=.5 2 Xt κ = κ = κ = t 3 α=.45 2 Xt t 3 α= Xt t Figure 2: Discretized trajectories of a BSS process, where g is the gamma kerel Example 2.3, λ = ad σt = for all t R. Trajectories cosistig of = 5 observatios o [, ] were geerated with the hybrid scheme κ =, 2 ad b = b ad Riemasum scheme κ = ad b = b solid lies, b = b FWD dashed lies, usig the same iovatios for the drivig Browia motio i all cases ad N = 5.5 = 353. The simulated processes were ormalized to have uit statioary variace. 7
18 3.2 Estimatio of the roughess parameter Suppose that we have observatios X i m, i =,,..., m, of the BSS process X, give by 2., for some m N. BardorffNielse et al. [6] ad Corcuera et al. [6] discuss how the roughess idex α ca be estimated cosistetly as m. The method is based o the chageoffrequecy COF statistics COFX, m = m k=5 m k=3 X k m 2X k 2 m + X k 4 2 m X k m 2X k m + X k 2 2, m 5, m which compare the realized quadratic variatios of X, usig secodorder icremets, with two differet lag legths. Corcuera et al. [6] have show that uder some assumptios o the process X, which are similar to A, A2 ad A3 albeit slightly more restrictive, it holds that ˆαX, m := log COFX, m P α, m log 2 2 A idepth study of the fiite sample performace of this COF estimator ca be foud i [2]. To examie how well the hybrid scheme reproduces the fie properties of the BSS process i terms of regularity/roughess, we apply the COF estimator to discretized trajectories of X, where the kerel fuctio g is agai the gamma kerel Example 2.3 with λ =, geerated usig the hybrid scheme with κ =, 2, 3 ad b = b. We cosider the case where the volatility process satisfies σt =, that is, the process X is Gaussia. This allows us to quatify ad cotrol for the itrisic bias ad oisiess, measured i terms of stadard deviatio, of the estimatio method itself, by iitially applyig the estimator to trajectories that have bee simulated usig a exact method based o the Cholesky factorizatio. We the study the behavior of the estimator whe applied to a discretized trajectory, while decreasig the step size of the discretizatio scheme. More precisely, we simulate ˆαX, m, where m = 5 ad X is the hybrid scheme for X with = ms ad s {, 2, 5}. This meas that we compute ˆαX, m usig m observatios obtaied by subsamplig every sth observatio i the sequece X i, i =,,...,. As a compariso, we repeat these simulatios substitutig the hybrid scheme with the Riemasum scheme, usig κ = with b {b FWD, b }. The results are preseted i Figure 3. We observe that the itrisic bias of the estimator with m = 5 observatios is egligible ad hece the bias of the estimates computed from discretized trajectories is the attributable to approximatio error arisig from the respective discretizatio scheme, where positive resp. egative bias idicates that the simulated trajectories are smoother resp. rougher tha those of the process X. Cocetratig first o the baselie case s =, we ote that the hybrid scheme produces essetially ubiased results whe α 2,, while there is moderate bias whe α, 2, which disappears whe passig from κ = to κ = 3, eve for values of α very close to 2. The largest value of α cosidered i our simulatios is α =.49; oe would expect the performace to weake as α approaches 2, cf. Figure, but this rage of parameter values seems to be of limited practical iterest. The stadard deviatios exhibit a similar patter. The correspodig results for the Riemasum scheme are clearly iferior, exhibitig sigificat bias, while usig optimal evaluatio poits b = b improves the situatio slightly. I particular, 8
19 Bias.4.2 s = exact κ =.2 κ = κ = 2 κ = α.2.4 s =2.4 Stadard deviatio s = exact κ = κ = κ = 2 κ = α.2.4 s =2 Bias.2.2 Stadard deviatio α.2.4 s = α.2.4 s =5 Bias.2.2 Stadard deviatio α α Figure 3: Bias ad stadard deviatio of the COF estimator 3.8 of the roughess idex α, whe applied to discretized trajectories of a BSS process with the gamma kerel Example 2.3, λ = ad σt = for all t R. Trajectories were geerated usig a exact method based o the Cholesky factorizatio, the hybrid scheme κ =, 2, 3 ad b = b ad Riemasum scheme κ = ad b = b solid lies, b = b FWD dashed lies. I the experimet, = ms observatios were geerated, where m = 5 ad s {, 2, 5}, o [, ] usig N =.5. Every sth observatio was the subsampled, resultig i m = 5 observatios that were used to compute the estimate ˆαX, m of the roughess idex α. Number of Mote Carlo replicatios:. 9
20 the bias i the case α 2, is positive, idicatig too smooth discretized trajectories, which is coected with the failure of the Riemasum scheme with α ear 2, illustrated i Figure 2. With s = 2 ad s = 5, the results improve with both schemes. Notably, i the case s = 5, the performace of the hybrid scheme eve with κ = is o a par with the exact method. However, the improvemets with the Riemasum scheme are more meager, as cosiderable bias persists whe α is ear Optio pricig uder rough volatility As aother experimet, we study Mote Carlo optio pricig i the rough Bergomi rbergomi model of Bayer et al. []. I the rbergomi model, the logarithmic spot variace of the price of the uderlyig is modelled by a rough Gaussia process, which is a special case of 2.4. By virtue of the rough volatility process, the model fits well to observed implied volatility smiles [, pp ]. More precisely, the price of the uderlyig i the rbergomi model with time horizo T > is defied, uder a equivalet martigale measure idetified with P, as t St := S exp usig the spot variace process vt := ξ t exp η 2α + vsdzs 2 t t t s α dw s }{{} =:Y t vsds, t [, T ], η2 2 t2α+, t [, T ]. Above, S >, η > ad α 2, are determiistic parameters, ad Z is a stadard Browia motio give by Zt := ρw t + ρ 2 W t, t [, T ], 3.9 where ρ, is the correlatio parameter ad {W t} t [,T ] is a stadard Browia motio idepedet of W. The process {ξ t} t [,T ] is the socalled forward variace curve [, p. 89], which we assume here to be flat, ξ t = ξ > for all t [, T ]. We aim to compute usig Mote Carlo simulatio the price of a Europea call optio struck at K > with maturity T, which is give by CS, K, T := E[ST K + ]. 3. The approach suggested by Bayer et al. [] ivolves samplig the Gaussia processes Z ad Y o a discrete time grid usig exact simulatio ad the approximatig S ad v usig Euler discretizatio. We modify this approach by usig the hybrid scheme to simulate Y, istead of the computatioally more costly exact simulatio. As the hybrid scheme ivolves simulatig icremets of the Browia motio W drivig Y, we ca coveietly simulate the icremets of Z, eeded for the Euler discretizatio of S, usig the represetatio
21 Table : Parameter values used i the rbergomi model. S ξ η α ρ T =.4 exact κ = κ = κ = T = exact κ = κ = κ = 2 IVk,T.4 IVk,T k.5.5 k Figure 4: Implied volatility smiles correspodig to the optio price 3., computed usig Mote Carlo simulatio 5 time steps, replicatios, with two maturities: T =.4 left ad T = right. The spot variace process v was simulated usig a exact method, the hybrid scheme κ =, 2 ad b = b ad Riemasum scheme κ = ad b = b solid lies, b = b FWD dashed lies. The parameter values used i the rbergomi model are give i Table. We map the optio price CS, K, T to the correspodig Black Scholes implied volatility IVS, K, T, see, e.g., [9]. Reparameterizig the implied volatility usig the logstrike k := logk/s allows us to drop the depedece o the iitial price, so we will abuse otatio slightly ad write IVk, T for the correspodig implied volatility. Figure 4 displays implied volatility smiles obtaied from the rbergomi model usig the hybrid ad Riemasum schemes to simulate Y, as discussed above, ad compares these to the smiles obtaied usig a exact simulatio of Y via Cholesky factorizatio. The parameter values are give i Table. They have bee adopted from Bayer et al. [], who demostrate that they result i realistic volatility smiles. We cosider two differet maturities: short, T =.4, ad log, T =. We observe that the Riemasum scheme κ =, b {b FWD, b } is able capture the shape of the implied volatility smile, but ot its level. Alas, the method eve breaks dow with more extreme logstrikes the prices are so low that the rootfidig algorithm used to compute the implied volatility would retur zero. I cotrast, the hybrid scheme with κ =, 2 ad b = b yields implied volatility smiles that are idistiguishable from the bechmark smiles obtaied usig exact simulatio. Further, there is o discerible differece betwee the smiles obtaied usig κ = ad κ = 2. As i the previous sectio, we observe that the hybrid scheme is ideed capable of producig very accurate trajectories of T BSS processes, i particular i the case α 2,, eve whe κ =. 2
22 4 Proofs Throughout the proofs below, we rely o two useful iequalities. The first oe is the Potter boud for slow variatio at, which follows immediately from the correspodig result for slow variatio at [5, Theorem.5.6]. Namely, if L :, ], is slowly varyig at ad bouded away from ad o ay iterval u, ], u,, the for ay δ > there exists a costat C δ > such that { Lx x Ly C δ max y δ, x y The secod oe is the elemetary iequality } δ, x, y, ]. 4. x α y α α mi{x, y} α x y, x, y,, α,, 4.2 which ca be easily show usig the mea value theorem. Additioally, we use the followig variat of Karamata s theorem for regular variatio at. Its proof is similar to the oe of the usual Karamata s theorem for regular variatio at [5, Propositio.5.]. Lemma 4. Karamata s theorem. If α, ad L :, ] [, is slowly varyig at, the y x α Lxdx α + yα+ Ly, y. 4. Proof of Propositio 2.2 Proof of Propositio 2.2. i By the covariace statioarity of the volatility process σ, we may express the variogram V h for ay h as V h = E[ Xh X 2 ] = h gh u g u, u 2 E[σu 2 ]du h = E[σ 2 ] gx 2 dx + gx + h gx 2 dx. 4.3 Ivokig A ad Lemma 4., we fid that h gx 2 dx 2α + h2α+ L g h 2, h. 4.4 We may clearly assume that h <, which allows us to work with the decompositio where gx + h gx 2 dx = A h + A h, A h := h gx + h gx 2 dx, A h := gx + h gx 2 dx. h 22
23 Accordig to A2, there exists M > such that x g x is oicreasig o [M,. Thus, usig the mea value theorem, we deduce that gx + h gx = g sup y h,m] g y h, x h, M, ξ h g x h, x [M,. where ξ = ξx, h [x, x + h]. It follows the that lim sup h which i tur implies that A h M sup g y 2 + h2 y [,M] g x 2 dx <, A h = Oh2, h. 4.5 where Makig a substitutio y = x h, we obtai A h = h G h y := gx + h gx 2 dx = h /h ghy + ghy 2dy = h 2α+ L g h 2 G h ydy, y + α L ghy + y α L 2 ghy L g h L g h,/h y, y,. By the defiitio of slow variatio at, lim G hy = y + α y α 2, h y,. We shall show below that the fuctios G h, h,, have a itegrable domiat. Thus, by the domiated covergece theorem, A h h 2α+ L g h 2 y + α y α 2 dy, h. 4.6 Sice α < 2, we have lim h h gx 2 dx + A h h 2α+ L gh 2 = by 2.2 ad 4.5, so we get from 4.4 ad 4.6 gx + h gx 2 dx 2α + + y + α y α 2 dy h 2α+ L g h 2, h, which, together with 4.3, implies the assertio. It remais to justify the use of the domiated covergece theorem to deduce 4.6. For ay y, ], we have by the Potter boud 4. ad the elemetary iequality u + v 2 2u 2 + 2v 2, G h y 2y + 2α Lg hy + L g h 2C 2 δ y + 2α+δ + y 2α δ, 2 + 2y 2α Lg hy L g h 23 2
24 where we choose δ, α+ 2 to esure that 2α δ >. Cosider the y [,. By addig ad substractig the term y + α L ghy L gh ad usig agai the iequality u + v 2 2u 2 + 2v 2, we get G h y = y + α L ghy + y + α L ghy L g h L g h + y + α L ghy L g h L 2 ghy yα L g h,/h y 2y + 2α Lg hy + L g hy 2 L g h,/h y + 2 y + α y α 2 Lg hy 2 L g h,/h y. We recall that L g := if x,] L g x > by A, so L g hy + L g hy L g h,/h y L g hy + L g hy L,/h y. g Usig the mea value theorem ad the boud for the derivative of L g from A, we observe that L g hy + L g hy = L gξ hy + hy hc + C h +, ξ y where ξ = ξy, h [hy, hy + ]. Notig that the costrait y < h we obtai further L g hy + L g hy L g h,/h y C h+ L g y is equivalet to h < y+,,/h y C L g y + + C 3 y L g y +, as y, which we the use to deduce that 2y + 2α Lg hy + L g hy 2 L g h,/h y 8C2 y + 2α. Additioally, we observe that, by 4. ad 4.2, 2 y + α y α 2 Lg hy 2 L g h,/h y 2Cδ 2 2 α 2 y 2α +δ2, where we choose δ 2, 2 α, esurig that 2α +δ 2 <. We may fially defie a fuctio 2C 2 δ Gy := y + 2α+δ + y 2α δ, y, ], 8C 2 y + 2α + 2C L 2 δ 2 g 2 α 2 y 2α +δ2, y,, which satisfies G h y Gy for ay y, ad h,, ad is itegrable o, with the aforemetioed choices of δ ad δ 2. ii To show existece of the modificatio, we eed a localizatio procedure that ivolves a acillary process F t := g s 2 σt s 2 ds, t R. 24 L 2 g
25 We check first that F is locally bouded uder A ad A2, which is essetial for localizatio. To this ed, let T,, ad write for ay t [ T, T ], where F t = F t + F t, F t := M+2T g s 2 σt s 2 ds, F t := M+2T g s 2 σt s 2 ds, ad M > is such that x g x is oicreasig o [M,, as i the proof of i. Sice g is cotiuous o, ad σ locally bouded, we have for ay t [ T, T ], F t M + 2T sup g y 2 y [,M+2T ] Further, whe t [ T, T ], F t = t M+2T g t u 2 σu 2 du, sup u [ M 3T,T ] where g t u 2 g T u 2 sice the argumets satisfy Thus, t u T u T t M + 2T M. F t M+T g T u 2 σu 2 du σu 2 <. g s 2 σ T s 2 ds < for ay t [ T, T ] almost surely, as we have [ ] E g s 2 σ T s 2 ds = g s 2 E[σ T s 2 ]ds = E[σ 2 ] g s 2 ds <, where we chage the order of expectatio ad itegratio relyig o Toelli s theorem ad where the fial equality follows from the covariace statioarity of σ. So we ca coclude that F is ideed locally bouded. Let ow m N ad, for localizatio, defie a sequece of stoppig times τ m, := if{t [ m, : F t > or σt > }, N, that satisfies τ m, almost surely as sice both F ad σ are locally bouded. We follow the usual covetio that if =. Cosider ow the modified BSS process X m,t := t gt sσmi{s, τ m, }dw s, t [ m,, that coicides with X o the stochastic iterval m, τ m,. The process X m, satisfies the assumptios of [5, Lemma ], so for ay p > there exists a costat Ĉp > such that E[ X m,s X m,t p ] ĈpV s t p/2, s, t [ m,
26 Usig the upper boud i 2.2, we ca deduce from i that for ay δ > there are costats C δ > ad h δ > such that V h C δ h 2α+ δ, h, h δ. 4.8 Applyig 4.8 to 4.7, we get E[ X m,s X m,t p ] Ĉp C p/2 δ s t +pα+ 2 δ 2 p, s, t [ m,, s t < h δ. We may ote that pα + 2 δ 2 p > for small eough δ ad large eough p ad, i particular, pα + 2 δ 2 p α + p 2, as δ ad p. Thus it follows from the Kolmogorov Chetsov theorem [22, Theorem 3.22] that X m, has a modificatio with locally φhölder cotiuous trajectories for ay φ, α + 2. Moreover, a modificatio of X o R, havig locally φhölder cotiuous trajectories for ay φ, α + 2, ca the by costructed from these modificatios of X m,, m N, N, by lettig first ad the m. 4.2 Proof of Theorem 2.5 As a preparatio, we shall first establish a auxiliary result that deals with the asymptotic behavior of certai itegrals of regularly varyig fuctios. Lemma 4.2. Suppose that L :, ] [, is bouded away from ad o ay set of the form u, ], u,, ad slowly varyig at. Moreover, let α 2, ad k. If b [k, k] \ {}, the i lim k x α Lx/ Lb/ bα L/ L/ 2 dx = x α b α 2 dx <, k Lx/ ii lim x 2α k L/ Lb/ 2 dx =. L/ Proof. We oly prove i as ii ca be show similarly. By the defiitio of slow variatio at, the fuctio f x := x α Lx/ Lb/ 2 bα, x [k, k] \ {}, L/ L/ satisfies lim f x = x α b α 2 for ay x [k, k]\{}. I view of the domiated covergece theorem, it suffices to fid a itegrable domiat for the fuctios f, N. The costructio of the domiat is quite similar to the oe see i the proof of Propositio 2.2, but we provide the details for the coveiece of the reader. Usig the Potter boud 4. ad the iequality u + v 2 2u 2 + 2v 2, we fid that for ay x [k, k] \ {}, Lx/ f x 2x 2α L/ 2C 2 δ 2 Lb/ 2 + 2b 2α L/ x 2α max { x δ, x δ} 2 + b 2α max { b δ, b δ} 2 =: fx, 26
27 where we choose δ, α + 2. Whe k 2, we have x ad b, so fx = 2Cδ 2 x 2α+δ + b 2α+δ is a bouded fuctio of x o [k, k]. Whe k =, we have x ad b, implyig that fx = 2C 2 δ x 2α δ + b 2α δ, where 2α δ > with our choice of δ, so f is a itegrable fuctio o, ]. Proof of Theorem 2.5. Let t R be fixed. It will be coveiet to write X t as X t = κ t k k= t k N t k + k=κ+ t k t s α L g k g bk σ t k dw s σ t k dw s. Moreover, we itroduce a acillary approximatio of Xt, amely, N X t = k= t k t k By Mikowski s iequality, we have gt sσ t k t N dw s + gt sσsdw s. E [ X t Xt 2] 2 E [ X t X t 2] 2 E [ X t Xt 2] 2, E [ X t Xt 2] 2 E [ X t X t 2] 2 + E [ X t Xt 2] 2, which together, after takig squares, imply that E E 2 + E E [ X t Xt 2] E E E, 4.9 E E E E where E := E [ X t X t 2], E := E [ Xt X t 2]. Usig the Itō isometry, ad recallig that σ is covariace statioary, we obtai N E = k= t k t k [ gt s 2 E σ t k 2 ] σs ds sup E [ σu σ 2] u, ] gs 2 ds 27
Rafa l Kulik and Marc Raimondo. University of Ottawa and University of Sydney. Supplementary material
Statistica Siica 009: Supplemet 1 L p WAVELET REGRESSION WITH CORRELATED ERRORS AND INVERSE PROBLEMS Rafa l Kulik ad Marc Raimodo Uiversity of Ottawa ad Uiversity of Sydey Supplemetary material This ote
More information1 Estimating sensitivities
Copyright c 27 by Karl Sigma 1 Estimatig sesitivities Whe estimatig the Greeks, such as the, the geeral problem ivolves a radom variable Y = Y (α) (such as a discouted payoff) that depeds o a parameter
More information5. Best Unbiased Estimators
Best Ubiased Estimators http://www.math.uah.edu/stat/poit/ubiased.xhtml 1 of 7 7/16/2009 6:13 AM Virtual Laboratories > 7. Poit Estimatio > 1 2 3 4 5 6 5. Best Ubiased Estimators Basic Theory Cosider agai
More informationStatistics for Economics & Business
Statistics for Ecoomics & Busiess Cofidece Iterval Estimatio Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for the mea ad the proportio How to determie
More informationpoint estimator a random variable (like P or X) whose values are used to estimate a population parameter
Estimatio We have oted that the pollig problem which attempts to estimate the proportio p of Successes i some populatio ad the measuremet problem which attempts to estimate the mea value µ of some quatity
More information18.S096 Problem Set 5 Fall 2013 Volatility Modeling Due Date: 10/29/2013
18.S096 Problem Set 5 Fall 2013 Volatility Modelig Due Date: 10/29/2013 1. Sample Estimators of Diffusio Process Volatility ad Drift Let {X t } be the price of a fiacial security that follows a geometric
More informationx satisfying all regularity conditions. Then
AMS570.01 Practice Midterm Exam Sprig, 018 Name: ID: Sigature: Istructio: This is a close book exam. You are allowed oepage 8x11 formula sheet (sided). No cellphoe or calculator or computer is allowed.
More informationLecture 9: The law of large numbers and central limit theorem
Lecture 9: The law of large umbers ad cetral limit theorem Theorem.4 Let X,X 2,... be idepedet radom variables with fiite expectatios. (i) (The SLLN). If there is a costat p [,2] such that E X i p i i=
More information0.1 Valuation Formula:
0. Valuatio Formula: 0.. Case of Geeral Trees: q = er S S S 3 S q = er S S 4 S 5 S 4 q 3 = er S 3 S 6 S 7 S 6 Therefore, f (3) = e r [q 3 f (7) + ( q 3 ) f (6)] f () = e r [q f (5) + ( q ) f (4)] = f ()
More informationSampling Distributions and Estimation
Cotets 40 Samplig Distributios ad Estimatio 40.1 Samplig Distributios 40. Iterval Estimatio for the Variace 13 Learig outcomes You will lear about the distributios which are created whe a populatio is
More information14.30 Introduction to Statistical Methods in Economics Spring 2009
MIT OpeCourseWare http://ocwmitedu 430 Itroductio to Statistical Methods i Ecoomics Sprig 009 For iformatio about citig these materials or our Terms of Use, visit: http://ocwmitedu/terms 430 Itroductio
More informationA random variable is a variable whose value is a numerical outcome of a random phenomenon.
The Practice of Statistics, d ed ates, Moore, ad Stares Itroductio We are ofte more iterested i the umber of times a give outcome ca occur tha i the possible outcomes themselves For example, if we toss
More informationLecture 4: Parameter Estimation and Confidence Intervals. GENOME 560 Doug Fowler, GS
Lecture 4: Parameter Estimatio ad Cofidece Itervals GENOME 560 Doug Fowler, GS (dfowler@uw.edu) 1 Review: Probability Distributios Discrete: Biomial distributio Hypergeometric distributio Poisso distributio
More informationSequences and Series
Sequeces ad Series Matt Rosezweig Cotets Sequeces ad Series. Sequeces.................................................. Series....................................................3 Rudi Chapter 3 Exercises........................................
More informationChapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1
Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1 Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for
More informationEstimating Proportions with Confidence
Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter
More informationCombining imperfect data, and an introduction to data assimilation Ross Bannister, NCEO, September 2010
Combiig imperfect data, ad a itroductio to data assimilatio Ross Baister, NCEO, September 00 rbaister@readigacuk The probability desity fuctio (PDF prob that x lies betwee x ad x + dx p (x restrictio o
More informationAsymptotics: Consistency and Delta Method
ad Delta Method MIT 18.655 Dr. Kempthore Sprig 2016 1 MIT 18.655 ad Delta Method Outlie Asymptotics 1 Asymptotics 2 MIT 18.655 ad Delta Method Cosistecy Asymptotics Statistical Estimatio Problem X 1,...,
More informationUnbiased estimators Estimators
19 Ubiased estimators I Chapter 17 we saw that a dataset ca be modeled as a realizatio of a radom sample from a probability distributio ad that quatities of iterest correspod to features of the model distributio.
More informationMonetary Economics: Problem Set #5 Solutions
Moetary Ecoomics oblem Set #5 Moetary Ecoomics: oblem Set #5 Solutios This problem set is marked out of 1 poits. The weight give to each part is idicated below. Please cotact me asap if you have ay questios.
More informationSUPPLEMENTAL MATERIAL
A SULEMENTAL MATERIAL Theorem (Expert pseudoregret upper boud. Let us cosider a istace of the ISG problem ad apply the FL algorithm, where each possible profile A is a expert ad receives, at roud, a
More informationThe Valuation of the Catastrophe Equity Puts with Jump Risks
The Valuatio of the Catastrophe Equity Puts with Jump Risks ShihKuei Li Natioal Uiversity of Kaohsiug Joit work with ChiaChie Chag Outlie Catastrophe Isurace Products Literatures ad Motivatios Jump Risk
More informationAMS Portfolio Theory and Capital Markets
AMS 69.0  Portfolio Theory ad Capital Markets I Class 6  Asset yamics Robert J. Frey Research Professor Stoy Brook iversity, Applied Mathematics ad Statistics frey@ams.suysb.edu http://www.ams.suysb.edu/~frey/
More informationWe analyze the computational problem of estimating financial risk in a nested simulation. In this approach,
MANAGEMENT SCIENCE Vol. 57, No. 6, Jue 2011, pp. 1172 1194 iss 00251909 eiss 15265501 11 5706 1172 doi 10.1287/msc.1110.1330 2011 INFORMS Efficiet Risk Estimatio via Nested Sequetial Simulatio Mark Broadie
More informationAn Empirical Study of the Behaviour of the Sample Kurtosis in Samples from Symmetric Stable Distributions
A Empirical Study of the Behaviour of the Sample Kurtosis i Samples from Symmetric Stable Distributios J. Marti va Zyl Departmet of Actuarial Sciece ad Mathematical Statistics, Uiversity of the Free State,
More information1 Random Variables and Key Statistics
Review of Statistics 1 Radom Variables ad Key Statistics Radom Variable: A radom variable is a variable that takes o differet umerical values from a sample space determied by chace (probability distributio,
More information. (The calculated sample mean is symbolized by x.)
Stat 40, sectio 5.4 The Cetral Limit Theorem otes by Tim Pilachowski If you have t doe it yet, go to the Stat 40 page ad dowload the hadout 5.4 supplemet Cetral Limit Theorem. The homework (both practice
More informationOnline appendices from The xva Challenge by Jon Gregory. APPENDIX 10A: Exposure and swaption analogy.
APPENDIX 10A: Exposure ad swaptio aalogy. Sorese ad Bollier (1994), effectively calculate the CVA of a swap positio ad show this ca be writte as: CVA swap = LGD V swaptio (t; t i, T) PD(t i 1, t i ). i=1
More informationSubject CT1 Financial Mathematics Core Technical Syllabus
Subject CT1 Fiacial Mathematics Core Techical Syllabus for the 2018 exams 1 Jue 2017 Subject CT1 Fiacial Mathematics Core Techical Aim The aim of the Fiacial Mathematics subject is to provide a groudig
More informationMaximum Empirical Likelihood Estimation (MELE)
Maximum Empirical Likelihood Estimatio (MELE Natha Smooha Abstract Estimatio of Stadard Liear Model  Maximum Empirical Likelihood Estimator: Combiatio of the idea of imum likelihood method of momets,
More informationMinhyun Yoo, Darae Jeong, Seungsuk Seo, and Junseok Kim
Hoam Mathematical J. 37 (15), No. 4, pp. 441 455 http://dx.doi.org/1.5831/hmj.15.37.4.441 A COMPARISON STUDY OF EXPLICIT AND IMPLICIT NUMERICAL METHODS FOR THE EQUITYLINKED SECURITIES Mihyu Yoo, Darae
More informationThe material in this chapter is motivated by Experiment 9.
Chapter 5 Optimal Auctios The material i this chapter is motivated by Experimet 9. We wish to aalyze the decisio of a seller who sets a reserve price whe auctioig off a item to a group of bidders. We begi
More informationINTERVAL GAMES. and player 2 selects 1, then player 2 would give player 1 a payoff of, 1) = 0.
INTERVAL GAMES ANTHONY MENDES Let I ad I 2 be itervals of real umbers. A iterval game is played i this way: player secretly selects x I ad player 2 secretly ad idepedetly selects y I 2. After x ad y are
More informationr i = a i + b i f b i = Cov[r i, f] The only parameters to be estimated for this model are a i 's, b i 's, σe 2 i
The iformatio required by the meavariace approach is substatial whe the umber of assets is large; there are mea values, variaces, ad )/2 covariaces  a total of 2 + )/2 parameters. Siglefactor model:
More informationLecture 4: Probability (continued)
Lecture 4: Probability (cotiued) Desity Curves We ve defied probabilities for discrete variables (such as coi tossig). Probabilities for cotiuous or measuremet variables also are evaluated usig relative
More informationSolutions to Problem Sheet 1
Solutios to Problem Sheet ) Use Theorem.4 to prove that p log for all real x 3. This is a versio of Theorem.4 with the iteger N replaced by the real x. Hit Give x 3 let N = [x], the largest iteger x. The,
More informationBinomial Model. Stock Price Dynamics. The Key Idea Riskless Hedge
Biomial Model Stock Price Dyamics The value of a optio at maturity depeds o the price of the uderlyig stock at maturity. The value of the optio today depeds o the expected value of the optio at maturity
More informationFINM6900 Finance Theory How Is Asymmetric Information Reflected in Asset Prices?
FINM6900 Fiace Theory How Is Asymmetric Iformatio Reflected i Asset Prices? February 3, 2012 Referece S. Grossma, O the Efficiecy of Competitive Stock Markets where Traders Have Diverse iformatio, Joural
More informationIntroduction to Probability and Statistics Chapter 7
Itroductio to Probability ad Statistics Chapter 7 Ammar M. Sarha, asarha@mathstat.dal.ca Departmet of Mathematics ad Statistics, Dalhousie Uiversity Fall Semester 008 Chapter 7 Statistical Itervals Based
More informationStandard Deviations for Normal Sampling Distributions are: For proportions For means _
Sectio 9.2 Cofidece Itervals for Proportios We will lear to use a sample to say somethig about the world at large. This process (statistical iferece) is based o our uderstadig of samplig models, ad will
More informationModels of Asset Pricing
4 Appedix 1 to Chapter Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
More informationStochastic Processes and their Applications in Financial Pricing
Stochastic Processes ad their Applicatios i Fiacial Pricig Adrew Shi Jue 3, 1 Cotets 1 Itroductio Termiology.1 Fiacial.............................................. Stochastics............................................
More informationMath 312, Intro. to Real Analysis: Homework #4 Solutions
Math 3, Itro. to Real Aalysis: Homework #4 Solutios Stephe G. Simpso Moday, March, 009 The assigmet cosists of Exercises 0.6, 0.8, 0.0,.,.3,.6,.0,.,. i the Ross textbook. Each problem couts 0 poits. 0.6.
More informationCAPITAL ASSET PRICING MODEL
CAPITAL ASSET PRICING MODEL RETURN. Retur i respect of a observatio is give by the followig formula R = (P P 0 ) + D P 0 Where R = Retur from the ivestmet durig this period P 0 = Curret market price P
More informationLimits of sequences. Contents 1. Introduction 2 2. Some notation for sequences The behaviour of infinite sequences 3
Limits of sequeces I this uit, we recall what is meat by a simple sequece, ad itroduce ifiite sequeces. We explai what it meas for two sequeces to be the same, ad what is meat by the th term of a sequece.
More informationModels of Asset Pricing
APPENDIX 1 TO CHAPTER 4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
More informationModels of Asset Pricing
APPENDIX 1 TO CHAPTER4 Models of Asset Pricig I this appedix, we first examie why diversificatio, the holdig of may risky assets i a portfolio, reduces the overall risk a ivestor faces. The we will see
More informationBootstrapping highfrequency jump tests
Bootstrappig highfrequecy jump tests Prosper Dovoo Departmet of Ecoomics, Cocordia Uiversity Sílvia Goçalves Departmet of Ecoomics, Uiversity of Wester Otario Ulrich Houyo CREATES, Departmet of Ecoomics
More informationBayes Estimator for Coefficient of Variation and Inverse Coefficient of Variation for the Normal Distribution
Iteratioal Joural of Statistics ad Systems ISSN 0973675 Volume, Number 4 (07, pp. 773 Research Idia Publicatios http://www.ripublicatio.com Bayes Estimator for Coefficiet of Variatio ad Iverse Coefficiet
More informationA New Constructive Proof of Graham's Theorem and More New Classes of Functionally Complete Functions
A New Costructive Proof of Graham's Theorem ad More New Classes of Fuctioally Complete Fuctios Azhou Yag, Ph.D. Zhuqi Lu, Ph.D. Abstract A valued twovariable truth fuctio is called fuctioally complete,
More informationEXERCISE  BINOMIAL THEOREM
BINOMIAL THOEREM / EXERCISE  BINOMIAL THEOREM LEVEL I SUBJECTIVE QUESTIONS. Expad the followig expressios ad fid the umber of term i the expasio of the expressios. (a) (x + y) 99 (b) ( + a) 9 + ( a) 9
More informationHopscotch and Explicit difference method for solving BlackScholes PDE
Mälardale iversity Fiacial Egieerig Program Aalytical Fiace Semiar Report Hopscotch ad Explicit differece method for solvig BlacScholes PDE Istructor: Ja Röma Team members: A Gog HaiLog Zhao Hog Cui 0
More informationSimulation Efficiency and an Introduction to Variance Reduction Methods
Mote Carlo Simulatio: IEOR E4703 Columbia Uiversity c 2017 by Marti Haugh Simulatio Efficiecy ad a Itroductio to Variace Reductio Methods I these otes we discuss the efficiecy of a MoteCarlo estimator.
More informationSELECTING THE NUMBER OF CHANGEPOINTS IN SEGMENTED LINE REGRESSION
1 SELECTING THE NUMBER OF CHANGEPOINTS IN SEGMENTED LINE REGRESSION HyueJu Kim 1,, Bibig Yu 2, ad Eric J. Feuer 3 1 Syracuse Uiversity, 2 Natioal Istitute of Agig, ad 3 Natioal Cacer Istitute Supplemetary
More informationAnomaly Correction by Optimal Trading Frequency
Aomaly Correctio by Optimal Tradig Frequecy Yiqiao Yi Columbia Uiversity September 9, 206 Abstract Uder the assumptio that security prices follow radom walk, we look at price versus differet movig averages.
More informationAY Term 2 Mock Examination
AY 2067 Term 2 Mock Examiatio Date / Start Time Course Group Istructor 24 March 207 / 2 PM to 3:00 PM QF302 Ivestmet ad Fiacial Data Aalysis G Christopher Tig INSTRUCTIONS TO STUDENTS. This mock examiatio
More informationProceedings of the 5th WSEAS Int. Conf. on SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 1719, 2005 (pp )
Proceedigs of the 5th WSEAS It. Cof. o SIMULATION, MODELING AND OPTIMIZATION, Corfu, Greece, August 79, 005 (pp48849 Realized volatility estimatio: ew simulatio approach ad empirical study results JULIA
More information5 Statistical Inference
5 Statistical Iferece 5.1 Trasitio from Probability Theory to Statistical Iferece 1. We have ow more or less fiished the probability sectio of the course  we ow tur attetio to statistical iferece. I statistical
More informationParametric Density Estimation: Maximum Likelihood Estimation
Parametric Desity stimatio: Maimum Likelihood stimatio C6 Today Itroductio to desity estimatio Maimum Likelihood stimatio Itroducto Bayesia Decisio Theory i previous lectures tells us how to desig a optimal
More informationASYMPTOTIC MEAN SQUARE ERRORS OF VARIANCE ESTIMATORS FOR USTATISTICS AND THEIR EDGEWORTH EXPANSIONS
J. Japa Statist. Soc. Vol. 8 No. 1 1998 1 19 ASYMPTOTIC MEAN SQUARE ERRORS OF VARIANCE ESTIMATORS FOR USTATISTICS AND THEIR EDGEWORTH EXPANSIONS Yoshihiko Maesoo* This paper studies variace estimators
More informationSubject CT5 Contingencies Core Technical. Syllabus. for the 2011 Examinations. The Faculty of Actuaries and Institute of Actuaries.
Subject CT5 Cotigecies Core Techical Syllabus for the 2011 Examiatios 1 Jue 2010 The Faculty of Actuaries ad Istitute of Actuaries Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical
More informationEVEN NUMBERED EXERCISES IN CHAPTER 4
Joh Riley 7 July EVEN NUMBERED EXERCISES IN CHAPTER 4 SECTION 4 Exercise 4: Cost Fuctio of a CobbDouglas firm What is the cost fuctio of a firm with a CobbDouglas productio fuctio? Rather tha miimie
More informationInferential Statistics and Probability a Holistic Approach. Inference Process. Inference Process. Chapter 8 Slides. Maurice Geraghty,
Iferetial Statistics ad Probability a Holistic Approach Chapter 8 Poit Estimatio ad Cofidece Itervals This Course Material by Maurice Geraghty is licesed uder a Creative Commos AttributioShareAlike 4.0
More informationWe learned: $100 cash today is preferred over $100 a year from now
Recap from Last Week Time Value of Moey We leared: $ cash today is preferred over $ a year from ow there is time value of moey i the form of willigess of baks, busiesses, ad people to pay iterest for its
More informationInstitute of Actuaries of India Subject CT5 General Insurance, Life and Health Contingencies
Istitute of Actuaries of Idia Subject CT5 Geeral Isurace, Life ad Health Cotigecies For 2017 Examiatios Aim The aim of the Cotigecies subject is to provide a groudig i the mathematical techiques which
More informationLecture 5: Sampling Distribution
Lecture 5: Samplig Distributio Readigs: Sectios 5.5, 5.6 Itroductio Parameter: describes populatio Statistic: describes the sample; samplig variability Samplig distributio of a statistic: A probability
More informationMixed and Implicit Schemes Implicit Schemes. Exercise: Verify that ρ is unimodular: ρ = 1.
Mixed ad Implicit Schemes 3..4 The leapfrog scheme is stable for the oscillatio equatio ad ustable for the frictio equatio. The Euler forward scheme is stable for the frictio equatio but ustable for the
More informationThese characteristics are expressed in terms of statistical properties which are estimated from the sample data.
0. Key Statistical Measures of Data Four pricipal features which characterize a set of observatios o a radom variable are: (i) the cetral tedecy or the value aroud which all other values are buched, (ii)
More informationExam 1 Spring 2015 Statistics for Applications 3/5/2015
8.443 Exam Sprig 05 Statistics for Applicatios 3/5/05. Log Normal Distributio: A radom variable X follows a Logormal(θ, σ ) distributio if l(x) follows a Normal(θ, σ ) distributio. For the ormal radom
More informationREVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS. Guangwu Liu L. Jeff Hong
Proceedigs of the 2008 Witer Simulatio Coferece S. J. Maso, R. R. Hill, L. Möch, O. Rose, T. Jefferso, J. W. Fowler eds. REVISIT OF STOCHASTIC MESH METHOD FOR PRICING AMERICAN OPTIONS Guagwu Liu L. Jeff
More informationBootstrapping highfrequency jump tests
Bootstrappig highfrequecy jump tests Prosper Dovoo Departmet of Ecoomics, Cocordia Uiversity Sílvia Goçalves Departmet of Ecoomics, McGill Uiversity Ulrich Houyo Departmet of Ecoomics, Uiversity at Albay,
More informationForecasting bad debt losses using clustering algorithms and Markov chains
Forecastig bad debt losses usig clusterig algorithms ad Markov chais Robert J. Till Experia Ltd Lambert House Talbot Street Nottigham NG1 5HF {Robert.Till@uk.experia.com} Abstract Beig able to make accurate
More informationThe Limit of a Sequence (Brief Summary) 1
The Limit of a Sequece (Brief Summary). Defiitio. A real umber L is a it of a sequece of real umbers if every ope iterval cotaiig L cotais all but a fiite umber of terms of the sequece. 2. Claim. A sequece
More informationEconomic Computation and Economic Cybernetics Studies and Research, Issue 2/2016, Vol. 50
Ecoomic Computatio ad Ecoomic Cyberetics Studies ad Research, Issue 2/216, Vol. 5 KyougSook Moo Departmet of Mathematical Fiace Gacho Uiversity, GyeoggiDo, Korea Yuu Jeog Departmet of Mathematics Korea
More informationA point estimate is the value of a statistic that estimates the value of a parameter.
Chapter 9 Estimatig the Value of a Parameter Chapter 9.1 Estimatig a Populatio Proportio Objective A : Poit Estimate A poit estimate is the value of a statistic that estimates the value of a parameter.
More informationAppendix 1 to Chapter 5
Appedix 1 to Chapter 5 Models of Asset Pricig I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy a asset, we are
More informationEQUIVALENCE OF FLOATING AND FIXED STRIKE ASIAN AND LOOKBACK OPTIONS
EQUIVALENCE OF FLOATING AND FIXED STIKE ASIAN AND LOOKBACK OPTIONS ENST EBELEIN AND ANTONIS PAPAPANTOLEON Abstract. We prove a symmetry relatioship betwee floatigstrike ad fixedstrike Asia optios for
More informationEstimation of Population Variance Utilizing Auxiliary Information
Iteratioal Joural of Statistics ad Systems ISSN 0973675 Volume 1, Number (017), pp. 303309 Research Idia Publicatios http://www.ripublicatio.com Estimatio of Populatio Variace Utilizig Auxiliary Iformatio
More informationFourier Transform in L p (R) Spaces, p 1
Ge. Math. Notes, Vol. 3, No., March 20, pp.425 ISSN 229784; Copyright c ICSS Publicatio, 200 www.icsrs.org Available free olie at http://www.gema.i Fourier Trasform i L p () Spaces, p Devedra Kumar
More informationOverlapping Generations
Eco. 53a all 996 C. Sims. troductio Overlappig Geeratios We wat to study how asset markets allow idividuals, motivated by the eed to provide icome for their retiremet years, to fiace capital accumulatio
More informationii. Interval estimation:
1 Types of estimatio: i. Poit estimatio: Example (1) Cosider the sample observatios 17,3,5,1,18,6,16,10 X 8 X i i1 8 17 3 5 118 6 16 10 8 116 8 14.5 14.5 is a poit estimate for usig the estimator X ad
More informationof Asset Pricing R e = expected return
Appedix 1 to Chapter 5 Models of Asset Pricig EXPECTED RETURN I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy
More informationLecture 5 Point Es/mator and Sampling Distribu/on
Lecture 5 Poit Es/mator ad Samplig Distribu/o Fall 03 Prof. Yao Xie, yao.xie@isye.gatech.edu H. Milto Stewart School of Idustrial Systems & Egieerig Georgia Tech Road map Poit Es/ma/o Cofidece Iterval
More informationTopic7. Large Sample Estimation
Topic7 Large Sample Estimatio TYPES OF INFERENCE Ò Estimatio: É Estimatig or predictig the value of the parameter É What is (are) the most likely values of m or p? Ò Hypothesis Testig: É Decidig about
More informationToday: Finish Chapter 9 (Sections 9.6 to 9.8 and 9.9 Lesson 3)
Today: Fiish Chapter 9 (Sectios 9.6 to 9.8 ad 9.9 Lesso 3) ANNOUNCEMENTS: Quiz #7 begis after class today, eds Moday at 3pm. Quiz #8 will begi ext Friday ad ed at 10am Moday (day of fial). There will be
More informationPricing Asian Options: A Comparison of Numerical and Simulation Approaches Twenty Years Later
Joural of Mathematical Fiace, 016, 6, 810841 http://www.scirp.org/joural/jmf ISSN Olie: 1644 ISSN Prit: 16434 Pricig Asia Optios: A Compariso of Numerical ad Simulatio Approaches Twety Years Later Akos
More informationA Bayesian perspective on estimating mean, variance, and standarddeviation from data
Brigham Youg Uiversity BYU ScholarsArchive All Faculty Publicatios 00605 A Bayesia perspective o estimatig mea, variace, ad stadarddeviatio from data Travis E. Oliphat Follow this ad additioal works
More informationNOTES ON ESTIMATION AND CONFIDENCE INTERVALS. 1. Estimation
NOTES ON ESTIMATION AND CONFIDENCE INTERVALS MICHAEL N. KATEHAKIS 1. Estimatio Estimatio is a brach of statistics that deals with estimatig the values of parameters of a uderlyig distributio based o observed/empirical
More informationCAPITAL PROJECT SCREENING AND SELECTION
CAPITAL PROJECT SCREEIG AD SELECTIO Before studyig the three measures of ivestmet attractiveess, we will review a simple method that is commoly used to scree capital ivestmets. Oe of the primary cocers
More informationECON 5350 Class Notes Maximum Likelihood Estimation
ECON 5350 Class Notes Maximum Likelihood Estimatio 1 Maximum Likelihood Estimatio Example #1. Cosider the radom sample {X 1 = 0.5, X 2 = 2.0, X 3 = 10.0, X 4 = 1.5, X 5 = 7.0} geerated from a expoetial
More informationDESCRIPTION OF MATHEMATICAL MODELS USED IN RATING ACTIVITIES
July 2014, Frakfurt am Mai. DESCRIPTION OF MATHEMATICAL MODELS USED IN RATING ACTIVITIES This documet outlies priciples ad key assumptios uderlyig the ratig models ad methodologies of RatigAgetur Expert
More informationFaculdade de Economia da Universidade de Coimbra
Faculdade de Ecoomia da Uiversidade de Coimbra Grupo de Estudos Moetários e Fiaceiros (GEMF) Av. Dias da Silva, 65 3005 COIMBRA, PORTUGAL gemf@fe.uc.pt http://www.uc.pt/feuc/gemf PEDRO GODINHO Estimatig
More informationCHAPTER 8 Estimating with Confidence
CHAPTER 8 Estimatig with Cofidece 8.2 Estimatig a Populatio Proportio The Practice of Statistics, 5th Editio Stares, Tabor, Yates, Moore Bedford Freema Worth Publishers Estimatig a Populatio Proportio
More informationOnline appendices from Counterparty Risk and Credit Value Adjustment a continuing challenge for global financial markets by Jon Gregory
Olie appedices from Couterparty Risk ad Credit Value Adjustmet a APPENDIX 8A: Formulas for EE, PFE ad EPE for a ormal distributio Cosider a ormal distributio with mea (expected future value) ad stadard
More informationof Asset Pricing APPENDIX 1 TO CHAPTER EXPECTED RETURN APPLICATION Expected Return
APPENDIX 1 TO CHAPTER 5 Models of Asset Pricig I Chapter 4, we saw that the retur o a asset (such as a bod) measures how much we gai from holdig that asset. Whe we make a decisio to buy a asset, we are
More informationMODIFICATION OF HOLT S MODEL EXEMPLIFIED BY THE TRANSPORT OF GOODS BY INLAND WATERWAYS TRANSPORT
The publicatio appeared i Szoste R.: Modificatio of Holt s model exemplified by the trasport of goods by ilad waterways trasport, Publishig House of Rzeszow Uiversity of Techology No. 85, Maagemet ad Maretig
More information1 The BlackScholes model
The BlacScholes model. The model setup I the simplest versio of the BlacScholes model the are two assets: a risless asset ba accout or bod)withpriceprocessbt) at timet, adarisyasset stoc) withpriceprocess
More informationNORMALIZATION OF BEURLING GENERALIZED PRIMES WITH RIEMANN HYPOTHESIS
Aales Uiv. Sci. Budapest., Sect. Comp. 39 2013) 459 469 NORMALIZATION OF BEURLING GENERALIZED PRIMES WITH RIEMANN HYPOTHESIS WeBi Zhag Chug Ma Pig) Guagzhou, People s Republic of Chia) Dedicated to Professor
More informationOptimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models
Optimally Thresholded Realized Power Variatios for Lévy Jump Diffusio Models José E. FigueroaLópez Jeffrey Nise March 8, 13 Abstract: Thresholded Realized Power Variatios (TPV) are oe of the most popular
More informationEstimation of integrated volatility of volatility with applications to goodnessoffit testing
Beroulli 21(4, 215, 2393 2418 DOI: 1.315/14BEJ648 arxiv:126.5761v2 [math.st] 29 Sep 215 Estimatio of itegrated volatility of volatility with applicatios to goodessoffit testig MATHIAS VETTER 1 Fakultät
More information