High Frequency最新文献

{"title":"The dixie cup problem and FKG inequality","authors":"Leopold Flatto","doi":"10.1002/hf2.10048","DOIUrl":"10.1002/hf2.10048","url":null,"abstract":"<div>\u0000 \u0000 <p>Let <math>\u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> be the number of purchases required to obtain <math>\u0000 <mi>m</mi></math> copies of <math>\u0000 <mi>n</mi></math> given items, each purchase choosing at random one of the <math>\u0000 <mi>n</mi></math> items. <math>\u0000 <mrow>\u0000 <msub>\u0000 <mi>E</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is the expected value of the random variable <math>\u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math>. The problem of obtaining a formula for <math>\u0000 <mrow>\u0000 <msub>\u0000 <mi>E</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is known as the dixie cup problem. The problem is easy for <math>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow></math>, but difficult for <i>m</i> &gt; 1. Newman and Shepp solve the problem for all <i>m</i>,<i> n</i>. From the formula, they obtain the asymptotics of <math>\u0000 <mrow>\u0000 <msub>\u0000 <mi>E</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> for each fixed <math>\u0000 <mi>m</mi></math> and <math>\u0000 <mi>n</mi></math> tending to infinity. Later, Erdös and Rényi obtain the limit law for <math>\u0000 <mrow>\u0000 <msub>\u0000 <mi>T</mi>\u0000 <mi>m</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>n</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math>, for each fixed <math>\u0000 <mi>m</mi></math> and <math>\u0000 <mi>n</mi></math> tending to infinity. From the limit law, they also derive and improve on the result of New","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 3-4","pages":"169-174"},"PeriodicalIF":0.0,"publicationDate":"2020-01-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10048","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74148417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Market making under a weakly consistent limit order book model","authors":"Baron Law,&nbsp;Frederi Viens","doi":"10.1002/hf2.10050","DOIUrl":"10.1002/hf2.10050","url":null,"abstract":"<p>We develop a new market-making model, from the ground up, which is tailored toward high-frequency trading under a limit order book (LOB), based on the well-known classification of order types in market microstructure. Our flexible framework allows arbitrary order volume, price jump, and bid-ask spread distributions as well as the use of market orders. It also honors the consistency of price movements upon arrivals of different order types. For example, it is apparent that prices should never go down on buy market orders. In addition, it respects the price-time priority of LOB. In contrast to the approach of regular control on diffusion as in the classical Avellaneda and Stoikov (Quantitative Finance, 8, 217, 2008) market-making framework, we exploit the techniques of optimal switching and impulse control on marked point processes, which have proven to be very effective in modeling the order book features. The Hamilton-Jacobi-Bellman quasi-variational inequality (HJBQVI) associated with the control problem can be solved numerically via finite-difference method. We illustrate our optimal trading strategy with a full numerical analysis, calibrated to the order book statistics of a popular exchanged-traded fund (ETF). Our simulation shows that the profit of market-making can be severely overstated under LOBs with inconsistent price movements.</p>","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 3-4","pages":"215-238"},"PeriodicalIF":0.0,"publicationDate":"2020-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10050","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76884403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Barndorff-Nielsen and Shephard model for hedging energy with quantity risk","authors":"William Wilson,&nbsp;William Nganje,&nbsp;Semere Gebresilasie,&nbsp;Indranil SenGupta","doi":"10.1002/hf2.10049","DOIUrl":"10.1002/hf2.10049","url":null,"abstract":"<p>In this paper, the Barndorff-Nielsen and Shephard (BN-S) model is implemented to find an optimal hedging strategy in the presence of quantity risk for oil produced in the Bakken, a new region of oil extraction that is benefiting from fracking technology. Hedging and price risk management become much more involved with the inclusion of quantity risk. Explorers and drillers have uncertainty on the quantity of oil that would be extracted, and governments have uncertainty of the quantity of oil that will be extracted, sold, and available for imposing tax regimes. One of the main assumptions typically made in a portfolio model of hedging is that the quantity of inventory or demand is known. This is inappropriate in many hedging situations. Quantity risk compounds the difficulty of determining the optimal size of the position under both price and production risk. In this paper, we provide a novel way of handling the quantity risk in connection with the BN-S model. The model is analyzed as a quadratic hedging problem and related analytical results are developed. The results indicate that oil can be optimally hedged with a combination of variance swaps and options. For various quantity risks, the model is implemented to analyze hedging decisions numerically for managing price risk in the Bakken oil commodities.</p>","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 3-4","pages":"202-214"},"PeriodicalIF":0.0,"publicationDate":"2019-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10049","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91251080","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On multilateral incomplete information decision models","authors":"Krzysztof Szajowski,&nbsp;Marek Skarupski","doi":"10.1002/hf2.10047","DOIUrl":"10.1002/hf2.10047","url":null,"abstract":"<p>This paper treats the decision problem related to theobservation of a Markov process by decision makers. The informationdelivered to the players is based on the aggregation of thehigh-frequency data by some functions. Admissible strategies arestopping moments related to the available information. The paymentsare defined by the state at the time of stopping. The players' decision to stop has various effects which depend on the decisionmakers' type. The type β player's stopping decision assignsthe state of the process with chance β, and it offers thisstate to the opponent with probability <math>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>-</mo>\u0000 <mi>β</mi>\u0000 </mrow></math>. The knowledgeabout the type of the players is not public and in this way, thepayers have also different information. The details of thedescription allow to formulate the problem as a Bayesian game withsets of strategies based on the stopping times. It is an extensionof Dynkin's game related to the observation of a Markov process withthe random assignment mechanism of states to the players. Some examples related to the best choice problem (BCP) are analyzed. MSC (2000) Primary: 90D15; Secondary: 93C30.</p>","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 3-4","pages":"158-168"},"PeriodicalIF":0.0,"publicationDate":"2019-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10047","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75005653","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fluid limit for a genetic mutation model","authors":"Carlos Bajo Caraballo,&nbsp;Ilie Grigorescu","doi":"10.1002/hf2.10046","DOIUrl":"10.1002/hf2.10046","url":null,"abstract":"<p>We trace the time evolution of the number <i>U_t</i> of nondeleterious mutations, present in a gene modeled by a word of length <i>L</i> and DNA fragments by characters labeled 0, 1,…, <i>N</i>. For simplification, deleterious mutations are codified as equal to 0. The discrete case studied in Grigorescu (<i>Stochastic Models</i>, 29, 2013 p. 328), is a modified version of the Pólya urn, where the two types are exactly the zeros and nonzeros. A random continuous-time binary mutation model, where the probability of creating a deleterious mutation is 1/<i>N</i>, while the probability of recovery <math>\u0000 <mrow>\u0000 <mi>γ</mi>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mrow>\u0000 <mo>-</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <msub>\u0000 <mi>U</mi>\u0000 <mi>Lt</mi>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow></math>, <i>γ</i> continuous, is studied under a Eulerian scaling <math>\u0000 <mrow>\u0000 <msubsup>\u0000 <mi>u</mi>\u0000 <mi>t</mi>\u0000 <mi>L</mi>\u0000 </msubsup>\u0000 <mo>=</mo>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mrow>\u0000 <mo>-</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 </msup>\u0000 <msub>\u0000 <mi>U</mi>\u0000 <mi>Lt</mi>\u0000 </msub>\u0000 </mrow></math>, <i>L</i> → ∞. The fluid limit <i>u_t</i>, emerging due to the high frequency scale of mutations, is the solution of a deterministic generalized logistic equation. The power law <i>γ</i>(<i>u</i>) = <i>cu^a</i> captures important features in both genetical and epidemiological interpretations, with <i>c</i> being the intensity of the intervention, <i>a</i> the strength/virulence of the disease, and 1/<i>N</i> the decay rate/infectiousness. Among other applications, we obtain a quantitative study of Δ<i>T</i>, the maximal interval between tests. Several stochastic optimization problems, including a generalization of the Shepp urn (<i>The Annals of Mathematical Statistics</i>, 40, 1969 p. 993), are proposed.</p>","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 3-4","pages":"147-157"},"PeriodicalIF":0.0,"publicationDate":"2019-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10046","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73882550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A numerical investigation on the high-frequency geometry of spherical random eigenfunctions","authors":"Yabebal Fantaye,&nbsp;Valentina Cammarota,&nbsp;Domenico Marinucci,&nbsp;Anna Paola Todino","doi":"10.1002/hf2.10044","DOIUrl":"10.1002/hf2.10044","url":null,"abstract":"<p>A lot of attention has been drawn over the last few years by the investigation of the geometry of spherical random eigenfunctions (random spherical harmonics) in the high-frequency regime, that is, for diverging eigenvalues. In this paper, we present a review of these results and we collect for the first time a comprehensive numerical investigation, focussing on particular on the behavior of Lipschitz-Killing curvatures/Minkowski functionals (i.e., the area, the boundary length, and the Euler-Poincaré characteristic of excursion sets) and on critical points. We show in particular that very accurate analytic predictions exist for their expected values and variances, for the correlation among these functionals, and for the cancellation that occurs for some specific thresholds (the variances becoming an order of magnitude smaller—the so-called Berry's cancellation phenomenon). Most of these functionals can be used for important statistical applications, for instance, in connection to the analysis of cosmic microwave background data.</p>","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 3-4","pages":"184-201"},"PeriodicalIF":0.0,"publicationDate":"2019-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10044","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84805270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cauchy's functional equation and Gaussian measures","authors":"Daniel W. Stroock","doi":"10.1002/hf2.10043","DOIUrl":"10.1002/hf2.10043","url":null,"abstract":"<p>This article summarizes several examples of the way in which Gaussian measures can be used to obtain results about solutions to Cauchy's functional equation in both finite and infinite dimensions.</p>","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 3-4","pages":"142-146"},"PeriodicalIF":0.0,"publicationDate":"2019-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10043","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85800367","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Modeling leverage and long memory in volatility in a pure-jump process","authors":"Meng-Chen Hsieh,&nbsp;Clifford Hurvich,&nbsp;Philippe Soulier","doi":"10.1002/hf2.10042","DOIUrl":"10.1002/hf2.10042","url":null,"abstract":"<p>We propose a model for log asset prices in which the underlying transactions and price changes are governed by a marked Cox process. We derive tractable analytical expressions for the autocovariances of the returns, the squared returns, and the covariance of current returns and subsequent squared returns measured at fixed calendar-time frequency. We further prove that statistical properties of the derived return process match the stylized facts observed in empirical financial data such as short memory in returns, long memory in the counts (number of trades), long memory in the realized variance, and the leverage effect. Finally, we provide procedures for estimating the model parameters based on the transaction-level data for a special case of our model.</p>","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 3-4","pages":"124-141"},"PeriodicalIF":0.0,"publicationDate":"2019-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10042","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76016313","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"In memory of Larry Shepp: An editorial","authors":"Philip Ernst,&nbsp;Frederi Viens","doi":"10.1002/hf2.10035","DOIUrl":"10.1002/hf2.10035","url":null,"abstract":"<p>This special issue of <i>High Frequency</i> is dedicated to the memory of Professor Larry Shepp, who passed on April 23, 2013. We are humbled to honor his immense scientific impact through this issue. The authors who have written for this special issue contributed their work from a call which we circulated at a conference organized in Prof. Shepp's honor, at Rice University, in June 2018. The conference spanned a number of topics in the areas of applied probability, as influenced by Shepp, including optimal stopping, stochastic control, financial mathematics, Gaussian, and Lévy processes.</p><p>While Prof. Shepp himself did not explicitly describe any of his work as part of high-frequency (HF) data analysis, any methodology which follows the stopping and control rules which he pioneered, would necessarily run into the use of high-frequency data to ensure a close match with continuous-time modeling, and avoid overly sub-optimal approximations. In this issue, we see examples of this interpretation through four thought-provoking papers, some where the data analysis is implicit in the continuous-time framework of stochastic processes, some which invoke HF data analysis explicitly, all which honor the influence of Larry Shepp by proposing problems in optimal stopping, mathematical finance, and the theory of stochastic processes.</p><p>This issue's first paper, by Pavel Gapeev, revisits Larry Shepp's famous Russian option (see below) in a situation where the continuation region for optimal stopping has at least two disconnected components, specifically the dual Russian option pricing problem he proposed with Albert Shiryaev in 1993, now with a positive exponential discount rate. The implementation of the corresponding stopping scheme would easily miss this feature of a so-called double-continuation region unless decisions are made with sufficiently high frequency. This issue's paper by Paolo Guasoni and Ali Sanjari also treats an optimal trading strategy problem, but one of a very different nature: the liquidation of a large position in the financial market. Assuming high-frequency data and trading is available, the liquidator and the other market actors engage in competing objectives, which can lead to severely reduced liquidity in the late stages of liquidation. The paper explores this and other sources of nonlinear price impact, in the context of stochastic control and utility maximization. Switching gears, this issue's paper by Guodong Pang and Murad Taqqu honors the themes of Gaussian and Poisson stochastic processes, including the study of path regularity features, another area influenced by Larry Shepp. They study how the assumption of a power law in the shot distribution of a Poisson-based shot noise process leads to self-similar limiting Gaussian processes which extend fractional Brownian motion by dissociating the high-frequency regularity behavior from the self-similarity property. Finally, keeping within the study of stochastic processes","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 2","pages":"74-75"},"PeriodicalIF":0.0,"publicationDate":"2019-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10035","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87582886","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal stopping of Brownian motion with broken drift","authors":"Ernesto Mordecki,&nbsp;Paavo Salminen","doi":"10.1002/hf2.10034","DOIUrl":"10.1002/hf2.10034","url":null,"abstract":"We solve an optimal stopping problem where the underlying diffusion is Brownian motion on $bf R$ with a positive drift changing at zero. It is assumed that the drift $mu_1$ on the negative side is smaller than the drift $mu_2$ on the positive side. The main observation is that if $mu_2-mu_1>1/2$ then there exists values of the discounting parameter for which it is not optimal to stop in the vicinity of zero where the drift changes. However, when the discounting gets bigger the stopping region becomes connected and contains zero. This is in contrast with results concerning optimal stopping of skew Brownian motion where the skew point is for all values of the discounting parameter in the continuation region.","PeriodicalId":100604,"journal":{"name":"High Frequency","volume":"2 2","pages":"113-120"},"PeriodicalIF":0.0,"publicationDate":"2019-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1002/hf2.10034","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76705500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}