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In this paper, we conduct a systematic large-scale analysis of order book-driven predictability in high-frequency returns by leveraging deep learning techniques. First, we introduce a new and robust representation of the order book, the volume representation. Next, we carry out an extensive empirical experiment to address various questions regarding predictability. We investigate if and how far ahead there is predictability, the importance of a robust data representation, the advantages of multi-horizon modeling, and the presence of universal trading patterns. We use model confidence sets, which provide a formalized statistical inference framework particularly well suited to answer these questions. Our findings show that at high frequencies predictability in mid-price returns is not just present, but ubiquitous. The performance of the deep learning models is strongly dependent on the choice of order book representation, and in this respect, the volume representation appears to have multiple practical advantages.

In this article we will introduce the realised semicovariance for Brownian semistationary (BSS) processes, which is obtained from the decomposition of the realised covariance matrix into components based on the signs of the returns, and study its in-fill asymptotic properties. More precisely, a weak convergence in the space of c\`adl\`ag functions endowed with the Skorohod topology for the realised semicovariance of a general Gaussian process with stationary increments is proved first. The methods are based on Breuer-Major theorems and on a moment bound for sums of products of Gaussian vector's functions. Furthermore, we establish a corresponding stable convergence. Finally, a weak law of large numbers and a central limit theorem for the realised semicovariance of multivariate BSS processes are established. These results extend the limit theorems for the realised covariation to a result for non-linear functionals.

Renewal equations are a popular approach used in modelling the number of new infections, i.e., incidence, in an outbreak. We develop a stochastic model of an outbreak based on a time-varying variant of the Crump-Mode-Jagers branching process. This model accommodates a time-varying reproduction number and a time-varying distribution for the generation interval. We then derive renewal-like integral equations for incidence, cumulative incidence and prevalence under this model. We show that the equations for incidence and prevalence are consistent with the so-called back-calculation relationship. We analyse two particular cases of these integral equations, one that arises from a Bellman-Harris process and one that arises from an inhomogeneous Poisson process model of transmission. We also show that the incidence integral equations that arise from both of these specific models agree with the renewal equation used ubiquitously in infectious disease modelling. We present a numerical discretisation scheme to solve these equations, and use this scheme to estimate rates of transmission from serological prevalence of SARS-CoV-2 in the UK and historical incidence data on Influenza, Measles, SARS and Smallpox., 35 pages, 4 figures, v4: major revision, including a new argument for the equivalence of incidence equations

In this paper, we develop a generalized method of moments approach for joint estimation of the parameters of a fractional log-normal stochastic volatility model. We show that with an arbitrary Hurst exponent an estimator based on integrated variance is consistent. Moreover, under stronger conditions we also derive a central limit theorem. These results stand even when integrated variance is replaced with a realized measure of volatility calculated from discrete high-frequency data. However, in practice a realized estimator contains sampling error, the effect of which is to skew the fractal coefficient toward "roughness". We construct an analytical approach to control this error. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show that the bias correction attenuates any systematic deviance in the estimated parameters. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in integrated variance.

This article studies the finite sample behaviour of a number of estimators for the integrated power volatility process of a Brownian semistationary process in the non semi-martingale setting. We establish three consistent feasible estimators for the integrated volatility, two derived from parametric methods and one non-parametrically. We then use a simulation study to compare the convergence properties of the estimators to one another, and to a benchmark of an infeasible estimator. We further establish bounds for the asymptotic variance of the infeasible estimator and assess whether a central limit theorem which holds for the infeasible estimator can be translated into a feasible limit theorem for the non-parametric estimator., 21 pages, 7 figures

Stochastic processes provide a mathematically elegant way model complex data. In theory, they provide flexible priors over function classes that can encode a wide range of interesting assumptions. In practice, however, efficient inference by optimisation or marginalisation is difficult, a problem further exacerbated with big data and high dimensional input spaces. We propose a novel variational autoencoder (VAE) called the prior encoding variational autoencoder ($\pi$VAE). The $\pi$VAE is finitely exchangeable and Kolmogorov consistent, and thus is a continuous stochastic process. We use $\pi$VAE to learn low dimensional embeddings of function classes. We show that our framework can accurately learn expressive function classes such as Gaussian processes, but also properties of functions to enable statistical inference (such as the integral of a log Gaussian process). For popular tasks, such as spatial interpolation, $\pi$VAE achieves state-of-the-art performance both in terms of accuracy and computational efficiency. Perhaps most usefully, we demonstrate that the low dimensional independently distributed latent space representation learnt provides an elegant and scalable means of performing Bayesian inference for stochastic processes within probabilistic programming languages such as Stan.

We study statistical aspects of state-dependent Hawkes processes, which are an extension of Hawkes processes where a self- and cross-exciting counting process and a state process are fully coupled, interacting with each other. The excitation kernel of the counting process depends on the state process that, reciprocally, switches state when there is an event in the counting process. We first establish the existence and uniqueness of state-dependent Hawkes processes and explain how they can be simulated. Then we develop maximum likelihood estimation methodology for parametric specifications of the process. We apply state-dependent Hawkes processes to high-frequency limit order book data, allowing us to build a novel model that captures the feedback loop between the order flow and the shape of the limit order book. We estimate two specifications of the model, using the bid-ask spread and the queue imbalance as state variables, and find that excitation effects in the order flow are strongly state-dependent. Additionally, we find that the endogeneity of the order flow, measured by the magnitude of excitation, is also state-dependent, being more pronounced in disequilibrium states of the limit order book

We study the empirical properties of realized volatility of the E-mini S&P 500 futures contract at various time scales, ranging from a few minutes to one day. Our main finding is that intraday volatility is remarkably rough and persistent. What is more, by further studying daily realized volatility measures of close to two thousand individual US equities, we find that both roughness and persistence appear to be universal properties of volatility. Inspired by the empirical findings, we introduce a new class of continuous-time stochastic volatility models, capable of decoupling roughness (short-term behavior) from long memory and persistence (long-term behavior) in a simple and parsimonious way, which allows us to successfully model volatility at all intraday time scales. Our prime model is based on the so-called Brownian semistationary process and we derive a number of theoretical properties of this process, relevant to volatility modeling. As an illustration of the usefulness our new models, we conduct an extensive forecasting study; we find that the models proposed in this paper outperform a wide array of benchmarks considerably, indicating that it pays off to exploit both roughness and persistence in volatility forecasting.

We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method and in simulations we observe that the bootstrap-based hypothesis test provides considerable finite-sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data; we illustrate this by applying the bootstrap method to two empirical data sets: we assess the roughness of a time series of high-frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data. We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method and in simulations we observe that the bootstrap-based hypothesis test provides considerable finite-sample improvements over an existing test that is based on a central limit theorem. This is important when studying the roughness properties of time series data; we illustrate this by applying the bootstrap method to two empirical data sets: we assess the roughness of a time series of high-frequency asset prices and we test the validity of Kolmogorov's scaling law in atmospheric turbulence data.

We show that a trader, who starts with no initial wealth and is not allowed to borrow money or short sell assets, is theoretically able to attain positive wealth by continuous trading, provided that she has perfect foresight of future asset prices, given by a continuous semimartingale. Such an arbitrage strategy can be constructed as a process of finite variation that satisfies a seemingly innocuous self-financing condition, formulated using a pathwise Riemann-Stieltjes integral. Our result exemplifies the potential intricacies of formulating economically meaningful self-financing conditions in continuous time, when one leaves the conventional arbitrage-free framework.

We introduce a simulation scheme for Brownian semistationary processes, which is based on discretizing the stochastic integral representation of the process in the time domain. We assume that the kernel function of the process is regularly varying at zero. The novel feature of the scheme is to approximate the kernel function by a power function near zero and by a step function elsewhere. The resulting approximation of the process is a combination of Wiener integrals of the power function and a Riemann sum, which is why we call this method a hybrid scheme. Our main theoretical result describes the asymptotics of the mean square error of the hybrid scheme and we observe that the scheme leads to a substantial improvement of accuracy compared to the ordinary forward Riemann-sum scheme, while having the same computational complexity. We exemplify the use of the hybrid scheme by two numerical experiments, where we examine the finite-sample properties of an estimator of the roughness parameter of a Brownian semistationary process and study Monte Carlo option pricing in the rough Bergomi model of Bayer et al. (2015), respectively.

Motivated by the construction of the Itô stochastic integral, we consider a step function method to discretize and simulate volatility modulated Lévy semistationary processes. Moreover, we assess the accuracy of the method with a particular focus on integrating kernels with a singularity at the origin. Using the simulation method, we study the finite sample properties of some recently developed estimators of realized volatility and associated parametric estimators for Brownian semistationary processes. Although the theoretical properties of these estimators have been established under high frequency asymptotics, it turns out that the estimators perform well also in a low frequency setting. Motivated by the construction of the Ito stochastic integral, we consider a step function method to discretize and simulate volatility modulated Lévy semistationary processes. Moreover, we assess the accuracy of the method with a particular focus on integrating kernels with a singularity at the origin. Using the simulation method, we study the finite sample properties of some recently developed estimators of realized volatility and associated parametric estimators for Brownian semistationary processes. Although the theoretical properties of these estimators have been established under high frequency asymptotics, it turns out that the estimators perform well also in a low frequency setting.

We prove functional central and non-central limit theorems for generalized variations of the anisotropic d-parameter fractional Brownian sheet (fBs) for any natural number d. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and on the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application, we derive functional limit theorems for power variations of the fBs and discuss what is a proper way to interpolate them to ensure functional convergence. We prove functional central and non-central limit theorems for generalized variations of the anisotropic d-parameter fractional Brownian sheet (fBs) for any natural number d. Whether the central or the non-central limit theorem applies depends on the Hermite rank of the variation functional and on the smallest component of the Hurst parameter vector of the fBs. The limiting process in the former result is another fBs, independent of the original fBs, whereas the limit given by the latter result is an Hermite sheet, which is driven by the same white noise as the original fBs. As an application, we derive functional limit theorems for power variations of the fBs and discuss what is a proper way to interpolate them to ensure functional convergence.

We introduce the notion of relative volatility/intermittency and demonstrate how relative volatility statistics can be used to estimate consistently the temporal variation of volatility/intermittency even when the data of interest are generated by a non-semimartingale, or a Brownian semistationary process in particular. While this estimation method is motivated by the assessment of relative energy dissipation in empirical data of turbulence, we apply it also to energy price data. Moreover, we develop a probabilistic asymptotic theory for relative power variations of Brownian semistationary processes and Ito semimartingales and discuss how it can be used for inference on relative volatility/intermittency.

We study the asymptotic behavior of lattice power variations of two-parameter ambit fields that are driven by white noise. Our first result is a law of large numbers for such power variations. Under a constraint on the memory of the ambit field, normalized power variations are shown to converge to certain integral functionals of the volatility field associated to the ambit field, when the lattice spacing tends to zero. This law of large numbers holds also for thinned power variations that are computed by only including increments that are separated by gaps with a particular asympotic behavior. Our second result is a related stable central limit theorem for thinned power variations. Additionally, we provide concrete examples of ambit fields that satisfy the assumptions of our limit theorems.

Frictions are factors that hinder trading of securities in financial markets. Typical frictions include limited market depth, transaction costs, lack of infinite divisibility of securities, and taxes. Conventional models used in mathematical finance often gloss over these issues, which affect almost all financial markets, by arguing that the impact of frictions is negligible and, consequently, the frictionless models are valid approximations. This dissertation consists of three research papers, which are related to the study of the validity of such approximations in two distinct modeling problems. Models of price dynamics that are based on diffusion processes, i.e., continuous strong Markov processes, are widely used in the frictionless scenario. The first paper establishes that diffusion models can indeed be understood as approximations of price dynamics in markets with frictions. This is achieved by introducing an agent-based model of a financial market where finitely many agents trade a financial security, the price of which evolves according to price impacts generated by trades. It is shown that, if the number of agents is large, then under certain assumptions the price process of security, which is a pure-jump process, can be approximated by a one-dimensional diffusion process. In a slightly extended model, in which agents may exhibit herd behavior, the approximating diffusion model turns out to be a stochastic volatility model. Finally, it is shown that when agents' tendency to herd is strong, logarithmic returns in the approximating stochastic volatility model are heavy-tailed. The remaining papers are related to no-arbitrage criteria and superhedging in continuous-time option pricing models under small-transaction-cost asymptotics. Guasoni, Rásonyi, and Schachermayer have recently shown that, in such a setting, any financial security admits no arbitrage opportunities and there exist no feasible superhedging strategies for European call and put options written on it, as long as its price process is continuous and has the so-called conditional full support (CFS) property. Motivated by this result, CFS is established for certain stochastic integrals and a subclass of Brownian semistationary processes in the two papers. As a consequence, a wide range of possibly non-Markovian local and stochastic volatility models have the CFS property. Rahoitusteorian keskeisiin tutkimuskohteisiin kuuluu optioiden hinnoittelu tilanteessa, jossa kohde-etuutena olevan arvopaperin hintakehityksen tilastolliset ominaisuudet ovat ennalta tunnetut. Tavanomaiset rahoitusteoreettiset mallit eivät kuitenkaan yleensä ota huomioon rahoitusmarkkinoilla esiintyviä kitkaksi kutsuttuja arvopaperien kauppaa vaikeuttavia tekijöitä. Kitkaa aiheuttavat esim. arvopaperien rajallinen tarjonta ja kysyntä, transaktiokulut, kaupoille säädetyt minimikoot sekä verot. Kitkan sivuuttamista perustellaan usein väittämällä, että sen käytännön merkitys olisi vähäinen, jolloin kitkatonta asetelmaa voisi pitää pätevänä likimääräisenä mallina. Väitöskirja koostuu kolmesta artikkelista, jotka liittyvät kitkan vaikutuksia arvioivaan matemaattisen rahoitusteorian tutkimukseen. Diffuusioprosesseja (esim. Brownin liikettä, johon perustuu Blackin ja Scholesin tunnettu optionhinnoittelumalli) käytetään usein kitkattomien markkinoiden hintakehityksen malleina. Ensimmäisessä artikkelissa osoitetaan, että diffuusioprosessit voidaan todellakin tulkita approksimaatioina kitkallisten markkinoiden hintakehityksestä. Artikkelin perustana on kitkallisia markkinoita kuvaava mikrorakennemalli, jossa arvopaperin hinta määräytyy markkinoilla syntyvien kauppojen perusteella. Kun markkinoilla olevien sijoittajien määrä kasvaa, mallin hintakehitys lähestyy diffuusioprosessia, joka voidaan määritellä tarkemmin sijoittajia koskevien käyttäytymisoletusten perusteella. Mikrorakennemallia laajennetaan lisäksi ottamaan huomioon sijoittajien mahdollinen laumakäyttäytyminen. Tällöin hintadynamiikkaa kuvaa likimäärin diffuusioprosessi, jonka volatiliteetti vaihtelee satunnaisesti. Jos sijoittajien taipumus laumakäyttäytymiseen on hyvin vahva, kyseinen diffuusioprosessi heilahtelee voimakkaasti ja sillä on paksuhäntäinen jakauma, eli suurien hinnanmuutosten todennäköisyys on huomattavan korkea. Viimeiset kaksi artikkelia liittyvät optioiden hinnoitteluun tilanteessa, jossa option kohde-etuuden kaupasta aiheutuu transaktiokuluja. Guasoni, Rásonyi ja Schachermayer ovat osoittaneet, että tällöin kohde-etuuden kaupalla ei voi saavuttaa riskitöntä voittoa, mikäli kohde-etuuden hintaprosessilla on nk. CFS-ominaisuus, joka oleellisesti edellyttää, että hintakehitystä ei voi ennustaa tarkasti. Tällöin lisäksi tavanomaisen option myynnistä aiheutuvalta vastuulta ei voi suojautua taloudellisesti mielekkäin kustannuksin, toisin kuin kitkattomissa malleissa. Artikkeleissa tutkitaan CFS-ominaisuuden teoriaa tarkemmin ja osoitetaan, että useilla satunnaisen volatiliteetin omaavilla hintakehityksen malleilla on CFS-ominaisuus.