## Publications

108 results found

Cont R, Degond P, Xuan L, 2023, A mathematical framework for modelling order book dynamics

We present a general framework for modelling the dynamics of limit orderbooks, built on the combination of two modelling ingredients: the order flow,modelled as a general spatial point process, and market clearing, modelled viaa deterministic mass transport operator acting on distributions of buy and sellorders. At the mathematical level, this corresponds to a natural decompositionof the infinitesimal generator describing the evolution of the limit order bookinto two operators: the generator of the order flow and the clearing operator.Our model provides a flexible framework for modelling and simulating order bookdynamics and studying various scaling limits of discrete order book models. Weshow that our framework includes previous models as special cases and yieldsinsights into the interplay between order flow and price dynamics.

Cont R, Das P, 2023, Quadratic variation and quadratic roughness

We study the concept of quadratic variation of a continuous path along a sequence of partitions and its dependence with respect to the choice of the partition sequence. We introduce the concept of quadratic roughness of a path along a partition sequence and show that for Hölder-continuous paths satisfying this roughness condition, the quadratic variation along balanced partitions is invariant with respect to the choice of the partition sequence. Typical paths of Brownian motion are shown to satisfy this quadratic roughness property almost-surely along any partition with a required step size condition. Using these results we derive a formulation of the pathwise Föllmer-Itô calculus which is invariant with respect to the partition sequence. We also derive an invarience of local time under quadratic roughness.

Cont R, Cucuringu M, Glukhov V,
et al., 2023, Analysis and modeling of client order flow in limit order markets, *QUANTITATIVE FINANCE*, ISSN: 1469-7688

Cont R, 2023, In memoriam: Marco Avellaneda (1955-2022), *MATHEMATICAL FINANCE*, Vol: 33, Pages: 3-15, ISSN: 0960-1627

Cont R, Rossier A, Xu R, 2022, Asymptotic Analysis of Deep Residual Networks

We investigate the asymptotic properties of deep Residual networks (ResNets)as the number of layers increases. We first show the existence of scalingregimes for trained weights markedly different from those implicitly assumed inthe neural ODE literature. We study the convergence of the hidden statedynamics in these scaling regimes, showing that one may obtain an ODE, astochastic differential equation (SDE) or neither of these. In particular, ourfindings point to the existence of a diffusive regime in which the deep networklimit is described by a class of stochastic differential equations (SDEs).Finally, we derive the corresponding scaling limits for the backpropagationdynamics.

Cont R, Das P, 2022, Quadratic variation along refining partitions: Constructions and examples, *Journal of Mathematical Analysis and Applications*, Vol: 512, Pages: 126173-126173, ISSN: 0022-247X

Cont R, Das P, 2022, Quadratic variation along refining partitions: Constructions and examples, *JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS*, Vol: 512, ISSN: 0022-247X

Cont R, Jin R, 2021, Fractional Ito calculus

We derive It\^o-type change of variable formulas for smooth functionals ofirregular paths with non-zero $p-$th variation along a sequence of partitionswhere $p \geq 1$ is arbitrary, in terms of fractional derivative operators,extending the results of the F\"ollmer-Ito calculus to the general case ofpaths with 'fractional' regularity. In the case where $p$ is not an integer, weshow that the change of variable formula may sometimes contain a non-zero a'fractional' It\^o remainder term and provide a representation for thisremainder term. These results are then extended to paths with non-zero$\phi-$variation and multi-dimensional paths. Finally, we derive an isometryproperty for the pathwise F\"ollmer integral in terms of $\phi$ variation.

Cohen A-S, Cont R, Rossier A, et al., 2021, Scaling Properties of Deep Residual Networks

Residual networks (ResNets) have displayed impressive results in patternrecognition and, recently, have garnered considerable theoretical interest dueto a perceived link with neural ordinary differential equations (neural ODEs).This link relies on the convergence of network weights to a smooth function asthe number of layers increases. We investigate the properties of weightstrained by stochastic gradient descent and their scaling with network depththrough detailed numerical experiments. We observe the existence of scalingregimes markedly different from those assumed in neural ODE literature.Depending on certain features of the network architecture, such as thesmoothness of the activation function, one may obtain an alternative ODE limit,a stochastic differential equation or neither of these. These findings castdoubts on the validity of the neural ODE model as an adequate asymptoticdescription of deep ResNets and point to an alternative class of differentialequations as a better description of the deep network limit.

Cont R, Kotlicki A, Xu R, 2021, Modelling COVID-19 contagion: risk assessment and targeted mitigation policies, *ROYAL SOCIETY OPEN SCIENCE*, Vol: 8, ISSN: 2054-5703

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- Citations: 2

Cont R, Mueller MS, 2021, A Stochastic Partial Differential Equation Model for Limit Order Book Dynamics, Publisher: SIAM PUBLICATIONS

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- Citations: 1

Cont R, Kotlicki A, Xu R, 2020, Modelling COVID-19 contagion: Risk assessment and targeted mitigation policies

<jats:title>Abstract</jats:title><jats:p>We use a spatial epidemic model with demographic and geographic heterogeneity to study the regional dynamics of COVID-19 across 133 regions in England.</jats:p><jats:p>Our model emphasises the role of variability of regional outcomes and heterogeneity across age groups and geographic locations, and provides a framework for assessing the impact of policies targeted towards sub-populations or regions. We define a concept of efficiency for comparative analysis of epidemic control policies and show targeted mitigation policies based on local monitoring to be more efficient than country-level or non-targeted measures. In particular, our results emphasise the importance of shielding vulnerable sub-populations and show that targeted policies based on local monitoring can considerably lower fatality forecasts and, in many cases, prevent the emergence of second waves which may occur under centralised policies.</jats:p>

Cont R, Kotlicki A, Valderrama L, 2020, Liquidity at risk: Joint stress testing of solvency and liquidity, *JOURNAL OF BANKING & FINANCE*, Vol: 118, ISSN: 0378-4266

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- Citations: 6

Cont R, Kalinin A, 2019, On the support of solutions to stochastic differential equations with path-dependent coefficients, *Stochastic Processes and their Applications*, ISSN: 0304-4149

Given a stochastic differential equation with path-dependent coefficientsdriven by a multidimensional Wiener process, we show that the support of thelaw of the solution is given by the image of the Cameron-Martin space under theflow of the solutions of a system of path-dependent (ordinary) differentialequations. Our result extends the Stroock-Varadhan support theorem fordiffusion processes to the case of SDEs with path-dependent coefficients. Theproof is based on the Functional Ito calculus and interpolation estimates forstochastic processes in Holder norm.

Cont R, Schaanning E, 2019, Monitoring indirect contagion, *Journal of Banking & Finance*, Vol: 104, Pages: 85-102, ISSN: 0378-4266

We propose two indicators for quantifying the potential exposure of financial institutions to indirect contagion arising from deleveraging of assets in stress scenarios. The first indicator, the Endogenous Risk Index (ERI) captures spillovers across portfolios arising from deleveraging in stress scenarios. The second indicator, the Indirect Contagion Index (ICI) measures the systemic importance of a bank by quantifying the loss its distressed liquidation would inflict on other institutions. Both are computable from portfolio holdings of financial institutions and measures of market depth for the assets held in the portfolio. We discuss the micro-foundation of these indicators and apply them to the analysis of the vulnerability of the European banking system to indirect contagion.Using data on portfolio holdings of European banks, we show that our indicators correlate to the magnitude of fire-sales losses in simulated stress scenarios, thus providing a simple to compute proxy for the outcome of stress tests. We also show that the information provided by our indicators on the systemic importance of banks is different from indicators based on size, thereby providing a measure of interconnectedness complementary to those currently used by supervisors.

Cont R, Perkowski N, 2019, Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, *Transactions of the American Mathematical Society*, Vol: 6, Pages: 161-186, ISSN: 0002-9947

We construct a pathwise integration theory, associated with a change of variable formula, for smooth functionals of continuous paths with arbitrary regularity defined in terms of the notion of p-th variation along a sequence of time partitions. For paths with finite p-th variation along a sequence of time partitions, we derive a change of variable formula for p times continuously differentiable functions and show pointwise convergence of appropriately defined compensated Riemann sums. Results for functions are extended to regular path-dependent functionals using the concept of vertical derivative of a functional. We show that the pathwise integral satisfies an `isometry' formula in terms of p-th order variation and obtain a `signal plus noise' decomposition for regular functionals of paths with strictly increasing p-th variation. For less regular (Cp−1) functions we obtain a Tanaka-type change of variable formula using an appropriately defined notion of local time. These results extend to multidimensional paths and yield a natural higher-order extension of the concept of `reduced rough path'. We show that, while our integral coincides with a rough-path integral for a certain rough path, its construction is canonical and does not involve the specification of any rough-path superstructure.

Chiu H, Cont R, 2018, On pathwise quadratic variation for cadlag functions, *Electronic Communications in Probability*, Vol: 23, Pages: 1-12, ISSN: 1083-589X

We revisit Föllmer’s concept of quadratic variation of a càdlàg function along a sequence of time partitions and discuss its relation with the Skorokhod topology. We show that in order to obtain a robust notion of pathwise quadratic variation applicable to sample paths of càdlàg processes, one must reformulate the definition of pathwise quadratic variation as a limit in Skorokhod topology of discrete approximations along the partition. One then obtains a simpler definition which implies the Lebesgue decomposition of the pathwise quadratic variation as a result, rather than requiring it as an extra condition.

Cont R, Sirignano J, 2018, Universal features of price formation in financial markets: perspectives from Deep Learning

Using a large-scale Deep Learning approach applied to a high-frequency database containing billions of electronic market quotes and transactions for US equities, we uncover nonparametric evidence for the existence of a universal and stationary price formation mechanism relating the dynamics of supply and demand for a stock, as revealed through the order book, to subsequent variations in its market price. We assess the model by testing its out-of-sample predictions for the direction of price moves given the history of price and order flow, across a wide range of stocks and time periods. The universal price formation model is shown to exhibit a remarkably stable out-of-sample prediction accuracy across time, for a wide range of stocks from different sectors. Interestingly, these results also hold for stocks which are not part of the training sample, showing that the relations captured by the model are universal and not asset-specific.

Chiu H, Cont R, 2018, On pathwise quadratic variation for càdlàg functions, *Electronic Communications in Probability*, Vol: 23

Cont R, Sirignano JA, 2018, Universal Features of Price Formation in Financial Markets: Perspectives From Deep Learning

Using a large-scale Deep Learning approach applied to a high-frequency database containing billions of electronic market quotes and transactions for US equities, we uncover nonparametric evidence for the existence of a universal and stationary price formation mechanism relating the dynamics of supply and demand for a stock, as revealed through the order book, to subsequent variations in its market price. We assess the model by testing its out-of-sample predictions for the direction of price moves given the history of price and order flow, across a wide range of stocks and time periods. The universal price formation model is shown to exhibit a remarkably stable out-of-sample prediction accuracy across time, for a wide range of stocks from different sectors. Interestingly, these results also hold for stocks which are not part of the training sample, showing that the relations captured by the model are universal and not asset-specific.The universal model --- trained on data from all stocks --- outperforms, in terms of out-of-sample prediction accuracy, asset-specific linear and nonlinear models trained on time series of any given stock, showing that the universal nature of price formation weighs in favour of pooling together financial data from various stocks, rather than designing asset or sector-specific models as commonly done. Standard data normalizations based on volatility, price level or average spread, or partitioning the training data into sectors or categories such as large/small tick stocks, do not improve training results. On the other hand, inclusion of price and order flow history over many past observations is shown to improve forecasting performance, showing evidence of path-dependence in price dynamics.

Cont R, Gordy M, 2017, Special Issue: Monitoring Systemic Risk: Data, Models and Metrics, *Statistics and Risk Modeling*, Vol: 34, ISSN: 2193-1402

Cont R, Ananova A, 2016, Pathwise integration with respect to paths of finite quadratic variation, *Journal de Mathematiques Pures et Appliquees*, Vol: 107, Pages: 737-757, ISSN: 0021-7824

We study a pathwise integral with respect to paths of finite quadratic variation, defined as the limit of non-anticipative Riemann sums for gradient-type integrands.We show that the integral satisfies a pathwise isometry property, analogous to the well-known Ito isometry for stochastic integrals. This property is then used to represent the integral as a continuous map on an appropriately defined vector space of integrands. Finally, we obtain a pathwise 'signal plus noise' decomposition for regular functionals of an irregular path with non-vanishing quadratic variation, as a unique sum of a pathwise integral and a component with zero quadratic variation.

Cont R, Duffie D, Glasserman P,
et al., 2016, Preface to the special issue on systemic risk: Models and mechanisms, *Operations Research*, Vol: 64, Pages: 1053-1055, ISSN: 0030-364X

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- Citations: 1

Cont R, Kukanov A, 2016, Optimal order placement in limit order markets, *Quantitative Finance*, Vol: 17, Pages: 21-39, ISSN: 1469-7696

To execute a trade, participants in electronic equity markets may choose to submit limit orders or market orders across various exchanges where a stock is traded. This decision is influenced by the characteristics of the order flow and queue sizes in each limit order book, as well as the structure oftransaction fees and rebates across exchanges. We propose a quantitativeframework for studying this order placement problem by formulating it as a convex optimization problem. This formulation allows to study how the interplay between the state of order books, the fee structure, order flow properties and preferences of a trader determine the optimal placement decision. In the case of a single exchange, we derive an explicit solution for the optimal split between limit and market orders. For the general problem of order placement across multiple exchanges, we propose a stochastic algorithm for computing the optimal policy and study the sensitivity of the solution to various parameters using a numerical implementation of the algorithm.

Cont R, Wagalath L, 2016, Risk management for whales, *Risk -London- Risk Magazine Limited-*, ISSN: 0952-8776

We propose framework for modeling portfolio risk which integrates market risk with liquidation costs which may arise in stress scenarios. Our model provides a systematic method for computing liquidation-adjusted risk measures for a portfolio. Calculation of Liquidation-adjusted VaR (LVaR) for sample portfolios reveals a substantial impact of liquidation costs on portfolio risk for portfolios with large concentrated positions.

Amini H, Cont R, Minca A, 2016, Resilience to Contagion in Financial Networks, *Mathematical Finance*, Vol: 26, Pages: 329-365, ISSN: 0960-1627

Propagation of balance-sheet or cash-flow insolvency across financialinstitutions may be modeled as a cascade process on a network representingtheir mutual exposures. We derive rigorous asymptotic results for the magnitudeof contagion in a large financial network and give an analytical expression forthe asymptotic fraction of defaults, in terms of network characteristics. Ourresults extend previous studies on contagion in random graphs to inhomogeneousdirected graphs with a given degree sequence and arbitrary distribution ofweights. We introduce a criterion for the resilience of a large financialnetwork to the insolvency of a small group of financial institutions andquantify how contagion amplifies small shocks to the network. Our resultsemphasize the role played by "contagious links" and show that institutionswhich contribute most to network instability in case of default have both largeconnectivity and a large fraction of contagious links. The asymptotic resultsshow good agreement with simulations for networks with realistic sizes.

Cont R, Wagalath L, 2016, INSTITUTIONAL INVESTORS AND THE DEPENDENCESTRUCTURE OF ASSET RETURNS, *International Journal of Theoretical & Applied Finance*, Vol: 19, ISSN: 1793-6322

We propose a model of a financial market with multiple assets that takes into accountthe impact of a large institutional investor rebalancing its positions so as to maintaina fixed allocation in each asset. We show that feedback effects can lead to significantexcess realized correlation between asset returns and modify the principal componentstructure of the (realized) correlation matrix of returns. Our study naturally links, ina quantitative manner, the properties of the realized correlation matrix — correlationbetween assets, eigenvectors and eigenvalues — to the sizes and trading volumes oflarge institutional investors. In particular, we show that even starting with uncorrelated“fundamentals”, fund rebalancing endogenously generates a correlation matrix of returnswith a first eigenvector with positive components, which can be associated to the market,as observed empirically. Finally, we show that feedback effects flatten the differencesbetween the expected returns of assets and tend to align them with the returns of theinstitutional investor’s portfolio, making this benchmark fund more difficult to beat, notbecause of its strategy but precisely because of its size and market impact.

Cont R, Bally V, Caramellino L, 2016, Stochastic Integration by Parts and Functional Itô Calculus, Publisher: Birkhäuser, ISBN: 978-3-319-27128-6

This volume contains lecture notes from the courses given by Vlad Bally and Rama Cont at the Barcelona Summer School on Stochastic Analysis (July 2012).The notes of the course by Vlad Bally, co-authored with Lucia Caramellino, develop integration by parts formulas in an abstract setting, extending Malliavin's work on abstract Wiener spaces. The results are applied to prove absolute continuity and regularity results of the density for a broad class of random processes.Rama Cont's notes provide an introduction to the Functional Itô Calculus, a non-anticipative functional calculus that extends the classical Itô calculus to path-dependent functionals of stochastic processes. This calculus leads to a new class of path-dependent partial differential equations, termed Functional Kolmogorov Equations, which arise in the study of martingales and forward-backward stochastic differential equations.This book will appeal to both young and senior researchers in probability and stochastic processes, as well as to practitioners in mathematical finance.

Cont R, LU Y, 2015, Weak approximation of martingale representations, *Stochastic Processes and Their Applications*, Vol: 126, Pages: 857-882, ISSN: 0304-4149

We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are obtained by computing a directional derivative of the weak Euler scheme and yield a consistent estimator for the integrand in the martingale representation formula for any square-integrable functional of the solution of an SDE with path-dependent coefficients. Explicit convergence rates are derived for functionals which are Lipschitz-continuous in the supremum norm. Our results require neither the Markov property, nor any differentiability conditions on the functional or the coefficients of the stochastic differential equations involved.

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