CONT Rama

We propose an analytically tractable class of models for the dynamics of a limit order book, described as the solution of a stochastic partial differential equation (SPDE) with multiplicative noise. We provide conditions under which the model admits a finite dimensional realization driven by a (low-dimensional) Markov process, leading to efficient methods for estimation and computation. We study two examples of parsimonious models in this class: a two-factor model and a model in which the order book depth is mean-reverting. For each model we perform a detailed analysis of the role of different parameters, study the dynamics of the price, order book depth, volume and order imbalance, provide an intuitive financial interpretation of the variables involved and show how the model reproduces statistical properties of price changes, market depth and order flow in limit order markets.

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. The definition then simplifies and one obtains the Lebesgue decomposition of the pathwise quadratic variation as a result, rather than requiring it as an extra condition.

Given a stochastic differential equation with path-dependent coefficients driven by a multidimensional Wiener process, we show that the topological support in Holder norm of the law of the solution is given by the image of the Cameron-Martin space under the flow of the solutions of a system of path-dependent (ordinary) differential equations. Our result extends the Stroock-Varadhan support theorem for diffusion processes to the case of SDEs with path-dependent coefficients. The proof is based on the Functional Ito calculus and interpolation estimates in Holder norm.

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 exhibits 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 normal-izations 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 improves forecasting performance, showing evidence of path-dependence in price dynamics.

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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.

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.

We present a systematic method for computing explicit approximations to martingale representations for a large class of Brownian functionals. The approximations are based on a notion of pathwise functional derivative 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. The approximation and the proof of its convergence are based on the Functional Ito calculus, and require neither the Markov property, nor any differentiability conditions on the coefficients of the stochastic differential equations involved.

We present a quantitative methodology for analyzing the potential for contagion and systemic risk in a network of interlinked financial institutions, using a metric for the systemic importance of institutions: the Contagion Index. We apply this methodology to a data set of mutual exposures and capital levels of financial institutions in Brazil in 2007 and 2008, and analyze the role of balance sheet size and network structure in each institution's contribution to systemic risk. Our results emphasize the contribution of heterogeneity in network structure and concentration of counterparty exposures to a given institution in explaining its systemic importance. These observations plead for capital requirements which depend on exposures, rather than aggregate balance sheet size, and which target systemically important institutions.