Talks scheduled on Tuesday 7th afternoon:

BLANC Pierre (Ecole Nationale des Ponts et Chaussées)

FODRA Pietro (Paris 7 Denis Diderot)

HUANG Weibing (LPMA, University Pierre et Marie Curie)

IUGA Adrian (Paris-Dauphine, CREST & Université Paris-Est)

JAISSON Thibault (Ecole Polytechnique)

KRUSE Thomas (Université d’Evry)

LARUELLE Sophie (UPEC)

ROYER Guillaume (Ecole Polytechnique)

SADOGHI Amirhossein (Frankfurt School of Finance & Management)

SAGNA Abass (ENSIIE/UEVE)

Talks scheduled on Thursday 9th afternoon:

ABBAS-TURKI Lokman (TU Berlin)

BOMPIS Romain (Ecole Polytechnique)

DUBOIS Mathieu (London School of Economics)

KHARROUBI Idris (CEREMADE, Université Paris Dauphine)

REYGNER Julien (Ecole Nationale des Ponts et Chaussées)

TURKEDJIEV Plamen (Ecole Polytechnique)

L. Abbas-Turki: Toward a coherent Monte Carlo simulation of CVA

Abstract: This work is devoted to the simulation of the Credit Valuation Adjustment (CVA) using a pure Monte Carlo technique with Malliavin Calculus (MCM). The procedure presented is based on a general theoretical framework that includes a large number of models as well as various contracts, and allows both the computation of CVA and its sensitivity with respect to the different assets. Moreover, we provide the expression of the backward conditional density of assets vector that can be simulated off-line in order to reduce the variance of the CVA estimator. Regarding computational aspects, both complexity and accuracy are studied for MCM and regression methods and compared to the square Monte Carlo benchmark.

P. Blanc: Overnight and intra-day effects in volatility feedback modeling

Abstract: We decompose, within an ARCH framework, the daily volatility of stocks into overnight and intra-day contributions. We find, as perhaps expected, that the overnight and intra-day returns behave completely differently. For example, while past intra-day returns affect equally the future intra-day and overnight volatilities, past overnight returns have a weak effect on future intra-day volatilities (except for the very next one) but impact substantially future overnight volatilities. The exogenous component of overnight volatilities is found to be close to zero, which means that the lion’s share of overnight volatility comes from feedback effects. The residual kurtosis of returns is small for intraday returns but infinite for overnight returns. We provide a plausible interpretation for these findings, and show that our Intra-day/Overnight model significantly outperforms the standard ARCH framework based on daily returns for Out-of-Sample predictions.

R. Bompis: Price expansions for regular down barrier options in local volatility models

Abstract: We introduce option price expansions for regular barrier options focusing on the down and out case. In the framework of time-dependent local volatility models and martingale assets, we derive new formulas using a Gaussian proxy model and a method mixing Ito calculus and PDE approach. We also provide error estimates and we illustrate the accuracy of our formulas throughout numerical experiments.

M. Dubois: Portfolio allocation under parameter uncertainty

Abstract: We consider an investor who faces parameter uncertainty in a continuous- time financial market. We model the investor preference by a power utility function leading to constant relative risk aversion. We show that the loss in expected utility is large when using a simple plug-in strategy for unknown parameters. We also show that the loss due to estimation depends crucially on the coefficient of relative risk aversion. We provide theoretical results that show the trade-off between holding a well-diversified portfolio and a portfolio that is robust against estimation errors. To reduce the effect of estimation, we constrain the weights of the risky assets with an L1-norm leading to a sparse portfolio. We provide analytical results that show how the sparsity of the constrained portfolio depends on the coefficient of relative risk aversion. Based on a simulation study, we demonstrate the existence of an optimal bound on the L1-norm for each level of relative risk aversion.

P. Fodra: A semi Markov model for market microstructure and high-frequency trading

Abstract: The aim of this work is to construct a model for the asset price in a limit order book which captures the main stylized facts of the market microstructure, and that is tractable for dealing with optimal high frequency trading by stochastic control methods. For this purpose, we introduce a model describing the fluctuations of a tick-by-tick single asset price based on Markov renewal processes. We consider a point process associated to the timestamps of the price jumps, and the marks associated to the price increments. By modeling the marks with a suitable Markov chain, we can reproduce the strong mean-reversion of price returns known as microstructure noise. Moreover, using a Markov renewal process, we can model the presence of spikes in the intensity of the market activity, i.e. the volatility clustering. We also provide simple parametric and nonparametric statistical procedures for the estimation of our model. We obtain closed-form formulae for the mean signature plot, and show the diffusive behavior of our model at large scale limit. We illustrate our results by numerical simulations, and find that our model is consistent with empirical data on futures Euribor and Eurostoxx. In a second part, we use a dynamic programming approach for a problem of optimal high frequency trading where the stock price is a Markov renewal process and the modelling of market orders takes into account the adverse selection risk. We show a reduced-form for the value function of the associated control problem, and provide a convergent and computational scheme for solving the problem. Numerical tests display the shape of optimal policies for the market making problem. We decompose, within an ARCH framework, the daily volatility of stocks into overnight and intra-day contributions. We find, as perhaps expected, that the overnight and intra-day returns behave completely differently. For example, while past intra-day returns affect equally the future intra-day and overnight volatilities, past overnight returns have a weak effect on future intra-day volatilities (except for the very next one) but impact substantially future overnight volatilities. The exogenous component of overnight volatilities is found to be close to zero, which means that the lion’s share of overnight volatility comes from feedback effects. The residual kurtosis of returns is small for intraday returns but infinite for overnight returns. We provide a plausible interpretation for these findings, and show that our Intra-day/Overnight model significantly outperforms the standard ARCH framework based on daily returns for Out-of-Sample predictions. (in collaboration with Huyen Pham)

W. Huang: Simulating and analyzing order book data: The queue-reactive model

Abstract: Through the analysis of a dataset of ultra high frequency order book updates, we introduce a model which accommodates the empirical properties of the full order book together with the stylized facts of lower frequency financial data. To do so, we split the time interval of interest into periods in which a well chosen reference price, typically the mid price, remains constant. Within these periods, we view the limit order book as a Markov queuing system. Indeed, we assume that the intensities of the order flows only depend on the current state of the order book. We establish the limiting behavior of this model and estimate its parameters from market data. Then, in order to design a relevant model for the whole period of interest, we use a stochastic mechanism that allows for switches from one period of constant reference price to another. Beyond enabling to reproduce accurately the behavior of market data, we show that our framework can be very useful for practitioners, notably as a market simulator or as a tool for the transaction cost analysis of complex trading algorithms.

A. Iuga: Reflexions on Market Impact

Abstract: We empirically study the market impact of a set of orders executed by CA Cheuvreux trading desk during 2010 in the European equity markets. We are specically interested in large trading orders that are executed incrementally, which we call meta-orders or hidden orders. We are focusing on three important aspects:

– the evolution of the market impact from the beginning to the completion of the meta-order,

– the total market impact when the trade is completed,

– price reversion after the completion of a trade and permanent impact.

Finally, we explain some of the style facts we find using Hawkes model intoduced by Bacry et al. (2010) (joint work with Emmanuel Bacry et Charles-Albert Lehalle)

T. Jaisson Limit theorems for nearly unstable Hawkes processes

Abstract: Because of their tractability and their natural interpretations in term of market quantities, Hawkes processes are nowadays widely used in high frequency finance. However, in practice, the statistical estimation results seem to show that very often, only nearly unstable Hawkes processes are able to fit the data properly. By nearly unstable, we mean that the L1 norm of their kernel is close to unity. We study in this work such processes for which the stability condition is almost violated. Our main result states that after suitable rescaling, they asymptotically behave like integrated Cox Ingersoll Ross models. Thus, modeling financial order flows as nearly unstable Hawkes processes may be a good way to reproduce both their high and low frequency stylized facts.

I. Kharroubi: Discrete time approximation of fully nonlinear HJB equations via BSDEs with nonpositive jumps

Abstract: We propose a new probabilistic numerical scheme for fully nonlinear equation of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problem, which is based on a recent Feynman-Kac representation by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approximation for the minimal solution to this class of BSDE when the time step goes to zero, which provides both an approximation for the value function and for an optimal control in feedback form. We obtained a convergence rate without any ellipticity condition on the controlled diffusion coefficient. This is a joint work with Nicolas Langrene and Huyen Pham.

T. Kruse: Optimal position closure in a market with stochastic price impact

Abstract: Liquidity in financial markets usually is not constant – it varies randomly in time and sometimes faces shocks. We consider the problem of closing a large asset position in a market with stochastic temporary price impact. We provide a probabilistic solution of the associated control problem by means of a Backward Stochastic Differential Equation (BSDE). The novelty of the solution approach is that the BSDE possesses a singular terminal condition. We prove that a solution of the BSDE exists and perform a verification. For special cases we determine optimal trading strategies explicitly. The talk is based on joint work with S. Ankirchner and M. Jeanblanc.

S. Laruelle: Optimal posting price of limit orders: learning by trading

Abstract: Considering that a trader or a trading algorithm interacting with markets during continuous auctions can be modeled by an iterating procedure adjusting the price at which he posts orders at a given rhythm, this paper proposes a procedure minimizing his costs. We prove the a.s. convergence of the algorithm under assumptions on the cost function and give some practical criteria on model parameters to ensure that the conditions to use the algorithm are fulfilled (using notably the co-monotony principle). We illustrate our results with numerical experiments on both simulated data and using a financial market dataset.

J. Reygnier: Capital distribution and portfolio performance in the mean-field Atlas model

Abstract: We study a mean-field version of rank-based models of equity markets, introduced by Fernholz in the framework of stochastic portfolio theory. We first obtain an asymptotic description of the market when the number of companies grows to infinity. We then discuss the long-term capital distribution in this asymptotic model, as well as the performance of simple portfolio rules. In particular, we higlight the influence of the volatility structure of the model on the growth rates of portfolios.

G. Royer: A new proof of Strassen’s theorem

Abstract: This result concerns the existence of a martingale measure with given marginals. It stands that two measures admit such a probability law if and only if they are in convex order. Its proof is due to Strassen and is based on an application of Hahn-Banach theorem. We derive a new proof, by using utility maximisation in the context of martingale optimal transport. This technic is an adaptation of the proof of the fundamental theorem of asset pricing of Rogers (1994)

A. Sadoghi: Measuring Systemic Risk: Robust Ranking Techniques Approach

Abstract: The recent economic crisis has raised a wide awareness that the financial system should be considered as a complex network with financial institutions and financial dependencies respectively as nodes and links between these nodes. Systemic risk is defined as the risk of default of a large portion of financial exposures among institution in the network. Indeed, the structure of this network is an important element to measure systemic risk and there is no widely accepted methodology to determine the systemically important nodes in a large financial network. In this research, we interduce a metric for systemic risk measurement with taking into account both common idiosyncratic shock as well as contagion through counterparty exposures. Our focus is on application of eigenvalue problems, as a robust approach to the ranking techniques, to measure systemic risk. Recently, the efficient algorithm has been developed for robust eigenvector problem to reduce to a nonsmooth convex optimization problem. We applied this technique and studied the performance and convergence behavior of the algorithm with different structure of the financial network.

A. Sagna: Marginal quantization of an Euler diffusion process and its application to finance

Abstract: We propose a new approach to quantize the marginals of the discrete Euler process resulting from the discretization of a brownian diffusion process using the Euler scheme. The method is built recursively using the distribution of the marginals of the discrete Euler process. The quantization error associated to the marginals is computed. In the one dimensional setting we illustrate how to perform the optimal grids using the Newton algorithm and show how to estimate the associated weights from a recursive formula. Numerical tests are carried out for the pricing of European options in a local volatility model and a comparison with the Monte Carlo simulations shows that the proposed method is more efficient than the Monte Carlo method.

P. Turkedjiev: Discretization and empirical regression methods for quadratic backward stochastic differential equations

Abstract: After discussing a priori estimates and the time-discretization of backward stochastic differential equations (BSDEs) using the Euler scheme and a novel Malliavin weights scheme, I will present three algorithms for the numerical resolution of these schemes using empirical regression methods. Firstly, I present the usual Euler scheme in multi-step foward dynamical programming form, for which I give novel, tight error estimates. Secondly, I present a novel dynamical programming algorithm based on Malliavin weights and give tight error estimates. Thanks to a priori bonds particularly on the Z part of the solution, this algorithm yields an improved theoretical complexity compared to the Euler scheme. Both these algorithms are capable of handling general conditions on the driver and terminal condition, and in particular can handle a large class of quadratic BSDEs. Finally, I present a novel multilevel type algorithm in the uniformly Lipschitz setting. This algorithm is based on adaptive variance reduction, leading to improved theoretical complexity compared to the Euler scheme.