Papers by charles-albert lehalle
The invention relates to a method for initializing a neural network in order to emulate a non-lin... more The invention relates to a method for initializing a neural network in order to emulate a non-linear function, as realized by an initialization box receiving available function data in the form of an input, i.e. an output sample and an input sample, performing the following: automatic selection of the number of required linear zones, taking into account available non-linear function data for linear approximation of said function; automatic calculation of the characteristics of the linear forms on each function zone, i.e. slope; design automation of the neural network associating the corresponding linear form with each function zone; calculation of the initial values of the parameters of the neural network, i.e. the weights and biases of the hidden neuron layer and the output neuron layer.
Piecewise affine neural networks can be constructed to emulate any continuous piecewise affine fu... more Piecewise affine neural networks can be constructed to emulate any continuous piecewise affine function in any hypercube of its input space. This property can be used to initialize such a network with a set of linear controllers, where each of them is known to be efficient locally. This paper expose and illustrate this properties of piecewise affine neural networks.
Stochastic differential equations are the bread and butter of financial mathematics. But they are... more Stochastic differential equations are the bread and butter of financial mathematics. But they are not always the most appropriate tool. In these brief musings I train the spotlight on one of their lesser known cousins: stochastic convolution equations. How and why can stochastic convolutions add value in financial modeling? Read on and find out. Stochastic differential equations (SDEs) are the bread and butter of financial mathematics. Just think of the Black–Scholes–Merton model, where the price of a stock or an index is modeled by geometric Brownian motion, dSt/St = μdt + σdWt. Or, think of the spot variance in the Heston model, dvt = λ(θ − vt)dt+ η √ vtdBt. In fact, the majority of all continuous-time stochastic models used in financial mathematics—many of them highly complex— are specified in terms SDEs. This is not always a natural choice. Models based on SDEs share properties that are sometimes at odds with the reality they are meant to describe. A prime example is the charact...
arXiv: Trading and Market Microstructure, 2018
In this paper we investigate the endogenous information contained in four liquidity variables at ... more In this paper we investigate the endogenous information contained in four liquidity variables at a five minutes time scale on equity markets around the world: the traded volume, the bid-ask spread, the volatility and the volume at first limits of the orderbook. In the spirit of Granger causality, we measure the level of information by the level of accuracy of linear autoregressive models. This empirical study is carried out on a dataset of more than 300 stocks from four different markets (US, UK, Japan and Hong Kong) from a period of over five years. We discuss the obtained performances of autoregressive (AR) models on stationarized versions of the variables, focusing on explaining the observed differences between stocks. Since empirical studies are often conducted at this time scale, we believe it is of paramount importance to document endogenous dynamics in a simple framework with no addition of supplemental information. Our study can hence be used as a benchmark to identify exoge...
This paper presents a stochastic recursive procedure under constraints to find the optimal distan... more This paper presents a stochastic recursive procedure under constraints to find the optimal distance at which an agent must post his order to minimize his execution cost. 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 principle of opposite monotony). We illustrate our results with numerical experiments on simulated data but also by using a financial market dataset.
The efficient frontier is a core concept in Modern Portfolio Theory. Based on this idea, we will ... more The efficient frontier is a core concept in Modern Portfolio Theory. Based on this idea, we will construct optimal trading curves for different types of portfolios. These curves correspond to the algorithmic trading strategies that minimize the expected transaction costs, i.e. the joint effect of market impact and market risk. We will study five portfolio trading strategies. For the first three (single-asset, general multi-asseet and balanced portfolios) we will assume that the underlyings follow a Gaussian diffusion, whereas for the last two portfolios we will suppose that there exists a combination of assets such that the corresponding portfolio follows a mean-reverting dynamics. The optimal trading curves can be computed by solving an N-dimensional optimization problem, where N is the (pre-determined) number of trading times. We will solve the recursive algorithm using the "shooting method", a numerical technique for differential equations. This method has the advantage...
With the rise of the electronic trading, corporate bond traders have access to data information o... more With the rise of the electronic trading, corporate bond traders have access to data information of past trades. As a first step to automation, they have to start monitoring their own trades, and using past data to build a benchmark for the expected transaction costs with given bond characteristics and market conditions. Given the limited liquidity of corporate bonds which are traded few times daily, a statistical model is the only way to benchmark effective costs. It brings focused attention of the dealing desk of an institutional investor on the most costly trades, and enables identifying and improving business practices such as the market timing for selection counterparties. Unlike existing literature which focuses on general measurements using OLS, this paper takes the viewpoint of a given investor, and provides an analytical approach to establish a benchmark for transaction cost analysis in corporate bond tradings. Regularized methods are used to improve the selection of explana...
Almgren and Chriss ("Optimal execution of portfolio transactions", Journal of Risk, Vol... more Almgren and Chriss ("Optimal execution of portfolio transactions", Journal of Risk, Vol. 3, No. 2, 2010, pp. 5-39) and Lehalle ("Rigorous strategic trading: balanced portfolio and mean reversion", Journal of Trading, Summer 2009, pp. 40-46.) developed optimal trading algorithms for assets and portfolios driven by a brownian motion. More recently, Gatheral and Schied ("Optimal trade execution under geometric brownian motion in the Almgren and Chriss framework", Working paper SSRN, August 2010) addressed the same problem for the geometric brownian motion. In this article we extend these ideas for assets and portfolios driven by a discrete version of a selfsimilar process of exponent H in (0,1), which can be either a fractional brownian motion of Hurst exponent H or a truncated Levy distribution of index 1/H. The cost functional we use is not the classical expectation-variance one: instead of the variance, we use the p-variation, i.e. the Lp equivalent of ...
With MiFID, the recent changes in the European landscape modified the shape of intra-day volume c... more With MiFID, the recent changes in the European landscape modified the shape of intra-day volume curves. These, in combination with market risk through market impact function, are a key element in finding the balance between trading fast (to avoid market risk) and trading slow (to decrease market impact of the transactions). As these curves have a central role in quantitative trading, those changes have to be taken into account. The proposed statistical analysis of tick data allows us to infer some typical investor behavior in terms of intra-day activity, and to use them to optimize multi-destination trading.
We use a deep neural network to generate controllers for optimal trading on high frequency data. ... more We use a deep neural network to generate controllers for optimal trading on high frequency data. For the first time, a neural network learns the mapping between the preferences of the trader, i.e. risk aversion parameters, and the optimal controls. An important challenge in learning this mapping is that in intraday trading, trader's actions influence price dynamics in closed loop via the market impact. The exploration--exploitation tradeoff generated by the efficient execution is addressed by tuning the trader's preferences to ensure long enough trajectories are produced during the learning phase. The issue of scarcity of financial data is solved by transfer learning: the neural network is first trained on trajectories generated thanks to a Monte-Carlo scheme, leading to a good initialization before training on historical trajectories. Moreover, to answer to genuine requests of financial regulators on the explainability of machine learning generated controls, we project the ...
In this paper we propose a mathematical framework to address the uncertainty emergingwhen the des... more In this paper we propose a mathematical framework to address the uncertainty emergingwhen the designer of a trading algorithm uses a threshold on a signal as a control. We rely ona theorem by Benveniste and Priouret to deduce our Inventory Asymptotic Behaviour (IAB)Theorem giving the full distribution of the inventory at any point in time for a well formulatedtime continuous version of the trading algorithm.Since this is the first time a paper proposes to address the uncertainty linked to the use of athreshold on a signal for trading, we give some structural elements about the kind of signals thatare using in execution. Then we show how to control this uncertainty for a given cost function.There is no closed form solution to this control, hence we propose several approximation schemesand compare their performances.Moreover, we explain how to apply the IAB Theorem to any trading algorithm drivenby a trading speed. It is not needed to control the uncertainty due to the thresholding of...
We study the simplest discrete-time finite-maturity model in which default arises when the firm i... more We study the simplest discrete-time finite-maturity model in which default arises when the firm is not able to pay its debt obligation using the current cash-flow plus the corporate liquidity. An important distinction is made between liquidity and solvency of the firm. The corporate financial policy is simultaneously defined by the dividend rate (or policy), the coupon and the principal of the bond. In our model, the dividend rate both aects the default probability and the bondholders’ recovery rate. When the corporate financial policy implies no credit risk, we find the famous Modigliani-Miller propositions. When the corporate financial policy implies credit risk, we show that the value of the firm is a decreasing piecewise function of the dividend rate. As a consequence, zero is an optimal dividend rate: the value of the firm is not invariant with respect to the dividend rate, as suggested by Modigliani-Miller (1961). However, shareholders may not always have the incentives to imp...
This paper goes beyond the optimal trading Mean Field Game model introduced by Pierre Cardaliague... more This paper goes beyond the optimal trading Mean Field Game model introduced by Pierre Cardaliaguet and Charles-Albert Lehalle in [Cardaliaguet, P. and Lehalle, C.-A., Mean field game of controls and an application to trade crowding, Mathematics and Financial Economics (2018)]. It starts by extending it to portfolios of correlated instruments. This leads to several original contributions: first that hedging strategies naturally stem from optimal liquidation schemes on portfolios. Second we show the influence of trading flows on naive estimates of intraday volatility and correlations. Focussing on this important relation, we exhibit a closed form formula expressing standard estimates of correlations as a function of the underlying correlations and the initial imbalance of large orders, via the optimal flows of our mean field game between traders. To support our theoretical findings, we use a real dataset of 176 US stocks from January to December 2014 sampled every 5 minutes to analyze...
Machine Learning for Asset Management
SSRN Electronic Journal
Using a large database of US institutional investors' trades in the equity market, this paper exp... more Using a large database of US institutional investors' trades in the equity market, this paper explores the effect of simultaneous executions on trading cost. We design a Bayesian network modelling the inter-dependencies between investors' transaction costs, stock characteristics (bid-ask spread, turnover and volatility), meta-order attributes (side and size of the trade) and market pressure during execution, measured by the net order fow imbalance of investors meta-orders. Unlike standard machine learning algorithms, Bayesian networks are able to account for explicit inter-dependencies between variables. They also prove to be robust to missing values, as they are able to restore their most probable value given the state of the world. Order fow imbalance being only partially observable (on a subset of trades or with a delay), we show how to design a Bayesian network to infer its distribution and how to use this information to estimate transaction costs. Our model provides better predictions than standard (OLS) models. The forecasting error is smaller and decreases with the investors' order size, as large orders are more informative on the aggregate order fow imbalance (R 2 increases out-of-sample from-0.17% to 2.39% for the smallest to the largest decile of order size). Finally, we show that the accuracy of transaction costs forecasts depends heavily on stock volatility, with a coeffcient of 0.78.
SSRN Electronic Journal
We explore in this paper the drivers of equity portfolio selection with an active strategy, to be... more We explore in this paper the drivers of equity portfolio selection with an active strategy, to be understood as the combination of the use of a rewarded factor as an expected return, and a risk management made thanks to a risk model. The main message of the paper is that quantitative long only portfolios (built in a Markowitz-driven way) are high conviction portfolios, with few strong bets, and hence few effective (or non-zero) positions. In this respect, they share some similarity with discretionary stock pickers. This conclusion is valid either the objective of the fund is to follow the rewarded factor, either it is to target a risk strongly different from the risk of the market. We derive theoretical results and show that: (i) the long only constraint induce naturally a high concentration of the portfolio which becomes naturally parsimonious; (ii) closed-form formulas may be derived for the weights of the portfolio either for a Minimum Variance portfolio, either for a Managed Volatility portfolio with a rewarding factor; (iii) in the case of the Managed Volatility portfolio with a rewarded factor, the stocks that are selected are those that realize a trade-off between a low β and a high factor loading, both relatively to (respectively) a risk threshold and a factor threshold; (iv) those thresholds are endogenous in the sense that they depend on the risk structure and implicitly of the stocks finally kept in the portfolio, leading to a recursive procedure to select the stocks and the weights of the final portfolio. In particular, this means that the portfolio selection problem may be solved linearly instead of using an optimizer. A strong message following our results is also the essential role played by low β stocks, and by the interaction of the factor with the risk model, as the selectivity effect is higher for factors with a lower co-linearity with the risk model.
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Papers by charles-albert lehalle