Multivariate transient price impact and matrix-valued positive definite functions

We consider a model for linear transient price impact for multiple assets that takes cross-asset impact into account. Our main goal is to single out properties that need to be imposed on the decay kernel so that the model admits well-behaved optimal trade execution strategies. We first show that the existence of such strategies is guaranteed by assuming that the decay kernel corresponds to a matrix-valued positive definite function. An example illustrates, however, that positive definiteness alone does not guarantee that optimal strategies are well-behaved. Building on previous results from the one-dimensional case, we investigate a class of nonincreasing, nonnegative and convex decay kernels with values in the symmetric $K\times K$ matrices. We show that these decay kernels are always positive definite and characterize when they are even strictly positive definite, a result that may be of independent interest. Optimal strategies for kernels from this class are well-behaved when one requires that the decay kernel is also commuting. We show how such decay kernels can be constructed by means of matrix functions and provide a number of examples. In particular we completely solve the case of matrix exponential decay

Optimal pair-trade execution with generalized cross-impact (Financial Modeling and Analysis)

We examine a discrete-time optimal pair-trade execution problem with generalized cross-impact. This research is an extension of [14], which consider the price impact of aggregate random orders posed by small traders with a Markovian dependence. We focus on how a risk-averse large trader optimally executes two correlated assets to maximize his/her expected utility from the final wealth over a finite horizon. A stochastic dynamic programming modeling constitutes the basis for the formulation of the optimal pair-trade execution problem. Then, under some regularity conditions, the backward induction method of dynamic programing enables us to derive the optimal pair-trade execution strategy and its associated optimal value function. Besides, we reveal that the trading orders of each risky asset posed by small traders do affect the optimal execution volume of both risky assets

From macro to micro: causal inference, firm valuation and trading conditions

The aim of the Ph.D. thesis is twofold. First, we investigate possible stock market mispricing to eventually build profitable investment strategies. Second, we analyze how the microscopic interactions among agents influence trading conditions thereby leading to market instabilities. As regards the study of possible mispricing, we identify via the vector autoregressive approach revenues as the primary driver process of firm growth. To do so, we employ the recent Independent Component Analysis (ICA) technique which allows us to identify contemporaneous causal relations among the considered variables. In particular, the first original contribution of the thesis is to extend the ICA methodology for singular and noisy structural vector autoregressive models; see Chapter 2. As a second original contribution, starting from the revenues, we propose a firm valuation framework incorporating the associated intrinsic uncertainty. We derive a probability distribution of fair values, we construct a market factor capturing misvaluation comovements and we propose two stock recommendation systems that hinge on the fair value distribution; see Chapters 3, 4 and 5. Finally, in the last contribution, we analyze asymptotically market stability as the number of assets and traders increase. Market instability is defined as a result of oscillating equilibrium strategies of optimal execution problems in market impact games, where the dynamical equilibrium between the activity of simultaneously trading agents generates the price dynamics. One of the main results is the connection of market instability to the market cross-impact structure when portfolios execution orders are considered; see Chapter 7, 8 and 9