Discrete homotopy analysis for optimal trading execution with nonlinear transient market impact

Highlights

• Optimal execution problem in presence of a concave instantaneous market impact.

• Application of the homotopy analysis method to a weakly singular Urysohn equation of the first kind.

• Development of a discrete homotopy analysis to deal with the no-closed form of definite integrals under exam.

Abstract

Optimal execution in financial markets is the problem of how to trade a large quantity of shares incrementally in time in order to minimize the expected cost. In this paper, we study the problem of the optimal execution in the presence of nonlinear transient market impact. Mathematically such problem is equivalent to solve a strongly nonlinear integral equation, which in our model is a weakly singular Urysohn equation of the first kind. We propose an approach based on Homotopy Analysis Method (HAM), whereby a well behaved initial trading strategy is continuously deformed to lower the expected execution cost. Specifically, we propose a discrete version of the HAM, i.e. the DHAM approach, in order to use the method when the integrals to compute have no closed form solution. We find that the optimal solution is front loaded for concave instantaneous impact even when the investor is risk neutral. More important we find that the expected cost of the DHAM strategy is significantly smaller than the cost of conventional strategies.

Introduction

The optimization of trading strategies has long been an important goal for investors in financial markets. As demonstrated in the context of a linear equilibrium model by Kyle thirty years ago [33], the optimal strategy for an investor with insider information on the fundamental price of an asset is to trade incrementally through time. This strategy allows the trader to minimize costs whilst also minimizing the revelation of information to the rest of the market. The precise way in which it is optimal to split the large order (herein called metaorder) [55] depends on the objective function and on the market impact model, i.e. the change in price conditioned on signed trade size. In part due to the increasing tendency toward a full automation of exchanges and in part due to the discovery of new statistical regularities of the microstructure of financial markets, the problem of optimal execution is receiving growing attention from the academic and practitioner communities [6], [25], [29].

As pointed out in Gatheral et al. [26], a first generation of market impact models, studied by Bertsimas, Lo and Almgren [12], [13], [14], [17], distinguishes between two impact components. The first component is temporary and only affects the individual trade that has triggered it. The second component is permanent and affects all current and future trades equally. These models can be either in discrete or in continuous time and can assume either linear or nonlinear market impact for individual trades. The second generation of market impact models focuses on the transient nature of market impact [18], [20], [29], [44]. In such models, market impact is typically assumed to factorize into two components: instantaneous market impact and a decay component. The instantaneous component models the reaction of price to traded volume. The decay component describes how the market price relaxes on average after the execution of an order. In such models, each trade affects future price dynamics with an intensity that decays with time.

The problem of optimal execution in the presence of transient impact has been considered in a series of recent studies. In the case of linear instantaneous market impact [9], [21], [26], the problem has been completely solved by showing that the cost minimization problem is equivalent to solving an integral equation. In particular Gatheral et al. [26] proved that optimal strategies can be characterized as measure-valued solutions of a Fredholm integral equation of the first kind. They show that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time. This extends the result of Alfonsi et al. [10] on the non-existence of transaction triggered price manipulation, i.e. strategies where the expected execution costs of a sell (buy) program can be decreased by intermediate buy (sell) trades.

However, a series of empirical studies [16], [20], [38] has clearly shown that the instantaneous market impact is a strongly concave function of the volume, well approximated by a power law function. The resulting optimal execution problem in the presence of nonlinear and transient impact is mathematically much more complicated than the linear case. Some important results in the nonlinear transient case were established by Gatheral [29] who showed that under certain conditions, the model admits price manipulation, i.e. the existence of round trip strategies with positive expected revenues. This money machine should of course be avoided in the modeling of market impact. In particular Gatheral set some necessary conditions for the absence of price manipulation (see below for details and Ref. [23]). A step toward the solution of the optimal execution problem under nonlinear transient impact has been made recently by Dang [24]. In this paper, Dang suggests a way to convert the cost minimization problem into a nonlinear integral equation and proposes a numerical fixed point method on a discretization of the trading time interval to solve this equation.

In this paper we propose the Homotopy Analysis Method (HAM) [34], [35], [37] to solve the integral equation proposed by Dang [24]. This method is now widely used in dealing with non-linear equations. Nonlinear equations are difficult to solve, especially analytically. Perturbation techniques are widely applied in science and engineering, and give a great contribution to help us understand nonlinear phenomena [43]. However, it is well known that perturbation methods are strongly dependent upon small/large physical parameters, and therefore are valid in principle only for weakly nonlinear problems. The so-called non-perturbation techniques, such as the Lyapunov’s artificial small parameter method [40], the δ-expansion method [15], Adomian’s decomposition method [7], are formally independent of small/large physical parameters. The problem is that all of these traditional non-perturbation methods cannot ensure the convergence of the solution series: they are in fact only valid for weakly nonlinear problems too. The homotopy analysis method [37] is a general analytic approach to get series solutions of various types of nonlinear equations, including algebraic equations, ordinary differential equations, partial differential equations, recently linear and nonlinear integral equations [30], [53] and coupled version of them. Unlike perturbation methods, the HAM is independent of small/large physical parameters, and thus it is valid no matter whether a nonlinear problem contains small/large physical parameters or not. More important, differently from all perturbation and traditional non-perturbation methods, the HAM provides us a simple way to ensure the convergence of solution series, and therefore, the HAM is valid even for strongly nonlinear problems [34], [35]. More and more researchers have successfully applied this method to various nonlinear problems in science and engineering, such as the viscous flows of non-Newtonian fluids [52], the KdV-type equations [1], nonlinear heat transfer [2], projectile motion [54], magneto-hydrodynamics [8], Burgers–Huxley equation [42]. The HAM has been successfully applied to solve a few problems in finance [45], [56], e.g. in the case of American Put Option (Ref. [22] and Chapter 13 in [37]). Abbasbandy and Shivanian [3] described the usage of HAM for solving the nonlinear Fredholm integral equation of the second kind. In this case the uniqueness of solution was proven and the sufficient condition for convergence of the created series was given.

We apply the HAM to solve a weakly singular Urysohn integral equation of the first kind. The particular form of nonlinearity, described by a nonlinear transient market impact, requires homotopy-derivatives relative to a bi-dimensional system [37]. Moreover, this approach implies the computation of several definite integrals, which cannot be solved analytically. This technical problem requires the use of a discretization of integrals involved in computations and led us to discretize the deformation equations relative to HAM. A similar approach was followed by Allahviranloo and Ghanbari [11] to solve nonlinear Fredholm integral equations of the second kind. The method starts from an initial guess for the trading strategy and deforms it continuously in order to find better and better approximations of the solution of the integral equation. In doing this, we are implicitly restricting the space of solutions to continuous nonvanishing functions of the trading rate. We find that the optimal solution is a non time-symmetric U-shape; in the case of concave instantaneous impact, it is optimal to trade more at the beginning of the metaorder in presence of a buy program. A comparative cost analysis shows that our solution outperforms conventional strategies.

The paper is organized as follows. In Section 2, we state the problem and explain why it is difficult to solve. In Section 3, we present the HAM approach to the solution of the cost minimization problem. In Section 4 we show the HAM solutions for three different cases of study. In Section 5, we summarize and conclude. In the Appendix A.1 we report the details on the computation of the homotopy-derivative for a bi-dimensional system.