Transient Linear Price Impact and Fredholm Integral Equations

Abstract

We consider the linear-impact case in the continuous-time market impact model with transient price impact proposed by Gatheral. In this model, the absence of price manipulation in the sense of Huberman and Stanzl can easily be characterized by means of Bochner's theorem. This allows us to study the problem of minimizing the expected liquidation costs of an asset position under constraints on the trading times. We prove that optimal strategies can be characterized as measure-valued solutions of a generalized Fredholm integral equation of the first kind and analyze several explicit examples. We also prove theorems on the existence and nonexistence of optimal strategies. We show in particular that optimal strategies always exist and are nonalternating between buy and sell trades when price impact decays as a convex function of time. This is based on and extends a recent result by Alfonsi, Schied, and Slynko on the nonexistence of transaction-triggered price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest.

Optimal execution with non-linear transient market impact

“…The optimization problem of minimizing the expected cost of equation (2.2) under the constraint of equation (2.3) in the case of linear impact, f (ẋ) ∝ẋ, has been solved and widely studied [23]. In what follows, we use the symbol v(t) to indicate the rate of tradingẋ (t).…”

“…In the case of linear instantaneous market impact [23,5,19], 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 [23] 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.…”

Quintet Volume Projection

“…Papers in the literature include Clark (1973), Epps and Epps (1976), Tauchen and Pitts (1983), Karpoff and Boyd (1987), Easley and O'Hara (1987), Admati and Pfleiderer (1988), Jain and Joh (1988), Foster and Viswanathan (1990), Gallant, Rossi, and Tauchen (1992), Chan, Christie, and Schultz (1995), Kaastra andBoyd (1995), Andersen (1996), Gouriéroux, Jasiak, and Fol (1999), Chan and Fong (2000), Lo and Wang (2000), Manganelli (2005), Darat, Rahman, and Zhong (2003), Giot, Laurent, and Petitjean (2010), Manchaldore, Palit, and Soloviev (2010), and Brownlees, Cipollini, and Gallo (2011 Estimating market impact from trades empirically is discussed in the papers by Bouchaud, Gefen, Potters, and Wyart (2004), Almgren et al (2005), Engle, Furstenberg, and Russell (2008), Obizhaeva and Wang (2006), Gatheral, Schied, and Slynko (2012), and reviewed in Gatheral and Schied (2013). A theoretical foundation for the permanent impact from a trade is provided in the paper by Kyle (1985), who derives a linear equilibrium from fundamental principles.…”

Mean-Field Games and Ambiguity Aversion

“…In optimal execution problems, an agent is required to choose an efficient trading strategy for liquidating/acquiring a portfolio containing a large quantity of a given security in order to maximize her expected wealth (by minimizing cost or maximizing profit). In this problem formulation, there is a tradeoff between trading slowly to reduce market impact, or execution cost, and trading fast to reduce the risk of future uncertainty in prices (see e.g., Almgren and Chriss (2001); ; Gatheral et al (2012); Guéant and Lehalle (2013);Jaimungal and Kinzebulatov (2014); Cartea and Jaimungal (2016) among many others). Optimal limit order placement aims instead to benefit from buying low and selling high and benefit from the bid-ask spread without being adversely selected.…”