# Optimal trading dynamic stock liquidation strategies

Then the form of satisfies When is not an integer, the solution of 55 is When is an integer, the solution of 55 is where and is the logarithmic derivative of the gamma function. Whenthe result is also right. The key element is that the variable temporary market impact is analyzed. There have been allegations that high-frequency traders prey on other participants making profits taking no or minimal risk in the process by predicting with near certainty where orders will be routed. The model gives an optimal investment portfolio method when optimal trading dynamic stock liquidation strategies want to get the highest return specifying their acceptable risk level.

From the concavity of 21 and the solution of 31we get Then, we obtain where and. For, andthere exists a unique Nash equilibrium for mean-variance optimization. Sasha Stoikov Cornell University The Micro-Price I define the micro-price to be the limit of a sequence of expected mid-prices and provide conditions for this limit to exist. The existence and uniqueness of the Nash equilibria optimal trading dynamic stock liquidation strategies equivalent to the existence and uniqueness of solutions for the second differential equation with variable coefficients.

From the strict concavity of optimal trading dynamic stock liquidation strategies Lagrangian and the convexity of the setthere exists at most one maximizer in. The model gives an optimal investment portfolio method when investors want to get the highest return specifying their acceptable risk level. Whenthe agent thinks that the price process is unperturbed. The equilibrium strategy is the unique solution of the following second-order system of differential equation:

Carmona and Joseph Yang [ 18 ] use 6 to deal with the problem of the maximization. Let be any subfiltration of. Under the assumption and from Definition 2 and Lemma 3the Lagrangian equation becomes then Let the right-hand side of 31 be represented by, where, and. Next, we discuss a special situation in which.

We suppose that all participators are active in the market. Huberman and Stanzl [ 12 ] show that permanent market impact must be linear in the trade quantity and symmetry between buyers and sellers in order to rule out any price manipulation strategies. The concrete form of optimal strategy in special cases is given.

For, andthere exists a unique Nash equilibrium for mean-variance optimization. Under assumption of Corollaries 8 and 9the equilibrium strategy satisfies the form When is not an integer, one has When is an integer, one gets. The initial optimization frameworks were based optimal trading dynamic stock liquidation strategies mean-variance minimization for the trading costs. The information flow is considered as a filtration on a given probability space. We incorporate a Markovian signal in the optimal trading framework which was initially proposed by Gatheral, Schied, and Slynko and provide results on the existence and uniqueness of an optimal trading strategy.

Therefore, we get forwhich implies Then, it becomes Since we get which contradicts Forlet and define Assume that maximization of the functional is given by for. At the same time, the optimal strategies are adapted to.

Table of Contents Alerts. Except the above two decay functions, Schied [ 22 ] further discusses the capped linear decay and Gaussian decay and gives some useful properties. Suppose that the unique Nash equilibrium with mean-variance optimization in Theorem 4 is.

When the temporary market impact is decreasing linearly, the optimal problem is described by a Nash equilibrium in finite time horizon. As an interesting outcome of this approach, intra-day patterns are recovered without the need of any time or cross-sectional averaging, allowing, for instance, to estimate the real-time response of the market covariances to macro-news announcements. We study the dependence of the optimal solution on the choice of the risk aversion criterion. In the past 15 years, finer optimal trading dynamic stock liquidation strategies of price dynamics, more realistic control variables and different cost functionals were developed.

On the one hand, we investigate the changes in HFTs behaviour related to the level of market stress at a daily scale. The mean-variance optimization is equivalent to the maximization of CARA utility. Therefore, we have where is the initial price, is the standard Brownian motion, the positive constant is the volatility of unaffected price process, and is an extra drift which is deterministic and continuous.