We consider the optimal trade execution strategies for a large portfolio of single stocks proposed by Almgren (2003). This framework accounts for a nonlinear impact of trades on average market prices. The results of Almgren (2003) are based on the assumption that no shares of assets per unit of time are trade at the beginning of the period. We propose a general solution method that accomodates the case of a positive stock of assets in the initial period. Our findings are twofold. First of all, we show that the problem admits a solution with no trading in the opening period only if additional parametric restrictions are imposed. Second, with positive asset holdings in the initial period, the optimal execution time depends on trading activity at the beginning of the planning period.
The execution of large trades in financial markets requires the balance between risks and costs. The main risk concerns the lack of availability of a counterparty, which can lead to a delay in the execution of a transaction. In order to guarantee a fast trade execution, a trader may incur additional costs. As clarified by Hasbruock and Schwartz (1988), a trader faces a choice between a 'passive' and an 'actice' execution strategy.
Given this background, the available models of optimal execution assume that the trading activity of individual investors has an impact on the average price prevailing in the market. The transaction costs are characterized by parametric forms that replicate stylized facts documented in the market microstructure literature (e.g. see Kraus and Stoll, 1972). Almgren andChriss (1999, 2000) and Konishi and Makimoto (2001) provide examples of optimal strategies for the execution problem in the stock market. Their models assume that the transaction cost per share is a linear function of the number of shares of assets traded. The only source of uncertianty consists in the volatility of the stock price. Almgren (2003) suggests that the linearity assumption is largely at odds with reality.
First, the average liquidity premium on stocks tends to be either a convex or a concave function of the traded size. This depends on the counterparty’s perception about the reason for the trade, namely on whether it is driven by liquidity or information needs (see Huang and Stoll, 1997). Moreover, the liquidity premium is related to the risk of finding a counterparty. In other words, the lower the probability of finding a counterparty in the market, the higher the liquidity premium.
In this paper, we review the optimal transaction strategy proposed by Almgren (2003).
We show that the solution method used by Almgren (2003) is ill-posed. The reason is that it is based on the assumption that no shares per unit of time are exchanged at the beginning of the period. We use an approach based on the Gauss hypergeometric function to solve for the case of positive initial trades. Our results differ strongly from those of Almgren (2003). First of all, the problem admits a solution with no trading in the opening period only if additional parametric restrictions are imposed. Second, with positive initial trading, the optimal execution time depends on trading activity in the initial period. This note is organized as follows. Section 2 provides a selected discussion of the literature on optimal trade execution. Section 3 outlines the structure of the problem.
Section 4 proposes a general solution method for positive initial values of the velocity. Section 5 concludes. Finally, in Appendix A, we discuss the general method for the solution of second order differential equations with a Gauss hypergeometric function.
Rebalancing portfolios of assets requires executing trades in the marketplace. With the advent of algorithmic trading and access availability to many alternative trading venues, investors have dedicated increasing resources to the scheduling of trades. The practical setup of the problem is rather intuitive. An investor has a target number of, say shares that it intends to sell or buy within a given time frame. The decision problem requires to compute how many shares to place or demand in the market at each point in time within the trade horizon. The aim of the investor is to minimize the execution costs. These are typically measured as the difference between the price obtained from the market and a benchmark price for the transaction.
There are multiple relevant dimensions to the execution problem. Several contributions have showed that the liquidity premium is time-varying. The reason is that it is determined by the availability of traders willing to act as counterparties, namely traders willing to buy or sell a given quantity of an asset at a desired price. However, as the presence of traders willing to ’take the other side’ of a trade is uncertain, any trading is characterized by execution risk.
Another relevant aspect is related to the fact that market illiquidity generates transaction costs. This typically takes the form of a large spread between bid and ask prices (Huang and Stoll, 1997). Therefore, as noticed by Wagner and Banks (1992), the minization of transaction costs is a key aspect of the portfolio optimization problem. As documented in various studies including Chakravarty (2001), Holthausen, Leftwich and Mayers (1990) and Kraus and Stoll (1972), large trades do impact market prices and, thus affecting the bid-ask spreads.
Asset price volatility is a source for execution risk. The reason is that it affects the probability of finding a suitable counterparty. Hence, it affects the sucessfulness of a trading strategy execution. The recent literature has focused on the specific aspect of volatility, namely the increased uncertainty in execution price incurred by rapid execution of large share
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