# Overview of Contemporary Portfolio Management Models and Their Evolution

Davis and Norman (1990) introduced a consumption-investment problem for a CRRA agent with proportional transaction costs and obtained a closed-form solution for it. Another advantage of the model was allowing for discontinuous strategy. For this purpose, the original Merton framework had to be upgraded to semimartingale dynamics. Portfolio value in each of the assets is described by the following equations:

dXt = (rtXt — Ct) dt — (1 + A) dLt + (1 — p) dMt, X0 = x,

dY t = aYt dt + oYt dwt + dLt — dMt, Y0 = y,

where coefficients A, p define proportional transaction costs, Lt, Mt are cumulative amounts of bought and sold risky asset respectively. Results demonstrated the existence of three behavioral regions for portfolio managers, and are presented in Fig. 3.

Unlike Merton’s case, the so-called wait region appears due to transaction costs. That is, it is suboptimal to trade while in the area. Leaving the area leads to immediate buy or sell to get to the wait region’s border. Analogous results were also obtained for the infinite horizon problem by Shreve and Soner (1994).

Another extension of the Merton model was presented by Framstad et al. (2001) for jump diffusion price dynamics. It was shown that wait region is absent in this case. That is, this strategy’s structure is the same as for Merton’s continuous diffusion market.

A number of papers considered a price impact model instead of unrealistic ‘fundamental price’ dynamics. For example, Vath et al. (2007) presented the following complex price impact function, depending on current price and volume

of a triggering trade. Around that time, Zakamouline (2002) took another step toward a realistic market model that allowed both proportional and fixed transaction costs. The proportional component described costs due to insufficient liquidity of the market, while the fixed component represented the participation fee for each transaction. Both papers considered discrete trading and produced interesting results. Buy and sell borders were no longer straight lines, as seen in Fig. 3, but still could be obtained beforehand and then used for decision making during trading sessions.

Neither of the abovementioned models considered the form and dynamics of the limit book itself—only the dynamic of an aggregated of a deal, which was considered as price. Microstructure models of electronic limit order markets have become quite popular in literature devoted to the problem of optimal liquidation of a portfolio. This particular case differed from the consumption-investmentframework due to the terminal condition—predefined volume of the portfolio to be liquidated. The most notable results in this field are from Almgren and Chriss (1999) and Obizhaeva and Wang (2012). The framework has become quite popular in practice due to the simple models and intuitive results. Both approached considered discrete strategies and defined optimality functional not through utility function, but as a weighted sum of expected value and standard deviation of portfolio value.

The work of Obizhaeva and Wang first appeared as a draft in 2005 and considered a flat static structure of the limit book. Their approach has been adopted by many authors, evolving into several directions. The most realistic models were presented by Predoiu et al. (2011) and Fruth et al. (2011). Predoui et al. consider a general form of order distribution inside a book and non-adaptive strategies of liquidation. Fruth et al. postulate a flat but dynamic form of order distribution while allowing for both discrete and continuous trading in the same framework, linear permanence and general temporary price impact; the described model does not allow several kinds of arbitrage and non-adaptive strategies, which proved to be optimal in the

framework. Analytical solutions have been obtained for discrete cases and for continuous trading.

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