Ebook Risk aversion and the dynamics of optimal liquidation strategies in illiquid markets
A standard service of investment banks is the execution of large trades. Unlike for small trades, the liquidation of a large portfolio is a very complex task: an immediate execution is often not possible or only at a very high cost due to insufficient liquidity. Significant added value therefore lies in the experience in exercising an order in a way that minimizes execution costs for the client. Triggered by the introduction of electronic trading systems by many exchanges, automatic order execution has become an alternative to manually worked orders.
Our goal in this paper is to determine the adaptive trading strategy that maximizes the expected utility of the proceeds of an asset sale1. We address this question in the continuous-time liquidity model introduced by Almgren (2003) with an infinite time horizon and linear price impact (see also Bertsimas and Lo (1998), Almgren and Chriss (1999), and Almgren and Chriss (2001) for discrete-time precursors of this model). Since we consider a wide range of utility functions, we cannot hope to find closed-form solutions for the optimal trading strategies. Instead, we pursue a stochastic control approach and show that the value function and optimal control satisfy certain nonlinear parabolic partial differential equations. These PDEs can be solved numerically, thus providing a computational solution of the problem. But perhaps even more importantly, the PDE characterization facilitates a qualitative sensitivity analysis of the optimal strategy and the value function.
It turns out that the absolute risk aversion of the utility function is the key parameter that determines the optimal strategy by defining the initial condition for the PDE of the optimal strategy. The optimal strategy thus inherits monotonicity properties of the absolute risk aversion. In particular, we show that investors with increasing absolute risk aversion (IARA) should sell faster when the asset price rises than when it falls.
The optimal strategy is hence “aggressive in-the-money” (AIM). On the other hand, investors with decreasing absolute risk aversion (DARA) should sell slower when asset prices rise, i.e., should pursue a strategy that is “passive in-the money” (PIM). In general, adaptive liquidation strategies can realize higher expected utility than static liquidation strategies which do not react to asset price changes: static strategies are optimal only for investors with constant absolute risk aversion.
The preceding characterization of AIM and PIM strategies is a consequence of the more general fact that the optimal trading strategy is increasing in the absolute risk aversion of the investor. Surprisingly, however, very few monotonicity relation exists with respect to the other model parameters. For example, a larger asset position can lead to a reduced liquidation speed. Moreover, reducing liquidity by increasing the temporary price impact can result in an increased liquidation speed. The occurrence of the preceding anomaly, however, depends on the risk profile of the utility function, and we show that it cannot happen in the IARA case.
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