Abstracts

Black and Litterman (1991, 1992) developed a successful one-period mean-variance optimization model in which the expected risk premia of the assets returns incorporate views formulated by securities analysts. The model has proved extremely popular although most discussions take place in the original one-period setting and hardly any dynamic extensions exist.
We developed a continuous-time analogue to the Black-Litterman model using a standard linear filtering argument to incorporate analyst views and stochastic control to solve the asset allocation problem [QFL, 1 (2013)]. The key in our approach is that the filtering problem and the stochastic control problem are effectively separable. Hence we can incorporate analyst views and non-investable assets as observations in our filter even though they are not present in the portfolio optimization.
In this paper, we generalize the continuous time Black-Litterman model in three significant ways. First, we examine the selection of the prior, that is the initial uninformed vector of risk premia. Black and Litterman’s choice of prior through a reverse optimization process is an important reason for the success of their model. However, the question of the prior is often overlooked in filtering theory, where the prior is drawn from a given distribution. We propose a method to construct the prior in a continuous time setting.
Second, we use jump-diffusion processes to model the observations. An obvious motivation is to model asset price jumps and the higher moments of the return distribution. A less obvious but equally important reason is to define non-Gaussian confidence intervals around the analyst views. The literature on expert opinions suggests that the Gaussian distribution may not generate appropriate confidence intervals. Lévy processes give us access to a wider class of distributions and enable us to develop a more accurate probabilistic characterization of the analyst views.
Finally, we discuss applications to stochastic control problems in general and more specifically to risk-sensitive investment management models. We consider three examples: portfolio management, benchmarked asset management and asset and liability management. Although the filtering step and the stochastic control problems are effectively separable, the nature of the investment problem has an impact on both. These three examples show show the impact of a replicable benchmark and of an unhedgeable liability on the filtering process.