Forward optimization and real-time model adaptation with applications to portfolio management, indifference valuation and optimal liquidation

The goal of this thesis is to introduce a new, alternative approach to deal with model uncertainty and “real-time” model revisions and, in turn, develop a comparative study with existing approaches in the context of various applications in financial mathematics. This new approach is based on the forward performance criteria which adapt in a time-consistent way to “real-time” model revisions. The novelty is that these revisions are genuinely “model-free” in that they occur in “real-time”, without any modeling pre-commitment. For example, in the context of optimal liquidation (see Chapter 3 and Chapter 4), there is no a priori model for the evolution of the market impact parameter λ. It is rather assumed that this parameter switches at predictable times, to values only observable at the switching times. As such, the model revisions capture the evolving reality and allow for considerable flexibility. This forward approach thus incorporates “real-time” model revisions and is, therefore, close to adaptive optimization. On the other hand, it produces, by construction, time-consistent policies and is, thus, close to the classical optimization with model(s) pre-commitment. In other words, it can be thought as a hybrid approach that accommodates dynamic model changes while preserving time-consistency. We apply the forward approach with “real-time” model revisions in four distinct problems: portfolio management in discrete and continuous settings (binomial and lognormal, respectively), indifference valuation in lognormal models and optimal liquidation in the continuous time Almgren-Chriss model. We produce closed form solutions and characterize the optimal policies and optimal criteria. As the analysis shows, one needs to solve various sequential “inverse” optimal investment problems with random coefficients, corresponding to model revisions in real-time. We develop a comparative study with the classical settings. A main novelty is the introduction of two performance metrics which measure the discrepancies between the actual performance, and the projected or the true optimal performances under the various criteria and behavior. We study these metrics for various scenaria, related to favorable and non-favorable market changes, and compare their performance. These metrics resemble the notion of “regret”, which is now considered in a more dynamic and “real-time” manner. Among others, we show that the regret of the forward decision maker is always zero, independently of the upcoming model changes. In what follows, we describe each application separately. For each application, we introduce the model, the forward and classical criteria, construct the corresponding solutions and policies, and compare them in detail