Essays on forward portfolio theory and financial time series modeling
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This dissertation contains four self-contained essays that explore the application of stochastic and statistical modeling techniques to the problem of optimal portfolio choice and financial time series analysis. The first essay presents turnpike-type results for the risk tolerance function in an incomplete Ito-diffusion market setting under time-monotone for- ward performance criteria. We show that, contrary to the classical case, the temporal and spatial limits do not coincide. Rather, we establish that they depend directly on the left- and right-end of the support of an underlying measure, used to construct the forward performance criterion. We provide examples with discrete and continuous measures, and discuss the asymptotic behavior of the risk tolerance for each case. The second essay examines the long term behavior of the optimal wealth and optimal portfolio weights processes in an Ito-diffusion market under the time-monotone forward performance criteria. We show that the underlying measure [mu] associated with the forward performance criterion defines the risk profile of the investor, and in turn determines the optimal portfolio strategy and optimal wealth in the long run. The third essay considers two fund managers who trade under relative performance concerns, depending on each other’s strategies, in an Ito-diffusion market, We analyze both the passive and the competitive cases, and under both asset specialization and diversification. To allow for dynamic model re- vision and flexible investment horizons, we introduce the concept of relative forward performance for the passive case, and the notion of forward Nash equilibrium for the competitive one. For homothetic forward criteria, we provide explicit solutions for all cases. In the fourth essay, we assess the dynamics of realized betas, relative to the dynamics in the underlying market variance and covariances with the market, using 5-minute high-frequency asset prices of the DJIA component stocks from January 1, 2010 to December 31, 2014. We find that, unlike the realized variances and covariances which fluctuate widely and are highly persistence, the realized beta series, on the other hand, display much less persistence. We then construct a simple autoregressive plus noise DLM time series model for the realized beta, where the measurement error follows a normal distribution centered at zero with asymptotically valid variance given in Barndorff-Nielsen & Shepherd (2004). This approach helps us obtain samples from filtered and smoothed true underlying beta series and forecast future betas.