Browsing by Subject "Stochastic control"
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Item Asymptotics for optimal investment with high-water mark fee(2015-08) Kontaxis, Andrew; Sîrbu, Mihai; Gamba, Irene M; Mendoza-Arriaga, Rafael; Zariphopoulou, Thaleia; Zitkovic, GordanThis dissertation studies the problem of optimal investment in a fund charging high-water mark fees. We consider a market consisting of a riskless money-market account and a fund charging high-water mark fees at rate λ, with share price given exogenously as a geometric Brownian motion. A small investor invests in this market on an infinite time horizon and seeks to maximize expected utility from consumption rate. Utility is taken to be constant relative risk aversion (CRRA). In this setting, we study the asymptotic behavior of the value function for small values of the fee rate λ. In particular, we determine the first and second derivatives of the value function with respect to λ. We then exhibit for each λ explicit sub-optimal feedback investment and consumption strategies with payoffs that match the value function up to second order in λ.Item Computational methods for stochastic control problems with applications in finance(2014-05) Mitchell, Daniel Allen; Muthuraman, KumarStochastic control is a broad tool with applications in several areas of academic interest. The financial literature is full of examples of decisions made under uncertainty and stochastic control is a natural framework to deal with these problems. Problems such as optimal trading, option pricing and economic policy all fall under the purview of stochastic control. These problems often face nonlinearities that make analytical solutions infeasible and thus numerical methods must be employed to find approximate solutions. In this dissertation three types of stochastic control formulations are used to model applications in finance and numerical methods are developed to solve the resulting nonlinear problems. To begin with, optimal stopping is applied to option pricing. Next, impulse control is used to study the problem of interest rate control faced by a nation's central bank, and finally a new type of hybrid control is developed and applied to an investment decision faced by money managers.Item Essays on achieving investment targets and financial stability(2013-05) Monin, Phillip James; Zariphopoulou, Thaleia, 1962-This dissertation explores the application of the techniques of mathematical finance to the achievement of investment targets and financial stability. It contains three self-contained but broadly related essays. Sharpe et al. proposed the idea of having an expected utility maximizer choose a probability distribution for future wealth as an input to her investment problem rather than a utility function. They developed the Distribution Builder as one way to elicit such a distribution. In a single-period model, they then showed how this desired distribution for terminal wealth can be used to infer the investor's risk preferences. In the first essay, we adapt their idea, namely that a desired distribution for future wealth is an alternative input attribute for investment decisions, to continuous time. In a variety of scenarios, we show how the investor's desired distribution, combined with her initial wealth and market-related input, can be used to determine the feasibility of her distribution, her implied risk preferences, and her optimal policies throughout her investment horizon. We then provide several examples. In the second essay, we consider an investor who must a priori liquidate a large position in a primary risky asset whose price is influenced by the investor's liquidation strategy. Liquidation must be complete by a terminal time T, and the investor can hedge the market risk involved with liquidation over time by investing in a liquid proxy asset that is correlated with the primary asset. We show that the optimal strategies for an investor with constant absolute risk aversion are deterministic and we find them explicitly using calculus of variations. We then analyze the strategies and determine the investor's indifference price. In the third essay, we use contingent claims analysis to study several aggregate distance-to-default measures of the S&P Financial Select Sector Index during the years leading up to and including the recent financial crisis of 2007-2009. We uncover mathematical errors in the literature concerning one of these measures, portfolio distance-to-default, and propose an alternative measure that we show has similar conceptual and in-sample econometric properties. We then compare the performance of the aggregate distance-to-default measures to other common risk indicators.