Meng, Fei2020-09-212020-09-212020-09-212020-09-10http://hdl.handle.net/10012/16333Hedge funds have specialized fee structures, often including performance fees designed to align the incentives of investors and fund managers. However, hedge funds have faced intense scrutiny since the financial crisis, as the fees they charge investors have been outsized compared to the returns. Consequently, innovative fee structures have emerged aiming at better alignment between investors' interests and the hedge fund business objective. In this thesis, we present mathematical and numerical analyses of many aspects of hedge fund investments with three fee structures, first-loss, shared-loss and negative fee structure. The motivation for this is to understand an important new investment type and its implications for investors and managers, as well as the mathematical problems that it poses. In Chapter 2 and 3, we investigate the optimal withdrawal time of a first-loss or shared-loss hedge fund fee structure from an investor's perspective. Given that a hedge fund dynamic follows a geometric Brownian, calculating the optimal withdrawal time entails solving an optimal stopping problem with a continuous piece-wise linear payoff function. In particular, we explicitly solve the problem in the infinite horizon case. Next, we show that there exist two monotonic and continuous early exercise boundaries and derive an early exercise premium integral representation in the finite horizon case. Finally, we analyze the asymptotic behavior of the early exercise boundaries near maturity. In Chapter 4, we test the hypothesis of fee diversification. In particular, we study the optimization problem of an investor who may choose any combination of the first loss and classical fee structures, and seeks to maximize either the Sharpe ratio or the Sortino ratio of their final payoff, evaluated using real-world probabilities. We demonstrate that for the vast majority of fund mean returns and volatilities, there is no fee diversification effect: either the first-loss structure or the classical structure is optimal for the investor. In Chapter 5, we present an analysis of the value and risks for negative fee structure. We begin by employing risk-neutral valuation, using both Black-Scholes and regime-switching models. We then proceed to analyze the risks inherent in investments in hedge funds with negative fee structures. Given the resemblance of these investments to asset-backed securities, we in particular study probability of default and loss given default under the real-world measure for both geometric Brownian motion and regime-switching models.enMathematical FinanceOptimal StoppingDoctoral Thesis