High Water Mark、Convex Compensation And Portfolio Choice In Hedeg Funds

With the development of hedge fund, more and more scholars began to study the problem of the optimal investment of hedge funds. The hedge fund issue has been one of the frontiers of financial mathematics, and received the extensive attention from researchers. In recent years, the researchers have considered the impact of the high water mark and incentive fees on fund managers, making the model more realistic. However, in reality, the inflation and the Knightian uncertainty affect an manager’s decisions. So, in order to theoretical modeling of investment strategies making more in line with today’s economic situations, we introduce the two factors of the Knightian uncertainty of volatility of asset prices and the inflation into our model.The difference of Knightian uncertainty and the classical risk is remarkable. The so-called risk refers to the probabilistic uncertainty while the so-called Knightian uncertainty refers to the uncertainty that the probability of occurrence of each outcome for events is still unknown. Because the impact of the Knightian uncertainty on rewards is often largely higher than risks, it is necessary to consider Knightian uncertainty.On the other hand, if we make the economic analysis of the existing data, we find that there is the impact of inflation or deflation on portfolio of hedge fund managers in the real economy. In our country, a cycle of the comprehensive inflation has emerged. For this, we will consider the inflationary factor in the portfolio of hedge fund managers.The main work of this thesis is the problem of the optimal portfolio of an hedge fund manager under high water mark and convex compensations.On one hand, we first study the strategy of the fund manager’s portfolio as the volatility of the asset price has Knightian uncertainty. We discuss a model in which a fund manager invests in a riskless asset and a risky asset disturbed by a G-Brownian motion. On the other hand, for the hedge fund with the contract of high water mark, the fund manager’s target is to maximize the expected net present value of the cumulative incentive fees. Moreover, we deduce the corresponding G-HJB (G-Hamilton-Jacobi-Bellman) equation of the value function with specific boundary conditions through the stochastic calculus and the stochastic dynamic programming method under nonlinear expectations, and the corresponding optimal portfolio strategy of the fund manager is obtained. And, the static economic analyses are given.On the other hand, we study the strategy of the fund manager’s portfolio under the inflationary environment. We discuss a model in which a fund manager invests in a riskless asset and a risky asset disturbed by a Brownian motion. For the hedge fund with the contract of high water mark, the fund manager’s target is to maximize the expected net present value of the cumulative incentive fees. Moreover, we deduce the corresponding HJB (Hamilton-Jacobi-Bellman) equation of the value function with specific boundary conditions through the stochastic calculus and the stochastic dynamic programming method, and the corresponding optimal portfolio strategy of the fund manager is obtained. And we give the static economic analyses of our results.Finally, we summarize the study results and introduce the future work plan.