Incentive Fees in Hedge Fund Management

This GitHub repository contains all the work related to my masters thesis.

Introduction

We consider an expected utility maximising manager who controls the allocation of wealth between a riskless and risky asset. The manager has power utility describing their terminal wealth preference. The manager’s compensation includes both a management fee and a performance fee based on exceeding the high-water mark of the fund. Under assumptions about the continuous-time financial market and the asset prices, we present a numerical solution of the optimal trading strategy.

Problem Formulation

Using a dynamic programming method, similar of that to solving the Merton problem, we show that a solution to the manager's optimal control problem is also a solution to a partial differential equation. We find a solution to this equation, known as the Hamilton-Jacobi-Bellman (HJB) equation, using finite difference. To understand the details of this contruction, please read the document titled MA4K9_Project_U2007120.pdf. The equation for the value function $V^c(t, x)$ is $$-\frac{\partial V^c}{\partial t}(t, x)-\sup_{\pi\in\mathcal{A}(t, x)}\left[(rx+\sigma\lambda\pi_t)\frac{\partial V^c}{\partial x}(t, x)+\frac{1}{2}\sigma^2\pi_t^2\frac{\partial^2 V^c}{\partial x^2}(t, x)\right]=0,$$ with terminal condition $$V^c(T, x)=\Phi^c(x, B_T)\quad\forall x\in\mathbb{R}.$$ Maximising over $\pi$ yields the optimal feedback control function $$\hat\pi(t, x)=-\frac{\lambda}{\sigma}\frac{V_x^c(t, x)}{V_{xx}^c(t, x)},$$ and the partial differential equation $$V_t^c(t, x) + rxV_x^c(t, x) - \frac{\lambda^2}{2}\frac{\left(V_x^c(t, x)\right)^2}{V^c_{xx}(t, x)} = 0.$$

Numerical Approximation

We use a naive finite difference method to numerically approximate these partial differential equations. Using forward Euler approximations in time and central Euler approximations in space we obtain $$V_i^{j+1} = V_i^j+\frac{rh}{2}\left(V_{i+1}^j-V_{i-1}^j\right)-\frac{\lambda^2h}{8}\frac{\left(V_{i+1}^j-V_{i-1}^j\right)^2}{V_{i+1}^j - 2V_i^j + V_{i-1}^j}\quad0\leq j\leq N\quad 1\leq i\leq M,$$ with boundary conditions $$V_i^0=\Phi^c(ik, B_T)\quad0\leq i\leq M,$$ $$V_0^j=U(K)\quad0\leq j\leq N.$$ For the approximation of the optimal control, we obtain $$\hat\pi_i^j=-\frac{\lambda k}{2\sigma}\frac{V_{i+1}^j-V_{i-1}^j}{V_{i+1}^j-2V_i^j+V_{i-1}^j}\quad0\leq j\leq N\quad 1\leq i\leq M-1.$$

The document 22052024_MA4K9_code.ipynb is a workbook that contructs these finite difference grids using dynamic programming.

Results

Please read Chapter 6.4 in MA4K9_Project_U2007120.pdf to understand the economic significance of this workbook.