Analysis of Real Options and Wealth Management Problems Using Non-smooth Variational Inequalities and Asymptotic Methods
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We consider investment problems involving the continuous time stochastic optimization models. The dissertation consists of two parts. In the first part, we consider a real options problem, which is posed as a stochastic optimal control problem. The investment strategy, which plays the role of control, involves a onetime option to expand (invest) and a one-time option to abandon (terminate) the project. The timing and amount of the investment and the termination time are parameters to be optimized in order to maximize the expected value of the profit. This stochastic optimization problem amounts to solving a deterministic variational inequality in dimension one, with the associated obstacle problem. Because we consider both cessation and expansion options and fixed and variable costs of expansion, the obstacle is non-smooth. Due to the lack of smoothness, we use the concept of a weak solution. However, such solutions may not lead to a straightforward investment strategy. Therefore, we further consider strong (C1) solutions based on thresholds. We propose sufficient conditions for the existence of such solutions to the variational inequality with a non-smooth obstacle in dimension one. When applied to the real options problem, these sufficient conditions yield a simple optimal investment strategy with the stopping times defined in terms of the thresholds. In the second part, we develop a dynamic wealth management model for risk-averse investors displaying present bias in the form of hyperbolic discounting. The investor chooses an optimal consumption policy and allocates her funds between a risk-free asset, a traded liquid asset, and a non-traded illiquid asset. We characterize these policies for both sophisticated and naive present-biased investors. There are three results. First, sophisticated investors over-consume more than their naive counterparts if and only if their coefficient of relative risk-aversion is smaller than one. As a result, sophistication is welfare reducing (increasing) when risk-aversion is low (high). Second, increasing asset illiquidity always benefits the sophisticated investor more than the naive investor. Thus, the welfare gap between sophisticated and naive investors is increasing in the proxy for asset illiquidity. Finally, present-biased investors accumulate a larger share of their wealth in the non-traded illiquid asset than in the traded risky stock compared to the neoclassical exponential discounter investor. As a consequence, from the perspective of present-biased investors, the equity premium puzzle and the private equity puzzle are two sides of the same coin.