Simulation Methods for Problems with Nested Structures in Risk Measurement and R&D Projects Evaluation
風險度量與研發項目評估中嵌套結構問題的仿真方法研究
Student thesis: Doctoral Thesis
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Award date  27 Sept 2024 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(20d7e80482904f8cb491f0940af08113).html 

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Abstract
A wide range of problems in financial engineering and operations management require the estimation of conditional expectations. In financial risk measurement, efficient methods are essential for measuring the risk associated with financial assets to facilitate effective portfolio management, where the loss of the portfolio at a future risk horizon is usually expressed as the expected value of its payoff conditional on a set of risk factors. Many commonly used risk measures, such as ValueatRisk (VaR) and conditional ValueatRisk (CVaR), are typically functionals (e.g., quantiles or tail expectations) of the portfolio loss expressed as a conditional expectation. As a result, estimating these risk measures entails the estimation of the functionals of a conditional expectation that may not have a closedform formula. This type of problem is often referred to as estimation with nested structures in the literature. Similar problem formulation also finds applications in operations management. For instance, during the evaluation of investment opportunities for research and development (R&D) projects, an essential task for enterprises to make informed decisions is to determine the optimal investment strategy by comparing the continuation value and the termination value of the multistage project, where the former is often expressed as the expected value of the project payoff at the next stage conditional on the current state. Evaluating the R&D project thus requires estimating functionals of conditional expectations, and fits into the class of problems with nested structures. However, it should be noted that the nested structure in this multistage setting requires the estimation of multiple nested conditional expectations, a feature much more complicated compared to the financial risk measurement application.
As closedform expressions of conditional expectations may not be available, one often resorts to Monte Carlo simulation to estimate them. The theme of this thesis is the investigation of efficient Monte Carlo simulation methods for estimation problems involving conditional expectations, with two important applications to portfolio risk measurement and evaluation of investment opportunities in R&D projects.
First, this thesis studies the estimation of the VaR of a portfolio. In the existing literature, methods for estimating VaR were primarily developed through order statistics or by taking the inverse of the estimated distribution function of losses. Our thesis offers a new perspective on VaR estimation, exploring the fact that VaR can be estimated by solving a sample counterpart of an optimization problem. Leveraging the convexity of the objective function, we construct its lower and upper bounds, which transform the nested problem into a nonnested one, thereby enhancing the efficiency of the estimation process. To further improve estimation efficiency, importance sampling, a variance reduction method, is employed in the proposed algorithms. Numerical experiment results validate the applicability of the proposed algorithms for VaR estimation, demonstrating that the importance sampling technique may enhance the efficiency of the algorithms, particularly when the confidence level of the VaR is close to 1.
Second, this thesis investigates nested simulation methods for estimating the CVaR of the portfolio with discrete losses. While discrete losses are highly relevant to realworld scenarios, research on this aspect is notably less developed in the existing literature compared to studies on the case with continuous losses. Tailored to the discrete nature of portfolio losses in practice, a rounded estimator is proposed for CVaR estimation in this thesis. When the portfolio loss follows a subGaussian distribution or has a sufficiently highorder moment, the mean squared error (MSE) of the resulting CVaR estimator decays to zero at a rate close to Γ^{−1}, which is significantly faster than the rate of the CVaR estimator in the continuous case which is Γ^{−2/3}, where Γ denotes the sampling budget required by the nested simulation procedure. The results of numerical experiments are consistent with the proposed theory, demonstrating the effectiveness of the CVaR estimator based on discrete losses introduced in this thesis.
Third, this thesis explores efficient algorithms for evaluating the investment opportunities of R&D projects with multiple stages, which can assist investors or enterprises in making prudent investment decisions to optimize resource allocation, minimize risks, and maximize potential returns. The investment opportunities of R&D projects can be viewed as compound real options, typically evaluated using real options analysis (ROA). R&D projects may involve multiple stages, where investors make use of the continuation value and the termination value at each stage to determine the optimal investment strategy. Since the continuation value can be expressed as a conditional expectation based on the current state of the project, the structure of the problem involves multiple conditional expectations nested within each other, making it more intricate than standard nested estimation problems. Additionally, ROA in the existing literature usually assumes that the price dynamics of the underlying asset follow a specific stochastic process, such as Geometric Brownian Motion (GBM), which limits their applicability in more general practical scenarios. This thesis proposes novel simulationbased algorithms that are not constrained by the dynamic process of the underlying asset or the number of stages, providing a more flexible approach for estimating the compound real options of R&D projects and making the proposed algorithms more suitable for practical applications. By introducing the lower and upperbound estimation methods, the nested structure of the problem is transformed into a nonnested one, significantly enhancing estimation efficiency. Theoretical analysis demonstrates the convergence properties of the proposed estimators under mild conditions. Numerical experiments and case studies confirm the effectiveness of the proposed simulation algorithms.
As closedform expressions of conditional expectations may not be available, one often resorts to Monte Carlo simulation to estimate them. The theme of this thesis is the investigation of efficient Monte Carlo simulation methods for estimation problems involving conditional expectations, with two important applications to portfolio risk measurement and evaluation of investment opportunities in R&D projects.
First, this thesis studies the estimation of the VaR of a portfolio. In the existing literature, methods for estimating VaR were primarily developed through order statistics or by taking the inverse of the estimated distribution function of losses. Our thesis offers a new perspective on VaR estimation, exploring the fact that VaR can be estimated by solving a sample counterpart of an optimization problem. Leveraging the convexity of the objective function, we construct its lower and upper bounds, which transform the nested problem into a nonnested one, thereby enhancing the efficiency of the estimation process. To further improve estimation efficiency, importance sampling, a variance reduction method, is employed in the proposed algorithms. Numerical experiment results validate the applicability of the proposed algorithms for VaR estimation, demonstrating that the importance sampling technique may enhance the efficiency of the algorithms, particularly when the confidence level of the VaR is close to 1.
Second, this thesis investigates nested simulation methods for estimating the CVaR of the portfolio with discrete losses. While discrete losses are highly relevant to realworld scenarios, research on this aspect is notably less developed in the existing literature compared to studies on the case with continuous losses. Tailored to the discrete nature of portfolio losses in practice, a rounded estimator is proposed for CVaR estimation in this thesis. When the portfolio loss follows a subGaussian distribution or has a sufficiently highorder moment, the mean squared error (MSE) of the resulting CVaR estimator decays to zero at a rate close to Γ^{−1}, which is significantly faster than the rate of the CVaR estimator in the continuous case which is Γ^{−2/3}, where Γ denotes the sampling budget required by the nested simulation procedure. The results of numerical experiments are consistent with the proposed theory, demonstrating the effectiveness of the CVaR estimator based on discrete losses introduced in this thesis.
Third, this thesis explores efficient algorithms for evaluating the investment opportunities of R&D projects with multiple stages, which can assist investors or enterprises in making prudent investment decisions to optimize resource allocation, minimize risks, and maximize potential returns. The investment opportunities of R&D projects can be viewed as compound real options, typically evaluated using real options analysis (ROA). R&D projects may involve multiple stages, where investors make use of the continuation value and the termination value at each stage to determine the optimal investment strategy. Since the continuation value can be expressed as a conditional expectation based on the current state of the project, the structure of the problem involves multiple conditional expectations nested within each other, making it more intricate than standard nested estimation problems. Additionally, ROA in the existing literature usually assumes that the price dynamics of the underlying asset follow a specific stochastic process, such as Geometric Brownian Motion (GBM), which limits their applicability in more general practical scenarios. This thesis proposes novel simulationbased algorithms that are not constrained by the dynamic process of the underlying asset or the number of stages, providing a more flexible approach for estimating the compound real options of R&D projects and making the proposed algorithms more suitable for practical applications. By introducing the lower and upperbound estimation methods, the nested structure of the problem is transformed into a nonnested one, significantly enhancing estimation efficiency. Theoretical analysis demonstrates the convergence properties of the proposed estimators under mild conditions. Numerical experiments and case studies confirm the effectiveness of the proposed simulation algorithms.