Upper Confidence Bound: Methods for Order Statistics with Applications to Risk Measurement

上置信界算法: 次序統計量方法及其在風險管理中的應用

Student thesis: Doctoral Thesis

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Award date30 Mar 2022


Portfolio risk measurement has attracted increasing attention these days. Due to the importance and ubiquity of risk in financial activities, it is necessary for financial institutions to analyze and manage the risks entailed by a financial decision. In this thesis, we focus mainly on two kinds of risk measures, which are value-­at-­risk (VaR) and conditional value­-at-­risk (CVaR), and we put forward an upper confidence bound (UCB) methodology to estimate the risk measures. This thesis consists of two topics.

Firstly, we solve the top 𝑀 identification problem via an upper confidence bound methodology, which is famous for balancing the exploration-­exploitation trade-­off in the multi­-armed bandit (MAB) problem. In the top 𝑀 identification problem, each arm provides an unknown random reward after a pull. The goal of the agent is to identify the top 𝑀 arms with the largest means and estimate the top 𝑀 means after a fixed number of pulls. During the course of top 𝑀 arms identification, estimators for the order statistics of top 𝑀 means are constructed at the same time. Strong consistency, rate of MSE, central limit theorem and confidence intervals are provided, which offers statistical guarantees for these estimators.

Secondly, we apply the upper confidence bound methodology to estimate VaR and CVaR. Different from standard nested simulation, the upper confidence bound methodology sequentially allocates computational effort to inner step simulations nonuniformly in a discrete scenario space. More specifically, we allocate additional computational budget to scenarios entailing larger losses. For any 𝛼 ∈ (0, 1), the VaR of level 𝛼 is the 𝛼­-quantile of loss 𝐿. The 𝛼­-VaR is just the smallest value of large losses, i.e., the first order statistic of large losses, where large losses denote the upper (1 − 𝛼)­-tail losses. The 𝛼­-CVaR is the mean of all 𝛽-­VaR with 𝛽 ∈ (𝛼, 1). In other words, the 𝛼-­CVaR is just the average of large losses, where large losses denote the upper (1 − 𝛼)­-tail losses. The upper confidence bound methodology identifies and estimates the large losses among a fixed number of portfolio losses at the same time, thus the estimation of VaR and CVaR is solved. Besides, we build statistical guarantees for the estimators of VaR and CVaR. When Γ is the total simulation budget, the convergence rate of the MSE of our method is of order Γ−1, which outperforms nested simulation. A numerical study is consistent with the theory we present.