Abstract
Portfolio risk measurement has received increasing attention among researchers in recent years. In this thesis, we focus mainly on portfolio risk measurement in the nested setting and multiple-period risk measurement. This thesis consists of three topics.First, we study portfolio risk measurement in the nested setting via standard nested simulation, where a two-level simulation proceeds. The outer-level simulation generates financial scenarios, whereas the inner-level estimates future portfolio values in each scenario and then computes portfolio losses. We provide central limit theorems and then construct asymptotically valid confidence intervals for portfolio risk via standard nested simulation. The central limit theorems specifically demonstrate that the convergence rate of the estimator is n-1/2 only if m = O(n1/2.a(n)) with a function a(n) of n such that a(n) →∞ as n →∞, where n and m are the outer-and inner-level sample sizes, respectively.
Second, we study portfolio risk measurement in the nested setting via the stochastic mesh method. In this part, the mean squared error (MSE) of the portfolio risk's stochastic mesh estimators is analyzed, and its convergence rate is Γ-1 which is the same as the rate by non-nested simulation, where Γ measures computational effort. We also provide central limit theorems and then construct asymptotically valid confidence intervals for portfolio risk via the stochastic mesh method. The method we introduce is specifically feasible for a portfolio consisting of path-dependent derivatives. A numerical study is consistent with the theory we present.
Third, we investigate multi-period risk measures which serve as an important tool in the financial industry. To manage the risk in multiple periods, we need to identify a suitable risk measure and know how to calculate its value so that it can be used in practice. In this paper, we introduce a coherent and time consistent multi-period risk measure which is constructed by an average value at risk (AVaR), called the conditional average value at risk, and we study its estimation. We demonstrate that this risk measure can be written as a dynamic programming problem which can then be decomposed into subproblems of estimating one-period AVaRs in each period. With this dynamic programming structure, the estimation of a conditional AVaR involves conditional expectations and conditional quantiles, for which we propose the least-squares method (LSM) and quantile regression respectively. A numerical study demonstrates that the method works well for large portfolios. This suggests that the proposed methodology may serve as a viable tool for multi-period risk measurement practices.
| Date of Award | 16 Nov 2017 |
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| Original language | English |
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| Supervisor | Guangwu LIU (Supervisor) |