Essays on Estimation of Risk Measures and their Sensitivities

Abstract

This thesis studies issues in two problems related to risk measures in a simulation environment. One is the estimation of risk measure when the loss of each scenario can not be evaluated directly, which is known as the nested simulation problem. The other one is the estimation of the sensitivity of risk measure when both risk and performance measure contains discontinuity.

In the first essay, we propose a new algorithm that gives a lower-bound estimator to convex risk measures in nested simulation problems. In such problems, after we generate a number of scenarios, evaluating the loss of each scenario requires further simulating, typically time-consuming, inner samples. We construct lower bounds for convex risk measures exploiting the conjugate theory of convex function and the existing approximation methods, such as least squares regression (LSR). Then we estimate this lower bound with the Monte Carlo method. We analyze the rate of convergence of our estimator and show that our estimator converges at the canonical rate before it reaches a certain bias, which is inherent in the LSR method and is dependent on the choice of basis functions. To further improve efficiency, we also developed a cross-estimation method, which originated from the cross-validation method in machine learning.

In the second essay, we investigate the problem of estimating the sensitivity of risk measures when the loss itself contains discontinuity with respect to the parameter of interest. Such problems arise in portfolio management with complex derivatives and stochastic activity networks. When the target risk measure also exhibits discontinuity, the problem consists of two levels of discontinuities. This kind of problem has not been studied in literature yet. We derive highly intuitive, closed-form expressions of the associated derivatives and present an estimation algorithm that is efficient and easy to implement.

The numerical illustrations show that both methods are robust in performance across a range of examples.

Date of Award	29 Nov 2023
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Jeff HONG (Supervisor) & Guangwu LIU (Supervisor)