Applications of constrained non-parametric smoothing methods in computing financial risk

The aim of this thesis is to improve risk measurement estimation by incorporating extra information in the form of constraint into completely non-parametric smoothing techniques. A similar approach has been applied in empirical likelihood analysis. The method of constraints incorporates bootstrap resampling techniques, in particular, biased bootstrap. This thesis brings together formal estimation methods, empirical information use, and computationally intensive methods. In this thesis, the constraint approach is applied to non-parametric smoothing estimators to improve the estimation or modelling of risk measures. We consider estimation of Value-at-Risk, of intraday volatility for market risk, and of recovery rate densities for credit risk management. Firstly, we study Value-at-Risk (VaR) and Expected Shortfall (ES) estimation. VaR and ES estimation are strongly related to quantile estimation. Hence, tail estimation is of interest in its own right. We employ constrained and unconstrained kernel density estimators to estimate tail distributions, and we estimate quantiles from the fitted tail distribution. The constrained kernel density estimator is an application of the biased bootstrap technique proposed by Hall & Presnell (1998). The estimator that we use for the constrained kernel estimator is the Harrell-Davis (H-D) quantile estimator. We calibrate the performance of the constrained and unconstrained kernel density estimators by estimating tail densities based on samples from Normal and Student-t distributions. We find a significant improvement in fitting heavy tail distributions using the constrained kernel estimator, when used in conjunction with the H-D quantile estimator. We also present an empirical study demonstrating VaR and ES calculation. A credit event in financial markets is defined as the event that a party fails to pay an obligation to another, and credit risk is defined as the measure of uncertainty of such events. Recovery rate, in the credit risk context, is the rate of recuperation when a credit event occurs. It is defined as Recovery rate = 1 - LGD, where LGD is the rate of loss given default. From this point of view, the recovery rate is a key element both for credit risk management and for pricing credit derivatives. Only the credit risk management is considered in this thesis. To avoid strong assumptions about the form of the recovery rate density in current approaches, we propose a non-parametric technique incorporating a mode constraint, with the adjusted Beta kernel employed to estimate the recovery density function. An encouraging result for the constrained Beta kernel estimator is illustrated by a large number of simulations, as genuine data are very confidential and difficult to obtain. Modelling high frequency data is a popular topic in contemporary finance. The intraday volatility patterns of standard indices and market-traded assets have been well documented in the literature. They show that the volatility patterns reflect the different characteristics of different stock markets, such as double U-shaped volatility pattern reported in the Hang Seng Index (HSI). We aim to capture this intraday volatility pattern using a non-parametric regression model. In particular, we propose a constrained function approximation technique to formally test the structure of the pattern and to approximate the location of the anti-mode of the U-shape. We illustrate this methodology on the HSI as an empirical example.