The measurement of market risk. Modelling of risk factors, asset pricing, and approximation of portfolio distributions (Q5938835)

This book is a revised version of the author's doctoral dissertation submitted to the University of St. Gallen in October 1999. Aim is to study some of the quantitative tools necessary for the modelling of uncertainty. The methodology supports an integrated view of risk measurement according to the three \(P\)'s described by Lo (1999): preferences, probabilities and pricing. The author's objective is to set up a quantitative economic model for the assessment of financial market risks. After a brief introduction, motivating the need and the nature of the measurement of financial risks, and describing the formal framework, Chapter 2 discusses risk and risk measures for the decision-making under uncertainty. Uncertainty is summarized here by the portfolio distribution, i.e. the distribution of the value function of the portfolio at a terminal date \(T\). Two types of decisions are relevant from the investor's viewpoint, the investment decision and the capital requirement decision, the latter of which may be based on various types of risk measures (VaR, lower partial moments etc.). Chapter 3-5 deal with the modelling and possible approximation of the portfolio distribution. Under the efficient market hypothesis, risk factors and their distributions are modelled in Chapter 3 via a time series approach. It is shown that log-returns are preferable, from a statistical point of view, to the prices themselves. However, the assumption of joint normality is not supported by empirical evidence which is illustrated by an alternative analysis of the returns on Swiss stocks under a normal model and a Student-\(t\) model, respectively. Chapter 4 is devoted to the price information, i.e. the valuation of financial instruments under the given distribution information from the risk factors. Based on the no arbitrage-principle, the valuation of cash instruments (like equities and fixed-income instruments), futures and forwards, and options is discussed in some detail. Since, typically, pricing formulae or not given in a closed form, analytical approximations of the value function such as a global Taylor approximation for option pricing, or piecewise Taylor approximations are also considered. Chapter 5 reviews various methods for the numerical approximation of the portfolio distribution like the Delta or Delta-Gamma approximations, generation of risk scenarios, and Monte Carlo simulation. A new scenario-based method, called Barycentric Discretization with Piecewise Quadratic Approximation (BDPQA) is introduced and illustrated by numerical examples. With respect to the tested environments it turus out the BDPQA outperforms the Delta-Gamma approximation consistently and provides accurate estimates. Moreover, the estimates are robust over the holding period. As a consequence of a scenario evaluation of risk factors, the approximation of the portfolio distribution is no longer given analytically or numerically, but is based on a large sample drawn from the possible scenarios. Hence, a sample estimation of risk measures is required which typically involves an extreme value analysis based on order statistics. Chapter 6 discusses suitable quantile estimators and, as a smoothed alternative, also kernel-based estimators. A comparison, however, shows that a possible variability reduction under the kernel-based smoothers is often linked to supplementary bias. Moreover, the choice of the bandwidth influences significantly the accuracy of the estimation. Chapter 7 concludes with a summary of main results and briefly touches other important issues such as e.g. credit risk or stochastic volatility models. In a final appendix, the author has summarized the probabilistic concepts and results used in the book for the modelling of uncertainty. In addition, about 200 references give a comprehensive picture of the recent development in the field of financial market risk measurement.