Following the financial crisis of 2008, the need for more robust techniques to quantify the capital charge for risk management has become a pressing problem. Under Basel II/III, banks are allowed to calculate the capital charge using internally developed models subject to regulatory approval. An interesting problem for the regulator is to compare the resulting figures against the required capital under worst case scenarios. The existing literature on the latter problem, which is based on the marginal problem, assumes that no a-priori information is known about the dependencies of contributing risks. These problems are linear optimisation problems over a constrained set of probability measures, discretisation of which leads to large scale LPs. But this approach is very conservative and cannot be implemented robustly in practice, due to the scarcity of historical data. In our approach, we take a less conservative strategy by incorporating dependence information contained in the data in a form that still leads to LPs, an important feature of such problems due to their high dimensionality. Conceptually, our model is the discretisation of an infinite dimensional linear optimisation problem over a set of probability measures. For some specific cases we can prove strong duality, opening up the approach of discretising the dual instead of the primal. This approach is preferable, as it yields better numerical results. In this work we also apply our model to model-free path-dependent option pricing. Use of delayed column generation techniques allows us to solve problems several orders of magnitude larger than via the standard simplex algorithm. For high-dimensional LPs we also implement Nesterov's smoothing technique to solve the problems.

Thesis (MEng)--Stellenbosch University, 2015. / ENGLISH ABSTRACT: In the current economic climate where credit crises, fluctuating commodity prices, poor governance, rising unemployment and declining consumer spending exist, risk management is of utmost importance. Proclaiming the existence of a risk management strategy is not enough to ensure that an enterprise achieves its objectives. The implementation of a holistic enterprise-wide risk management framework is required in order to execute strategies and achieve objectives effectively and efficiently Two types of risk management have emerged in industry, namely quantitative and qualitative risk management. On the one hand, qualitative analysis of risk can be done quickly and with minimal effort. However, these methods rely on the opinion of an individual or group of individuals to analyse the risks. The process may be highly subjective and does not fully consider the characteristics of the enterprise. This renders qualitative risk analysis as an ineffective singular strategy although it has been shown to be effective when the risks are well understood. Quantitative analysis, on the other hand, is particularly effective when the risks are not well understood. These methods have been shown to provide substantially more information regarding risks compared to qualitative analysis. However, many quantitative risk management methods presented in literature are studied in isolation and not within the context of a holistic risk management process. Furthermore, quantitative methods tend to be complex in nature and require a reasonable understanding of mathematical and statistical concepts in order to be used effectively. In view of this, there is a need for an enterprise risk management framework that emphasises the use of qualitative methods when the risks are well understood and quantitative methods when in-depth analyses of the risks are required. In this study, a systematic enterprise-wide risk management framework that incorporates both quantitative and qualitative methods was developed. The framework integrates these methods in a logical and holistic manner. The quantitative methods were found be to be largely practical while the qualitative methods presented are simple and easy to understand. / AFRIKAANSE OPSOMMING: In die huidige ekonomiese klimaat waar krediet krisisse, wisselende kommoditeitspryse, swak bestuur, stygende werkloosheid en dalende verbruikersbesteding bestaan, is risikobestuur van die uiterste belang. Die verkondiging van die bestaan van 'n risiko bestuurstrategie is nie genoeg om te verseker dat 'n onderneming sy doelwitte bereik nie. Die implementering van 'n holistiese ondernemings- breë risikobestuursraamwerk is nodig om strategieë en doelwitte doeltreffend en effektief te bereik. Twee tipe risikobestuur het na vore gekom in die bedryf, naamlik kwantitatiewe en kwalitatiewe risikobestuur. Aan die een kant , kan kwalitatiewe ontleding van risiko vinnig en met minimale inspanning gedoen word. Hierdie metode is gewoontlik die mening van 'n individu of 'n groep individue wat die risiko ontleed. Die proses kan hoogs subjektief wees en nie ten volle die eienskappe van die onderneming in ag neem nie. Kwalitatiewe risiko-analise kan dan gesien word as 'n ondoeltreffende enkelvoud strategie maar dit is wel doeltreffend wanneer daar verstaan word wat die onderneming se risiko is. Kwantitatiewe analise, aan die ander kant, is veral effektief wanneer die risiko's nie goed verstaanbaar is nie. Hierdie metode het getoon dat daar aansienlik meer inligting oor die risiko's, in vergelyking met kwalitatiewe ontleding, verskaf word. Daar is egter baie kwantitatiewe risikobestuur metodes wat in literatuur verskaf word, wat in isolasie bestudeer word en nie binne die konteks van 'n holistiese risikobestuur proses nie. Verder is, kwantitatiewe metodes geneig om kompleks van aard te wees en vereis 'n redelike begrip van wiskundige en statistiese konsepte sodat kwantitatiewe analise effektief kan wees. In lig hiervan, is daar 'n sterk behoefte vir 'n onderneming om 'n risikobestuursraamwerk in plek te het. Die risikobestuursraamwerk sal beide die gebruik van kwalitatiewe metodes, wanneer die risiko goed verstaan word, en kwantitatiewe metodes, wanneer daar in diepte-ontledings van die risiko is, beklemtoon. In hierdie studie was 'n sistematiese onderneming-breë risikobestuursraamwerk ontwikkel wat beide kwantitatiewe en kwalitatiewe metodes insluit. Die raamwerk integreer hierdie metodes in 'n logiese en holistiese wyse. Die kwantitatiewe metodes is gevind om grootliks prakties te wees, terwyl die kwalitatiewe metodes wat aangebied word, eenvoudig en maklik is om te verstaan.

It is well known that for highly skewed distributions the standard method of using the t statistic for the confidence interval of the mean does not give robust results. This is an important problem for importance sampling (IS) as its final distribution is often skewed due to a heavy tailed weight distribution. In this paper, we first explain Hall's transformation and its variants to correct the confidence interval of the mean and then evaluate the performance of these methods for two numerical examples from finance which have closed-form solutions. Finally, we assess the performance of these methods for credit risk examples. Our numerical results suggest that Hall's transformation or one of its variants can be safely used in correcting the two-sided confidence intervals of financial simulations.(author's abstract) / Series: Research Report Series / Department of Statistics and Mathematics