Thesis (MSc)--Stellenbosch University, 2015 / ENGLISH ABSTRACT : In recent years, the use of Lévy processes as a modelling tool has come to be viewed more favourably than the use of the classical Brownian motion setup. The reason for this is that these processes provide more flexibility and also capture more of the ’real world’ dynamics of the model. Hence the use of Lévy processes for financial modelling is a motivating factor behind this research presentation. As a starting point a framework for the LIBOR market model with dynamics driven by a Lévy process instead of the classical Brownian motion setup is presented. When modelling LIBOR rates the use of a more realistic driving process is important since these rates are the most realistic interest rates used in the market of financial trading on a daily basis. Since the financial crisis there has been an increasing demand and need for efficient modelling and management of risk within the market. This has further led to the motivation of the use of Lévy based models for the modelling of credit risky financial instruments. The motivation stems from the basic properties of stationary and independent increments of Lévy processes. With these properties, the model is able to better account for any unexpected behaviour within the market, usually referred to as "jumps". Taking both of these factors into account, there is much motivation for the construction of a model driven by Lévy processes which is able to model credit risk and credit risky instruments. The model for LIBOR rates driven by these processes was first introduced by Eberlein and Özkan (2005) and is known as the Lévy-LIBOR model. In order to account for the credit risk in the market, the Lévy-LIBOR model with default risk was constructed. This was initially done by Kluge (2005) and then formally introduced in the paper by Eberlein et al. (2006). This thesis aims to present the theoretical construction of the model as done in the above mentioned references. The construction includes the consideration of recovery rates associated to the default event as well as a pricing formula for some popular credit derivatives. / AFRIKAANSE OPSOMMING : In onlangse jare, is die gebruik van Lévy-prosesse as ’n modellerings instrument baie meer gunstig gevind as die gebruik van die klassieke Brownse bewegingsproses opstel. Die rede hiervoor is dat hierdie prosesse meer buigsaamheid verskaf en die dinamiek van die model wat die praktyk beskryf, beter hierin vervat word. Dus is die gebruik van Lévy-prosesse vir finansiële modellering ’n motiverende faktor vir hierdie navorsingsaanbieding. As beginput word ’n raamwerk vir die LIBOR mark model met dinamika, gedryf deur ’n Lévy-proses in plaas van die klassieke Brownse bewegings opstel, aangebied. Wanneer LIBOR-koerse gemodelleer word is die gebruik van ’n meer realistiese proses belangriker aangesien hierdie koerse die mees realistiese koerse is wat in die finansiële mark op ’n daaglikse basis gebruik word. Sedert die finansiële krisis was daar ’n toenemende aanvraag en behoefte aan doeltreffende modellering en die bestaan van risiko binne die mark. Dit het verder gelei tot die motivering van Lévy-gebaseerde modelle vir die modellering van finansiële instrumente wat in die besonder aan kridietrisiko onderhewig is. Die motivering spruit uit die basiese eienskappe van stasionêre en onafhanklike inkremente van Lévy-prosesse. Met hierdie eienskappe is die model in staat om enige onverwagte gedrag (bekend as spronge) vas te vang. Deur hierdie faktore in ag te neem, is daar genoeg motivering vir die bou van ’n model gedryf deur Lévy-prosesse wat in staat is om kredietrisiko en instrumente onderhewig hieraan te modelleer. Die model vir LIBOR-koerse gedryf deur hierdie prosesse was oorspronklik bekendgestel deur Eberlein and Özkan (2005) en staan beken as die Lévy-LIBOR model. Om die kredietrisiko in die mark te akkommodeer word die Lévy-LIBOR model met "default risk" gekonstrueer. Dit was aanvanklik deur Kluge (2005) gedoen en formeel in die artikel bekendgestel deur Eberlein et al. (2006). Die doel van hierdie tesis is om die teoretiese konstruksie van die model aan te bied soos gedoen in die bogenoemde verwysings. Die konstruksie sluit ondermeer in die terugkrygingskoers wat met die wanbetaling geassosieer word, sowel as ’n prysingsformule vir ’n paar bekende krediet afgeleide instrumente.