helping me out when I most needed it. His continuous support and encouragement has been invaluable, and this thesis would not have been realised without his profound understanding of mathematics, finance and supervision.Many thanks go out to my co-authors, Paul C. Kettler, Rodwell Kufakunesu, Dr. Carl Lindberg and Olli Wallin, who have shared their time and knowledge with me. Without their help I would not have accomplished this. I am obliged to Docent Roger Pettersson at Växjö University for his enthusiastic efforts in the first years. The encouragement from Docent Magnus Wiktorsson, Lund University, the opponent at my Licentiate defense, was much appreciated.The friendly and inspiring group of colleagues at the Centre of Mathematics for Applications is responsible for making work a pleasure. All facilitated by the excellent Administrative director Helge Galdal. I am grateful to all the participants at the Fourth Scandinavian Ph.D. workshop in Mathematical Finance for accepting my invitation. I also want to thank the Ph.D. committee at the Department of Mathematics for their unconditional support.On a personal level I want to thank all my friends for their unceasing pursuits to make my days include more than research. Proofreading is an unavoidable but unrewarding task and I am in debt to Camilla Malm for her kindness to carefully and courteously correct my mistakes. I owe everything to my parents for their endless devotion, and my siblings with families for always caring for me. Finally, Christina for all her support and love during the final year.

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The statistical distribution of financial returns plays a key role in evaluating Value-at-Risk using parametric methods. Traditionally, when evaluating parametric Value-at-Risk, the statistical distribution of the financial returns is assumed to be normally distributed. However, though simple to implement, the Normal distribution underestimates the kurtosis and skewness of the observed financial returns. This article focuses on the evaluation of the South African equity markets in a Value-at-Risk framework. Value-at-Risk is estimated on four equity stocks listed on the Johannesburg Stock Exchange, including the FTSE/JSE TOP40 index and the S & P 500 index. The statistical distribution of the financial returns is modelled using the Normal Inverse Gaussian and is compared to the financial returns modelled using the Normal, Skew t-distribution and Student t-distribution. We then estimate Value-at-Risk under the assumption that financial returns follow the Normal Inverse Gaussian, Normal, Skew t-distribution and Student t-distribution and backtesting was performed under each distribution assumption. The results of these distributions are compared and discussed.

The aim of this paper is to solve an optimal investment, consumption and life insurance problem when the investor is restricted to capital guarantee. We consider an incomplete market described by a jump-diffusion model with stochastic volatility. Using the martingale approach, we prove the existence of the optimal strategy and the optimal martingale measure and we obtain the explicit solutions for the power utility functions. ] in an optimal investment-consumption problem considering a stochastic volatility model described by diffusion processes. Similar works include (Liang and Guo [17], Michelbrink and Le [19] and references therein).The optimal solution to the restricted problem is derived from the unrestricted optimal solution, applying the option based portfolio insurance (OBPI) method developed by El Karoui et al. [6]. The OBPI method consists in taking a certain part of capital and invest in the optimal portfolio of the unconstrained problem and the remaining part insures the position with American put. We prove the admissibility and the optimality of the strategy.The structure of this paper is organized as follows. In Section 2, we introduce the model and problem formulation of the Financial and the Insurance markets. Section 3, we solve the unconstrained problem. In Section 4, we solve the constrained problem and prove the admissibility of our strategy. Finally, in Section 5 we give a conclusion. The Financial ModelWe consider two dimensional Brownian motion W = {W 1 (t); W 2 (t), 0 ≤ t ≤ T } associated to the complete filtered probability space (Moreover, we consider a Poisson process N = {N(t), F N (t), 0 ≤ t ≤ T } associated to the complete filtered probability space (Ω N , F N , {F N t }, P N ) with intensity λ(t) and a P Nmartingale compensated poisson process N (t) := N(t) − t 0