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.

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We consider the representation of forward entropic risk measures using the theory of ergodic backward stochastic differential equations in a jump-diffusion framework. Our paper can be viewed as an extension of the work considered by Chong et al. (2019) in the diffusion case. We also study the behavior of a forward entropic risk measure under jumps when a financial position is held for a longer maturity.

In this paper, we consider a risk-based optimal investment problem of an insurer in a regime-switching jump-diffusion model with noisy memory. Using the model uncertainty modeling, we formulate the investment problem as a zero-sum, stochastic differential delay game between the insurer and the market, with a convex risk measure of the terminal surplus and the Brownian delay surplus over a period [T − ̺, T ]. Then, by the BSDE approach, the game problem is solved. Finally, we derive analytical solutions of the game problem, for a particular case of a quadratic penalty function and a numerical example is considered.

In this paper, we provide a representation theorem for dynamic capital allocation under Itô-Lévy model. We consider the representation of dynamic risk measures defined under Backward Stochastic Differential Equations (BSDE) with generators that grow quadratic-exponentially in the control variables. Dynamic capital allocation is derived from the differentiability of BSDEs with jumps. The results are illustrated by deriving a capital allocation representation for dynamic entropic risk measure and static coherent risk measure.