Calculation of an optimal tariff is a principal challenge for pricing actuaries. In this contribution we are concerned with the renewal insurance business discussing various mathematical aspects of calculation of an optimal renewal tariff. Our motivation comes from two important actuarial tasks, namely a) construction of an optimal renewal tariff subject to business and technical constraints, and b) determination of an optimal allocation of certain premium loadings. We consider both continuous and discrete optimisation and then present several algorithmic sub-optimal solutions. Additionally, we explore some simulation techniques. Several illustrative examples show both the complexity and the importance of the optimisation approach.

The skew processes have recently received much attention, owing to their capacity to describe controlled dynamics. In this paper, we employ the skew geometric Brownian motion (SGBM) to depict nine major stock index markets. The skew process not only shows us where the “support” and “resistance” levels are, but also how strong the force is. However, the densities of the skew processes make it challenging to estimate the parameters in a convenient manner. For the sake of overcoming this challenge, we adopt a Bayesian approach, which plays an important role in allowing us to estimate the parameters by conditional probability densities without having to evaluate complex integrals. Furthermore, we also propose the likelihood ratio tests and significance tests for the skew probability. In the empirical study, our findings reveal that skew phenomenon exists in the global stock markets and that the SGBM model works better than the traditional GBM model, as well as performing competitively, compared to the GBM-jump model (GBM-J) and Markov regime switching GBM model (GBM-MRS). In addition, we explore the possible reasons behind the skew phenomenon in stock markets, the price clustering phenomenon and herd behaviors can help to explain the skew phenomenon.

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This paper describes the regulated agricultural commodity futures market of China, focusing on six actively traded futures: corn, strong gluten wheat, No. 1 soybean, soymeal, cotton, and white sugar. A novel skew Ornstein-Uhlenbeck model is employed to characterize price dynamics with government controls. The empirical analysis reveals significant skew phenomena in these six futures and indicates that the price dynamics are influenced by state policy. The observed skew phenomena are most notable in grain futures, with relatively weaker, but statistically significant, evidence of skew phenomena in oilseed and soft futures markets. In addition, generalized quasilikelihood ratio tests show that the skew Ornstein-Uhlenbeck model is superior to the Ornstein-Uhlenbeck model.