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Mortality forecasting plays an essential role in designing welfare policies and pricing aged-related financial derivatives. However, most prevailing models do not perform well in mortality forecasting particularly for the elder people. Indeed, the problem of missing category for the elderly is a typical feature in developing countries, because people are shorter-lived in earlier times and hence the mortality is recorded up to fewer age categories. For example, in Taiwan, the mortality is recorded up to an age of 95 before 1997, but as the improvement of life expectancy, the mortality is recorded up to an age of 100 afterwards. This paper proposes several approaches for data imputation to alleviate this systematic missing data problem of the mortality data. Motivated by Lee, and Carter. 1992. “Modelling and Forecasting the Time Series of US Mortality.”

This article deals with the pricing of double-barrier options monitored discretely. A continuity correction method is established to provide an analytical approximation for the price of such discrete options under the Black–Scholes model. We achieve this by applying the smooth-fit principle simultaneously to the two flat boundaries (barriers) associated. The resulting correction form still involves adjustments in the levels of barriers, but the amounts adjusted can be different for different boundaries. More interestingly, the shift for each boundary can also be in different directions, which depends largely on the position of the current level relative to the two boundaries. Numerical examples are provided as well which support our theoretical achievements.

We study a dynamic mean-variance portfolio selection problem subject to possible limit of market risk. Three measures of market risk are considered: value-at-risk, expected shortfall, and median shortfall. They are all calculated in a dynamic consistent sense. After applying the technique of delta-normal approximation, we can explicitly solve for the optimal solution and calculate the economic loss brought by the risk budget constraint. With the analytical results obtained, influential factors of economic loss are then explored by which some guidelines on trading practice are proposed. The guidelines are independent of risk measures, and are valuable to both institutions and regulators, for they suggest that an institutional investor would spontaneously obey good investment discipline to avoid potential impact of risk constraint. This result meets the purpose of external regulation from the perspective of market discipline.

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