The worldwide market of variable annuities (VAs) has been rapidly growing since their introduction in the mid-1980s in the United States. These fund-linked annuity products, which have become an essential part of the retirement plans in many countries, are often combined with additional living and death benefits. Since they are usually of a complex nature, consistent pricing of variable annuities becomes a difficult task. As there is often a trade-off between a realistic model and analytical tractability, several studies in the literature either focus on closed-form solutions, by simplifying the contract setups and the modeling assumptions, or propose numerical methods for the multi-factor models. This work aims to fill this gap by showing how the explicit representations for prices of some of the VA products can be derived in a hybrid model for insurance and market risks.