1. Introduction 

Variable annuities (VAs) are unit-linked investment policies providing a post-retirement income, which is generated by the returns on a suitably managed financial portfolio. Various guarantees are applied with the aim of providing protection of the policyholders’ saving accounts. VAs are popular insurance products in the USA, Japan, the UK, and are increasingly present in the other European markets as well. According to the Life Insurance and Market Research Association (LIMRA) Secure Retirement Institute and the Insured Retirement Institute (IRI), VA sales for 2018 in the USA were more than $100 billion—a 2% increase compared to 2017. Common types of guarantees offered by VA contracts are the so-called Guaranteed Minimum Accumulation Benefit (GMAB), the Guaranteed Minimum Death Benefit (henceforth DB)—which applies in case of early death—the Guaranteed Minimum Income Benefit and the Guaranteed Minimum Withdrawal Benefit. The former two offer protection during the accumulation period, i.e. up to the expiration of the contract, whilst the latter two provide payouts after expiration, *Corresponding author. Email: thorsten.schmidt@stochastik.unifreiburg.de during the so-called ‘annuitization’ period. For an extensive overview and classification of these products we refer to Bacinello et al. (2011) and references therein. Due to the construction of these contracts, the underwriting insurance companies are exposed to financial and mortality risk, as well as surrender risk originated by the policyholder behaviour. Indeed, the option to leave the contract prior to maturity is a common additional feature of insurance contracts, which might cause significant cash outflows for the insurer, and negatively impact on the market growth of VAs (LIMRA Secure Retirement Institute). The study of the pricing of life insurance contracts in the presence of financial risk has been pioneered by Brennan and Schwartz (1976) and Boyle and Schwartz (1977); an extensive literature has developed since, from the seminal contributions of Albizzati and Geman (1994), Bacinello and Ortu (1996) and Grosen and Jørgensen (2002), to the more recent ones of Bacinello et al. (2011), Deelstra and Rayée (2013), Giacinto et al. (2014) and Gudkov et al. (2018), to mention a few. These contributions distinguish themselves in terms of the specific product under consideration, although they are all based on diffusion-driven market models. Extensions to financial dynamics driven by Lévy processes are considered