ABSTRACT
In the chapter, the notions and facts from the theory of stochastic processes needed in what follows are presented. We provide facts from the general theory of stochastic processes and information about Wiener, Poisson, and Levy processes, together with a short sketch of the stochastic calculus of semimartingales. These materials are reflected in many books; see, for example, [Protter (2003)], [Cont and Tankov (2004)], [Cohen and Elliott (2015)]. Our exposition of the most important ingredients, for this work of financial markets and option pricing theory (completeness and incompleteness, hedging and superhedging, methodology finding for martingale measures, etc.) follows [Melnikov et al. (2002)]. We introduce in this chapter equity-linked life insurance contracts as innovative insurance instruments to combine both financial and insurance risks and incorporate a new randomness in actuarial calculations, stemming from the financial randomness of insurance guarantees. These contracts have been studied since the middle of the 1970s, and the first quantitative treatment of such innovative insurance products were given by [Brennan and Schwartz (1976)], [Brennan and Schwartz (1979)] and [Boyle and Schwartz (1977)]. In these papers the traditional actuarial principle of equivalence was transformed so that the Black–Scholes formula (see [Black and Scholes (1973)], [Merton (1973)]) appeared in the actuarial science context. Later, in the papers by [Delbaen (1986)], [Bacinello and Ortu (1993)], [Aase and Pearson (1994)], [Nielsen and Sandmann (1995)], and [Ekern and Persson (1996)] the pricing problem of equity-linked life insurance contracts was analyzed with the help of martingale measures. In some sense, the monograph by [Hardy (2003)] can be regarded as a result of studies in the area for this period. In the chapter, we present a leading idea to investigate such contracts based on partial/imperfect hedging techniques developed in the last two decades in mathematical finance. The essence of our approach is partial/imperfect hedging: quantile hedging, efficient hedging, CVaR hedging (see [Föllmer and Leukert (1999)], [Föllmer and Leukert (2000)], [Spivak and Cvitanic (1999)], [Rockafellar and Uryasev (2002)], [Melnikov and Smirnov (2012)]), developed for different financial markets and for deterministic and stochastic insurance 188guarantees. Equity-linked life insurance now is a rather wide area of research and applications. Therefore, it is difficult to include all exploited methods without loss of a homogeneous character of the book. This is a reason why the well-developed approaches (quadratic hedging and utility maximization) are not presented here (see, for example, [Moeller (1998)], [Young (2003)], [Biagini and Schreiber (2013)]).
