Variable Annuities with embedded guarantees are very popular in the US market. There exists a great variety of products with both, guaranteed minimum death benefits (GMDB) and guaranteed minimum living benefits (GMLB). Although several approaches for pricing some of the corresponding guarantees have been proposed in the academic literature, there is no general framework in which the existing variety of such guarantees can be priced consistently. The present paper fills this gap by introducing a model, which permits a consistent and extensive analysis of all types of guarantees currently offered within Variable Annuity contracts. Besides a valuation assuming that the policyholder follows a given strategy with respect to surrender and withdrawals, we are able to price the contract under optimal policyholder behavior. Using both, Monte-Carlo methods and a generalization of a finite mesh discretization approach, we find that some guarantees are overpriced, whereas others, e.g. guaranteed annuities within guaranteed minimum income benefits (GMIB), are offered significantly below their risk-neutral value.

We define a non-commutative product for arbitrary gauge and B-field backgrounds in terms of correlation functions of open strings. While off-shell correlations are, of course, not conformally invariant, it turns out that, at least to first derivative order, our product has the trace property and is associative up to surface terms if the background fields are put on-shell. No on-shell conditions for the inserted functions are needed, but it is essential to include the full contribution of the Born-Infeld measure. We work with a derivative expansion and avoid any topological limit, which would effectively constrain H.

Traditional life insurance policies in many markets are sold with minimum interest rate guarantees. This paper concentrates on the risk cliquet-style guarantees impose on the insurer, measured by shortfall probabilities under the so-called "real-world probability measure P". We develop a general model and analyze the impact of interest rate guarantees on the risk of an insurance company. Furthermore the paper is concerned with how default risk depends on characteristics of the contract, on the insurer's reserve situation and asset allocation, and on management decisions as well as on regulatory parameters. In particular, the interaction of the parameters is analyzed yielding results that should be of interest for insurers as well as regulators.

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