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The authors use risk-neutral option pricing theory to value the guaranteed minimum death benefit (GMDB) in variable annuities (VAs) and some recently introduced mutual funds. A variety of death benefits, such as returnof-premium, rising floors, and "ratches," are analyzed. Specifically, the authors compute the fair insurance risk fee, charged to assets, that funds the embedded option. The authors derive analytic option prices for a simplified exponential mortality model and robust numerical estimates in the case of a properly calibrated Gompertz model. The authors label this contingent claim a Titanic option because its payoff structure is in between European and American style but is triggered by death. The authors' main objective is to compare theoretical estimates against a cross-section of insurance risk charges, as reported by Morningstar, Inc. The authors' main conclusion is that a simple return-of-premium death benefit is worth between one and ten basis points, depending on gender, purchase age, and asset volatility. In contrast, the median Mortality and Expense risk charge for return-of-premium variable annuities is 115 basis points. Presumably, the remaining markup can be attributed to profits, model imperfections, or, more cynically, to an implicit payment for the tax-deferral privilege.

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The no-arbitrage valuation .f basket options is complicated by the fact that the sum of lognormal random variables is not lognormal. This problem is shared with arithmetic Asian options as well. Various ad hoc approximation techniques have been proposed, none of them very satisfactory or accurate.In this article we suggest using the rec&rocal gamma distribution as an approximation for the state-price density (SPD) function .f the underlying basket stochastic variable. This, in turn, allows us to obtain a closed-jorm expression for the price of a basket option. The technique, when compared against a simple lognormal approximation, perjorms at its best when the correlation structure of the underlying basket exhibits a spec$c decaying pattern.A s a by-product, we introduce a formal approach for assessing the goodness of _fit of candidate distributions for approximating the SPD. Finally, we present a numerical example in which we apply our formula to value (G-7) index-linked guaranteed investment certificates, which can be decomposed into a zero-coupon bond and a basket option. asket options have become extremely popular over the last few years as part of many structured index-linked products offered to B institutional and retail investors. A basket option, as its name implies, is an option on a collection, or basket, of assets, typically stocks. A basket call option gives the holder the right, but not the obligation, to purchase a prespecified fixed portfolio of stocks at a fixed strike price.As a drect consequence of the linear summation, the no-arbitrage valuation of basket options is complicated by the fact that the sum of lognormal dstributions is not lognormal. Consequently, there is no known closed-form solution to the problem of pricing and hedging these products.

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