established almost stochastic dominance to reveal preferences for most rather than all decision makers with an increasing and concave utility function. In this paper, we first provide a counterexample to the main theorem of Leshno and Levy related to almost seconddegree stochastic dominance. We then redefine this dominance condition and show that the newly defined almost second-degree stochastic dominance is the necessary and sufficient condition to rank distributions for all decision makers excluding the pathological concave preferences. We further extend our results to almost higher-degree stochastic dominance.

This study investigates the convergence patterns and the rates of convergence of binomial Greeks for the CRR model and several smooth price convergence models in the literature, including the binomial Black-Scholes (BBS) model of Broadie M and Detemple J (1996), the flexible binomial model (FB) of Tian YS (1999), the smoothed payoff (SPF) approach of Heston S and Zhou G (2000), the GCRR-XPC models of Chung SL and Shih PT (2007), the modified FB-XPC model, and the modified GCRR-FT model. We prove that the rate of convergence of the CRR model for computing deltas and gammas is of order O(1/n), with a quadratic error term relating to the position of the final nodes around the strike price. Moreover, most smooth price convergence models generate deltas and gammas with monotonic and smooth convergence with order O(1/n). Thus, one can apply an extrapolation formula to enhance their accuracy. The numerical results show that placing the strike price at the center of the tree seems to On the Rate of Convergence of Binomial Greeks 563Journal of Futures Markets DOI: 10.1002/fut enhance the accuracy substantially. Among all the binomial models considered in this study, the FB-XPC and the GCRR-XPC model with a two-point extrapolation are the most efficient methods to compute Greeks.

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This paper studies the optimal insurance contract under disappointment theory. We show that, when the individuals anticipate disappointment, there are two types of optimal insurance contract. The first type contains a deductible and a coinsurance above the deductible. We find that zero marginal cost is just a sufficient but not a necessary condition for a zero deductible. The second type has no deductible and the optimal insurance starts with full coverage for small losses and includes a coinsurance above an upper value of the full coverage. The Geneva Risk and Insurance Review (2012) 37, 258-284. doi:10.1057/grir.2012.2; published online 31 July 2012Keywords: disappointment; optimal insurance contract; deductible; coinsurance IntroductionSince Borch 1 , many researchers have studied the optimal risk sharing rules between individuals and insurance companies. A well-established result comes from Arrow's 2 findings. He showed that full coverage will be optimal when the insurance companies are risk neutral and the premium is actuarially fair. He also found that if the premium includes an insurance loading, then the optimal insurance contract will contain a deductible and a coinsurance above the deductible. Raviv 3 adopted the assumption of risk-averse insurers, solved the optimal insurance contract, and found the necessary and sufficient conditions for the existence of a deductible. He showed that the deductible is positive if and only if the marginal cost of insurance is positive.1 Borch (1962). 2 Arrow (1971). 3 Raviv (1979). The Geneva Risk and Insurance Review, 2012, 37, (258-284) r 2012 The International Association for the Study of Insurance Economics 1554-964X/12 www.palgrave-journals.com/grir/ Following Raviv's 3 paper, many researchers have studied optimal insurance contracts in alternative situations. For example, Schlesinger 4 and Gollier and Schlesinger 5 examined the optimal level of a deductible. Gollier and Breuer 6 discussed the optimal insurance contract by relaxing the non-negativity constraint. They found the conditions for a contract with negative coverage being optimal. Huberman et al. and Garratt and Marshall 7 respectively found that a contract with an upper-limit of indemnity could be optimal when the individual is able to file for bankruptcy and the insured has the option to convert the insured property. In addition, Gollier and Eeckhoudt et al. 8 indicated that a fixed reimbursement contract will be optimal if the indemnity can only be contingent upon an approximate loss amount, while Huang and Tzeng 9 found that a fixed reimbursement contract could be optimal if the insured commodities are irreplaceable. Furthermore, Dana and Scarsini 10 discussed optimal risk sharing under the assumption that individuals face a non-insurable background risk in addition to an insurable risk. They found that the correlation between the non-insurable and insurable risk is crucial to the form of the optimal insurance contract.Although the literature on the optimal insurance contract has provided many findi...

Real Option Investment, Network Externality, Consumer’s Waiting-to-Buy Effect, G12, O31, O34,