This paper deals with the construction of “smooth good deals” (SGD), i.e., sequences of self-financing strategies whose global risk diverges to minus infinity and such that every security in every strategy of the sequence is a “smooth” derivative with a bounded delta. Since delta is bounded, digital options are excluded. In fact, the pay-off of every option in the sequence is continuos (and therefore jump-free) with respect to the underlying asset price. If the selected risk measure is the value at risk, then these sequences exist under quite weak conditions, since one can involve risks with both bounded and unbounded expectation, as well as non-friction-free pricing rules. Moreover, every strategy in the sequence is composed of a short European option plus a position in a riskless asset. If the chosen risk measure is a coherent one, then the general setting is more limited. Indeed, though frictions are still accepted, expectations and variances must remain finite. The existence of SGDs will be characterized, and computational issues will be properly addressed. It will be shown that SGDs often exist, and for the conditional value at risk, they are composed of the riskless asset plus easily replicable short European puts. The ideas presented may also apply in some actuarial problems such as the selection of an optimal reinsurance contract.
Golden option Risk measure Smooth good deal Dual approach
Mathematics Subject Classification
91G10 91G20 91G80 91B06
G11 G13 C61 C65
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This research was partially supported by the University Carlos III of Madrid (Project 2009 / 00445 / 002). The authors sincerely thank the journal editor and the anonymous reviewer, whose comments led to significant improvements in this paper. The usual caveat applies.
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