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A note on utility based pricing and asymptotic risk diversification

Bouchard, Bruno; Elie, Romuald; Moreau, Ludovic (2012), A note on utility based pricing and asymptotic risk diversification, Mathematics and Financial Economics, 6, 1, p. 59-74. http://dx.doi.org/10.1007/s11579-011-0055-0

Type
Article accepté pour publication ou publié
Date
2012
Journal name
Mathematics and Financial Economics
Volume
6
Number
1
Publisher
Springer
Pages
59-74
Publication identifier
http://dx.doi.org/10.1007/s11579-011-0055-0
Metadata
Show full item record
Author(s)
Bouchard, Bruno
Elie, Romuald
Moreau, Ludovic
Abstract (EN)
In principle, liabilities combining both insurancial risks (e.g. mortality/longevity, crop yield,...) and pure financial risks cannot be priced neither by applying the usual actuar- ial principles of diversification, nor by arbitrage-free replication arguments. Still, it has been often proposed in the literature to combine these two approaches by suggesting to hedge a pure financial payoff computed by taking the mean under the historical/objective probability measure on the part of the risk that can be diversified. Not surprisingly, simple examples show that this approach is typically inconsistent for risk adverse agents. We show that it can nevertheless be recovered asymptotically when the number of sold claims goes to infinity and the absolute risk aversion of the agent goes to zero simultaneously. This follows from a general convergence result on utility indifference prices which is valid for both complete and incomplete financial markets. In particular, if the underlying financial market is complete, the limit price corresponds to the hedging cost of the mean payoff. If the financial market is incomplete but the agent behaves asymptotically as an exponential utility maximizer with vanishing risk aversion, we show that the utility indifference price converges to the expectation of the discounted payoff under the minimal entropy martingale measure.
Subjects / Keywords
entropy; diversification; risk aversion; Utility indifference pricing
JEL
D52 - Incomplete Markets
D53 - Financial Markets
D81 - Criteria for Decision-Making under Risk and Uncertainty

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