A note on utility based pricing and asymptotic risk diversification |
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Authors: | Bruno Bouchard Romuald Elie Ludovic Moreau |
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Institution: | 1. CEREMADE, CNRS, UMR 7534, Universit?? Paris-Dauphine CREST, Paris Cedex 16, France
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Abstract: | 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 actuarial 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 if we consider a sequence of agents whose absolute risk aversions
go to zero and if the number of sold claims goes to infinity 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 agents behave asymptotically as exponential utility maximizers 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. |
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