### Abstract

We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no-free-lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure, which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further, we investigate conditions under which any good deal valuation is relevant.

Original language | English |
---|---|

Pages (from-to) | 464-484 |

Number of pages | 21 |

Journal | Mathematical Finance |

Volume | 24 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2014 |

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### Keywords

- Convex risk measure
- Fundamental theorem of asset pricing
- Good deal bound
- Orlicz space
- Risk indifference price

### ASJC Scopus subject areas

- Applied Mathematics
- Finance
- Accounting
- Economics and Econometrics
- Social Sciences (miscellaneous)

### Cite this

*Mathematical Finance*,

*24*(3), 464-484. https://doi.org/10.1111/mafi.12020

**Convex risk measures for good deal bounds.** / Arai, Takuji; Fukasawa, Masaaki.

Research output: Contribution to journal › Article

*Mathematical Finance*, vol. 24, no. 3, pp. 464-484. https://doi.org/10.1111/mafi.12020

}

TY - JOUR

T1 - Convex risk measures for good deal bounds

AU - Arai, Takuji

AU - Fukasawa, Masaaki

PY - 2014

Y1 - 2014

N2 - We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no-free-lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure, which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further, we investigate conditions under which any good deal valuation is relevant.

AB - We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized by several equivalent properties and in particular, we see that a convex risk measure is a good deal valuation only if it is given as a risk indifference price. An application to shortfall risk measure is given. In addition, we show that the no-free-lunch (NFL) condition is equivalent to the existence of a relevant convex risk measure, which is a good deal valuation. The relevance turns out to be a condition for a good deal valuation to be reasonable. Further, we investigate conditions under which any good deal valuation is relevant.

KW - Convex risk measure

KW - Fundamental theorem of asset pricing

KW - Good deal bound

KW - Orlicz space

KW - Risk indifference price

UR - http://www.scopus.com/inward/record.url?scp=84901986674&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84901986674&partnerID=8YFLogxK

U2 - 10.1111/mafi.12020

DO - 10.1111/mafi.12020

M3 - Article

VL - 24

SP - 464

EP - 484

JO - Mathematical Finance

JF - Mathematical Finance

SN - 0960-1627

IS - 3

ER -