Convex Risk Measures for Good Deal Bounds

21 Pages Posted: 11 Jun 2014

See all articles by Takuji Arai

Takuji Arai

Keio University - Faculty of Economics

Masaaki Fukasawa

Osaka University

Date Written: July 2014


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.

Keywords: convex risk measure, good deal bound, Orlicz space, risk indifference price, fundamental theorem of asset pricing

Suggested Citation

Arai, Takuji and Fukasawa, Masaaki, Convex Risk Measures for Good Deal Bounds (July 2014). Mathematical Finance, Vol. 24, Issue 3, pp. 464-484, 2014, Available at SSRN: or

Takuji Arai (Contact Author)

Keio University - Faculty of Economics ( email )

2-15-45 Mita, Ninato-ku
Tokyo 1088345

Masaaki Fukasawa

Osaka University

1-1 Yamadaoka
Osaka, 565-0871

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