Convex Risk Measures for Good Deal Bounds

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.

Hedging and Pricing in Imperfect Markets Under Non-Convexity

“…This theorem extends Theorem 3.4 in Arai and Fukasawa (2014) to the case where no convexity assumption is made neither on % nor . The following corollary states the conditions under which % is a pricing rule.…”

“…This non-parametric or robust hedging approach 1 is fairly general and can be used for various purposes such as hedging contingent claims and economic risk variables. While it encompasses the methods developed in Jaschke and Küch-ler (2001), Staum (2004), Xu (2006), Assa and Balbás (2011), Balbás, Balbás, and Heras (2009), Balbás, Balbás, and Garrido (2010), Mayoral (2009), andArai andFukasawa (2014) for sub-additive risk measures and pricing rules, the main novelty of this paper lies in incorporating possibly non-convex risk measures which are extensively used in practice. For example, the celebrated Value at Risk and risk measures related to Choquet expected utility (Bassett, Koenker, and Kordas (2004)) are, in general, non-convex.…”

“…Proof See Appendix A. Garrido (2010), Balbás, Balbás, and Mayoral (2009) and Arai and Fukasawa (2014). Furthermore, in the existing literature, the set of stochastic discount factors is constructed either parametrically (using, for example, a semi-martingale process) or empirically, and a pricing rule is then obtained by taking supremum of prices over a closed convex subset R .…”

Existence, uniqueness, and stability of optimal payoffs of eligible assets

“…In case several optimal payoffs exist, how robust is the choice of a specific portfolio? More specifically, we are interested in (1) assessing if an optimal allocation remains close to being optimal after a slight perturbation of the underlying position and (2) ensuring that the accuracy of the optimal allocation increases with the degree of approximation of the underlying position. This is equivalent to investigating suitable continuity properties of E: (1) corresponds to lower semicontinuity and (2) to upper semicontinuity.…”

Market consistent valuations with financial imperfection

“…In the following discussion, we demonstrate the strong relation between a market consistent valuation, of either type, and hedging strategies as used in the literature on pricing (e.g., see Jaschke and Küchler 2001;Staum 2004;Xu 2006;Assa and Balbás 2011;Balbás et al 2009aBalbás et al , b, 2010Arai and Fukasawa 2014). We assume that the value of a variable is equal to the sum of a best estimate and a risk margin.…”