Extending pricing rules with general risk functions

Abstract

a b s t r a c tThe paper addresses pricing issues in imperfect and/or incomplete markets if the risk level of the hedging strategy is measured by a general risk function. Convex Optimization Theory is used in order to extend pricing rules for a wide family of risk functions, including Deviation Measures, Expectation Bounded Risk Measures and Coherent Measures of Risk. Necessary and sufficient optimality conditions are provided in a very general setting. For imperfect markets the extended pricing rules reduce the bid ask spread. The findings are particularized so as to study with more detail some concrete examples, including the Condi tional Value at Risk and some properties of the Standard Deviation. Applications dealing with the valu ation of volatility linked derivatives are discussed.

Good Deals and compatible modification of risk and pricing rule: a regulatory treatment

“…This problem has been studied in Balbás et al [3,2,4]. The dual of problem (3.4) is found in Balbás et al [3] as…”

“…By means of the Hahn-Banach Separation Theorem, one can easily prove that if ρ ⊂ L q is convex, σ (L q , L p )-compact and ρ satisfie (2.3), then there exists a unique continuous ρ satisfying (1), (2), (3) and (4) such that (2.2) holds.…”

“…For further discussion we refer the reader to Balbás et al [2]. Now we give our definition of a Good Deal inspired by definitions inČerný and Hodges [7] and Cherny [9].…”

Hedging and Pricing in Imperfect Markets Under Non-Convexity

“…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.…”

“…The robust pricing and hedging strategies of Cox and Ob÷ ój (2011b) and Cox and Ob÷ ój (2011a) serve as an example of this approach. A di¤erent line of research in model-free hedging is based directly on the concepts of hedging and minimization of risk (see Xu (2006), Assa and Balbás (2011), Balbás, Balbás, and Heras (2009), Balbás, Balbás, and Garrido (2010), and Balbás, Balbás, and Mayoral (2009)). In this setting, the investor or portfolio manager minimizes the risk of a global position given the budget constraint on a set of manipulatable positions (a set of accessible portfolios, for instance).…”