Academic journal article Federal Reserve Bank of Atlanta, Working Paper Series

Hedging and Pricing in Imperfect Markets under Non-Convexity

Academic journal article Federal Reserve Bank of Atlanta, Working Paper Series

Hedging and Pricing in Imperfect Markets under Non-Convexity

Article excerpt

1 Introduction

Hedging and pricing financial and economic variables in imperfect markets (incomplete markets and/or markets with frictions) proves to be a challenging problem. While pricing and hedging in complete and frictionless markets are typically carried out by a unique perfect replication of a contingent claim at a horizon time, the presence of market imperfections renders no unique solution to this problem.

There are two main approaches to pricing and hedging in incomplete markets. The first approach is parametric in nature as it assumes that the market price follows a particular diffusion process. This approach includes the super-hedging of El Karoui and Quenez (1995), the efficient hedging of Follmer and Leukert (2000) and the intrinsic risk hedging of Schweizer (1992), to name a few. The second approach is model-free (nonparametric) since it does not make use of the structure of the model that drives the underlying price dynamics. The robust pricing and hedging strategies of Cox and Obloj (2011b) and Cox and Obloj (2011a) serve as an example of this approach. A different line of research in model-free hedging is based directly on the concepts of hedging and minimization of risk (see Xu (2006), Assa and Balbas (2011), Balbas, Balbas, and Heras (2009), Balbas, Balbas, and Garrido (2010), and Balbas, Balbas, 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).

Furthermore, when the no-arbitrage condition holds, the set of admissible stochastic discount factors for pricing financial variables is strictly positive, implying monotonicity of the pricing rules. By contrast, in the presence of market imperfections, the stochastic discount factors do not price the set of all possible payoffs (see Jouini and Kallal (1995a), Jouini and Kallal (1995b) and Jouini and Kallal (1999)). In this case, the main problem lies in the existence of pricing rules that can be extended to the whole set of possible variables.

The goal of this paper is to develop a unifying framework for hedging and pricing in imperfect markets that allows for non-convex (non-subadditive) risk measures and pricing rules. To this end, we account for market incompleteness and frictions by minimizing aggregate hedging costs that consist of costs associated with the risk of the non-hedged part and costs of purchasing the hedging strategy. This non-parametric or robust hedging approach 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 Kuchler (2001), Staum (2004), Xu (2006), Assa and Balbas (2011), Balbas, Balbas, and Heras (2009), Balbas, Balbas, and Garrido (2010), Balbas, Balbas, and Mayoral (2009), and Arai and Fukasawa (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 pricing rules in actuarial applications also tend to be non-convex (Wang, Young, and Panjer (1997)) which further reinforces the need for a framework that deals with non-convex risk and pricing rules. While the focus in this paper is on the pricing part of the hedging problem and the extension of the pricing rule to the space of all financial and economic variables in imperfect markets, we also construct a set of market principles that are used to determine the existence of a solution to the hedging problem.

The rest of the paper is organized as follows. Section 2 introduces the notation, provides some preliminary definitions and states the main problem. Section 3 uses market principles to characterize the solutions to the hedging and pricing problems under generalized spectral risk measures. …

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