Good deal indices in asset pricing: actuarial and financial implications

Abstract

We integrate into a single optimization problem a risk measure, beyond the variance, and either arbitrage-free real market quotations or financial pricing rules generated by an arbitrage-free stochastic pricing model. A sequence of investment strategies such that the couple (expected return, risk) diverges to (+∞, −∞) will be called a good deal (GD). The existence of such a sequence is equivalent to the existence of an alternative sequence of strategies such that the couple (risk, price) diverges to (−∞, −∞). Moreover, by appropriately adding the riskless asset, every GD may generate a new one only composed of strategies priced at one. We will see that GDs often exist in practice, and the main objective of this paper will be to measure the GD size. The provided GD indices will equal an optimal ratio between both risk and price, and there will exist alternative interpretations of these indices. They also provide the minimum relative (per dollar) price modification that prevents the existence of GDs. Moreover, they will be a crucial instrument to detect those securities or marketed claims that are over-or underpriced. Many classical actuarial and financial optimization problems may generate wrong solutions if the used market quotations or stochastic pricing models do not prevent the existence of GDs. This fact is illustrated in the paper, and we point out how the provided GD indices may be useful to overcome this caveat. Numerical experiments are also included. Keywords: risk measure; compatibility between prices and risks; good deal size measurement; actuarial and financial implications A. Balbás et al. / Intl. Trans. in Op. Res. 26 (2019) are combined in a single problem, one often faces the existence of sequences of investment strategies (good deals or GDs) whose pairs (expected return, risk) diverge to (+∞, −∞). The existence of GDs is equivalent to the existence of alternative sequences of investment strategies whose pairs (risk, price) diverge to (−∞, −∞). This pathological finding has been analyzed in Balbás and Balbás (2009) and Balbás et al. (2016a), where explicit examples of the sequences above have been constructed and their performance empirically tested. The main conclusion was that the divergence of (expected return, risk) to (+∞, −∞) is more theoretical than real, but the performance of the constructed GD was good enough. The GD were collections of options providing much better realized Sharpe ratios than their underlying assets.In this paper, we will deal with a couple (ρ, ) composed of the risk measure ρ and the pricing rule . The pair (ρ, ) will be called noncompatible if it implies the existence of a GD, and the main objective of this paper will be the measurement of the GD size by means of a new index denoted byÑ orÑ(ρ, ). An important precedent in financial theory is the notion of arbitrage. Although the absence of arbitrage always holds in theoretical approaches, real market quotations sometimes reflect the existence of arbitrage. For this reason, some years ago many authors defined severa...