Good deal indices in asset pricing: actuarial and financial implications

• Author: José Garrido,Ramin Okhrati,Alejandro Balbás
• Published date: 01 July 2019
• Date: 01 July 2019
• DOI: http://doi.org/10.1111/itor.12424

Intl. Trans. in Op. Res. 26 (2019) 1475–1503

DOI: 10.1111/itor.12424

INTERNATIONAL

TRANSACTIONS

IN OPERATIONAL

RESEARCH

Good deal indices in asset pricing: actuarial and financial

implications

Alejandro Balb´

asa,Jos

´

e Garridoband Ramin Okhratic

aUniversity Carlos III of Madrid, C/Madrid, 126, 28903 Getafe, Madrid, Spain

bDepartment of Mathematics and Statistics, Concordia University,1455 de Maisonneuve Blvd. W., Montreal, Canada

H3G 1M8

cMathematical Sciences, University of Southampton, Highfield, Southampton SO17 1BJ, UK

E-mail: alejandro.balbas@uc3m.es [Balb´

as]; jose.garrido@concordia.ca[Garrido]; r.okhrati@soton.ac.uk [Okhrati]

Received 23 September 2016; receivedin revised form 28 January 2017; accepted 14 April 2017

Abstract

Weintegrate into a single optimization problem a risk measure, beyondthe 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)divergesto(+∞,−∞)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)divergesto(−∞,−∞). Moreover, by appropriately

adding the riskless asset, every GD may generatea 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

1. Introduction

The use of risk functions beyond the variance is becoming more and morefrequent in both actuarial

and financial studies. Nevertheless, when the most important arbitrage-free pricing models of

financial economics (binomial, trinomial trees, Black and Scholes,stochastic volatility, etc.) and the

most important risk functions (VaR,CVaR,weight ed -CVaR,robust -CVaR, spectral measures, etc.)

C

2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation ofOperational Research Societies

Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,

USA.

1476 A. Balb´

as et al. / Intl. Trans. in Op. Res. 26 (2019) 1475–1503

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)divergeto(+∞,−∞). The existence of GDs

is equivalent to the existence of alternativesequences of investment strategies whosepairs (risk, price)

diverge to (−∞,−∞). This pathological finding has been analyzed in Balb´

as and Balb´

as (2009)

and Balb´

as 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 ρand the pricing

rule . The pair (ρ, ) will be called noncompatible if it implies the existence of a GD,andthe

main objective of this paper will be the measurement of the GD size by means of a new index

denoted by ˜

Nor ˜

N(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

several measures of the arbitrage size. This allowed them to address interesting questions such as

pricing and hedging issues under transaction costs, cross-market arbitrage, integration between

markets, trading systems, valuation of embedded derivatives, etc. Similarly, the existence of GD (or

the lack of compatibility) must be measured now, because in some sense it indicates an important

lack of balance between the risk that the investor is facing and the wealth that he/she is expecting.

As we will see, these unbalanced situations may lead to wrong decisions in several fields. For

instance, managers could pay too high prices or compose inefficient portfolios, and insurers could

buy nonoptimal reinsurance contracts or receive insufficient premiums.

The arbitrage measurement has been addressedfrom several perspectives. One of them wasrelated

to the capital profits generated by an arbitrage strategy (Balb´

as et al., 1999). Nevertheless, if the

arbitrage strategy can be repeated time and again, the arbitrage profit will be multiplied time and

again also, and therefore it will become infinity. To prevent this caveat, Balb´

as et al. (1999) measure

the arbitrage level as the maximum ratio between the arbitrage income and the value of the sold

assets, that is, these authors give a relative measure of the arbitrage degree. Similarly, when (risk,

price)divergesto(−∞,−∞)we will need to maximize the risk/pr ice ratios, otherwise we will face

unbounded optimization problems.

Arisk/price ratio is an objective function that does not satisfy manydesirable analytical properties

(continuity, convexity, differentiability, etc.), therefore its optimization can be simplified by dealing

instead with vector optimization problems involving both risk and price. Since Harry Markowitz

published his seminal results, it is known that multiobjective analyses are useful in many financial

topics. In particular, for portfolio selection several interesting approaches exist, such as Ballestero

and Romero (1996), Ballestero et al. (2012), Dash and Kajiji (2014), among others. With respect

to the simultaneous optimization of both risk and price, we apply well-known results to optimize

“risk” under constraints for “price” (Sawaragi et al., 1985).

The paper outline is as follows. Section 2 sets notations and lists assumptions. The index (or

measure) ˜

N(ρ, ) is derived in Section 3. The first approach will apply when no theoretical pricing

model is considered, and only a finite collection of available securities and their market quotations

are involved. The advantage of this approach is clear, since it is sufficient to choose a robust (or

ambiguous) risk function ρfor the value of ˜

Nto be model-independent. Beyond the optimal

C

2017 The Authors.

International Transactionsin Operational Research C

2017 International Federation ofOperational Research Societies