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33 pages, 456 KiB

Open AccessArticle

Set-Valued T-Translative Functions and Their Applications in Finance

by Andreas H. Hamel and Frank Heyde

Mathematics 2021, 9(18), 2270; https://doi.org/10.3390/math9182270 - 15 Sep 2021

Cited by 1 | Viewed by 1281

A theory for set-valued functions is developed, which are translative with respect to a linear operator. It is shown that such functions cover a wide range of applications, from projections in Hilbert spaces, set-valued quantiles for vector-valued random variables, to scalar or set-valued [...] Read more.

A theory for set-valued functions is developed, which are translative with respect to a linear operator. It is shown that such functions cover a wide range of applications, from projections in Hilbert spaces, set-valued quantiles for vector-valued random variables, to scalar or set-valued risk measures in finance with defaultable or nondefaultable securities. Primal, dual, and scalar representation results are given, among them an infimal convolution representation, which is not so well known even in the scalar case. Along the way, new concepts of set-valued lower/upper expectations are introduced and dual representation results are formulated using such expectations. An extension to random sets is discussed at the end. The principal methodology consisted of applying the complete lattice framework of set optimization. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Financial Risk Management)

20 pages, 393 KiB

Open AccessArticle

Expected Shortfall Reliability—Added Value of Traditional Statistics and Advanced Artificial Intelligence for Market Risk Measurement Purposes

by Santiago Carrillo Menéndez and Bertrand Kian Hassani

Mathematics 2021, 9(17), 2142; https://doi.org/10.3390/math9172142 - 02 Sep 2021

Cited by 5 | Viewed by 1934

Abstract

The Fundamental Review of the Trading Book is a market risk measurement and management regulation recently issued by the Basel Committee. This reform, often referred to as “Basel IV”, intends to strengthen the financial system. The newest capital standard relies on the use [...] Read more.

The Fundamental Review of the Trading Book is a market risk measurement and management regulation recently issued by the Basel Committee. This reform, often referred to as “Basel IV”, intends to strengthen the financial system. The newest capital standard relies on the use of the Expected Shortfall. This risk measure requires to get sufficient information in the tails to ensure its reliability, as this one has to be alimented by a sufficient quantity of relevant data (above the 97.5 percentile in the case of the regulation or interest). In this paper, after discussing the relevant features of Expected Shortfall for risk measurement purposes, we present and compare several methods allowing to ensure the reliability of the risk measure by generating information in the tails. We discuss these approaches with respect to their relevance considering the underlying situation when it comes to available data, allowing practitioners to select the most appropriate approach. We apply traditional statistical methodologies, for instance distribution fitting, kernel density estimation, Gaussian mixtures and conditional fitting by Expectation-Maximisation as well as AI related strategies, for instance a Synthetic Minority Over-sampling Technique implemented in a regression environment and Generative Adversarial Nets. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Financial Risk Management)

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14 pages, 2471 KiB

Open AccessArticle

Goodness-of-Fit of Logistic Regression of the Default Rate on GDP Growth Rate and on CDX Indices

by Kuang-Hua Hu, Shih-Kuei Lin, Yung-Kang Ching and Ming-Chin Hung

Mathematics 2021, 9(16), 1930; https://doi.org/10.3390/math9161930 - 13 Aug 2021

Cited by 1 | Viewed by 2230

Abstract

Under the Basel II and Basel III agreements, the probability of default (PD) is a key parameter used in calculating expected credit loss (ECL), which is typically defined as: PD

\times

Loss Given Default

\times

Exposure at Default. In practice or in regulatory [...] Read more.

Under the Basel II and Basel III agreements, the probability of default (PD) is a key parameter used in calculating expected credit loss (ECL), which is typically defined as: PD

\times

Loss Given Default

\times

Exposure at Default. In practice or in regulatory requirements, gross domestic product (GDP) has been adopted in the PD estimation model. Due to the problem of excessive fluctuation and highly volatile ECL estimation, models that produce satisfactory PD and thus ECL estimations in the context of existing risk management techniques are lacking. In this study, we explore the usage of the credit default swap index (CDX), a market’s expectation of future PD, as a predictor of the default rate (DR). By comparing the goodness-of-fit of logistic regression, several conclusions are drawn. Firstly, in general, GDP has considerable explanatory power for the default rate which is consistent with current models in practice. Secondly, although both GDP and CDX fit the DR well for rating B class, CDX has a significantly better fit of DR for ratings [A, Baa, Ba]. Thirdly, compared with low-rated companies, the relationship between the DR and GDP is relatively weak for rating A. This phenomenon implies that, in addition to using macroeconomic variables and firm-specific explanatory variables in the PD estimation model, high-rated companies exhibit a greater need to use market supplemental information, such as CDX, to capture the changes in the DR. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Financial Risk Management)

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9 pages, 7841 KiB

Open AccessArticle

Regulatory Estimates for Defaulted Exposures: A Case Study of Spanish Mortgages

by Marta Ramos González, Antonio Partal Ureña and Pilar Gómez Fernández-Aguado

Mathematics 2021, 9(9), 997; https://doi.org/10.3390/math9090997 - 28 Apr 2021

Cited by 1 | Viewed by 1773

Abstract

The capital requirements derived from the Basel Accord were issued with the purpose of deploying a transnational regulatory framework. Further regulatory developments on risk measurement is included across several documents published both by the European Banking Authority and the European Central Bank. Among [...] Read more.

The capital requirements derived from the Basel Accord were issued with the purpose of deploying a transnational regulatory framework. Further regulatory developments on risk measurement is included across several documents published both by the European Banking Authority and the European Central Bank. Among others, the referred additional documentation focused on the models’ estimation and calibration for credit risk measurement purposes, especially the Advanced Internal-Ratings Based models, which may be estimated both for non-defaulted and defaulted assets. A concrete proposal of the referred defaulted exposures models, namely the Expected Loss Best Estimate (ELBE) and the Loss Given Default (LGD) in-default, is presented. The proposed methodology is eventually calibrated on the basis of data from the mortgage’s portfolios of the six largest financial institutions in Spain. The outcome allows for a comparison of the risk profile particularities attached to each of the referred portfolios. Eventually, the economic sense of the results is analyzed. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Financial Risk Management)

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34 pages, 464 KiB

Open AccessArticle

Noether Theorem in Stochastic Optimal Control Problems via Contact Symmetries

by Francesco C. De Vecchi, Elisa Mastrogiacomo, Mattia Turra and Stefania Ugolini

Mathematics 2021, 9(9), 953; https://doi.org/10.3390/math9090953 - 24 Apr 2021

Cited by 1 | Viewed by 1950

Abstract

We establish a generalization of the Noether theorem for stochastic optimal control problems. Exploiting the tools of jet bundles and contact geometry, we prove that from any (contact) symmetry of the Hamilton–Jacobi–Bellman equation associated with an optimal control problem it is possible to [...] Read more.

We establish a generalization of the Noether theorem for stochastic optimal control problems. Exploiting the tools of jet bundles and contact geometry, we prove that from any (contact) symmetry of the Hamilton–Jacobi–Bellman equation associated with an optimal control problem it is possible to build a related local martingale. Moreover, we provide an application of the theoretical results to Merton’s optimal portfolio problem, showing that this model admits infinitely many conserved quantities in the form of local martingales. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Financial Risk Management)

23 pages, 320 KiB

Open AccessArticle

Modified Mean-Variance Risk Measures for Long-Term Portfolios

by Hyungbin Park

Mathematics 2021, 9(2), 111; https://doi.org/10.3390/math9020111 - 06 Jan 2021

Cited by 1 | Viewed by 1810

Abstract

This paper proposes modified mean-variance risk measures for long-term investment portfolios. Two types of portfolios are considered: constant proportion portfolios and increasing amount portfolios. They are widely used in finance for investing assets and developing derivative securities. We compare the long-term behavior of [...] Read more.

This paper proposes modified mean-variance risk measures for long-term investment portfolios. Two types of portfolios are considered: constant proportion portfolios and increasing amount portfolios. They are widely used in finance for investing assets and developing derivative securities. We compare the long-term behavior of a conventional mean-variance risk measure and a modified one of the two types of portfolios, and we discuss the benefits of the modified measure. Subsequently, an optimal long-term investment strategy is derived. We show that the modified risk measure reflects the investor’s risk aversion on the optimal long-term investment strategy; however, the conventional one does not. Several factor models are discussed as concrete examples: the Black–Scholes model, Kim–Omberg model, Heston model, and 3/2 stochastic volatility model. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Financial Risk Management)

Review

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24 pages, 403 KiB

Open AccessReview

Real-Valued Systemic Risk Measures

by Alessandro Doldi and Marco Frittelli

Mathematics 2021, 9(9), 1016; https://doi.org/10.3390/math9091016 - 30 Apr 2021

Cited by 2 | Viewed by 1599

Abstract

We describe the axiomatic approach to real-valued Systemic Risk Measures, which is a natural counterpart to the nowadays classical univariate theory initiated by Artzner et al. in the seminal paper “Coherent measures of risk”, Math. Finance, (1999). In particular, we direct our attention [...] Read more.

We describe the axiomatic approach to real-valued Systemic Risk Measures, which is a natural counterpart to the nowadays classical univariate theory initiated by Artzner et al. in the seminal paper “Coherent measures of risk”, Math. Finance, (1999). In particular, we direct our attention towards Systemic Risk Measures of shortfall type with random allocations, which consider as eligible, for securing the system, those positions whose aggregated expected utility is above a given threshold. We present duality results, which allow us to motivate why this particular risk measurement regime is fair for both the single agents and the whole system at the same time. We relate Systemic Risk Measures of shortfall type to an equilibrium concept, namely a Systemic Optimal Risk Transfer Equilibrium, which conjugates Bühlmann’s Risk Exchange Equilibrium with a capital allocation problem at an initial time. We conclude by presenting extensions to the conditional, dynamic framework. The latter is the suitable setup when additional information is available at an initial time. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Financial Risk Management)

13 pages, 319 KiB

Open AccessReview

Capital Allocation Rules and the No-Undercut Property

by Gabriele Canna, Francesca Centrone and Emanuela Rosazza Gianin

Mathematics 2021, 9(2), 175; https://doi.org/10.3390/math9020175 - 16 Jan 2021

Viewed by 1572

Abstract

This paper makes the point on a well known property of capital allocation rules, namely the one called no-undercut. Its desirability in capital allocation stems from some stability game theoretical features that are related to the notion of core, both for finite [...] Read more.

This paper makes the point on a well known property of capital allocation rules, namely the one called no-undercut. Its desirability in capital allocation stems from some stability game theoretical features that are related to the notion of core, both for finite and infinite games. We review these aspects, by relating them to the properties of the risk measures that are involved in capital allocation problems. We also discuss some problems and possible extensions that arise when we deal with non-coherent risk measures. Full article

(This article belongs to the Special Issue Applied Mathematical Methods in Financial Risk Management)

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