# Sustainable Financial Risk Modelling Fitting the SDGs: Some Reflections

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Neoclassical Financial Risk Modelling: “We Were Wrong”

#### 2.1. The Formula That Killed Wall Street

- −
- (Sarah Robertson) It’s legit… the kid killed it. The formula’s worthless.
- −
- (Jared Cohen) What does that mean?
- −
- (Sarah Robertson) It’s broken.
- −
- (Jared Cohen) There are 8 trillion dollars of paper around the world relying on that equation!
- −
- (Sarah Robertson) Well, we were wrong.

#### 2.2. The Performativity of Financial Risk Modelling: Dispositive and Discourse

#### 2.2.1. Socio-Technical Instruments: The Financialised Tools

#### 2.2.2. Discourse of Financialisation: The Financial Logos

#### 2.3. Sustainable Finance and Epistemic Ethics

## 3. From the Philosophy of Science to the Risk Culture in Finance

#### 3.1. Mental Models in Finance

#### 3.2. The Philosophy Hook: The Principle of Continuity and the Theory of Average

#### 3.3. From Philosophy to Mental Model of Financial Techniques: The Risk Culture of Finance

#### 3.4. Sustainable Risk Culture for Financial Risk Modelling

## 4. The Unsustainable Risk Culture of Brownian Finance and Its Cognitive Biases

#### 4.1. Brownian Regulation and Financial Black Swans

#### 4.2. The Status of Discontinuities in Brownian Risk Culture

#### 4.3. A Note from Keynes

## 5. A Possible Solution for the Sustainable Modelling of Financial Risk: Fractal Geometry

#### 5.1. Is Fractal Geometry a “Sustainable” Geometry of Nature?

#### 5.1.1. Fractals in Nature

#### 5.1.2. Fractals and Sustainability

#### 5.2. Fractalisation of Financial Risk Modelling

#### 5.2.1. The Greening of Financial Risk Models and the Green Premium Puzzles

#### 5.2.2. How to Reconcile Finance and Nature

## 6. Conclusions and Future Research

- (1)
- To verify how good are fractal and multifractal methods to model important features of physical and human geographies in the sense of SDGs.
- (2)
- If a relevant financial risk modelling should take these fractal characteristics into account, to seek to construct “green metrics” and tools in the fractal sense to lay the groundwork for ecological finance theory as an alternative of neoclassical finance.
- (3)
- To apply these new metrics to portfolio management and risk measurement techniques.
- (4)
- To introduce financial risk modelling issues and metrics into a philosophical reflection on epistemic ethics because financial tools and financial risk modelling contribute to shape the real world. This last issue is important, keeping in mind that ecological finance theory aims to move from “what is” to “what should be”.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Pareto Distributions, Scaling Laws and Irregularity

^{−α}. If the scale invariance property is validated, objects do not change if scales of length are multiplied by a common factor. It represents a kind of universality. Like Russian dolls, the shape of the object repeats itself at all sizes. It is said that there is a self-similarity of the phenomenon being studied. So there is a relationship between power law and self-similarity. Here again, in practice, there are several forms of self-similarity, but we do not want to complicate the presentation in the context of this introductory article.

^{α}for x > a. We find a power law. The name of this function comes from the fact that one of the quantities (here, the cumulative empirical frequency) varies as the power of another (here, the values exceeding the threshold). This is the Pareto type I distribution. A very large number of situations are possible, especially with other power laws, but we do not intend to make the presentation more complicated here. Turning now to the empirical data and empirical frequencies, a rank-frequency diagram allows us to visually capture a characteristic property of Pareto’s power laws. The straight line of the log-log plot has the slope −α. In fact, when logarithms are taken instead of raw values, if $Y=a{X}^{-\alpha}$, then ln Y = −α lnX + k. We deduce from this that the “signature” of a power law is a linear relationship between the two quantities in log-log diagram.

## Appendix B

#### Fractal Properties of Neoclassical Finance Theory and Beyond

## References

- Epstein, G.A. (Ed.) Financialization and the World Economy; Edward Elgar Publishing: Northampton, MA, USA, 2005. [Google Scholar]
- Lawson, T. The Nature of Social Reality; Routledge: London, UK, 2019. [Google Scholar]
- Lagoarde-Segot, T. Diversifying finance research: From financialization to sustainability. Int. Rev. Financ. Anal.
**2015**, 39, 1–6. [Google Scholar] [CrossRef] - Lagoarde-Segot, T. Sustainable finance. A critical realist perspective. Res. Int. Bus. Financ.
**2019**, 47, 1–9. [Google Scholar] [CrossRef] - Lagoarde-Segot, T.; Martinez, E. Ecological Finance Theory: New Foundations. SSRN Electron. J.
**2020**. [Google Scholar] [CrossRef] - Badurdeen, F.; Shuaib, M.; Liyanage, J.P. Risk Modeling and Analysis for Sustainable Asset Management. In Engineering Asset Management and Infrastructure Sustainability; Mathew, J., Ma, L., Tan, A., Weijnen, M., Lee, J., Eds.; Springer: London, UK, 2012. [Google Scholar] [CrossRef]
- Zerbib, O.D. The Green Bond Premium; Working Paper; Tilburg University: Tilburg, Nederland, 2017. [Google Scholar]
- Karpf, A.; Mandel, A. Does it Pay to be Green? Working paper; Paris School of Economics: Paris, France, 2017. [Google Scholar]
- Callon, M. (Ed.) Introduction: The embeddedness of economic markets in economics. In The Law of the Markets; Blackwell: Oxford, UK, 1998. [Google Scholar] [CrossRef]
- MacKenzie, D.; Millo, Y. Constructing a Market, Performing Theory: The Historical Sociology of a Financial Derivatives Exchange. Am. J. Sociol.
**2003**, 109, 107–145. [Google Scholar] [CrossRef] [Green Version] - Muniesa, F. The Provoked Economy. Economic Reality and the Performative Turn; Routledge: Abingdon, UK, 2015. [Google Scholar]
- Chambost, I.; Lenglet, M.; Tadjeddine, Y. (Eds.) The Making of Finance: Perspectives from the Social Sciences; Routledge: Oxon, UK, 2019. [Google Scholar] [CrossRef]
- Walter, C. The financial Logos: The framing of financial decision-making by mathematical modelling. Res. Int. Bus. Financ.
**2016**, 37, 597–604. [Google Scholar] [CrossRef] - Kuhn, T. The Structure of Scientific Revolutions; The University of Chicago Press: Chicago, IL, USA, 1962. [Google Scholar]
- Morin, E. La nature de la nature; Seuil: Paris, France, 1977. [Google Scholar]
- Dicks, H. Being Like Gaia: Biomimicry and Ecological Ethics. Environ. Values
**2019**, 28, 601–620. [Google Scholar] [CrossRef] - Dicks, H.; Blok, V. Can Imitating Nature Save the Planet? Environ. Values
**2019**, 28, 519–526. [Google Scholar] [CrossRef] - Kennedy, E.; Fecheyr-Lippens, D.; Hsiung, B.-K.; Niewiarowski, P.H.; Kolodziej, M. Biomimicry: A Path to Sustainable Innovation. Des. Issues
**2015**, 31, 66–73. [Google Scholar] [CrossRef] - Benyus, J. Biomimicry: Innovation Inspired by Nature; Harper Perennial: New York, NY, USA, 1997. [Google Scholar]
- Braje, T.J.; Lauer, M. A Meaningful Anthropocene? Golden Spikes, Transitions, Boundary Objects, and Anthropogenic Seascapes. Sustainability
**2020**, 12, 6459. [Google Scholar] [CrossRef] - Embrechts, P. Did a Mathematical Formula Really Blow up Wall Street? 2009. Available online: https://www.actuaries.org/ASTIN/Colloquia/Helsinki/Presentations/Embrechts.pdf (accessed on 20 September 2020).
- MacKenzie, D.; Spears, T.C. ‘The formula that killed Wall Street’: The Gaussian copula and modelling practices in investment banking. Soc. Stud. Sci.
**2014**, 44, 393–417. [Google Scholar] [CrossRef] [Green Version] - Salmon, F. The Formula that Killed Wall Street; Wired: San Francisco, CA, USA, 2009. [Google Scholar]
- Walter, C. La représentation brownienne du risque: Une faute morale collective? Financ. Bien Commun.
**2008**, 31, 137–144. [Google Scholar] [CrossRef] - Walter, C. Le Modèle de Marche au Hasard en Finance; Economica: Paris, France, 2013. [Google Scholar]
- Walter, C. Financial Black Swans: Unpredictable Threat or Descriptive Illusion? In Societies under Threat: A Pluri-Disciplinary Approach; Springer: Berlin/Heidelberg, Germany, 2020; pp. 173–186. [Google Scholar] [CrossRef]
- Cavell, S. Cities of Words—Pedagogical Letters on a Register of the Moral Life; Harvard University Press: Cambridge, MA, USA, 2005. [Google Scholar]
- Cassidy, R. ‘Casino capitalism’ and the financial crisis. Anthr. Today
**2009**, 25, 10–13. [Google Scholar] [CrossRef] - Engelen, E.; Ertürk, I.; Froud, J.; Johal, S.; Leaver, A.; Moran, M.; Williams, K. Misrule of experts? The financial crisis as elite debacle. Econ. Soc.
**2012**, 41, 360–382. [Google Scholar] [CrossRef] - Larroche, V. The Dispositif: A Concept for Information and Communication Sciences; Wiley: London, UK, 2019. [Google Scholar]
- MacKenzie, D. An Engine Not a Camera. How Financial Models Shape Markets; MIT Press: Cambridge, MA, USA, 2006. [Google Scholar]
- MacKenzie, D.; Muniesa, F.; Siu, L. Do Economists Make Markets? On the Performativity of Economics; Princeton University Press: Princeton, NJ, USA, 2007. [Google Scholar]
- Chiapello, E. Financialization of valuation. Hum. Stud.
**2015**, 38, 13–35. [Google Scholar] [CrossRef] - Chiapello, E. Financialization as a socio-technical process. In The Routledge International Handbook of Financialization; Taylor & Francis: New York, NY, USA, 2020; pp. 81–91. [Google Scholar] [CrossRef]
- Chiapello, E.; Gilbert, P. Management Tools. A Social Sciences Perspective; Cambridge University Press: Cambridge, UK, 2019. [Google Scholar]
- Svetlova, E. Financial Models and Society. Villains or Scapegoats; Edward Elgar Publishing: Cheltenham, UK, 2018. [Google Scholar]
- Desrosières, A. Pour une sociologie historique de la quantification; Presses de l’Ecole des Mines: Paris, France, 2008. [Google Scholar]
- Mennicken, A.; Espeland, W.N. What’s New with Numbers? Sociological Approaches to the Study of Quantification. Annu. Rev. Sociol.
**2019**, 45, 223–245. [Google Scholar] [CrossRef] [Green Version] - Knorr Cetina, K. Epistemic cultures. How the Sciences Make Knowledge; Harvard University Press: Cambridge, MA, USA, 1999. [Google Scholar]
- Chiapello, E.; Walter, C. The three ages of financial quantification: A conventionalist approach to the financier’s metrology. Hist. Soc. Res.
**2016**, 41, 155–177. [Google Scholar] [CrossRef] - Austin, J.L. How to do Things with Words; Clarendon Press: Oxford, UK, 1955. [Google Scholar]
- Brisset, N. Models as speech acts: The telling case of financial models. J. Econ. Methodol.
**2018**, 25, 21–41. [Google Scholar] [CrossRef] [Green Version] - De Bruin, B. Ethics and the Global Financial Crisis: Why Incompetence Is Worse than Greed; Cambridge University Press: Cambridge, UK, 2015. [Google Scholar]
- Lamy, E. Epistemic Responsibility in Business: An Integrative Framework for an Epistemic Ethics of Business; Working Paper; FMSH: Paris, France, 2020. [Google Scholar]
- Duhem, P. La théorie physique, son objet, sa structure (Eng. tr. The Aim and Structure of Physical Theory; Wiener, P.P., Ed.; Chevalier & Rivière: Paris, France, 1906. [Google Scholar]
- Aït-Sahalia, Y.; Cacho-Diaz, J.; Hurd, T.R. Portfolio choice with jumps: A closed-form solution. Ann. Appl. Probab.
**2009**, 19, 556–584. [Google Scholar] [CrossRef] - Barndorff-Nielsen, O.E. Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Stat.
**1997**, 24, 1–13. [Google Scholar] [CrossRef] - Bouchaud, J.P.; Sornette, D.; Walter, C.; Aguilar, J.P. Taming Large Events: Optimal Portfolio Theory for Strongly Fluctuating Assets. Int. J. Theor. Appl. Financ.
**1998**, 1, 25–41. [Google Scholar] [CrossRef] [Green Version] - Boudt, K.; Croux, C.; Laurent, S. Robust estimation of intraweek periodicity in volatility and jump detection. J. Empir. Financ.
**2011**, 18, 353–367. [Google Scholar] [CrossRef] - Eberlein, E.; Prause, U.K. New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model. J. Bus.
**1998**, 71, 371–405. [Google Scholar] [CrossRef] [Green Version] - Kou, S.G. A Jump-Diffusion Model for Option Pricing. Manag. Sci.
**2002**, 48, 1086–1101. [Google Scholar] [CrossRef] [Green Version] - Liu, G.; Hong, L.J. Kernel Estimation of the Greeks for Options with Discontinuous Payoffs. Oper. Res.
**2011**, 59, 96–108. [Google Scholar] [CrossRef] [Green Version] - Wang, X.; Tan, K.S. Pricing and hedging with discontinuous functions: Quasi–Monte Carlo methods and dimension reduction. Manag. Sci.
**2013**, 59, 376–389. [Google Scholar] [CrossRef] [Green Version] - Walter, C. The extreme value problem in finance: Comparing the pragmatic programme with the Mandelbrot programme. In Extreme Events in Finance: A Handbook of Extreme Value Theory and Its Applications; Wiley: London, UK, 2017; pp. 25–51. [Google Scholar] [CrossRef]
- Craik, K. The Nature of Explanation; Cambridge University Press: Cambridge, UK, 1943. [Google Scholar]
- Johnson-Laird, P.N.; Byrne, R.M.J. Conditionals: A theory of meaning, inference, and pragmatics. Psychol. Rev.
**2002**, 109, 646–678. [Google Scholar] [CrossRef] [PubMed] - Mantzavinos, C. Individuals, Institutions, and Markets; Cambridge University Press: Cambridge, UK, 2001. [Google Scholar]
- Granger, C.W.J.; Orr, D. “Infinite Variance” and Research Strategy in Time Series Analysis. J. Am. Stat. Assoc.
**1972**, 67, 275–285. [Google Scholar] [CrossRef] - De Bruin, B.; Walter, C. Research habits in financial modelling: The case of non-normality of market returns in the 1970s and the 1980s. In Methods and Finance. A Unifying View on Finance, Mathematics and Philosophy; Springer: Berlin/Heidelberg, Germany, 2017; pp. 79–93. [Google Scholar]
- Walter, C. The Brownian motion in finance: An epistemological puzzle. Topoi
**2019**, 1–17. [Google Scholar] [CrossRef] - Marshall, A. Principle of Economics; Macmillan: London, UK, 1890. [Google Scholar]
- Brian, E. Y a-t-il un objet Congrès ? Le cas du Congrès international de statistique (1853–1876). Mil Neuf Cent
**1989**, 7, 9–22. [Google Scholar] [CrossRef] - Armatte, M. Les Index-Numbers: Controverse sur une approche probabiliste. Cahiers du Centre d’analyse et de mathématiques sociales. In Série Histoire du Calcul des Probabilités et de la Statistique; Centre d’analyse et de mathématiques sociales: Paris, France, 1996. [Google Scholar]
- Martins, N. Sraffa, Marshall and the principle of continuity. Camb. J. Econ.
**2013**, 37, 443–462. [Google Scholar] [CrossRef] [Green Version] - Wiener, N. God and Golem; MIT Press: Cambridge, MA, USA, 1966. [Google Scholar]
- Sider, T. Writing the Book of the World; Oxford University Press: Oxford, UK, 2011. [Google Scholar]
- Walter, C. La gestion indicielle et la théorie des moyennes. Revue d’économie Financière
**2005**, 79, 113–136. [Google Scholar] [CrossRef] - Chen, P. Mathematical representation in Economics and Finance: Philosophical preference, mathematical simplicity and empirical evidence. In Methods and Finance. In A Unifying View on Finance, Mathematics and Philosophy; Springer: Berlin/Heidelberg, Germany, 2017; pp. 17–49. [Google Scholar] [CrossRef]
- Lawson, T. Economics and Reality; Routledge: London, UK; New York, NY, USA, 1997. [Google Scholar]
- Lawson, T. Reorienting Economics; Routledge: London, UK; New York, NY, USA, 2003. [Google Scholar] [CrossRef]
- Davis, M.; Etheridge, A. Louis Bachelier’s Theory of Speculation—The Origins of Modern Finance, Princeton; Princeton University Press: Princeton, NJ, USA, 2006. [Google Scholar]
- Roeschmann, A.Z. Risk Culture: What It Is and How It Affects an Insurer’s Risk Management. Risk Manag. Insur. Rev.
**2014**, 17, 277–296. [Google Scholar] [CrossRef] - Palermo, T.; Power, M.; Ashby, S. Navigating Institutional Complexity: The Production of Risk Culture in the Financial Sector. J. Manag. Stud.
**2016**, 54, 154–181. [Google Scholar] [CrossRef] [Green Version] - Bozeman, B.; Kingsley, G. Risk Culture in Public and Private Organisations. Public Adm. Rev.
**1998**, 58, 109–118. [Google Scholar] [CrossRef] - Power, M.; Ashby, S.; Palermo, T. Risk Culture in Financial Organisations: A Research Report; Analysis of Risk and Regulation: London, UK, 2013. [Google Scholar]
- PwC. The Risk Culture Survey. Delaware USA: PricewaterhouseCoopers. 2012. Available online: http://www.pwc.com/us/en/riskculture/index.jhtml (accessed on 20 September 2020).
- Institute of Risk Management. Risk Culture under the Microscope Guidance for Boards; Institute of Risk Management: London, UK, 2012. [Google Scholar]
- Dimon, J. Testimony of James Dimon, Chairman & CEO, JP Morgan Chase & Co before the US Senate Committee on Banking; Housing and Urban Affairs: Washington, DC, USA, 2012. [Google Scholar]
- Green, P.; Jennings-Mares, J. IIF’s Final Report on Market Best Practices for Financial Institutions and Financial Products. Bank. Financ. Serv. Policy Rep.
**2009**, 27, 1–5. [Google Scholar] - Levy, C.; Lamarre, E.; Twining, J. Taking Control of Organizational Risk Culture; McKinsey & Company: New York, NY, USA, 2010. [Google Scholar]
- Institute of International Finance. IIF’s Final Report on Market Best Practices for Financial Institutions and Financial Products; Institute of International Finance: Washington, DC, USA, 2008. [Google Scholar]
- Le Courtois, O.; Walter, C. Extreme Financial Risks and Asset Allocation; Series in Quantitative Finance; Imperial College Press: London, UK, 2014. [Google Scholar] [CrossRef] [Green Version]
- Le Courtois, O.; Lévy-Véhel, J.; Walter, C. Regulation Risk. N. Am. Actuar. J.
**2020**, 24, 463–474. [Google Scholar] [CrossRef] - Taleb, N. The Black Swan: The Impact of the Highly Improbable; Penguin: London, UK, 2007. [Google Scholar]
- Longin, F. Volatilité et mouvements extrêmes du marché boursier. Ph.D. Thesis, HEC, Paris, France, 1993. [Google Scholar]
- Longin, F. Extreme Events in Finance: A Handbook of Extreme Value Theory and Its Applications; Wiley: London, UK, 2017. [Google Scholar]
- Cont, R. Empirical properties of asset returns: Stylized facts and statistical issues. Quant. Financ.
**2001**, 1, 223–236. [Google Scholar] [CrossRef] - Guillaume, D.M.; Dacorogna, M.; Davé, R.; Muller, U.; Olsen, R.; Pictet, O.V. From the bird’s eye to the microscope: A survey of new stylized facts of the intra-day foreign exchange markets. Financ. Stoch.
**1997**, 1, 95–130. [Google Scholar] [CrossRef] [Green Version] - Longin, F.M. The Asymptotic Distribution of Extreme Stock Market Returns. J. Bus.
**1996**, 69, 383. [Google Scholar] [CrossRef] - Lux, T. Power-laws and long memory. Quant. Financ.
**2001**, 1, 560–562. [Google Scholar] [CrossRef] - Plerou, V.; Gopikrishnan, P.; Rosenow, B.; Amaral, L.; Stanley, H.E. Universal and non-universal properties of cross-correlations in financial time series. Phys. Rev. Lett.
**1999**, 83, 1471. [Google Scholar] [CrossRef] [Green Version] - Lo, A.W. Long-Term Memory in Stock Market Prices. Econometrica
**1991**, 59, 1279. [Google Scholar] [CrossRef] - Ding, Z.; Granger, C.W.; Engle, R.F. A long memory property of stock market returns and a new model. J. Empir. Financ.
**1993**, 1, 83–106. [Google Scholar] [CrossRef] - Mantegna, R.N.; Stanley, H.E. Scaling behavior of an economic index. Nature
**1995**, 376, 46–49. [Google Scholar] [CrossRef] - Lux, T. The stable Paretian hypothesis and the frequency of large returns: An examination of major German stocks. Appl. Financial Econ.
**1996**, 6, 463–475. [Google Scholar] [CrossRef] - Schmitt, F.; Schertzer, D.; Lovejoy, S. Multifractal analysis of foreign exchange data. Appl. Stoch. Models Data Anal.
**1999**, 15, 29–53. [Google Scholar] [CrossRef] - Mantegna, R.; Stanley, E. Introduction to Econophysics: Correlations and Complexity in Finance; Cambridge University Press: Cambridge, UK, 1999. [Google Scholar]
- Calvet, L.; Fisher, A. Multifractality in asset returns: Theory and evidence. Rev. Econ. Stat.
**2002**, 83, 381–406. [Google Scholar] [CrossRef] - Walter, C. The leptokurtic crisis and the discontinuous turn in financial modelling. In The Making of Finance. Perspectives from the Social Sciences; Chambost, I., Lenglet, M., Tadjeddine, Y., Eds.; Routledge: London, UK, 2019; pp. 77–89. [Google Scholar]
- Savoiu, G.; Simăn, I.I. History and Role of Econophysics in Scientific Research. In Econophysics: Background and Applications in Economics, Finance, and Sociophysics; Savoiu, G., Ed.; Elsevier: Amsterdam, The Netherlands, 2012; pp. 3–16. [Google Scholar] [CrossRef]
- Arnéodo, A.; Muzy, J.F.; Sornette, D. Causal cascade in the stock market from the infrared to the ultraviolet. Euro. Phys. J. B
**1998**, 2, 277–282. [Google Scholar] [CrossRef] [Green Version] - Borland, L.; Bouchaud, J.-P.; Muzy, J.-F.; Zumbach, G. The Dynamics of Financial Markets—Mandelbrot’s Cascades and Beyond. Wilmott Mag. Available online: https://www.cfm.fr/insights/the-dynamics-of-financial-markets-mandelbrots-multifractal-cascades-and-beyond/ (accessed on 14 September 2020).
- Bouchaud, J.-P.; Potters, M. Theory of Financial Risk and Derivative Pricing: From Statistical Physics to Risk Management; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Dacorogna, M.M.; Gençay, R.; Müller, U.A.; Olsen, R.B.; Pictet, O.V. An Introduction to High Frequency Finance; Academic Press: San Diego, CA, USA, 2001. [Google Scholar]
- Potters, M.; Cont, R.; Bouchaud, J.-P. Financial Markets as Adaptive Ecosystems. Europhys. Lett.
**1998**, 41, 239–244. [Google Scholar] [CrossRef] [Green Version] - Bacry, E.; Delour, J.; Muzy, J.F. Multifractal random walk. Phys. Rev. E
**2001**, 64, 026103–026106. [Google Scholar] [CrossRef] [Green Version] - Pochart, B.; Bouchaud, J.P. The skewed multifractal random walk with applications to option smiles. Quant. Financ.
**2002**, 24, 303–314. [Google Scholar] [CrossRef] - Calvet, L.E.; Fisher, A.J. Regime-Switching and the Estimation of Multifractal Processes. SSRN Electron. J.
**2003**, 2, 44–83. [Google Scholar] [CrossRef] [Green Version] - Segnon, M.; Lux, T. Multifractal Models in Finance: Their Origin, Properties, and Applications, In The Oxford Handbook of Computational Economics and Finance; Chen, S.-H., Kaboudan, M., Du, Y.-R., Eds.; Oxford University Press: Oxford, UK, 2018. [Google Scholar] [CrossRef]
- Mandelbrot, B.B. The Fractal Geometry of Nature; Freeman: New York, NY, USA, 1982. [Google Scholar]
- Gao, J.; Xia, Z.-G. Fractals in physical geography. Prog. Phys. Geogr. Earth Environ.
**1996**, 20, 178–191. [Google Scholar] [CrossRef] - Ghanbarian, B.; Hunt, A.G. Fractals: Concepts and Applications in Geosciences; Taylor & Francis: London, UK, 2017. [Google Scholar]
- Puente, C.E.; Maskey, M.L.; Sivakumar, B. Combining Fractals and Multifractals to Model Geoscience Records. In Fractals: Concepts and Applications in Geosciences; CRC Press: London, UK, 2017. [Google Scholar]
- Bak, P. How nature works. The Science of Self-Organized Criticality; Oxford University Press: Oxford, UK, 1997. [Google Scholar]
- West, G.S. The Universal Laws of Life, Growth, and Death in Organisms, Cities, and Companies; Penguin: New York, NY, USA, 2018. [Google Scholar]
- West, G.; Brown, J. Life’s Universal Scaling Laws. Phys. Today
**2007**, 57, 36–42. [Google Scholar] [CrossRef] - Mandelbrot, B. How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science
**1967**, 156, 636–638. [Google Scholar] [CrossRef] [Green Version] - Mandelbrot, B.B. Fractals. Form, Chance and Dimension; Freeman: San Francisco, SF, USA, 1977. [Google Scholar]
- Avnir, D.; Biham, O.; Lidar, D.; Malcai, O. Applied Mathematics: Is the Geometry of Nature Fractal? Science
**1998**, 279, 39–40. [Google Scholar] [CrossRef] [Green Version] - Shenker, O.R. Fractal geometry is not the geometry of nature. Stud. Hist. Philos. Sci. Part A
**1994**, 25, 967–981. [Google Scholar] [CrossRef] - Mandelbrot, B.; Pfeifer, P.; Biham, O.; Malcai, O.; Lidar, D.A.; Avnir, D. Is Nature Fractal? Science
**1998**, 5352, 783–789. [Google Scholar] [CrossRef] - Environmental Protection Agency. Available online: http://www.epa.gov/sustainability/basicinfo.htm (accessed on 20 September 2020).
- James, P. Urban Sustainability in Theory and Practice: Circles of Sustainability; Routledge: New York, NY, USA, 2015. [Google Scholar]
- Triple Pundit, Cracking the Code: The Essence of Sustainable Development. Available online: https://www.triplepundit.com/story/2011/cracking-code-essence-sustainable-development/80966 (accessed on 7 September 2020).
- Sustainable Land Development Initiative. The Fractal Frontier—Sustainable Development Trilogy. 2011. Available online: http://www.thegreenmarketoracle.com/2012/07/fractal-frontier-sustainable.html (accessed on 21 September 2020).
- Mock, T.; Wernke, T. Like Life Itself, Sustainable Development is Fractal. Triple Pundit. Available online: https://www.triplepundit.com/story/2011/life-itself-sustainable-development-fractal/81586 (accessed on 20 September 2020).
- Perey, R. Organizing Sustainability and the Problem of Scale. Organ. Environ.
**2014**, 27, 215–222. [Google Scholar] [CrossRef] - Summerhayes, G. Financial exposure: The role of disclosure in addressing the climate data deficit. In Proceedings of the United Nations, Sustainable Insurance Forum, London, UK, 22 February 2019. [Google Scholar]
- European Banking Federation Report. Towards a Green Finance Framework. Available online: https://www.ebf.eu/wp-content/uploads/2017/09/Geen-finance-complete.pdf (accessed on 7 September 2020).
- Kunreuther, H.; Heal, G.; Allen, M.; Edenhofer, O.; Field, C.B.; Yohe, G. Risk management and climate change. Nat. Clim. Chang.
**2013**, 3, 447–450. [Google Scholar] [CrossRef] [Green Version] - Weitzman M., L. Fat tails and the social cost of carbon. Am. Econ. Rev.
**2014**, 104, 544–546. [Google Scholar] [CrossRef] [Green Version] - Solomon, S.; Plattner, G.K.; Knutti, R.; Friedlingstein, P. Irreversible climate change due to carbon dioxide emissions. Proc. Natl. Acad. Sci. USA
**2009**, 106, 1704–1709. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Knutti, R. The end of model democracy? Clim. Chang.
**2010**, 102, 395–404. [Google Scholar] [CrossRef] - Nordhaus, W.D. A review of the Stern review on the economics of climate change. J. Econ. Lit.
**2007**, 45, 686–702. [Google Scholar] [CrossRef] - Stern, N. The economics of climate change. Am. Econ. Rev.
**2008**, 98, 1–37. [Google Scholar] [CrossRef] [Green Version] - Pindyck, R.S. Climate change policy: What do the models tell us? J. Econ. Lit.
**2013**, 51, 860–872. [Google Scholar] [CrossRef] [Green Version] - Fernandez, P. Is It Ethical to Teach That Beta and CAPM Explain Something? Available online: https://ssrn.com/abstract=2980847 (accessed on 28 May 2019).
- Lopez de Prado, M.; Fabozzi, F. Who Needs a Newtonian Finance? J. Portf. Manag.
**2017**, 44, 1–4. [Google Scholar] [CrossRef] - Bachelet, M.J.; Becchetti, L.; Manfredonia, S. The Green Bonds Premium Puzzle: The Role of Issuer Characteristics and Third-Party Verification. Sustainability
**2019**, 11, 1098. [Google Scholar] [CrossRef] [Green Version] - Kob, J. Realising Natural Disaster: A Financial Ontology of Catastrophe. In Proceedings of the SASE 32nd Annual Meeting, Amsterdam, The Netherlands, 18–21 July 2020. [Google Scholar]
- Mandelbrot, B.B. The inescapable need for fractal tools in finance. Ann. Financ.
**2005**, 1, 193–195. [Google Scholar] [CrossRef] - Bouchaud, J.-P. Power laws in economics and finance: Some ideas from physics. Quant. Financ.
**2001**, 1, 105–112. [Google Scholar] [CrossRef] - Müller, U.A.; Dacorogna, M.M.; Dave, R.D.; Olsen, R.B.; Pictet, O.V.; Von Weizsäcker, J.E. Volatilities of different time resolutions—Analyzing the dynamics of market components. J. Empir. Financ.
**1997**, 4, 213–239. [Google Scholar] [CrossRef] - Muzy, J.; Delour, J.; Bacry, E. Modelling fluctuations of financial time series: From cascade process to stochastic volatility model. Eur. Phys. J. B
**2000**, 17, 537–548. [Google Scholar] [CrossRef] [Green Version] - Walter, C. Lévy-stability-under-addition and fractal structure of markets: Implications for the investment management industry and emphasized examination of MATIF notional contract. Math. Comput. Model.
**1999**, 29, 37–56. [Google Scholar] [CrossRef] - Walter, C. Research of scaling laws on stock market variations. In Scaling, Fractals and Wavelets; Wiley: London, UK, 2009; pp. 437–464. [Google Scholar] [CrossRef]
- Newman, M. Power laws, Pareto distributions and Zipf’s law. Contemp. Phys.
**2005**, 46, 323–351. [Google Scholar] [CrossRef] [Green Version] - Redner, S. How popular is your paper? An empirical study of the citation distribution. Eur. Phys. J. B
**1998**, 4, 131–134. [Google Scholar] [CrossRef] - Zipf, G.K. Human Behavior and the Principle of Least Effort; Addison-Wesley: Cambridge, MA, USA, 1949. [Google Scholar]
- Daníelsson, J.; Zigrand, J.-P. On time-scaling of risk and the square-root-of-time rule. J. Bank. Financ.
**2006**, 30, 2701–2713. [Google Scholar] [CrossRef] [Green Version] - Mandelbrot, B.B. Fractals and Scaling in Finance; Discontinuity, Concentration, Risk: Selecta Volume E; Springer: Berlin/Heidelberg, Germany, 1997. [Google Scholar] [CrossRef]
- Walter, C. Benoît Mandelbrot in finance. In Benoît Mandelbrot. A Life in Many Dimensions; World Scientific: Singapore, 2015; pp. 459–469. [Google Scholar] [CrossRef]

© 2020 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Walter, C.
Sustainable Financial Risk Modelling Fitting the SDGs: Some Reflections. *Sustainability* **2020**, *12*, 7789.
https://doi.org/10.3390/su12187789

**AMA Style**

Walter C.
Sustainable Financial Risk Modelling Fitting the SDGs: Some Reflections. *Sustainability*. 2020; 12(18):7789.
https://doi.org/10.3390/su12187789

**Chicago/Turabian Style**

Walter, Christian.
2020. "Sustainable Financial Risk Modelling Fitting the SDGs: Some Reflections" *Sustainability* 12, no. 18: 7789.
https://doi.org/10.3390/su12187789