Should We Disembed?
The trend towards the ‘mathematisation of finance and economics’ is a mis- nomer: the long shared history between mathematics and economics points rather to an embedding of parts of mathematical sciences in economics and financial economics. Positive economics, with its objective of transforming economics from the ‘camera’ embedded within political philosophy to an analytical ‘engine’ based on descriptively false assumptions has gained much by embedding mathematical sciences. So should we try to do away with the embedding? And if so, how?
The answer to the first question is that mathematical sciences applied to finance and economics need to grow out of this embedding for three main reasons. First, reference points in economics, such as the Market and con- gruence with existing theories, have replaced traditional mathematical reference points, for example Nature and data. As a result, the range of mathematical techniques applied in finance is getting narrower, and so is the range of mathematical cultures involved in finance. It is also becoming increasingly difficult to innovate and to propose new solutions that enrich both economic theories and mathematical sciences. Instead, the emphasis is on pursuing the present course of investigation with an increasingly limited ability to stray and discover something genuinely new. Disembedding would be beneficial to both mathematics and financial economics.
Second, the embedding of mathematics into financial economics feeds the embedding of the economy within the financial sphere. This second embedding has genuine economic and social consequences. Giving some independence to mathematics will undoubtedly weaken this trend, and might help reverse it.
Finally, the embedding of mathematics is a leading contributor to the Barnesian performativity of financial economics. Mathematical models and computer codes are easy to communicate and quick to implement.
In turn, performativity may lead to the procyclicality of financial models.
Danı´elson et al. (2001)already suggested that risk management models and regulations could play a procyclical role that would facilitate the growth of financial bubbles. The financial crisis of 20072009 has provided evidence to support this point of view. By amplifying bubbles and crashes, procycli- cality poses a further danger: turning models from Barnesian performative to counterperformative as was the case for the CDO pricing models based on the Gaussian copula.
How Can We Disembed?
There is no straightforward answer to the second question: disembedding is no easy task. A first, partial solution would be to open the academic field to complementary viewpoints and theories, such as behavioural finance and econophysics. Behavioural finance developed out of mathematical psychol- ogy and questions of decision under uncertainty. Over the past 30 years, behavioural finance has challenged and changed our view of how individuals and group make decisions. This has had profound implications for corporate finance, investment finance and risk management.
Econophysics, which was greatly inspired by Mandelbrot’s work on fractals and on the statistical analysis of prices (Mandelbrot, 1963, 1997;
Mandelbrot & Hudson, 2004) seeks to apply techniques used in physics to economics. Anecdotal evidence, from conference line-ups to publications in peer-reviewed journals, suggests that mathematical finance has opened up to econophysics and that it is becoming open to behavioural finance.
Another solution would be to shift the perspective we adopt when we teach and apply finance and financial economics (Fabozzi et al., 2014).
Anyone who works in the field should develop their critical thinking by understanding what works, what does not, when and why. Theories can be helpful as a structuring tool, as long as we do not convey the illusion that finance is a physical science in the strictest sense. The financial world is more complicated, with some hints of mathematical complexity. This idea seemed obvious to Keynes and Knight in the 1920s and 1930s with the notion of animal spirits (Keynes, 2007) and the distinction between risk and uncertainty (Knight, 1921), but got lost in the 1950s and 1960s.
115 Finance and Mathematics: Merger or Acquisitions?
Decision-theoretic foundations need once more place equal weight on eco- nomics, mathematical sciences and behavioural sciences.
A change in the methodological framework adopted in finance might also be warranted. To that end, we could look at epistemology and use Popper’s falsifiability or Kuhn’s work on the history of science (Kuhn, 1962) as a possible starting point.
We could also promote large public and private federating projects.
History shows that large multidisciplinary projects can have a structuring effect on the advancement of scientific knowledge. Examples include the birth of operations research in the 1940s and the development of the space programme in the 1960s. To be successful, these projects should be truly multidisciplinary and multicultural, have clear objectives and look for real- world problems. The recent economic and financial crisis is a missed oppor- tunity to rethink the concept of ‘risk’.
A final aspect that we have not addressed in this chapter is how the increas- ing use of mathematics in finance and financial economics has shaped mathe- matics to the point of becoming a primary motivation to do mathematics. In the authors’ opinion, this is an important question in itself. The study Measuring the Economic Benefits of Mathematical Science Research in the UK commissioned by the London Mathematical Society and EPSRC (Deloitte, 2013) and the Enqueˆte sur l’Impact Socio-E´conomique des Mathe´matiques en France(CMI, 2015) have started to address this important issue.
CONCLUSION
In recent years, mathematical models used in finance have come under severe criticism. A widespread opinion is that the ‘mathematisation’ of finance, meaning the increased use of mathematical models to shape finance theory, acted as a catalyst of the financial and economic crisis. Our explora- tion of the long historical relationship between finance and mathematics suggests that the fundamental issue has more to do with the embedding of mathematical finance within financial economics than with the mathemati- sation of finance.
This embedding process, which started in the 1950s, with the birth of posi- tive economics and financial economics, and accelerated in the early 1970s with the discovery of the Black-Scholes option pricing formula, has reached a mature stage. We argue that it is in the best interest of mathematics, eco- nomics and society to disembed mathematical finance from financial
economics and return to the intertwined relationship that existed through most of history. Disembedding will be a complex and slow process. We pro- pose a number of partial steps that can be implemented to facilitate this pro- cess, but we also acknowledge that the greatest challenges are still to come.
NOTES
1. Brown, R. (1828). A brief account of microscopical observations made in the months of June, July and August 1827 on the particles contained in the pollen of plants. Private communication.
2. The problem related to negative prices and negative rates, though unwelcome, can be addressed practically through a suitable parametrisation of the process. This is the case for the first published stochastic interest rate model, theVasicek (1977) model, which has remained extremely popular although it allows interest rates to become negative.
3. This represents Polya’s personal view of mathematics, which is shared by a large number of mathematicians, including and the first author of this chapter.
However, this point of view is not universally accepted. For example, the Bourbaki group held a very different view in which mathematics should be alpha and the omega, a self-centred and self-sufficient intellectual endeavour.
4. In a plenary session of the Global Derivatives, Trading and Risk Management Conference held in Amsterdam in 2013, Ricardo Rebonato recounted a conversation with Pat Hagan, the inventor of the SABR model. Hagan made the point that as his model became widely adopted, its ability to fit market data increased noticeably.